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Computer assisted tomography using microwave-induced thermoelastic waves

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Computer assisted tomography using microwave-induced
thermoelastic waves
Su, Jenn-Lung, Ph.D.
University of Illinois at Chicago, 1988
UMI
300 N. Zeeb Rd.
Ann Arbor, MI 48106
COMPUTER ASSISTED TOMOGRAPHY USING MlCROWAVE-INDUCED THERMOELASTIC
WAVES
BY
JENN-LUNG SU
B.S. Chung Yuan University, Taiwan, R.O.C., 1977
M.S. University of Illinois at Chicago, 1985
THESIS
Submitted in partial fulfillment of the requirements
for the degree of Doctor of Philosopy in Bioengineering
in the Graduate College of the
U n i v e r s i t y o f I l l i n o i s a t C h i c a g o , 1 988
Chicago, I 11inois
THE UNIVERSITY OF ILLINOIS AT CHICAGO
The Graduate College
July 26, 1988
I hereby recommend that the thesis prepared under my supervision by
Jenn-Lung Su
entitled
Computer Assisted Tomography Using Microwave-Induced
Thermoelastic Waves
be accepted in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
In Charge of Thesis
Recommendation concurred in
Committee
T
THE
UNIVERSITY
Of
ILLINOIS
AT
CHICAGO
s
D517 Rev. 1/87
Final Examination
This thesis
is dedicated
to my family, without their support
would never have been accomplished.
i i i
it
ACKNOWLEDGEMENT
I would like to express my deep appreciation to professor James C.
Lin for his guidance, advice, and support throughout the preparation of
this thesis.
I wish to thank Dr. Mahmood Mafee, Dr. William O'Neill, Dr. Larry
Reynolds, and Dr. Thomas Sandercock for their advice and being
in my
commi ttee.
Financial support for this research was provided, in part, by the
National Science Foundation and the Office of Naval Research.
It is also a pleasure to acknowledge my friends and colleagues Dr.
Karen
Wang
Chan, Dr.
for
Thanks
their
also
Kenneth
technical
due
to
Ginsburg, Mr. George
support
Saiu-Mei
and
and
Ashman, and
assistance
Diane
in
Tobias
the
for
Mr. Yujin
laboratory.
typing
and
proof-reading my thesis.
Finally, I would like to deeply thank my dear wife, Saiu-Mei, for
her
love, constant support, encouragement, and patience throughout my
stud i es.
JLS
iv
T A B L E OF CONTENTS
CHAPTER
1
2
3
PAGE
INTRODUCTION
1.1
General Review and Background
1.1.1
Computed Tomography and Medical
Imaging
1.1.2
Transmission Computed Tomography . .
1.1.2.1
X-ray Imag i ng
1.1.2.2 Ultrasound and Microwave Imaging . .
1.1.2.3 Magnetic Resonance Imaging
1.1.3
Emission Computed Tomography ....
1.1.3.1 Single Photon Emission Computed
Tomography
1.1.3-2 Positron Emission Tomography ....
1.1.3.3 Comparison between Transmission
Computed Tomography and Emission
Computed Tomography
1.1.4
Thermoelastic Waves Imaging
1 .2
Objective
1.3
Organization of This Thesis
THERMOELASTIC WAVE PROPAGATION IN
BIOLOGICAL TISSUE
2.1
Physical Properties of Biological
Tissue
2.1.1
Microwave Properties
2.1.2
Thermal Properties
2.1.3
Acoustic Properties
2.2
Microwave Induced Thermoelastic
Waves
RECONSTRUCTION ALGORITHMS
3.1
Transmission Computed Tomography
Algorithms
3.1.1
Projection Theorem
3.2
Formulation of Parallel Beam
Projection
3-2.1
Discretized Form for Parallel
Projection
3.3
Formulation of Fan Beam Projection . .
3.3.1
Equiangular Rays
3-3.2
Discretized Form for Equiangular
Rays Case
3.3.3
Equally Spaced Detectors Case ....
3.3.^
Discretized Form for Equally
Spaced Detectors Case
3.4 * Filters and Interpolation
v
1
3
3
7
7
9
12
13
14
16
18
20
24
25
26
26
26
29
34
36
41
42
46
49
51
52
53
56
58
6l
62
T A B L E OF CONTENTS
(continued)
CHAPTER
PAGE
3-5
4
5
Reconstruction Algorithm in
Emission Computed Tomography Case ...
COMPUTER SIMULATION FOR TRANSMISSION
COMPUTED TOMOGRAPHY
4.1
General Considerations
4.2
Parallel Projections of the
Phantoms
4.3
Fan-Beam Projections
4.4
Experimental Protocol
4.5
Results and Discussion
68
72
72
75
77
79
80
COMPUTER SIMULATION FOR EMISSION COMPUTED
TOMOGRAPHY
5.1
Projection Generation
5.2
Simulation Protocol
5.3
Simulation Results and Discussion . . .
95
95
101
102
6
SYSTEM DESIGN
6.1
Microwave Devices
6.2
Rotating Phantom
6.3
Hydrophone Transducer Array
6.4
Signal Conditioning and Storage ....
6.5
Microcomputer and Display
6.6
Image Processing Software
119
119
122
124
124
127
129
7
METHOD of EXPERIMENT
7.1
Preliminary Study of System
7.1.1
Thermoelastic Wave of Single
Transducer
7.1.2
Estimation of Phantom Attenuation
Coefficient
7.1.3
Phantoms
7.1.4
Array Signals and Projection ....
7.1.5
Image Reconstruction and
Processing
7.2
Experimental Protocol . . .
131
131
8
RESULTS and DISCUSSIONS
8.1
Computer Simulation Model
8.2
Experimental Results
8.2.1
Single Phantom
8.2.2
Double Phantoms
8.3
Discussion
vi
131
137
138
140
148
150
154
154
155
158
I67
184
T A B L E OF CONTENTS
CHAPTER
9
(continued)
PAGE
Conclusion
197
APPENDICES
Appendix A
Appendix B
201
202
209
CITED LITERATURE
237
VITA
247
vi i
L I S T OF T A B L E S
TABLE
I
II
III
IV
V
VI
VII
VIII
IX
X
XI
XI I
XIII
XIV
XV
PAGE
TOMOGRAPHY IN MEDICAL IMAGING
6
DEPTH OF PENETRATION OF MICROWAVE IN
BIOLOGICAL TISSUE
28
COMPLEX TRANSMISSION COEFFICIENTS FOR
SYSTEMS CONSISTING OF TWO BIOLOGICAL TISSUE
. .
31
...
32
THERMAL PROPERTIES OF BIOLOGICAL MATERIALS ...
33
MEASURED ACOUSTIC PROPERTIES OF BIOLOGICAL
MATERIALS
37
MICROWAVE PROPERTIES OF BIOLOGICAL TISSUE
PARAMETERS FOR EQUATIIONS (2.8) AND (2.9)
...
kO
PARAMETERS FOR EQUATION (3.1)
hS
PARAMETERS OF THE FOUR PHANTOMS
7^
RECONSTRUCTION SETUPS AND CPU TIMES
Sk
SPECIFICATION OF SCOPE kOSkl
128
READING ERROR TEST
T+7
IMAGE RECONSTRUCTION PROTOCOL
152
GRAY LEVEL SCALE OF IMAGES
161
SOURCE AND MECHANISM ERROR TEST
190
vi i i
L I S T OF F I G U R E S
FIGURE
PAGE
1
A comparison between ECT and TCT
19
2
Propagating plane wave
30
3
Acoustic attenuation of different biological
tissue
3&
Simplified diagram of the microwave-induced
thermoelastic imaging apparatus
k}
5
Typical hydrophone output signal
kb
6
Illustration of the projection measurement
system and the Fourier projection theorem. ...
h~]
7
Two types of fan-beam project system
53
8
Coordinate system for fan-beam with
equiangular case
55
Coordinate system for fan-beam with
equi spaced case
59
Discrete Fourier transforms of Ramachandra
filter and Shepp-Logan filter
64
Geometric description of projections for ECT
and TCT
69
12
Testing phantom in TCT case
73
13
Simulated projections in parallel rays case
14
Simulated projections in fan-beam case
78
15
Images of phantom 1, 2, 3» and k
81
16
The reconstructed images using different
numbers of projections
83
The reconstructed images for phantom 1 using
different algorithms and setups
85
The reconstructed images of phantom 1
86
k
9
10
11
17
18
ix
. .
76
L I S T OF F I G U R E S
(continued)
FIGURE
PAGE
19
The reconstructed images of phantom 2
87
20
The reconstructed images of phantom 3
88
21
The reconstructed images of phantom 4
89
22
The reconstructed images using different
f iIters
91
The reconstructed images for testing
resolution
92
24
Testing phantom in ECT case
96
25
A description of projections for different
cases
100
A comparison between the original phantom
and its reconstructed image
104
A comparison between ECT and TCT for a more
complicated case
106
28
Reconstructed images for case 1
108
29
Reconstructed images for case 2
110
30
Reconstructed image for case 3
Ill
31
Reconstructed images for case 4
113
32
Reconstructed images with high m
114
33
Reconstructed images with comparable m
115
34
Comparison of the accuracy of two methods
35
A block diagram of the thermoelastic imaging
system
120
36
Major components of imaging system
121
37
Arrangement for controlling the microwave
source with an external pulse
123
23
26
27
x
...
116
L I S T OF F I G U R E S
(continued)
FIGURE
PAGE
38
The setup of rotating phantom
125
39
An overall structure of ACQ4
130
1*0
Experimental result of thermoelastic wave
. . .
133
41
Frequency spectrum of thermoelastic wave ....
134
42
A sequence of thermoelastic waves
135
43
The least square regression of previous
figure
13&
Thermoelastic wave measured for different
phantoms
139
45
Relative location of the nine holes
141
46
Non-uniform gain of processing channels
47
Two rows of thermoelastic wave
48
Reading error from single triggering signal
49
One row of signals detected by same
transducer
149
50
A time-of-f 1 ight image
156
51
A comparison of simulation models
157
52
A sequence of images for image processing
53
Reconstructed images set 1
1 63
54
Reconstructed images set 4
164
55
Reccnstructed images set 5
1 65
56
Reconstructed images set 6
166
57
Reconstructed images set 11
168
58
Reconstructed images set 13
1 69
44
xi
.
....
142
144
. .
...
145
160
L I S T OF F I G U R E S
(continued)
PAGE
59
Reconstructed images set 15
60
Linear gray scale transformation method
61
Reconstructed images set 10
172
62
Reconstructed images set 12
173
63
Reconstructed images set 7
175
6U
Reconstructed images set 9
176
65
Reconstructed images set 8
177
66
Reconstructed images set 16
178
67
Reconstructed images set 17
179
68
Reconstructed images set 18
181
69
Reconstructed images set 19
70
Reconstructed images set 20
183
71
Reconstructed images set 21
185
72
Reconstructed images set 22
186
73
Reconstructed images set 23
187
7h
Reconstructed images set 2k
188
15
Reconstructed images and averaged image
76
The reconstructed images with/without
rotated
193
Thermoelastic waves at frequency of maximum
intensity
196
78
Frequency spectrum 1 of thermoelastic wave . . .
204
79
Frequency spectrum 2 of thermoelastic wave . . .
205
80
Frequency spectrum 3 of thermoelastic wave . . .
206
77
xi i
170
....
•
....
171
182
192
L I S T OF F I G U R E S
(continued)
FIGURE
PAGE
81
Frequency spectrum A of thermoelastic wave . . .
207
82
Frequency spectrum 5 of thermoelastic wave . . .
208
xi i i
L I S T OF A B B R E V I A T I O N S
CT
Computed tomography
ECT
Emission computed tomography
FWHM
Full-width half-maximum
MITW
Microwave-induced thermoelastic wave
MRI
Magnetic resonance imaging
PET
Positron emission tomography
SPECT
Single photon emission computed tomography
TCT
Transmission computed tomgraphy
xiv
SUMMARY
A computer assisted tomography (CT)
microwave-induced
two-dimensional
thermoelastic
system using a new modality,
waves,
is
presented.
The
images are reconstructed from the projections generated
from phantoms.
Moreover,
the system has the
ability to reconstruct
projections to three-dimensional images.
Based
on
algorithms
backprojection convolution
are derived
thermoe1astic waves.
study was done.
to
cover the
theorem,
possible
three
cases
Using those algorithms,
a
different
of CT,
using
computer simulation
The results show that in shape,
size,
and location,
the reconstructed images correspond very well to the original simulated
phantoms.
The gray levels of
the attenuation
reconstructed images correspond well to
constant of the
phantom in all
transmission computer
tomography (TCT) cases; however, they are proportional to the ratio
the
emitter
concentration
and the
attenuation
constant
which
of
are
distributed quantities in emission CT case.
The hardware,
consist
of,
transducers,
used to generate the
microwave
devices,
projections of the test object,
rotating
phantom,
interface electronics and a computer.
pulsed microwave
is used
to generate
The
corresponds
to
setups
limitations
have been
those
A 2 ms,
thermoelastic waves.
different setups with various phantom size, location,
are used in this study.
hydrophone
2^50 MHz
Several
and test objects
preliminary results show that this system
found in
very
consistently.
this system,
xv
Although
the results
some
encourage
SUMMARY
further
exploration
of
(continued)
microwave-induced thermoelastic
possible medical imaging modality.
xvi
waves
Chapter 1
INTRODUCTION
The advent of computer science in the last decade has made notable
advances
in
medical
imaging.
Imaging
modalities
and
protocols
which
were in the experimental stage have become routine clinical procedures.
The
earliest
use
of
the
medical
imaging
information about the body without painful
system,
which
provided
and often life-threatening
surgery, goes back to the discovery of x-rays by William K. Roentgen in
1895>
Later, a variety of
specifying the regions of
procedures were developed for selectively
interest, such as intravenous (IV), catheter
and orally administered dyes.
performance of
the
often
to
invasive
invisible organs.
and
ultrasound,
noninvasive
Thus the radiologist, faced with limited
instrumentation, devised
the
body,
despite
visualization
of
of
procedures,
to facilitate visulization of
In 1950's, a
which
a variety
otherwise
revolution began with nuclear medicine
serious
a
imaging
diseases
limitations,
progress
which
provided
would
be
otherwise unavailable. The physical scientist and engineer have played
the dominant role in processing this large amount of data.
Typically, medical
three
basic
through
the
imaging systems
techniques:
body,
(2)
(1)
the
in clinical
measurement
the measurement
of
of
use evolves from
transmitted
reflected
energy
energy
from
the
body, (3) the measurement of emitted energy from selectively deposited
1
2
chemicals in the body (Macovski, 1983) -
In the 1970's, the development
of
is
x-rays
significant
computed
tomography
advancement
in
the
(CT)
unquestionably
diagnostic
imaging
the
field
most
since
the
discovery of x-rays (Kak, 1981; Brownell et al., 1982).
Different types of CT which use modalites other than x-ray's have
been developed
computer.
based
on
Because of
the same
principle, with
the assistance of
the ionizing radiation effect, x-ray tomography
is not suitable for long-term follow-up and monitoring situations. New
imaging schemes such as positron emission tomography (PET) and nuclear
magnetic
resonance
diagnostic
imaging
capabilities.
potentials and
to
biological
materials
explore
Although
these
physics,
mathematics,
process,
they
functional
Each
of
have
these
significantly
methods
has
improved
its
own
set
the
of
limitations. These techniques have prompted a number of
researchers
noninvasive
(MRI)
other
for
modalities
all
have
images of
process
and
in
physical
and
non-invasive
differ
tissue
somewhat
equipment
common
chemical
used
the
in
in
unique
properties
of
characterization.
the
details
the
image
capability
of
the
formation
to
produce
the distributions of various structures and/or
within
the
body.
Microwave-induced
thermoelastic
wave (MITW) appear to possess some unique features that may allow them
to become as useful
as these other methods and to permit non-invasive
imaging of tissue characteristics which
are not
t e c h n i q u e s ( O l s e n , 1 9 8 2 ; O l s e n a n d L i n , 1 983) -
identifiable by other
3
1.1
General Review and Background
In this section, we present a brief
literature review of relative
fields distributed among four subsections: section 1.1.1 is a synopsis
of
image
reconstruction
methods
and
those
applications
in
medical
imaging, section 1.1.2 is a review of transmission computed tomography
(TCT)
in
computed
different
tomography
modalites,
(ECT)
and
section
1.1.3
a comparison
is
emphasizes
emission
between TCT and
ECT, and
section 1.1.4 is a review for how investigators use thermoelastic waves
to obtain biological tissue images.
1.1.1
Computed Tomography and M e d i c a l Imaging
In general,
property
of
images are used
an
object
reconstruction from
years
in a
physical
system.
The
problem
projection has repeatedly arisen over
large number of medical, scientific, and
The range of
application
radioastronomy,
particle
or
to represent the distribution of some
is wide.
visualization
reactions,
satellite
of
Examples
tracks
mapping
of
last 30
technical
fields*
particles
the
image
the
include, space
of
of
earth's
in
imaging and
nuclear
and
resources,
and
industrial radiography. These seemingly different applications have the
same
mathematical
number of
and
computational
foundations.
Basically,
given
a
projections at different angles of view, an estimate is made
of the corresponding distribution within the object.
In
medicine,
radiology
the application
is computerized
that
has
tomography (CT),
revolutionized
diagnostic
in which X-rays are used to
generate the projection data for a cross section of the human body. The
b
reconstructed image can help us to discover the location and shape of a
suspected tumor inside a patient's brain and to plan a suitable course
of treatment.
A
large and
rapidly expanding body of
has accumulated since development of
1970 1 s,
by
Hounsfield.
Recently
literature relevant to CT
the first commercial
an overview of
computed
scanner
in
tomography
with emphasis on medical applications has been presented (Bates et al.,
1983) which includes definitions that are needed to assimilate the vast
amount of
literature produced
in this field. As they did, we
like to
reintroduce some relevant terminology:
Objectrany objects whose
interior
is desired for study, such as
head, heart etc.
Emanation: represent any physical
from outside.
Transducer:
For example: radiation, wave, field.
any
transmitting
process used to study the body
or
source
or
receiving
sink
of
the
antenna,
emanation.
x-ray
tube,
For
example:
scintillation
counter.
Characteristics: the spatial distribution (profile or density) of
whatever material property of the object one wishes to determine.
CT:
the
digital
reconstruction
computational
of
a
clean
operations
on
image
of
the
measurements
density
of
from
emanations
that have passed through the body.
Those definitions capture the essence of what technical scientists now
understand
by
the
term
CT. Table I
lists
a
selection of
emanations
5
which apply to medical
image
information
various modalities
parameters.
ionizing
is
imaging. From this table, we knows that medical
generated
are
However,
radiation.
amounts
tissue
this
imaging
modality
nonionizing
launch
of
both
x-ray
Where
as,
electric
and
as
and
nuclear
is
in
noisy
as
these
other
beam of
and
rely
on
on
permits
thermoelastic
that may
methods.
allow
It
uses
impinging microwave
tissue. Based
and acoustic properies of the body,
The
is slow and consumes
Microwave-induced
relies on a
an acoustic wavefront
medicine
some unique features
useful
sources.
measure different specific
ultrasound
power.
to offer
become
radiation
CT
different
bony structures, and NMR
appears
to
several
in competition and
interference by gas and
large
from
to
the electromagnetic
images are given to represent
the
tissues structure.
Based
on
the
location
of
the source emanations, we can
divided
medical CT into two types. When exterior sources are used, the term TCT
(Transmission Computed Tomography)
is used and
includes, for
example,
x-ray, microwave, Nuclear Magnetic Resonance/Zeugmatography (MRI), and
ultrasound
tomography.
When
interior
sources
(Emission Computed Tomography) is used and
(Single-Photon
Tomography).
as RCT
Emission
Tomography)
are
used,
the
term
ECT
includes, for example, SPECT
and
PET
(Positron
Emission
There also exists a subclass of problems within TCT known
(Reflection
Computed Tomography)
scattered field are taken
in which measurements of
at the same vicinity
in which
field was generated. For example, ultrasound, microwave.
the
the
incident
The MITW CT
6
TABLE I
TOMOGRAPHY IN MEDICAL IMAGING
Energy
Characteristics
Transducers
X-rays
X - r a y attenuation
X-ray tube;
Scintillation detector
coefficients
Gamma-rays
Concentration of radio­
labeled substance
Scintillation counters
Compton scattered
X and r rays
Electron density or
distribution of atomic
number
X-ray tube;
Scintillation counters
Heavy particles;
protons,pions.
Scatter i ng/absorpt ion
cross section
Linear accelerators;
NaI,BGO detectors
Ultrasound;
acoustic wave
Piezoelectric sensors
Attenuat ion;Refract ive
index; Acoustic impedance
Magnetic fields &
Radio frequency
Distribution of nuclear
spi ns
RF coils; Magnetometers
Magnetic fields &
microwave field
Distribution of electro
spi ns
Antennas;RF coi1s;
Magnetometers
Microwave field
Refractive index
Dipole; Horn; Antennas
Low-frequency
electric currents
Electrical conductivity
d i str i but i on
Electrodes
also could be classified into either TCT or ECT depending on the waves
which you pick up to reconstruct the image. We will explain this later.
In the next section, we will review TCT and ECT in different modalites,
then make a comparison between TCT and
ECT. Both of
them play a major
role later in this thesis.
T r a n s m i s s i o n Computed Tomography
1.1.2
X-ray Imaging
1.1.2.1
The
object
differential
is
the
radiology.
pictures
structure
main
This
whose
of
attenuation
parameter
that
characteristics
varying
an object
levels
and
X-ray tranmission tomography
any
coefficient
has
make
of
of
been
used
possible
grayness
departures
the
in
the
in
the
normality
the
diagnostic
production
display
from
X-ray
of
anatomical
within
is concerned with the provision of
it.
images
of sections or thin slices through the object at different depths.
There are two basic orientations relative to the body and
detector
in which
transverse. The
oriented
chosen
initial
longitudinally
and
detector
tomographic sections are obtained:
controlled
during
the
analog
images of
(Bocage,
blurred
197*0- This
relative
motion
exposure.
This
of
longitudinal
X-ray
X-ray
techniques
is
and
tomograms was
is achieved
the
imaging
by carefully
source
called
and
the
motion
tomography (Meredith and Massey, 1977)•
The
procedure
limited clinical
of
motion
tomography
is
use (Garcia et al., 1980).
basic disadvantages. The first, for
each
simple.
However,
it
has
Motion tomography has two
tomography
plane, the entire
8
volume of interest is exposed by X-rays.
Usually, a number of sections
are desired, then the radiation is extensive,. The second, the detailed
contrast
in the plane of
interest is not improved over a conventional
radiography. Specifically,
difference in recorded
if
a
lesion
intensity
continue to be 1% different
in
the plane results
in a
1%
in a conventional radiograph, it will
in the motion tomogram. The first of
the
disadvantages can be remedied by a system which was introduced by Grant
(1972).
Many
developed
other
but
techniques
the most
for
important
achieving
is
that of
tomography
have
computerized
been
tomography
(CT).
In
1973.
introduced
by
collection of a
measurements,
the
concept
EMI
Ltd,
of
of
computerized
England.
The
tomography
technique
large number of finite collimated
from
which
reconstructed. This
a
system
tomographic
provides
image
an
first
involves
be
mathematically
image
of
a
section
I98O;
within a volume completely eliminating all other planes (Herman,
Gordon,
197 1 *) •
successful
in
So
far,
clinical
computerized
tomography
designs
organs
and
the
X-ray transmission
can
isolated
was
has
that
been
were
extremely
heretofore
impossible to visualize are seen with remarkable clarity.
In a current clinical setting, an X-ray source is collimated
a
narrow
beam
transmitted
position
of
proportional
and
photons
the
to
scanned
are
scan.
the
through
detected
Each
absorbed
by
detector
X-ray
the
a
plane
scanning
element
energy,
of
interest.
detector
produces
which
at
a
into
The
each
signal
reflects
the
attenuation
along
the
ray
path
between
the
element
and
the
X-ray
source. This same procedure is repeated at intervals for 360° so that a
set
of
projections
are
applied to a digital
is
obtained. The
devised
projection
data
computer where an accurate two-dimensional
reconstructed, representing
the section of
resultant
interest. A
the
linear
are
image
attenuation coefficient
in
number of approximation methods have been
to provide a solution to the problem of
image reconstruction
from these data. Most of
the commercial
scanners used a variation of
technique referred
the "filtered
back
to as
projection".
A
detailed
mathematical derivation of this reconstruction process will be provided
in chapter 3> So far, X-ray computed tomography is capable of providing
spatial
resolution in a range from 1
to 1.5 mm and the thickness of a
slice in the range of l-15mm.
1.1.2.2
U1 trasound and Microwave Imaging
Many ultrasonic imaging techniques have been developed within the
last decade in an effort to produce images that are either quantitative
or at
least better than those obtained by B-scanning. A common feature
of all these techniques, which distinguishes them from B-scan, is that
a
computer
measured
is
data.
necessary
This
to
reconstruct
technique
is
the
called
image
from
"Ultrasonic
the
raw
or
computerized
tomography".
Ultrasonic computerized tomography (UCT) is accomplished using the
geometry of parallel
the first-generation
beam scanning. The scanning motion is similiar to
X-ray CT scanning, although unlike X-ray
tubes,
ultrasound
transducers
can
both
transmit
and
receive
energy.
This
duality results in the ability to acquire backscattered signals as well
as forward scattered signals in the same plane. Parameters that can be
mathematically calculated are acoustic attenuation, from the amplitude
measurements
of
the
transmitted
signal,
acoustic
speed,
from
measurements of travel time between the two transducers, and estimates
of
backscattering,
obtained
from
a
typical
resolution of transmission tomography
backscatter
compound
B-scans.
The
images is lower than that of the
images (Greenleaf et al., 1985; Greenleaf, 1984). Based on
different fundamental properties of imaging techniques and the physical
limitations, transmission tomography
images give quantitative data but
low resolution information, whereas the backscatter or compound B-scan
images give high resolution but little quantitative images.
Different reconstruction
approximations.
dispersion
in
Since
the
biological
images have been obtained with different
precise
media
mechanism
are
not
yet
for
absorption
completely
and
understood
(Sehgal and Greenleaf, 1982), an exact wave equation derived from first
principles
made
is
not
to obtain
a
known.
more
However,
tractable
several
assumptions
wave equation
are
(Mueller,
commonly
1980). Two
common approximations which are known as Born approximation and
approximation (Kaveh et al., 1982;
Rytov
Ishimaru, 1978) are used to solve
the wave equations.
Preliminary
results
have
been
performed
tomography and diffraction tomography. However, all
with
reflection
the techniques are
11
approximate, requiring various assumptions about tissue properties. The
degradations in image quality due to violations of the assumptions are
not
well
characterized
approximations for
and
clinical
so
the
ultrasonic
appropriateness
imaging
of
many
of
is uncertain (Robinson
and Greenleaf, 1985) Microwave imaging
imaging. Microwave
is another form of radiation used for biological
imaging makes
different tissues on
it possible to differentiate
the basis of
local
propagation parameters and
reconstruct a corresponding cross-sectional
bear
information
obtain by
the
other
water
biological
of
diagnostic
imaging.
molecule
systems
relevance,
image. These
which
are
(Larsen
and
the
interaction
Jacobi,
1978)
of
and
to
properties of
microwaves
thus
to
images could
not; possible
For example , the dielectric
dominate
between
the
and
state
of
hydration of an object is possible to image by interrogating the object
wi th mi crowave.
Several
reports (Larsen
and
Jacobi,
1979;
Lin,
1985)
show that
some electromagnetic material
parameters (relative dielectric constant
and electric
various biological
loss factor) of
tissues
in microwave
regions vary. The optimal frequency for microwave imaging of biological
tissues
is from
2 GHz
to 8 GHz.
Converting
the frequency
wavelength, resolutions in the range of 1 cm may be expected.
range
to
Like in
ultrasound cases, the effects of wave propagation such as refraction,
diffraction, reflection, and depolarization of microwaves are no longer
negligible,
and
imaging
concepts
based
on
geometrical
optics
like
12
transmission-mode
tomography
CT
concept
lead
to
applied
poor
in
the
results.
Again,
reconstruction
(Ermert and Dohlus, 1 986 ; Slaney et al., 1 986)
the
diffraction
algorithm
is
used
to obtain the microwave
imaging. Both of results show that under some limitations, such as, if
the degree of dielectric
inhomogenity is small and the cross-sectional
areas of objects are small, microwave imaging using diffraction CT is
possible. So far, in this field, the diffraction
imaging concept
has
not been investigated and no suitable imaging system has been presented
for cli nical use.
Magnetic Resonance Imaging
1.1.2.3
Nuclear
imaging
magnetic
(MR I)
is
a
non i nvas i vel y. NMR
resonance
new
has a
(NMR)
technique
imaging
for
or
magnetic
checking
long and successful
human
body
history reaching back to
the 1940's. In recent years, the application of
domain from the basic science into the medical
the
resonance
NMR has expanded
imaging area.
its
MRI holds
great clinical promise, and researchers are beginning to establish the
range of its possible applications. Five physical characteristics that
NMR can measure
such
as
in
the human
hydrogen,
carbon,
body are: (1) concentrations of
and
others
with
appropriate
nuclei
nuclear
characteristics; (2) T1 spin-lattice relaxation time; (3) T2 spin-spin
relaxation
time;
magnetized
nuclei
which
is the small
(4)
Volume
through
the
flow
information
resonance
region;
due
(5)
to
movement
chemical
of
shift,
change in the resonance frequency due to chemical
bonds with magnetic field perturbation (Brownel1 et al., 1982; Hill and
Hinshaw, 1985)- These five factors influence image contrast.
13
The
basic
principles
of
NMR
itself
are
covered
in
numerous
textbooks (Abragram, 1961; Farrar and Beckes, 1971; Slichter, 1978).
comprehensive
book
which
is closest
cross-sectional
Projections of
NMR
imaging
to existing diagnostic methods
images
in
detail
has
been
Morris (1 982) . The medical application of NMR
written by Marsfield and
that
describes
A
similar
to
those
is
the
generated
magnetic resonance can be generated
ways. In general, a higher static magnetic field
magnetic alignment of nuclei with an odd
production of
by
x-ray
CT.
in many different
is applied to set up
number of protons. A weak but
rapidly alternating magnetic field applied by a coil
near the subject
stimulates in the direction of the nucleus relative to the orientation
of the static magnetic field. This result in the absorption of energy,
which
is
then
emitted
when
the
nucleus
returns
to
a
state
of
equilibrium. The absorption and emission of energy takes place at
the
resonance frequency, known as the Larmor frequency and given by wo=yH,
where y is the gyromagnetic ratio and H is the external magnetic field.
Either
by
magnetic
varying
field
information
reflect
the
can
the
field
gradient
be
in
obtained.
distribution
of
in
known
manner
different
Comparied
hydrogen
or
by
oscillating
directions,
with
and
of
x-ray
T1
the
CT,
and
spatial
NMR
T2.
the
images
They
are
believed to carry more information about the diseased state of tissues.
1.1.3
Emission Computed Tomography (ECT)
Emission computed tomography is reviewed
in several books (Herman,
1979; Ell and Holman, 1982; Kouris et al., 1982) . Here we will describe
H
it briefly. Generally, emission tomography systems can be divided
two types according to the angles of viewing.
into
A system which views an
object over a narrow range of angles (typically less than 90°) produces
images which are somewhat ambiguous
in their
information. Such systems are commonly
tomographic
systems.
information
are
Systems
commonly
which
referred
representation of depth
referred to as "limited angle"
provide
to
as
3&0 0
complete
computed
projection
tomographic
(CT)
systems.
The limited angle tomography has been with us in nuclear medicine
since the early days. However, the systems provide tomograms which are
somewhat
vague
in
depth.
In
addition,
out-of-focus
data
from
other
planes is always present in these tomograms. This information tends to
reduce
image
transaxial
and
contrast
and
limit
quantitative
accuracy.
tomograms produced by ECT systems have good
potentially
measurements and
are
quantitatively
accurate
in
However,
image contrast
both
spatial
in their representation of activity. These advantages
are the main reasons for using an ECT system in our simulation study.
1.1.3-1
S i n g l e P h o t o n E m i s s i o n C o m p u t e d T o m o g r a p h y (SPECT)
Single
Photon
Emission
surprisingly early development
Computed
in nuclear
and Edwards demonstrated the theoretical
clinical
applicability of
Edwards, 1963)backprojection
Tomography
such systems
(SPECT)
medicine. The work
possibility and
in
the early
was
of
a
Kuhl
the practical
1960s
(Kuhl
and
The early systems relied on variations of the simple
technique
for
image
reconstruction,
which
severly
15
limited the quality and accuracy of the resulting images. In the 1970s,
following
the
outstanding
success
of
X-ray
CT,
images began to appear. The significant potential
and
the
number
ubiquitous
of
presence of
investigators
technologies
into
a
to
the
study
practical
Auger
the
quality
SPECT
advantages of
SPECT
scintillation
feasibility
tomographic
Wetzei, 1971; Keyes and Simon, 1973)-
good
of
system
camera
led
combining
a
these
(Muehllehner
and
Since multiple views of the body
from around its total circumference are an essential element of CT, the
major
problem was to develop a system which
Several
early
investigators
approached
holding the camera stationary and
1974; Budinger and Gullberg,
are
readily
apparent.
In
could obtain such views.
this
problem
initially
rotating the patient (Keyes et
1977)-
The
particular,
a
limitations of
patient with
with
these
systems
and
they
did
serve
to
al.,
this approach
I.V.
or
lines may be too ill to cooperate. Nevertheless, successful
obtained
by
cather
images were
illustrate
the
feasibility of the approach.
The
successful
rotating chair
which
the
camera
(Keyes et al.,
best of
and
demonstration
Auger
camera
detector
was
imaging
reali zat i ons.
tomography
to the development of
rotated
and
the camera
procedures
systems, the practicality of
demonstrated
led
transaxial
around
a
with
systems
stationary
the
in
patient
1977; Jaszczak et al., 1977; Murphy et al., 1979) • The
these designs allowed
conventional
of
this
led
as
well
ECT with
to
an
the
to be used
as
for
most or
tomography.
Auger camera was
development
of
With
all
these
repeatedly
commercial
16
The earliest SPECT imaging was of the brain. Imaging of other body
regions and organ systems, including abdominal
has
also
been
predominate
practical
and
SPECT
clinical
regional
tomography
lag
performed.
imaging.
tool
cerebral
developments
the
and
SPECT
flow.
has
Compared
the pharmaceutical
imaging
in SPECT will
technology.
thoracic scanning,
cardiac
the measurement of
blood
radiopharmaceuticals
brain
Recently,
for
(PET) system,
behind
However,
and
been
imaging
shown
cerebral
with
Further
to
blood
positron
developments
methods
be
volume
emission
in SPECT now
clinical
imaging
likely be driven by the development of
capable
of
measuring
a
metabolic
new
function
particularly in the brain and heart.
Positron Emission Tomography (PET)
1.1.3.2
Positron emission tomography
(PET)
is a nuclear
medicine
imaging
modality which yields transverse tomographic images of the distribution
of positron-emitting radionuclides which were systemically administered
to
the
subject
under
several
books
(Ell
study. The
and
Ter-Pogossian,
1985)•
intravenously
into
In
the
procedures
Holman,
general,
1982;
of
PET are
Kouris
et
radiopharmaceuticals
patient.
These
described
al.,
are
in
1982;
injected
radiopharmaceuticals
emit
positrons, which after travelling a distance from the nucleus, interact
with a negative electron and
emitted
in
opposite
results in the production of two photons
directions.
Detectors
are
made
of
mounted
scintillation crystals in which a decay occurs. The annihilation of the
radiation
consists
of
two
photons
travelling
nearly
colinear
in
17
opposite directions, the col 1imation of this radiation can be achieved
very efficiently electronically. The nearly colinear and simultaneous
emission of the annihilation photons also permits the localization of
the annihilation event by the photon time-of-f1ight method.
Positron emission tomography (PET) technique has the potential for
measurement
metabolism
(Browne 1 1
PET
has
of
as
sugar,
well
fatty
as
receptor
amino
acid,
concentration
and
any
other
where
substrate
in
the
body
et al., 1982). The compelling factors in the development of
been
the
recognition
specifically carbon-11,
fluoride-l8
particularly
fundamental
PET
acid,
has
possess
useful
that
a
small
number
nitrogen-13, oxygen-15, and
chemical
in
the
characteristics
study
of
greater
efficiency
or
radionuclides,
to a
which
biochemical
importance in biology and medicine.
much
of
less degree
render
them
processes
of
Compared with SPECT,
sensitivity
for
detecting
radiopharmaceutions. The distance the positron travels through tissue
before annihilation, the angular deviation of ±0.25° from 180° for the
two photons emitted on annihi11 ation, finite detector dimensions, and
statistical
aspects of
reconstruction are the most
important factors
that limit the ultimate resolution of the positron imaging system. Most
current instruments are capable of producing imaging with resolution of
10 to 20mm FWHM, which is the smallest separation at which two sign
sources
can
be
placed
and
still
be
detected.
In
spite
of
its
low
resolution, PET has found to be very useful
in measuring the blood flow
in
glucose
tissue,
bra i n.
determing
O2
consumption
and
metabolism
in
the
18
Comparison
1.1.3=3
between
Transmission
Computed
Tomography
and
Emission Computed Tomography
Two major difference separate TCT from ECT. First, TCT has a major
emphasis on anatomic description,
density sensitivity are very
the quantitative
and
flow
of
radioactive atoms. Just
from
that
of
TCT,
injected
the
intensity of
whereas
TCT
sources of
seeks
to
or
and
its major
changes
inhaled
mathematical
somewhat different (Figure 1).
and
momentary
as the objective of
thus
resolution
important; ECT has as
determination of
physiology
thus spatial
in
tissue
emphasis
the chemistry
compounds
labeled
with
ECT is somewhat different
and
instrument
methods
are
The ECT seeks to describe the location
emitted
determine
the
photons
in an attenuating
distribution
of
the
medium
attenuating
medium. In the ECT case, the photons emitted by sources in the body are
attenuated
in proportion to the amount of attenuating material
along a
line between the source, the body edge and the detector; however, both
the source and the attenuating function are unknown.
A
second
available
important
statistics.
transverse section
than TCT. This led
to reconstruct
each
difference
The
ECT
devices
ECT
collect
and
TCT
much
is
less
image than TCT. The expected errors will
investigators to prefer
using
that
data
of
per
be higher
iterative algorithms
the images. The ECT statistics are unfavorable because
photon must be detected
the current
between
from a
and
analyzed; whereas,
phototube excited
by many
the TCT measures
photons.
In a practical
case, the count-rate facts need to be kept in mind. Firstly, capability
19
ECT
Radionuclide source
in an organ
Profile of gamma rays
from organ isotope
distribution modified
by body attenuation
constant and thickness
from sources and edge
Source
Detectors
TCT
Prof i ie of X-ray
intensities modified by
attenuation constant
and thickness of body
IZ
Source
gure 1.
Detectors
The basic physical
A comparison between ECT and TCT.
difference between ECT and TCT is that both the source and
the attenuation functions are unknown in ECT, whereas only
the attenuation functions is unknown in TCT.
20
of
detector
crystals
and
minimun event processing
patient
imaging
collected.
associated
time of
1
limit
the
duration
Thirdly,
imaging
time
electronic
/JS.
dictates
a
Secondly, radiation dose and
number
must
circuits
be
of
events
short
which
because
can
either
be
the
isotope decays by one half-life or more during the 10-20 min study, or
the radiopharmaceutical distribution changes within a few minutes. The
latter fact of
greatest
nature represents both the greatest
potential
for
ECT; namely,
ECT
is a
limitation and
unique approach
the
to the
determination of flow and biochemical kinetics in the body.
1.1.4
Thermoelastic Waves Imaging
Thermoelastic
several
(Thermoacoustic)
different
microwave-induced
that is still
areas
(Hutchins
thermoelastic
imaging
and
Tarn,
imaging
system
is
1986a;
applies
applicable
1986b).
to a
in
However,
biomedical
area
in the experimental stage. Some researchers have applied
a slightly different approach to use thermoelastic waves as a medical
imaging modality. Their preliminary results are summarized below.
Olsen
and
Lin
(1981)
reported
acoustic pressure in a spherical
measurements of
microwave-induced
model of human and
animal brain. 1.1
GHz microwave with k kilowatt peak power and a 10 microsecond microwave
pulse was applied
detected by a
barium
through a waveguide
to the model. The signals were
1 cm diameter spherical
hydrophone which was made of a
titanate
applied on animal
piezoelectric
element.
Later,
brains, such as rats, guina
al., 1982; Olsen and
similar
setups
were
pigs and cats (Lin et
Lin, 1983) • The experimental
results agreed with
21
those predicted by the thermoelastic theory of interaction in regard to
pulse
width
absorbed
and
the
microwave energy and
1977a; 1977b; 1978).
spatial
frequency
power
of
impinging
microwaves,
frequency vibration
pattern
presented
of
in (Lin,
Another group also reported their measurement of
density
distribution
on
lossy
inhomogeneous
material
during microwave hyperthermia (Caspers and Conway, 1982) . The signals
were
recorded
either
by
a
50
KHz
reponse
or
a
500
KH2
reponse
microphone. The results show that the optimum microwave pulse width for
maximum
acoustic
results from
signal
Borth and
direct measurements of
brains
in
good
agreement
Cain (1977) -
with
the
theoretical
Recently, Lin's group presented
thermoelastic pressure wave propagation
in cat
irradiated with pulsed 2.^5 GHz microwaves (Lin et al., 1988) .
Hydrophone output
small
is
applicator,
signals detected
located
directly
show
the
pressure
wave
propagation.
at
at
the
characteristics
The
various distances away
occipital
of
pole
attenuation
measurements
of
and
clearly
pulsed microwave induced acoustic pressure waves which
an acoustic wave velocity of
lend further support for
1523 m/s
a
from
cat's
time
head,
delay
indicated
a
of
that
propagate with
in cat's brain. These
results
the thermoelastic theory of microwave-induced
auditory effects in humans and animals.
Several
groups present thermoelastic
the attenuation of pressure waves as
imaging system by measuring
it propagates through the tissue.
Olsen (1982) reported a thermoelastic image system which can obtain an
image in two-dimensional
form. 1.1 GHz pulsed microwaves were applied
22
on a
tank of water
through a waveguide. Output waveforms from a 8x8
transcuder array, which was submerged at the bottom of the water tank,
were observed
first
in oscilloscope and
acoustic
collected first
wave
of
each
the
peak-to-peak
transcuder
in the absence of a
was
amplitude of
recorded.
Data
the
were
phantom as a reference and
then
with a phantom placed on top of an array. After the second set of data
were subtracted from
those originally obtained, pixel
normalized to obtain a two-dimensional
the overall
features
Later, Olsen and
and
the
pixel, then
image. This crude image showed
respective
Lin (1983) attempted
by
phantoms were clearly
seen.
to improve the image resolution
of their system by using higher power microwave sources operating at a
higher
frequency
and
a
shorter
pulse width, and
a
higher
resolution
transducer array. In this study, a 20x20 piezoelectric transcuder array
was
made
of
lead
zirconate
titanate
material
and
with
free
field
voltage sensitivity of -201±0.1db at 100 KHz as acoustic wave detector.
A military radar transmitter at 5-66 GHz was used to generate 200 KW,
2 MS microwave pulse through a standard
repeated
exactly
oscilloscope,
as
then
keyed
processing. Resulted
matched
very
well
in
Olsen
into
horn.
(1982).
a
Data
PDP11/23
The same procedure was
were
read
minicomputer
images were displayed on a graphic
with
the
original
phantom.
Later,
from
for
further
terminal
Lin
the
and
and
Chan
(1984) reported a system design automating the data acquisition in the
previous
procedures.
thermoelastic
tissue
Results
imaging
from
this
prototype
system
show
that
microwave-induced
the system
can detect
23
objects of about 1 cm size with minimum separation of about 2 cm from
each
other,
contents
biological
can
be
materials with
differentiated
from
high, moderate, and
resulting
images
low water
(Chan,
1 988).
Their results led to this present research.
Another type of
thermoelastic wave was suggested
for soft tissue
imaging at about the same time (Bowen et al., 1981). In their study,
pulsed electrical current with 75 V to 1.3 KV was applied across metal
plates to a
signals
layer of soft tissue phantom and
were
detected
by
a
transducer
then generated acoustic
located
perpendicular
direction of the applied electrical current. In 1983»
observation of acoustic signals from a phantom
to
the
they reported an
in an 18 MeV electron
beam for cancer therapy (Bowen et al., 1984a; Bowen et al., 1984b). By
using signa1-averaging of successive pulses to differentiate and locate
regions
of
coefficient
different
and
heat
electrical
capacity
in
conductivity,
a
tissue
thermal
phantom.
expansion
Nasoni
et
al .
(1984) reported the measurement of thermoacoustic waves induced at the
interfaces of deliver microwave energy to the cavity. The signals were
received by a
1
MHz resonant PZT5 transducer, sampled at 20 MHz, 100
MHz and 5 MHz sampling frequencies and
samples.
Although, different
energy
then averaged over
level
and
sources
1
to 64000
were
applied.
Howev.er, the properties of the induced acoustic signals agreed with the
predictions of the theory of thermoacoustic emission reported in Bowen
(1981).
Moreover,
Su
(1985)
and
Su
and
Lin
(1987)
also
thermoacoustic wave to obtain the thermoelastic signatures of
used
a
tissue
2k
phantom
absorption
and
thermal
expansion.
A
pattern
extraction
algorithm was developed to analyze the acoustic wave patterns generated
from biological
sizes.
The
simulated
results
phantoms contained
suggested
that
in test tube of various
thermoelastic
waveform
would
be
proportional to the difference of energy absorbed between two different
med i a.
The theoretical calculation and experimental results from previous
studies
suggested
that
thermoelastic
imaging
may
appears
to
some unique features that may allow them to become as useful
other
methods
and
to
permit
non-invasive
imaging
possess
as these
of
tissue
characteristics which can not be identified by other techniques.
1.2
Objective
The
objective
of
this
project
tomography using microwe-induced
This
would
receiving
involve
system
(a)
to
implementation of signal
and
digitization;
(c)
design
obtain
is
to
investigate
thermoelastic waves for body tissues.
and
fabrication
the
of
projections
a
of
scanning
and
object;
(b)
conversion circuitry for signal
modification
obtain the image; (d) digital
computed
of
reconstruction
conditioning
algorithm
to
processing and display of thermoelastic
images of the irradiated region.
However, many limitations are imposed
in order to simplify the system.
The long-term goal of the project
to
develop
imaging
microwave-induced
techniques
for
body
thermoelastic
tissues
which
imaging
into
would
yield
information complementary to other imaging modalities.
is
specialized
diagnostic
25
1.3
Organization of This Thesis
The
background
assistant
imaging
information
tomography
are
properties
and
presented
of
review
in
biological
on
on
Chapter
tissues
different
types
of
microwave-induced
1.
which
In
Chapter
relates
thermoelastic
2,
to
computer
the
physical
microwave-induced
thermoelastic wave characteristics are described. Later theoretical and
experimental
data of
Chapter
reconstruction
3.
reconstruction
thermoelastic waves are given for comparison. In
theorem.
algorithms
are
These algorithms
derived
based
are suitable for
on
the
different
geometric cases.
Computer simulation results for two different cases (TCT and ECT)
are
presented
ideal
in
Chapters
conditions.
acquistion
and
image
imaging system are
phantoms
and
The
the
k
and
hardware
testing
The
system,
processing
introduced
S-
in
results were obtained
the
routines
implementation
for
protocols.
They
are
of
thermoelastic
Chapter 6. Chapter
under
data
tissue
7 describes
designed
to
test
the
the
limitation and capability of this system in tissue phantoms imaging.
Finally,
processed
in Chapters 8 and 9. data collected
with
limitations of
different
the system
reconstruction
are
by
algorithms.
the system
are
Advantages
and
then discussed. Some further
suggestions are made to conclude this thesis.
research
Chapter 2
THERMOELASTIC WAVE PROPAGATION IN BIOLOGICAL TISSUE
In
this chapter, a
brief
review of
thermoelastic waves will
be
presented.
Some parameters of the microwave source and properties of
biological
tissues directly related to the generation of thermoelastic
waves by pulsed microwave will be described.
2.1
Physical Properties of Biological Tissue
It
is
important
to review
properties of
biological
thermoelastic
waves
comparison among
materials for
imaging
physical
the microwave,
of
thermal, and
a better
biological
properties of
acoustic
understanding of
tissue.
biological
To
simplify
the
the
tissues, we would
like to classify biological materials into three major categories based
on their
percentage of water
content. The first group is "high water
content" (90 percent and more) such as blood and cerebrospinal fluids
etc..
The
second
percent) which
group
is
"moderate
includes skin, muscle,
water
content"
brain and
(less
internal
than
80
organs. The
third one is low water content group (about 50 percent) which consists
of bond, fat and tendon.
2.1.1
Microwave Properties
The
microwave
characterized
by
the
properties
of
dielectric
biological
constant,
er,
structures
and
the
.are
effective
conductivity, u. Due to the conduction currents and dielectric losses,
26
27
these two parameters together determine the amount of microwave energy
transmitted into and absorbed by tissue media.
The electric field E inside a given tissue medium is
_
-az-j(3z _
E = |Eo| e
X
where
Eo
is
the
magnitude
of
the
(2.1)
electric
field, a and
3
are
the
attenuation and propagation coefficients respectively, and are defined
as (Stratton, 19^1)
a = a>{(e r /2)[(1+cr 2 ) w 2 e r 2 e Q 2 ) 1 '2 -1]
}1'2 /C
(2.2)
0 = «{(e r /2)[(l+a 2 /« 2 e r 2 6 0 2 ) l ' 2 +l]} l ' 2 /C
where
C
and
eo
are
the
speed
of
light
and
(2.3)
permittivity
in
vaccum
respectively.
As the wave propagates through
the medium, energy
is dissipated
from the wave and absorbed by the medium. The absorption will result in
a
progressive
reduction
distance
is
in
Calculated
presented
reduction
quantified
which
depth
the
of
in Table II.
of
by
power
the
the
transmitted
depth
density
penetration
for
of
power
penetration
decreases
the
by
three
It can be seen that <5
a
density.
This
o~]/a as
factor
tissue
of
groups
the
14%.
is
is frequency dependent.
Less penetration occurs as frequency is increased in each group.
The transmission of microwave at
'sssue boundaries is determined
by the complex reflection and transmission coefficient. For example, in
28
TABLE I I
DEPTH OF PENETRATION OF MICROWAVE IN BIOLOGICAL TISSUE
a
Ti ssue
Group 1
Frequency (MHz)
Blood
Sali ne
Group 111
Group 11
Fat
(Bone)
Muse 1e
(Sk i n)
Depth of penetration (cm)
*•33
2.8
3-7
3-0
16.3
915
2.5
3.0
2-5
12.8
2U50
1.3
»-9
1.7
7-9
5800
0.7
0.7
0.8
k.l
10000
0.2
0.3
0.3
2-5
a Borrowed
from Lin (1978).
Figure 2, a plane wave polarized in the positive x direction propagates
through
medium
1
in
the
positive direction
encounters medium 2 at a planar
the
wave
propagates
characterized
thickness
of
by
an
the
from
one
complex
object
is
of
the z axis.
The wave
interface lying on the z=0 axis. When
medium
to
dielectric
greater
than
another,
constant
one
the
situation
(e).
Assuming
penetration
depth,
complex transmission coefficient is defined by (Stratton,19^1):
is
the
the
29
where e x = eo (e r -ja x /w6o) ; x=l,2 and the transmitted power density
related
(1-R 2 ),
to
Table III
where
presents
the
R
is
the
amount
complex
of
reflection
transmission
at
is
coefficient.
various
tissue
boundaries at different frequencies. The transmitted energy is strongly
frequency dependent.
Table IV presents the dielectric
(cr)
and conductivity
for brain, muscle, bone and fatty tissue summarized
others
(1953.195^»1955»1957)• The dielectric
increasing
frequency
while
the
attenuation coefficient, a, e r
higher
surface
frequencies.
in
any
case,
and a as
decreased
a function of
most
frequency
by Schwan and
increases.
in Table IV.
Although,
low
constant
conductivity
each type of tissue is also given
at
constants (e r )
computed
frequency
for
Attenuation increases
energy
energy
The
with
is
absorbed
penetrates
at
deeper
the
into
a
given type of tissue.
Thermal Properties
2.1.2
Thermal
heating
interaction
that
results
may
from
involve
either
absorption
of
general
microwave
or
local
energy
tissue
with
or
without measureable whole body temperature elevation.
The thermal
properties of biological structures are characterized
by the specific heat and
local
transient
and
thermal
steady
state
conductivity. These can predict the
temperature
distributions
and
heat
transfer due to microwave exposure. Table V is an abridged
collection
of
biological
existing
materials.
information
on
the
thermal
properties
of
The thermal conductivity values show that the body tissues
30
MEDIUM 1
Mi
MEDIUM 2
,crt
2
x
*
DIRECTION
OF
PROPAGATION
Z—0
Figure 2.
Propagating
plane
wave.
The
dielectric
constant,
conductivity and permeability determinate the behavior of
plane wave propagating through two media.
31
TABLE I I I
COMPLEX TRANSMISSION COEFFICIENTS FOR SYSTEMS CONSISTING OF TWO
BIOLOGICAL TISSUE
Group I
Material
Ai r
Frequency
(MHz)
*•33
915
2^50
5800
10000
Sa1i ne
*•33
915
2U50
5800
10000
Blood
*•33
915
2^50
5800
10000
Muse 1e
1 »33
915
2U50
5800
10000
Group I I
•
Muscle
(Skin)
Group I I I
Saline
Blood
0.311
0.36
0.36
0.392
0.392
0.3M
0.328
0.788
0.376
O. 8 1 5
0.1«07
0.1»22
0.^22
0-392
0.1»22
0.i»38
O.I»52
0.832
0.8i»8
O. 8 6 3
1
1
1
1
1
0.996
O. 9 9 8
0.997
0.997
0.996
0.996
0.995
0.99*4
0.992
0.992
0.6^
0.675
0.675
0.686
0.66^4
1
1
1
1
1
0.998
0.998
0.997
0.999
0.686
0.708
0.719
0.719
O. 7 0 8
1
1
1
1
l
0.686
0.73
0.75
0.75
0.73
0.998
Fat
(Bone)
TABLE IV
MICROWAVE PROPERTIES OF BIOLOGICAL TISSUE
Mater 1 a 1
Bra i n
Frequency
(MHz)
Dielectr i c
Contant
er
5000
10000
38.4
35-6
32.0
28.8
25.1
Muscle
^33
915
2450
5000
10000
53
51
47
44
40
Fat
(Bone)
433
915
2450
5000
10000
433
915
21*50
a
Borrowed from Lin (1978).
5.6
5.6
5-5
5-5
4.5
Conduct i vi ty
6
mho/m
3
Attenuat i on
Coeff i c i ent
ot .cm* 1
0.77
0.85
1.32
3.02
9.08
0.22
0.26
0.43
1 .04
3.06
1.43
1.60
2.21
3.92
10.3
0.33
0.^0
0.60
1 . 10
3.00
O . O 3 8 --O. 1 1 8
0.056—0. li»7
0.096--0.213
0.020--0.080
0.044—0. 113
0.077--0.169
0.162--0.309
0.130—0.247
0.324—0.549
0.287—0.481»
33
are relatively poor conductors. The coefficients of
thermal
expansion
for a number of materials are included in this table.
TABLE V
THERMAL PROPERTIES OF BIOLOGICAL MATERIALS
Mater i a 1
Therma1
Spec i f i c
Conductivi ty
Heat
Ca1/msec°C
Coeff i c i ent
of Thermal
Expans i on
Therma1
D i ffus i v i ty
Cal/g°C
a
J0" 7 m 2 /sec
lo
'Vcr1
Di sti1 led Water
0.15
0.998
1.50
6.9
Bra i n
Muse 1 e
0.126
0.122
0.88
0.75
1.38
1.52
U. lit
k.]k
Fat
Bone
0.0525
0.35
0.62
0.1»9
0.873
1». 20
2.76
2.76
a
Borrowed from Lin (1978).
In addition to thermal properties of the irradiated tissues, other
factors determine the amount of change
induced
in body temperature that can be
by microwave energy absorption. These
include
the actions of
the different .heat production and dissipation pathways of
the body and
the
in
body
surface
area.
Also,
the
conduction
process
the
body
involves heat flow through body tissue due to the temperature gradient.
3^
So, the thermodynamics of
tissue heating can be represented by (Emery
et al1976).
dT
- =
0.239 X10"3
J
(w a +w„-w c -w b -w s )
(2.5)
Where W a =ct E2 /2 p is the specific rate of absorption, S is the specific
heat
in Kcal/Kg°C, T is the temperature. W m
heat production, and W c , W^, and
is the rate of metabolic
Ws are the rate of
heat dissipation
due to thermal conduction, blood circulation and surface transfer.
When
energy
the
incident
power
density
absorption are high, T will
and
specific
increase with
a
rate of
microwave
linear
transient
initial period and is characterized by (Guy et al., 197^).
dT
0.239x10*3
- »
WA
dt
(2.6
a
S
Acoustic Properties
2.1.3
The acoustic properties which determine the propagation of sonic
energy in tissue are the sound velocity and
the absorption coefficient
of the medium. Also, we can express the acoustic properties in terms of
Lame's constants and the density of the materials by (Love, 1927):
V = [(X+2M)/P]1 /2
where
V
is
the
bulk
velocity
of
acoustic
(2.7)
wave
propagation
medium, X and m are Lame's constants, p is volume density.
in
the
35
A number of articles have reported the numerical
parameters
for
various
tissue
materials.
The
values of
most
these
comprehensive
compilation of the acoustic properties obtained with mammaliam tissues
is
presented
by
discrepancies
authors. At
(Goss et
between
least
al.,
the
1978;
value of
part of
1980). This survey
absorption
this variation
measurement techniques which were employed.
that tissues are not
homogeneous and
obtained
is caused
by
shows
by
large
different
the different
The root of the problem is
therefore consist of regions
in
which the ultrasound travels at slightly different velocities. Any two
portions of the incident ultrasonic beam will
cells and will
the detector
travel
through different
therefore arrive at detector slightly out of phase. If
is sensitive to the phase of the incident wave (e.g., a
piezoelectric
transducer)
then
the waves may
destructively
interfere
with each other and so an aritifactua11y low signal may be detected and
interpreted as a falsely high attenuation coefficient. A detector which
is
not
sensitive
to
phase
(e.g.,
acoustic-electric receiver) yield
a
radiation-force
receiver
or
an
higher signals which are interpreted
as a lower attenuation coefficient. However, as long as same method was
applied
during
the
measurement,
the
ratio
of
attenuation
between
different tissues is very close.
Table VI
presents
for five typical
(Goss
et
squares
al.,
linear
some
representative
mammalian tissues as a
1979)-
The
last
two
regression power fit
acoustic
function of
columns
attenuation
applied frequency
display
to the equation
data
the
best
least
A=af' 3 where f
is
36
frequency in MHz. The correlation coefficient R describes the goodness
of fit.
It can be seen that tendon has an attenuation coefficient at
low frequencies
which
Figure 3 presents
a
is
about
brief,
ten
highly
times
larger
selected
than
graphical
tissue attenuation values as a function of frequency.
brain
tissue.
compilation
of
It can be seen
that at any given frequency lung and bone have the highest attenuation
followed by tendon, muscle, kidney, liver/nerve/brain, and fat with the
fluid tissues such as blood having the least attenuation.
2.2
Mi crawave Induced Thermoelastic Waves
The overview of
microwave
induced
thermoelastic
waves
has
been
presented (Lin, 1978; Lin, 1980; Chou and Guy, 1982) which includes the
psychophysical
observations
and
the
interactive
mechanism.
Pulsed
microwaves have been heard as sound by radar operators since radar was
invented
beings
during
can
World
hear
War
II. Frey
pulse-modulated
(1961)
microwave
first
reported
energy
that
transmitted
human
through
the air. He initiated research by selecting people who have sensed the
phenomenon.
average
He
power
exposed
them
to
densities of O.k
1310MHz
and
2982MHz
microwaves.
to 2 mw/cm 2 , the auditory
At
sensations
perceived by the subjects were reported as buzzing or knockinmg sounds.
The
mechanism
of
"radio-frequency" (rf)
hearing
remained
obscure
for
more than a decade. Frey (1962; 1971) suggested that direct stimulation
of
the nervous tissue might be responsible. Subsequent studies using
psychological, physiobiological, behavioral, physical, and theoretical
approaches
have
revealed
what
most
investigators
now
believe
is
the
TABLE VI
MEASURED ACOUSTIC PROPERTIES OF BIOLOGICAL MATERIALS 9
Group
Material
Bulk
Velocity
(m/sec)
I
Blood
1550
11
Liver
Brain
Muscle
15321560
1520
1580
Fat
Tendon
Bone
1440
1750
3360
III
Frequency
(MHz)
0.8
Attenuation!Nepers/cm) at
Frequency (MHz)
0.5
1.0
3.0
0.031
0.069
0.038
0.080
0.290
0.032
0.120
0.070
0.160
0.240
0.450
0.330
0.070
0.560
1.500
0.180
1.300
'Summarized from Lin(1978), Goss et al.(1979), and Williams(1983).
Regression Analysis Fit
1 13
A=0.08r ,i,J
R=0.934
A=0.07f1,14
R=0.822
n n KCfU.763
A=0.56f
R=0.998
38
Lung
10
Bone
Tendon
Heart and muscle
Kidney and fat
Brain and liver
CL
—Blood
001
10
Frequency (MHz)
Figure 3*
Acoustic attenuation of different biological tissue.
This
graph shows the attenuation coefficient varies rapidly as
function of sonic frequency. For different water content,
different attenuation was measured in acoustic frequency
range.
39
mechanism
of
rf
hearing:
thermoelastic
expansion,
first
reported
by
Foster and Finch (197M •
According
to
the
thermoelastic
expansion
mechanism,
rapidly
absorbed energy from pulsed microwaves can not only be dissipated
by
conduction processes, but can cause expansions of volume and changes of
pressure (Love, 1927)- Whenever such forces are applied to media which
are not too lossy, acoustic waves will
propagate.
Theoretical studies
have been well covered in numerous publications, including the text by
Lin (1978). Complete solutions of
the thermodynamic equations for
one-dimensional case are presented by Borth and Cain (1977)-
the
According
to them, the spectrum of acoustic waves is described by
.
,
6/2y V 2 o: I
,[1-cos(wT)]1/2
0 {IF M I > —2^
, }
S
<j) 2 +lt\l 2 a 2
(2.8)
for free surfaces;
or by
!FM| ,
(2
a> 3 +4a)V 2 a 2
S
.9)
for constrained surfaces;
The parameters of these equation are explained
to (2.8) and (2.9) »
in Table VII. According
the spectra (appendix A) of acoustic waves depend
on the physical properties of the irradiated media.
In
the
absorption
preceding
of
sections,
microwave
in
we showed
biological
that
the
materials
propagation
are
governed
and
by
uo
TABLE VI I
PARAMETERS FOR EQUATIIONS (2.8) AND (2.9)
Symbol
Physical Property
T
Microwave pulse length
IQ
Microwave peak power density
to
Acoustic wave frequency
S
Specific heat
p
Displacement function
v
Sound velocity
Coefficient of thermal expansion
Attenuation coefficient
dielectric
constants,
Regardless of
conductivities
the geometry,
and
geometrical
the values of
a
arrangement.
dielectric constant
and
conductivity of tissues with low water content are an order of magitude
lower
than
the
corresponding
values
for
tissues
content. This is probably so because the polar
of
macromolecules
thermal
In
properties
electrolytes
of
biological
materials
the
generated
characteristics
lead
us
thermoe1astic
to
use
modality for biological tissue.
waves
also
water
The
acoustic
depend
on
and
water
to different biological
will
thermoelastic
higher
properties of water and
become dominant.
content. By applying the same amount of energy
tissues,
with
be
waves
different.
as
an
This
imaging
Chapter 3
RECONSTRUCTION ALGORITHMS
Techniques in CT can be divided
Methods and Transform Methods.
into two broad classes:
Iteration
An overview of these algorithms can be
found in (Herman, 1979; Hall, 1979; Deans, 1983; Lewitt, 1983; Censor,
1983).
Iteration methods are the most generally applicable techniques,
however, to avoid excessive error propagation and to speed convergence
of
numerical
"known"
methods,
before
they
require a
successive
characteristics.
The
"guesses"
Transform
imaging.
algorithms, such
use
the
methods
tool
of
The
Fourier
independent
characteristics.
any
Therefore,
that
made
the
for
rise
to
in commercial
and
analysis
of
knowledge
a
must
be
object's
variety
of
CT scanners for
implementations
of
those
inversion and convolution-backprojection,
possess advantages due
otherwise
be
gives
used
principles
as Fourier
can
method
algorithms, including those widely
medical
priori
we
to
which
their
prior
plan
are
familiar
computational
knowledge
to
use
about
the
to
us.
speed
the
transform
These
and
are
object's
methods
technique to develop the reconstruction algorithm.
Different
algorithms
could
be
implemented
to
reconstruct
the
images based on two types of signals. Those two types of signals that
could be generated at the same time use the setup in Figure k.
One is
from the surface between the microwave source and phantom, another one
b)
i+2
is due to inhomogeneity between two materials in the phantom.
discuss
these
two different
types of signals
later.
We will
Figure 5 shows
those two signals which can be separated using time-gating techniques.
Moreover, Those two type of signals could be simulated as two different
projections
which
(transimission
tomography)
are
used
computed
cases,
to
reconstruct
tomography)
respectively.
and
the
ECT
Therefore,
image
(emission
two
in
TCT
computer
different
types
algorithms which are used to generate tomography will be discussed.
3•1
Transmission Computed Tomography Alaor i thms
An
idealized
be described
in
the following way. Consider a microwave radiation system as shown
in
Figure 4.
When
projection-measurement
a
beam
of
process
microwave energy
may
impings
normally
on
the
boundary of
a semi-infinite region of homogeneous tissue material, a
portion
the
of
converted
incident
radiation
is
absorbed
by
the
tissue
into heat which generates a temperature gradient normal
the surface. As a result of thermal
and
to
expansion occurring within a few
microseconds, this temperature produces strains in the tissue material
and
leads to generation of stress waves which propagate away from the
surface.
describe
Several
the
articles
displacement
(Gournay,
and
1966;
pressure
Hu,
1969;
generated
expansion in detail. The elastic wave energy produced
by
Lin,
1978)
thermoelastic
is characterized
by the followimg equation:
AaC32|o (1-e
at ^°-a:tCToe a t^°)
Ea =
2pJ2S2
atC
(3-1)
*•3
|\^y\/
Hydrophone
(Project ions)
Applicator
Rotary
Tank
filled
w i t h
w a t e r
Figure 1».
Output
Phantom
Hydrophone
A r r a y
Simplified diagram
imaging apparatus
of
the
microwave-induced
thermoeI as t i c
1,1
'AV*r-,u'
*+
100 mv
Figure 5•
Typical hydrophone output signal.
Two different signals
obtained, since the distances from hydrophone to applicator
and to object are different. From time scale we can seperate
each other.
where the parameters are explained in Table VIII.
As
occurs
this acoustic wave
which
received
at
is
the
assumed
propagates
to
be
piezoelectric
inside
the
exponential.
transducer
can
tissue, attenuation
Therefore,
be
the
expressed
energy
by
the
equation:
E (1,4>) « Eaexp {-fn(x,y)ds)
Where v
is
the attenuation coefficient of
media, and the projection can
the
(3.2)
thermoe 1 ast i c
be represented simply by
wave
in
'•5
TABLE VI I I
PARAMETERS FOR EQUATION (3.1)
Symbol
Meaning
A
Area through which the elastic wave is propagated
ijAa
Power output from a microwave pulse generator
C
Velocity of elastic wave propagation through media
/3
Coefficient of thermal expansion
P
Densi ty of media
J
Mechanical equivalent of heat
S
specific heat of media
Attenuation coefficient of microwave in media
T0
Pulse width
(3.3)
p(l,<*>) = -ln~— = /sm(x,y) ds
s
Ea
There are several equivalent ways of describing the derivation of
the equation which will express m at a given point in the plane in term
of integrals over the projection value
p(l,<#>). Both Herman (1980) and
Barrett
the
and
sequential
Swindell
(1981)
application of
describe
the Abel
inversion
transform,
the
process
Fourier
as
the
transform,
i+6
and
the
Hankel
inversion
from
transform.
the
However,
Fourier
easily understood and will
it
transform
be useful
is
possible
alone.
This
for us at a
to
derive
approach
is
the
more
later stage of
our
development.
Once
the
can
be
modified to account for the finite number and size of sources used
in
collecting
integral
actual
scanner. There
inversion
data
are
from
multiple
a
real
ways
of
multiple ways of discretizing
Kak, 1982).
expansion of
equations.
expression
the
is
obtained,
Computerized
collecting
integral
it
Tomography
the
X-ray
expression
(CT)
data
and
(Rosenfeld
and
The pioneering publication by Cormack (19^4) used a series
harmonic functions to obtain a solution to the discrete
However,
most
CT
convolution-backprojection. This
scanners
use
a
method
known
as
involves a direct application of the
trapezoidal rule to the integral expressions, along with band limiting
of the functions in frequency domain. This method will
some detail
for
be derived
the two data collection geometries most common
in
to CT
scanners which are currently available, fan beam and parallel beam.
Project ion Theorem
3.1.1
The
Fourier
Fourier
Slice
transform
two-dimensional
of
Theorem
a
Fourier
which
projection
transform
of
of
relates
a
the
function
M(X,Y)
is
one-dimensional
M(X,Y)
the
to
basis
the
of
reconstruction techniques..
Let
M(Kx,Ky)
def i ni t ion
be
the
Fourier
transform
of
the
image /z(x,y).
By
^7
M(Kx,Ky) >= // n (x,y) exp[-2jri (Kxx+Kyy)]dxdy
(3-^)
Also let P(K,<#>) be the Fourier transform of the projection p(l,<£) that
is
P(K ,4>) "7 p (1 ,<£) exp (-2iri K1)d 1
—oo
if
M(K,4>) denotes
the values of
H(Kx,Ky) along
with the Kx-axis (Figure 6), and if P(K,<£)
(3-5)
a
line at an angle
is the Fourier
transform of
the projection p(lt<£) then
M (K ,<#>) - P (K ,<£) - M(Kx,Ky)
(3.6)
K,
K
Source
M(Kx.Ky)
Figure 6.
Illustration of the projection
Fourier projection theorem.
measurement
system
and
the
48
To
prove
this,
coordinate
let
system
MO.S)
(l,s)
in
be
the
function
Figure 6.
The
n{x,y)
in
coordinates
the
rotated
(l,s)
are
related to the (x,y) coordinates by
•
cos<t>
sin<£
x
-s i ncf>
cos4>
y
(3-7)
L
c 1 ear 1 y
P (114>) = J>(1 ,s)ds
(3.8)
Therefore
P(K,<£) = / p (1 ,4>) exp (-27T i Kl) dl
(3.9)
= // M (1.s)dsexp ((-27Ti K1) dl
Using the Jacobian to transform the right-hand side of (3-9) into (x,y)
coord i nates,
dx/31
dxdy =
dx/ds
dlds = dlds
(3-10)
we get
OO
P(K,<£) = //m(x,y) exp[-27ri K (xcos<^+-ys i n<j!>) ]dxdy=M(Kx> Ky)
= M(K ,4>)
which proves (3*6). This
, where Kx=Kcos# and K y =Ksin<£.
(3-11)
is usually known as the Projection Theorem.
Substitution of (3-11) into (3.4) yields the desired inversion relation
for ,u(x,y) in terms of the projections p(l,<£).
49
M(X,y) = f f P (K,<£) exp[27ri (Kxx+Kyy)]dKxdKy
where
P(K,<£)
is
given
by
(3*5) •
Using
the
polar
(3-12)
Fourier
space
coordinates K and <£, we can write (3.12) as
2iz °°
M(x,y) = I f P (K,<£) exp[27ri K (xcos^fys i n«£) ]KdKd<£
(3-13)
where <£=tan-1 (Ky/Kx)
It
is straightforward
to write the spatial
coordinates
in polar form
yielding an elegant formulation:
2TZ
°°
m(t,0) = J / P (K,<£) exp[27ri Krcos (0-0)]KdKd<£
(3-14)
where x=rcos0, y=rsin0.
3.2
F o r m u l a t i o n o f P a r a l l e l Beam P r o i e c t i o n
The early
CT scanners collected
which were separated from each other
data
in sets of
by the angle of
parallel
beams
<t> of Figure 7«
Equation (3-14) could be used with this geometry, but it easy to show
that it is only necessary to collect data over ir radius. (3-14) can be
rewitten as follows,
m((*,0) = ^ /|
|
K P (K,<£) exp[27ri Krcos (<t>-9) ]dKd<£
(3*15)
By using the property P(K,d+n)=P(-K,6) , then
M(r,0) =
1 S f p (1 ,<t>)|
|
K
exp[27ri K (rcos
-1)]dKd<£dl
(3-16)
50
The
parameter
integration
in
K
has
(3-16)
spatial frequencies.
the
must,
in
dimension
of
principle,
spatial
be
frequency.
carried
Thus the projections may
the width of
over
all
In practice, however, the energy contained in the
Fourier transform components above a certain frequency
both
out
The
be considered
to be band
the source beam and
is negligible.
limited. Obviously,
the distance between distinct
beams will place a maximum on the spatial frequencies obtainable from a
given
scanning
system.
If
the
beams
are assumed
to
be
sufficiently
narrow, then it is the incremental spacing of beams, Al , which provides
the Nyquist
limit,
=
1/2A1.
Also
if we assume
that
object lies within a circle of radius R, then the real
part of (3-16)
can be written as follows.
^
(3.14) simplifies to :
K
the scanned
p(1,0) Kcos[27rK (rcos (<^>-0) -l]dKd
51
(3-18) could be discretized as
it
singularity
of each fraction. However, it
in the denominator
is if care were taken to avoid
the
is much
more efficient computationally to evaluate the inner integral of (3.18)
for the particular values of the variable <t>
where h
is an
such that rcos (<£-0) = hAl
integer. This has the advantage of
reducing
the
inner
integral to a convolution which does not depend on angle <£. Using this
idea, each set of data p(l,<£)
is used to evaluate the inner integral at
as many points as there are elements in the data set. If these are not
the required
points for
a given
(r,#,<£),
the required
points can be
interpolated from the computed ones.
Discretized Form for Parallel Projection
3-2.1
The discretized (3.18) will have the following form:
ir/A£
n(r,6) =
2 AM.
J =1
j,H(j,r,0)
R/Al
where A j>h = , = _f /A)ah, i Pj, i
(3.19a)
(3.19b)
and
Pj , j = P(iAl,jAj>)
H (j>r,0) =
It is common to use a
linear
Crcos (jAtf>-0)]/Al
interpolation to find
computed Aj^ which described in the next section.
(3.1
(3.1
the Aj^ from the
The a^j of (3.19b)
are usually called the elements of the "convolution filter". They are
obtained by evaluating (3.18) with KN = 1/2A1.
For h * i
A j,h
=
i= _f /A]
+
2 P (iAl 'J A ^ ,){
(2tt) Mh-i) 2A1
kir (h- i) Al
For h = i, we can modify (3-17) to
M(r,0) = J jfp (1,<£) KN2dldc£
(3.21)
then
(3-22)
2
p (i Al ,jAtf>) /4A1
i =-R/Al
(3-18) and (3-19) define the convolution filter, a^j of (3.19b) with
, h=i
kA]
h-i,even,*0
(3.23)
7T2 (h-i) 2A1
This filter is called the "Ramachandran" filter after the man who first
derived it (Ramachandran and Lakshminarayanan, 1971)•
3•3
F o r m u l a t i o n o f F a n Beam P r o j e c t i o n
There are two types of fan projections. The projected beam can be
sampled
at
illustrated
equiangular
in Figure 7•
or
equispaced
intervals.
This
difference
The algorithms that reconstruct
is
images from
these two types of fan projections are different. Here we derive both
cases.
53
SO
SO
DS
DA
SO: Source
\
DA: Detectors in equiangular.
DS: Detectors for equispace.
Figure 7-
3•3-1
Two
types
of
fan-beam
project
system.
(a)Equiangular
raysrthe spacing between the detectors would be unequal if
they are arranged on a straight line, or equal if they are
arranged on a circular arc. (b)equispaced detectors arranged
on a stra i ght 1i ne.
Equiangular Rays
Obviously, the cartesian coordinate system, which is the basis for
parallel
beam data collection,
collection.
is
not
We must make a coordinate
appropriate
for
fan
transformation
the integral equations. Substituting (3.11) into (3. 1
transformation
Figure 8.
of
coordinates
to
The (a,0) coordinate system
be
used
is
data
beginning with
:
n ( r , 6 ) » ^ ^ / p(1 , < f > ) Kexp[27ri K (rcos (<£-0) -1]d I dKd0
The
beam
(3-21*)
illustrated
is not orthogonal
in
and does not
Sh
cover
all
space.
However, we can
make
the
transformation
if
it
is
assumed that ju(r,0) is nonzero only within a finite region of radius R
less than D. Here D
is the distance from the origin of coordinates to
the source. We limit
1
in
to a radius R. From Figure 8 we can
see the following relationship.
with
and
1 = Dsina , a = sin-1 (1/D)
<t> = o+3
, 3 = 4>~ot
Using the Jacobian to transform the volume element dld$ we have;
dl/da
d1d<£
?)</>/ bee
(3.25)
01/33
0</>/&3
dad3 = Dcosadad3
Then, (3.2U) becomes;
27f s i n * IK/U;»
f
/p(a,3) KDcosaexp[27ri K (rcos(a+3-0)
0 -sin -1 (R/D)®
-Ds i na]dadKd3
M(r,0) = J
It
is
desirable
Unfortunately, the power
to
implement
(3-26)
as
in the exponential factor
a
(3*26)
convolution.
in (3.26) does not
appear as a difference between the variable a and a function of 3 and
0.
However, It is possible to regroup the terms into such a form by
making the following definitions:
U(r,0,3) = [(D+rsi n(3-0)) 2+r2cos2 (3-0)]^
(3•27)
a1 (r,0,3) = tan"1[rcos(3-0)/(D+rs i n(3~0))]
(3-28)
SO
to
,
V
DA
b
y
so
Arbitrary ray
Figure 8.
Coordinate system for fan-beam with equiangular case.
(a)A
new coordinate system
(or,3) to be used with fan-beam
geometry. For reference, the parallel beam coordinates (!,<£)
are included, (b)A schematic representation of the fan-beam
geometry with designation of the quantities U(r,6, 3) and
a' (r,0,3) •
56
We can rewrite (3-26) as follows.
llcosap (a,3) /Kexp[27ri KUs i n (a 1 - a )]dKdad0
-sin"1 (R/D)
0
(3.29)
M (r,0) = J
0
where
U(r,0,(3):
the
distance
from
a
source
at
angle
3
to
the
reconstruction point (r, 6), and a': the angle made by the ray from the
source at angle 3 which goes through the point (r,6). (Ref. Figure 8)
As
interval
in
the parallel
projection case, we must
limit the frequency
in some physically meaningful way. Herman et al. (1977) chose
the following frequency limit:
Kjg
= l/[2Aa :L)s i n (a 1 -a) / (a 1 -a) ]
(3*30)
where Aa is the equi-angular spacing between the rays of the fan.
Using the limit of 3-30 in 3-29 and evaluating the frequency
integral
yields the expression;
COS I
sin"1 (R/D)
2IT
Dcosap(a,3)
M(r,0) = /
/
... .
.— r {•
-1
o — s!
n ~ 1 (R/D) U 2 s i n 2 (a' - a )
in
(a 1 -a)
+ -7—-—
kTT&OZ
3-3.2
7r {a' -a)
.i —in—— [ — 1
Aa
(2tt)
_7r(a'-a)
sin[
]}dad3
oa
(3-31)
Piscretized Form f o r Equiangular Rays Case
For equiangular case, the discretized form (3-31) for those values
of the variables 3 such that a' = hAa where h is an integer, is:
M r .6 ) =
2
J ^ % S B
J
f
H
(
J
( 3 . 3 2 a )
57
where;
sin-1(R/D)/Aa
. -1/D/nx /A bh» ipj. i
i =-s i n (R/D)/Aa
(3-32b)
Pj,
|
= p(iAa,jA|3)
(3.32c)
H(j,r,0) = (1/Aa)tan-1 [rcos (0-0)/(D+rsin (3-0))]
(3-32d)
B J« h
=.
and
From (3.31), we have, for h*i:
[s i n _1 (R/D)]/Aa Dcos (i Aa) p(iAa,jA(3)
Bj
' h i = - [ s i n~^(R/D)]/Aa
U 2 s i n2( ( h - i ) Aa)
C
cos (7r(h- i))-1
h-i
——
+ — - s i n ( ( h - i ) 7 r )]Aa
(27r) 2
1»7T
(3-33)
then;
(sin_1^R/D))/Aa DAacos (i Aa) p (i Aa,jA(3)
-l/27r2,h-i ,odd
(3.3*0
J
* V=-(si n_i (R/D))/Aa U2si n2 ((h-i)Aa)
For h = i, set a = a1
B:
V. 0 ,h-i .even
in (3*29). (3*30). Then
(s i n_1(R/D))/Aa DAacos (i Aa)Ap (i Aa,jA|3)
2
i =-(s i n"1 (R/D))/Aa
8U2 (Aa) 2
h=
Summarizing, the b^j of (3.32b) are given by;
,h=i
(3-35)
58
1
8 (Aa)
DAacos (i Aa)
^J » H
, h-i
2
0
[J 2
(3.36)
,h-i,even,#0
-1
,h-i,odd
27r2si n2 ((h-i)Ax)
This filter is called the Lakes Filter.
3• 3- 3
E q u a l l y S p a c e d D e t e c t o r s Case
The
transformation
illutrated in Figure 9»
(a,3)
coordinate
of
coordinates
to
be
used
in
this
case
is
The (d,3) coordinate system is very similar as
system
in
Equiangular
relationships between (1 ,4>) and
.(<2,3)
Rays
case.
From
Figure 9
are given by
dD
' *
dc ° 5a
'(
d .+o,).,.
•
<t> = a+3 = 3+tan_1d/D ,
Again,
using
the
Jacobian
to
transform
the
a=s i n"1(1/0)
(3.37)
3=</>-a
(3*38)
volume element
did<£ we
have;
dl/fcd
31/33
W&d
W?>3
dld<*> -
3=0-a
(3.39)
Substituting (3-37) (3-38) (3-39) into (3-210. we obtained
1 2ir d «
dD
M(r,0) = — T J fp(,a+3) Kexp[27ri K
2
2 & -d-» (d +D2) 1/2
dD
D2
(rcos(3+tan~ld/D-0))
]
dKdad3(3.^0)
(D2+d2) 1/2
(d2+D2) 3/2
59
y
so
DS
b
y
so
Arbitrary ray
Figure 9.
Coordinate system for fan-beam with equispaced case.
(a)
new coordinate system (d,0) to be used with this geometry.
For reference, the parallel
beam coordinate (!,</>)
are
included. (b)A schematic representation of the fan-beam
geometry with designation of the quantities U(r,0,0) and
d'(r,0,0).
60
In order to make the calculation in computer implementation easily, two
new variables were introduced. The first of these, denoted by U, is for
each pixel
(r,0) the ratio of SOP (Figure 9) to the source to origin
distance. Thus
U(r,0,3) = [D+rsi n (0-0)]/D
The other parameter
(3.M)
is the value of d for the ray that passed through
the pixel (r,0) under consideration. Let d1
denote this value of d. We
have
(3-^2)
d'(r,0,3) = Drcos (3-0)/[D+rs i n (3-0)]
Substituting (3-^1) (3-^2) in (3.*t0) , we get
2tz d°?
UD
D3
M (f,0)=J Jfp(3>d) Kexp[27ri K (d 1 -d)
]dKddd3
0 -dO
(D2+d2) 1/2 (D2+d2) 3/2
Using the transformation
UD
K' = K(D2+d2) 1/2
,
UD
, ,,,
(3.ifi»)
dK' = dK—
(D2+d2) l'2
We can rewrite the convolving kernel in (3-M) » then we get
M ( r , 0 ) = 2 J — f f p ( d , < t > ) Kexp[27riK(d1-d)]0 U2-dO
(D2+d2)1X2
The real part of (3•^+5) can be written as follows,
dKddd3
(3-^5)
61
kN
m(r,6)=//p(d,3)
where KN=l/2Ad
D
K
;
J —cos[27rK (d1 -d)]dKddd(3
(D2+d2) 1/2 0 U2
is the limit of frequency. Using KN and evaluating the
frequency integral yields the expression:
d 1 ~d)
r(
cos[
-i .
-1
M
Mr.W .!Tip(d.g)
°
[0 -d
U2 (D2+d2) 1/2
[2tt(d1 -d)]2
1
(d1 -d)
-sin(
7r)]ddd@
27r(d 1 —d)
Ad
2Ad
3.3•A
(3*^6)
Piscretized Form f o r Equally Spaced Detectors Case
The discretized form of (3-46) for those values of the variables (3
such that d'=h5d where h is an integer is :
M(r,« =
C J.h
"
46Cj,H(j,r,0)
c h.|Pj,i
Pjj = p(iAd,jA|3)
(3.47a)
(3.1.7b)
(3 •^7c)
rcos (3~0)
H (j,r,0) =
(3.47d)
AdU
Form (3.46), we have, for h * i:
d^Ad
J »h
Dp(iAd,jA3)
i =-d/Ad U2 (D2+(iAd) 2)*'2
cos[(h-i)7r]-l
kir2 (h-i) 2Ad
62
then;
^
d^Ad
J,h
Dp(iAd,jA|3)
" 1 / 2 7 T 2 (h-i) Ad,h-i ,odd
f
~i=-d/Ad U 2 [D 2 +(iAd) »] l '»
I
0
j
,h-i,even
For h=i , set d=d' in (3^6) then
^
d/Ad
J"»h
Summarizing, the
C H ,J
Op(i Ad,J A3)
~i =-d/Ad
of
(3 .48)
1
U2[D2+(iAd) 2]1'2
8Ad
are given by
1
,h-i,odd
2 tt
2 (h-i) 2 Ad
'h, i
0
,h-i,even
U 2[D 2 +(iAd) 2 ]x ' 2
1
(3.5D
,h=i
8Ad
2>.k
Filters and Interpolation
If
we
examine
a
plot
(see
Figure 10)
transform of the Ramachandran filter , it
emphasizes higher frequencies over
of
the
discrete
Fourier
is apparent that the filter
lower ones. This tends to increase
random noise in the convolved data. Numerous attempts have been made to
alter this high frequency emphasis by smoothing the convolution filter
in
Fourier
space.
One of
the
first
and
most
successful
attempts
to
smooth the Ramachandran filter was made by Shepp and Logan (197^)- They
suggested
that the | K|
in (3-16)
be replaced with any function which
approximates the value |K j for small K. Among the choices are low pass
63
sine,
low
pass
cosine,
Hamming,
HAN,
PARZN,
Gaussian
(Gullberg
Budlinger, 1981; Shepper and Logan, 197^*; Rowland, 1979;
1979) •
Gullberg,
Here, we use
the sine
function
Budlinger and
replace |k|
to
and
in
(3.16) to obtain a smoothed filter.
s i n (K7rAl)
M(r,0) = lJ7p(l ,<t>)
exp[27riK (rcos (<f)-6) -1)]dKdld<£
7rAl
As before, the frequency
(3-52)
integral
must be limited to the Nyquist
frequency 1/2A1 and the spatial integral to ±R top 1
M(r»0)=—^ H ^
p (1 .0)sin (7rKAl) cos[27rK (rcos<^>-0)-1)]dKdld^>
r r /i ,w2[rcos W>-0)-1][rr(rcos
-1)/Al]-<£1
-J Jp (1 ,<t>){
-——
=
—
}d 1 d<^>
2tt2A1
(Al/2) 2-[rcos (0-0)-1] 2
_1
(3-53)
Similarly,
we
assume
that
rcos (<t>-6)
=
hAl
where
h
is
an
integer.
(3-^3) reduces to the form of (3»19a).
7r/A/>
n ( r , 6 ) = ^2 ^ A j ,H(j,r, 0 )
(3.5^)
1
A! h =
J»h
Pj 11
and
H(j,r,0)
respectively.
We
R/Al
1
2
p (iAl,jA£)
J
2tt*(A1)* i—R/Ar
1-4(h-i) ^
have
also
the
can
same
apply
forms
the
as
filter
(3.19c)
for
a
fan
(3.54b)
^
and
(3.19d)
beam
case.
Replacing K in (3-29) by s i n (7tKAq;/Aq;) gives the following equation.
Fourier Transform of
D-O-O-O : nnmnchnrnlrnn Piltor
+H—I—h : Sfiepp-Logan Filter
A A A A • linearly Interpolated
Ramacliandran Filter
, „ , . Linearly Interpolated
t
" Shepp-Logan Filter
l/Al
K
SPATIAL FREQUENCY
Figure 10.
Discrete Fourier
transforms
Shepp-Logan filter
of
Ramachandra
filter
and
65
2ir
»s i n(tt^K)
M(r,0) = J
J
Dcosap(a,3)I
exp[27riKU
D -sin _1 (R/D)
B
***
s i n (a' - a ) ]dKdad|3
sin"1 (R/D)
(3-55)
or
2rt sin~J(R/D)
~s i n (7rAa:K)
/
Dcosap(a,3)/
cos[27rKU
0 -sin _1 (R/D)
0
s i n(a 1 -a) ]dKdad3
(3*56)
(r ,6) = J
Again,
the
frequency
procedure as was used
integral
is
limited
in the parallel
to
KN.
Using
the
same
projection case, we obtain the
discretized form.
»(r'e) -
WjMlj.r.O
(3-57a)
where;
B'j,h
J
Pj ,;
=
[si n_1j[R/D)]/Ao>
2
i =—[sin-1 (R/D)]/Aa
b '«,iPj,i
(3.57b)
J
and H(j,r,0) have the same forms as (3-32c) and (3.32d)
The convolution filter d^ ; will be
66
. h=i
—[1-cos(—)]
2U
(3-58)
Ax*
(h-i)Aa 7r
l-cos[.
J
*2Us i n((h- i)Ax)
AaDcos (i Ax)
•, h-i,even,*0
b h,i
(Ax) 2-[2Us i n ((h- i) Ax)] 2
1H-COS[
]
2Us i n ((h- i) Ax)
(Ax) 2-[2Us i n((h- i)Ax)] 2
, h-i,odd
U(r,0,3) = [(D+rs i n (3-0)) 2+(rcos (3-0)) z]^
For the equally spaced detectors case, following the same procedure we
get;
D
°?sin(7rAdK)
M(r,
|
0) =
5|p(d» 3)
(D2+d2) 1 /2 0
7rAd
exp[27ri K (d1 -d)]dKddd3
(3-59)
or
D
M(r,0)
»sin(7rAdK)
cos[27tK (d1 -d)]dKddd0
=2 ^_|p( d '3)
(D2+d2) 1'20
Again, the frequency integral
7rAd
(3.60)
is limited to K^. We can find that
D
1
m(i*,0) = y f p (d,3)
ddd3
2
2
1
2
2
2
0 -a
(D +d ) ' 7r Ad [l-if(h-i) 2 ]
(3-61)
The discretized form in this case will be
"<r.«
0)
(3.62a)
67
d/ Ad
-i'h
~
U27r2Ad
p(iAd,jA0)
(3-62b)
i=-d/Ad [D 2 +(iAd) 2 ]1'21- (h- i) 2
Comparison of the transforms of the Ramachandran and Shepp-Logan
filters shows that, while they are approximately
values
of
frequency
K,
the
higher
frequencies
smaller magnitude by Shepp-Logan filter
Ramachandran
filter,
discontinuity
in
the
its
are
given
small
relatively
(see Figure 10). Also, unlike
Shepp-Logan
the slope of
the same for
filter
Fourier
does
transform
not
at
have
the
a
Nyquist
frequency 1/2A1.
earlier that required Aj ^ H (j »r,0) values can be
It was mentioned
obtained
by
interpolating
linearly
between
the
two computed
values.
This has the effect of artifically producing a continuous filter from
the discrete one of (3-23) (or (3-55b)).
frequency response of
Interpolation also alters the
the filter. Obviously, it is the convolved data
which are actually interpolated. However, to demonstrate the effect of
interpolation
delta
simply, we will
function,
so
that
the
assume
that
convolution
the P(iAl,jA£)
is just
of
with
the
filter
a
this
function yields the filter itself. Let
A (hAl ,4>) «
2
(3.63)
A (1 A)5(1-hAl)
h = -co
The
linearly
interpolated
filter
is
obtained
by
convolving
the
interpolation function
(.) - {
1
H,|/A '
0
' j'(3.6«
, |1|>Al
68
with the sampled filter,
A, (1,0) = t}(l-|l |/A1) 2 A (1 -1 1 ,<f>) 5(1-1 1 - h Al)d 1
-Al
h=-°°
(3.65)
1
= [(1 -hAl) Al]A ((h+1) Al,0)+{[(h+1)Al -1 ]/Al} A (hAl ,<£)
We have derived the fundamentals of mathematical reconstruction of
two dimensional
pictures from
a complete set of
projections
some
The
discretized
commonly
in
detail.
theoretical
one dimensional
formulation
to accommodate the data collection modes of
used scanner
types.
algorithms on three sets of
In
the next chapter
has
TCT
been
the two most
we will
test
these
projections made with the three different
geometr i es.
3.5
Reconstruction Algorithm in Emission Computed Tomography Case
As
a
simple example, we calculate
the
projection
Pr (yd,<£) [ECT
case] using parallel beam geometry for a circular disc (see Figure 11)
containing
a
homogeneous
emitter
concentration
attenuation coefficient n. The projection Pr (yd, <j>)
p
and
a
constant
is defined by the
integral equations (Jaszczak et al, 1977)
Pr (Yd»^
~ SS
P (x,y) a (x,y ,yd,4>) 0 (yd+xsi n<£-ycos<£) dxdy
(3.66)
where;
detector
a(x,y,yd,<£) = exp[- jj M (X 1 ,y1) <5 (yd+x1 si n^-y1 cos<£) dxdy]
xy
The functions p(x,y) and n ( x , y ) are defined by the equations
(3• 67)
69
xJ+y2<r£
c ,
P (x
• » - u :
(3-68)
otherwi se
, x*+ya<r*
n
(x,y) = {
0
(3.69)
, otherwise
TCT
ECT
?xWd,0)
P f U.O)
Figure 11.
Geometric description of projections for ECT and TCT. The
projections of ECT(Pr ) and TCT (Px ) can be obtained using
(3.71) and (3.72). In the ECT case, for a circular disc
with constant attenuation and factor is exp(-Ail) where I is
the length of the line segment between the point (x,y) and
the edge of the disc. In the TCT case, exp(-/uwi) is the
atenuation factor where wi
is the length of the line
segment between two edges of the disc.
Substituting these conditions
to
determining
the
i 3 . & 7 ) . we see that the
problem
is
where
is the distance between the element (x,y) and
1
reduced
into (3-66) and
attenuation
factor
exp(-/ul),
the boundary of
the disc. For <j>=0, a(x,y,yd,0)»exp[-/u1 (x,y,yd ,0)] where l(x,y,yd,0) is
70
just -x+Vr o2-yd2.
independent of
Also, for a disc, the attenuated Radon transform
is
because of circular symmetry. The (3.66) reduces to
w
Pr(yd,0) =/
cexp[-/x(-x+W)]dx, where W = (r2-y^)^
"w
(3,70)
Integration of (29) gives
Pr(yd,0)
[1-exp(-2mW)]
(3-71)
For the TCT case, assuming the X-ray source was located at one end, and
detected
by
detectors
at
the
opposite
end
(see
Figure 11),
the
projections can be obtained as follows (Macovski, 1983):
P x (/d»
=
Id
"In— = /, Mds
lo
1
where lo is the incident X-ray beam intensity, Id
detector
pass,
plane, 1
and
m
is
(3.72)
is the intensity at
is the thickness of object through which the X-ray
the
linear
attenuation
coefficient
at
each
region,
respectively. For <j>=0, then the (3-72) reduces to
Px(yd,0) = m W]
where
Wj = 2W = 2(r02-yd2)l'2
(3-73)
71
Comparison of (3•71) with (3-73). shows that the projections of TCT and
ECT for
the same phantom are different. However, they could
identical by using the following approach.
P r (yd'°) = 1 {1-C1"2MW+2 / U 2 W j -
be made
(3-71) can be rewritten as:
]}
(3.7M
by taking a first order approximation, then
P r (yd'°) ~
where R=c/u
M
x
2mW « f P x(y d .O)
M
(3-75)
or
P r (y d ,0) a R P x (y d ,0)
(3-76)
From (3.76). we conclude that the TCT reconstruction algorithm can be
used for ECT in many situations.
Chapter k
COMPUTER SIMULATION FOR TRANSMISSION COMPUTED TOMOGRAPHY
The computer
Figure 12.
simulation
results will
be shown for
a
phantom
in
This phantom was used to test the ability of the algorithms
to reconstruct cross sections of the human tissue with CT. The phantom
is composed of a ring which encloses two circles. The ring and circles
boundaries between media whose attennution coefficients are different.
The parameters of these circles are variable.
Four sets of data which
represent four different phantoms are given in Table IX.
A major advantage of using a phantom such as the one as Figure 12
is that one can write analytical expressions for the projections, which
can
be
used
in
computer
simulations.
The
projections
of
an
image
composed of a number of circles is simply the sum of the projections of
each of the circles. This follows from the linearity of the Radon(1917)
transform.
We will
present
expressions
for
the
projections
of
this
phantom in the next section.
h.1
General Considerations
Let ,u(x,y) be as shown in Figure 12, i.e.,
Mj
,
x 2 +y 2 £R[
(inside the circle)
C| <x 2 +y J £Cf
(inside the ring)
,
otherwise
Mg 9
f
72
(4.1)
t
R j -a,b
73
/Mx.yJ
: (ra,da)
°b : ( r b, d J
°c: (r e ,d c )
Figure 12.
Testing phantom in TCT case. The phantom with the various
parameters were used in the computer simulation. 0 is the
center of the rotation.
TABLE IX
PARAMETERS OF THE FOUR PHANTOMS
Phantom
Parameter
Radius,
(cm)
1
2
3
4
a
2.0
2.0
2.0
2.0
b
1.0
1.0 .
2.0
3.0
6.5
7.0
7.5
7.0
6.0
6.0
6.0
5.0
(1.0,2.0)
C1
cs
Origin,
°a
(1.0,2.2)
(1.0,2.5)
(1.0,2.5)
(cm,cm)
°b
(-1.0,-2.0)
(-1.0,-1.5)
(-1.0,-2.5)
(1.0,-3.0)
°c
(0.5,0.0)
(0.5,0.0)
(-1.0,-2.5)
(1.0,-1.0)
Attenuation,
0.144
0.144
0.144
0.144
0.075
0.144
0.144
0.244
0.244
0.244
0.244
0.244
0.00025
0.00025
0.00025
0.00025
(cm
YTTY
75
It is clear that the origin of each circle is given by
0j (xi,yj) • (1;cos0j,1jsinflj) »
where
The
projection
rotated
i=a„b,c
.tan" 1(T i / d i ), 1; - ( «V +.<*;' )%
pi (tfi.yj)
generated
as
the
through a fixed angle 4> are equal
which are generated
source
(U.2)
and
detector
are
to the projections P2(tf>,yd)
by rotating the phantom through a fixed angle -<i>.
Thus the origin of each circle can be defined by
(x; f n»yi,n) " 0 fcos (0j-nc£) , 1 ;sin (0j-n<£))
n - l,2,3»
it. 2
Parallel Projections of
(k.l)
»N and N<£=27r
the Phantoms
The projections of the phantoms p (1 ,<t>)
is defined by:
«o
p(l.4>) • / M (x.y)ds = In—
—ray
l d (y d> <£)
Let
M(x,y)
be
as
shown
in
Figure 12.
It
is
easy
(b.U)
to
show
that
the
projections of the phantoms are given by
p(1.<t>) =
MWW
.region A
H w(W-W oc )+/xcWoc
.region B
M W (W-W oc -W oa )+M c
w oc +M a w oa
.region C
/xwOrf~ W oc~ W oa~ W ob) + ^c w oc +M a W oa +M b w ob .region D
' My(W~Woc~Wob)^"^c^oc^^b^ob
.region E
(^.5)
76
where W is the distance between the source and detectors. Woa, Wob, and
Woe will
be determined by the following equations. For each circle, we
know that
r (*~ x i,n > ,+(y-yi.n )' =f V
- yd
i • a,b,c
Rj = a,b,c],c s
(*».6)
too
SO
Figure 13.
Simulated projections in parallel rays case.
A schematic
representation of
the parallel
projection system with
designation of the quantities Woa.Wob.Woc, A,B,C,D.
Solving the above equations, we find that the nearest and farthest
points on the circle will be
*1.2 " *l,n ±»l*-(y d "/i.n >']"•
I"1-"
77
Then
Woi - *1 ~x2 =2DV-(yd ~yj ,n ) , 3 I ' / a
(4.8)
and for the ring,
{
2[c,*-(yd -yc.r,)2]'5 . ICsHlyd-YcnHIc, I
(M)
2 { [ c , > - ( y d - y c n ) 1 ^ - ^ ! - ( y d - y c ,n >
I yd-yc,nI<II
it.3
Fan-Beam Projections
Let M(x,y)
be as shown
phantoms can also be found
in
Figure 14.
using
The
projections of
these
(4.5). However, the values of Woa,
Wob, Woe will be different. They are to be determined by the following
equat ions:
Again, Woi can be obtained by solving the above equations.
For the ci rcles,
y i ,n) 2 U i s / C n - (yd / w ) ( 4 . n )
for i = a, b.
And for the ring,
78
Figure ll».
Simulated
projections
in
fan-beam
case.
A
schematic
representation of the fan beam projection system with
designation of the quantites: Woa,Wob,Woe,A,B,C,D,and E.
Those parameters are suited to both cases: Equiangular (DA)
and Equally spaced (OS) cases.
r 2(A 1 ) J V[l+(y d /WP ]
,A S <0<A,
w oc
^ 2[(A I ) J S-(A s ) J S]/[l+(y d /Wp] ,0<A,
where
|
A "C(y d Yc ,n^^ +x c.
* ~^
^
^
d
^
c ,n
+ x
c ,n " ^ I * ^
yc,n/ w > +x c,n^ a -C 1+ (yd/ w )'3 (y^,n +x c,n" c s)
In
equiangular
detectors will
rays
case,
the
be different. Based
on
distance
the
between
illustration
two
^- 12)
adjacent
in Figure |i»,
the y d which are required in the above calculation can be obtained.
79
Assuming there are N + 1 detectors per projection, a is the angle
between two detectors, then
a = tan" 1 [(y d>n -yd,0 )/W]/N
(4.13)
and
yd,n
=
Vd.O
+w
tan(na)
(A. 14)
In the case of equally spaced detectors, the distance between the
source and the detectors will
in
Figure 14,
we
need
be different. Based on the illustration
replacing
W
to
Wd
in
(4.5)
to
obtain
projections of the phantoms, where W d is equal to (w2+ycjz) 1 ^ 2 .
point,
we
can
generate
the
projections
for
various
angles
the
At this
in
both
parallel and fan-beam systems.
4.4
Experimental Protocol
Based
on
the
formation
in
programs (Appendix B) were written
previous
sections,
three
computer
in FORTRAN to simulate the various
reconstruction cases. The results of
the simulations will be shown
in
the next chapter. The simulations will demonstrate the function of the
filter
in
different
geometries,
as
well
as
the
effects
on
image
reconstruction of varying the number of views, the location, the size,
and the attenuation constant.
Ideally,
in the parallel
projection cases, the source should be
rotated in successive 1-degree angular steps around the phantom, until
180 views have been obtained. To do
this, we would
not only
need
a
80
large
amount
of
memory
to
store
the
data
phantom), we would also need excessive CPU
However
for
many
phantoms, fewer
(246k
words
for
a
time to obtain each
39x39
image.
projections are needed. We can
try
using 2, 4, 8, 16,..., 2^ projections to reconstruct the image; this
may suggest economical ways to do other simulations. The same procedure
can
be
applied
to
projections (M) and
the
fan-beam
projection
the number of rays
case.
The
number
of
in each projection (N) can be
increasd until artifacts and aliasing errors are eliminated. We chose
to use 2,32 projections and either 1 ray/cm or 2 ray/cm.
As mentioned
before, a suitable filter and
interpolation method
can reduce noise. In Figure 10 are shown the discrete Fourier transform
of the Ramachandran filter and the Shepp-Logan filter. The effects of
linear
interpolation,
with
both
filters,
will
be
shown
in
the
simulation.
Four
phantoms
constants were
three
with
used
different
to
different
test
projection
the
size,
three
geometries.
location,
and
programs which
These
will
attenuation
correspond
demonstrate
to
the
abilities and limitations of the system.
k.5
Results and Discussion
Figure 15
shows
the
four
digitized
phantoms.
The
different
characters inside the phantoms, represent gray levels. They depend on
the
attennuation
coefficients
of
objects.
For
convenience
discussion we will call the phantoms 1, 2, 3 and h.
of
later
PHANTOM 1
PHANTOM 2
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82
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shows
projection
the
case,
images
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reconstruction
as
done
of
using
to
the original
phantom
rotation
2
in
in
phantoms.
the
parallel
successive
7r /l6
angularsteps. We concluded that 16 views would be adequate to show the
performance of the algorithms.
A similar result was obtained for the
other phantoms.
Figure 17a
situations.
fan-beam
sampling
and
In
c
compare
general,
projections
can
the
images
be
parallel
reconstructed
expected
in the two geometries, as well
functions are different.
and
to
be
fan-beam
from
projection
parallel
different,
since
and
the
as the weight and convolving
Only if the following conditions hold
A£ = 40, A1 a DAa
(4.15)
are the images obtained from equations (3*54) and (3-57) expected to be
nearly the same (Herman, 1980).
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Ac£ = Aar+A|3, Al = DsinAa
If
is very small, then A-x as sinAa
(4.16)
(4.16) will become
Atf> ^ A0, Al ^ DAa
(4.17)
In Figure 17a and c, the images are slightly different, even though the
set
of
angles of
reason for
not spaced
rotation
(Atf>)
is
the same
in
the
two cases.
The
this may be that Aa is not small enough (the detectors are
closely
enough) for
(4.17)
to
hold.
The other
parts of
Figure 17 describe the fan beam situation. Figure 17c and d show that
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The
reconstructed
images
using
different
numbers
of
projections. The number of projections are (a) 2, (b) k,
(c) 8, and (d) 16 (from phantom 2 parallel beam case).
81+
decreasing
the
angle
resolution
without
increasing
the
number
of
projections causes unclear images and some artifacts. Comparing b with
d
shows
that
decreasing
the
number
of
views
without
changing
the
resolution reduces the image quality in a similar way.
Figures
18
reconstructed
for
four
to
by
21
compare
the
images
of
the
four
phantoms
the three algorithms at the same spatial
phantoms.
It
appears
from
the
results
projection method performs better at each case.
that
resolution
the
limitations
samples, and
concerning
CPU time, all
the
of
number
the
of
images
parallel
Moreover, some noise
appears around the object in the fan beam reconstructed images.
the
as
projections,
in those four
Within
number
of
figures seem
reasonable in comparison with the original phantom.
Many factors can influence the resolution of images. Some of them
can
be
ignored
in
a
computer
simulation
study.
However,
filtering,
interpolation, and the distance between adjacent detectors (the sizes
of detectors) play very
the
effect
images of
of
the
important roles.
filter
and
interpolation.
to obtain
the
linear
images, shown
interpolation
Figure 22b,d. The results for
expected
explained
in Figure 22 show
Figure 22a,b
are
the
phantom 2 without the ring; Figure 22c,d are the images of
phantom 2. The Ramachandran filter and
with
The images
from
as
the filter
follows.
as
was
linear
Figure 22a,c;
used
to
the Shepp-Logan filter
obtain
the
images
in
Figure 22a,c do not seem to be noisy as
character i ct i cs
The
interpolation were used
Fourier
in
Figure 10.
transform
of
This can
the
be
linearly
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case; (c) equiangular fan beam case; (d)equispace fan beam
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90
interpolated filter
the sine 2 function and the
is just the product of
continuous transform of the filter. This is shown in Figure 10 for the
linearly interpolated Ramachandran and Shepp-Logan filters'.
of
these
interpolated
transforms with
the original
some indication of the advantage of the smoothing.
in the slope of
still
evident
the Fourier
in the transform of
the 'bump' at K|g.
sharp boundaries
frequency
Loss
performance.
transforms
gives
The discontinuity
the Ramachandran filter
is
the interpolated filter, producing
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in a reconstructed
response of
smoothing.
transform of
Comparison
of
image. However, some of
the Ramachandran
resolution
has
filter
been
has been
exchanged
for
the high
lost
through
better
noise
In the case of phantom 2, there are no sharp boundaries,
so high resolution is not needed.
Figure 23
shows
the
effect
on
images
between adjacent detectors and the sizes of
of
the
phantoms
were
changed
while
the
of
changing
the
distance
phantoms. First, the sizes
distance
between
pairs
of
detectors was kept constant (0.5 cm). Figure 23a,b,c are reconstructed
images whose sizes
were
scaled
so
that
the
smallest
phantom
had
a
radius of 1.0, 0.5, 0.25 cm, respectively. Figure 23d is an image which
was reconstructed by holding the radius of the smallest phantom at 0.25
cm, and changing the distance between two adjacent detectors to 0.125
cm. The
results show that with
detectors,
obta i ned.
high
enough
a
resolution
smaller
distance between
to detect smaller
pairs of
phantoms can be
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and (d) show changing the sizes of phantoms and detectors
proportionally, a similar image can be obtained.
93
Previous results show that there are some advantages in collecting
data in the parallel
beam mode. However, speed of data collection
is
not among them, even where the range of projection is only 180 degrees.
Also, from a mechanical view point rotate-translate motions, which are
required in parallel beam mode, are not as fast as rotate-only motions,
which are sufficient to obtain a full set of data in the fan-beam mode.
On
the
faster
other
than
hand,
computation
in fan-beam
case were compared
in
projection.
in (Table X).
parallel
CPU
This
projection
is
10
times(IBM 308lk64) for
times
each
is not surprising, since the
fan-beam projects uses twice as many views
(360 degrees instead of 180
degrees) and requires more calculation during
its backprojection(refer
to chapter 3)- Moreover, in order to reconstruct the same region more
detectors
electronic
(56 instead of 18) are necessary
in the fan-beam case. Using
hardware to reduce the time of calculation
solve this problem.
is one way to
9^
TABLE X
RECONSTRUCTION SETUPS AND CPU TIMES
Setup No.
A Igor i thm
2222
Parallel Pro­
jection wi th
Lakes Filter
Total Detector
Uni t
18
# of Detectors
Per Unit
2
if of Views
1* 1 12
2122
2112
Fan-beam Projection with Modified
Smoothing Filter
56
1
6i<
16
(11/16)
(n/32)
32
(71/16)
32
(11/16)
Diameter of
Recontructed
Reg i on
18
18
18
18
Spatial (r)
Resolution (0)
0.5
10°
0.5
20°
0.5
10°
0.5
20°
1 10
1 10
55
(Rotated Angle)
CPU Time 3
(sec)
3 The
9
simulation was done in an IBM308lk61» computer.
Chapter 5
COMPUTER SIMULATION FOR EMISSION COMPUTED TOMOGRAPHY
The computer simulation results will
be shown for the phantom
i ri
Figure 2k, which was designed to test the performance of the algorithm.
The phantom
is similar
to the phantom used
in
the previous chapter,
except the concentration of each has to be added to simulate emission
case.
Again, the parameters of the ring and circles are adjustable.
This
phantom
looks
more
complicated
than
the
one
we
used
in
chapter three. However, the projection PrCy^,#) can be obtained using
the same equations, with the following conditions:
(1)
The
origins
of
the
circles
are
not
at
(0,0),
i.e.,
modification of (3-70) is neccessary.
(2) When the phantom is rotated, the attenuation Radon transform
will change, since the system is not circularly symmetric.
(3) The projections of the discs may overlap, this means, we need
to sum the projection of the discs.
(4) The attenuations of surrounding objects, mw» are not zero in
this phantom.
5.1
Projection Generation
Let M(x,y) and p(x,y) be as shown in Figure 24, i.e.,
n j , x 2 + y 2 < R j 2 (inside the circle)
M (x,y) =
r| £x 2 +y 2 <rj 2
,
otherwise
95
(inside the ring)
(5*1)
96
Mlx.y;
>x
• ( r o#d 0 )
°b : (r b dj
°c; (rc,dc)
gure 2k.
Testing phantom in ECT case. A phantom showing the various
parameters that were used in the computer simulation. (0,0)
is the origin of the rotation circle.
(x oa »yoa^» ^xob*
are
t
ie
or
ns
the
y 0 b)• (*oc'Yoc)
'
'9'
circles A, B and ring
C. n and c represent the attenuation constant and emitter
concentration, respectively.
97
CJ ,X2+Y 2<R:2
p(x,y) =
r| <x2+y2£r j2
J
0
,
(5.2)
otherwise
i=a,b,c,
Rj =r a ,r b
From Figure 2A, it is clear that the origin of each circle is given by
(5.3)
Oj (x; , y j )= (ljcos0j , 1 j s i n0j )
0, = tan"1 (yoi /xoi ), 1; = (xoi2 +yoi2
The projections Pi^.y^) generated
as the detectors are rotated
through a fixed angle </> are equal to the projections P2 (<^>»y^) which are
generated by rotating the phantom through a fixed angle ~4>. Thus the
origin of each circle can be defined by
(x n j,y ni ) = (1 jcos (0; -n<£) , 1 j s i n(0 j -n<£))
where n = 1,2,3.
N, and N = 2tt/4>
(5-^)
Using (3-66), the projections of the phantoms can be obtained. Let
M(x,y) and
projections
p(x,y) be as shown
are
different
in
in Figure 25.
different
The expression for
regions
of
the
phantom.
the
For
convenience, we define the following terms (refer to (3-71))
At .Bt ,Ct
C;
= -7ry{l-exp[-2Mi (Rj2-(yd-yi)
where i = a,b,c and
2)j2]}
Ri = '" a ,r b ,r cl
(5.5)
98
where
^=0.
Ai,Bi,Ci
If
represent
the
projections
for
a
single
object while
0, then we define
[A w i i B w]» C w i1 ~ [Ai .B t ,C 1 ]exp ( - e w W wi )
W WA = ( D " X A)-( AA ) JS
' WWB=(D-XB)-(AB)J2
(5«6)
, WWC= (D-XC)- (ACL)%
wi th
AA=r a 2 -(y d -y a ) 2 , AB=r£-(y d -y b ) 2 , ACL=r 2 r(y d -y c ) 2
The following are additional frequently used terms:
W wc
W WC
=
=
D-XC-(ACL)J2
D-XC~ (ACL)^+2 (ACS)%
WIA=2(AA)J5 , WIB=2(AB)J5 , WJ C= (ACL)
where
(ACS)^
ACS=r 2 s -(y d -y c ) 2
Finally, we define a set of parameters which will make the form of the
projections of the phantom clear.
(5.7)
= —{l-exp[-M c (ACL)JS-(ACS)JS]}
C^ 1 =—{exp[-Mc((ACL)J2-(ACS)J5)]-exp[-2MI (ACL)
"c
c w2
= ^exp(-MwW^)
c wi =
ci 1
exp("MwWj,I)
}
(5.8)
99
The
equations
(5*5)"(5-8)
represent
object only; however, the phantom of
the
projections
Figure 25 is
of
single
a superposition of
circles. Thus, different projections can be detected, according to the
location
of
the
projections,
detector
modification
(Figure 25).
of
the
In
equations
order
is
to
obtain
the
necessary.
The
projections on the detector line can be obtained as follow:
In region I (ACL< 0): no source exists, therefore
Pr (yd,<£) = 0
(5-9)
In region II (ACS<0<ACL): a source is located in object C, then
P r (y d ,<tf = C w l
(5-10)
In region III (ACS>0): Serveral cases occur,
Case
1
(AA<0,AB<0): only
object
C
contains a source.
It
is
separated into two parts in this case,
pr (yd'*>
=
cwi
+
cwi
(5.11)
Case 2 (AA>0,AB<0): sources are located in objects C.C" and A,
then
P r ( y d '^ =A wl ex P£- ^c" M w)w ic] +c w£ ex PE" ^a'^w)W ia^ +C wi
(5-1 2 )
Case 3 (AA<0,AB>0): Sources are located in objects C',C" and B,
then
P r (yd '^ =B wl ex P£-(Mc-Mw)W ic ]+C^exp[- (mjj-Mw) W; b ] + C w ^
(5-13)
Region I
Reg Ion I I
Region I I I
Reg ion I I
Region I
11 I
Case 1
Case 2
Case
Case 3
Case I
Case It
Case I»B
igure 25.
A description of projections for different cases.
(a) The
projections can be divided into three regions depending on
the location of detectors (y^). (b) The projections in
region I I I depend on the number of objects through which a
straight line passes. These could be one(case 1), two(case
2,3) and three(case It), (c) The sequence in which the line
passes through the objects can affect the projections. The
projections in case k can be divided to case 4A and 4B with
passing sequence C'-A-B-C" and C'-B-A-C" , resp.
101
Case k (AA>0,AB>0): sources are located in objects C',C", A and
B. Here the projections are different for different relative
positions of A and B. They could be either( case i»A)
PP (yd,^0 *~^wl ®^P^
^i b + ^c~ M W^ ^i c^^"^"®w 1 ®^P £*" (^*c~"^w^ ^i c
^ ^b _lU w^ ^i b +
^i a^"*"^w2
,for x a < x b (C"-A-B-C 1 )
(5.1*0
or (case hB)
P p ( yd » " ~ ® w l ® ^ ^( i " a " ^ w ) ^ i a + ( ^ c " " ^ W ^ ^ i c ^ ^ " ^ " ^ w 1 ® ^ P E"" ( m ^" " M ^ ) W j ^
+ ^wi ex P
C(^b~M w )^ i b"*"('"a^ i a-^ + ^w2
,for x a > x b (C'-A-B-C 1 )
5.2
Simulation Protocol
The results
in the previous chapter
indicated that the algorithm
which used the Shepp-Logan filter was superior, we chose to use it here
for our image reconstruction. The algorithm can be described briefly as
follows:
M (r ,6) =
^ SSp (1,4>)|
K|exp[27ri K(rcos
-1)]dKd<£dl
The discrete version of (5•15) will have following form:
M(r,S)'
J= f ^J.HU.r.O)
_!
%
A]
27t2(Al)2 i=-R/Al
(5-15)
102
pj f j = p(iAl,jA£)
H (j.r,0) =
where p(iAl,jAtf>)
is the sampled projection, Al
two adjacent detectors,
radius of
7r/l6,
R=9
our
is the distance between
is the angle of rotation per scan, R is the
the reconstruction
in
[rcos (jAtf>-0)]/Al
region. We set
simulation.
Note
that
the values
in
the
ECT
Al=0.5, A#=
case,
unlike
TCT,Pr (y^,<t>) *Pr (y^, <£-Hr). Therefore, we must use data obtained over 3&0
degrees instead of 180 degrees, to reconstruct the image.
We wrote a program
projection
generation
in FORTRAN 77 to implement the equations for
and
image
reconstruction
(Appendix
B).
The
the
four
program ran on an IBM3081k64 computer.
5.3
S i m u l a t i o n R e s u l t s and D i s c u s s i o n
Figure 26
shows
the
four
digitized
phantoms
and
corresponding reconstructed images. The projections which were used in
the reconstruction program were calculated directly from equations
the
previous section. The reconstruction algorithm was based
reconstruction
theroem
of
TCT.
We
assumed
that
the
on
objects
in
the
have
different attenuation constants, but the same emitter concentraction.
In
this case, except
images
are
similar
for
to
attenuation
the
original
information,
phantoms
in
the reconstructed
shape,
size
and
location. We explain this as follows.
Although
the
emitter
concentration
(c)
and
the
attenuation
constant ( m ) of an object are generally both unknown, their ratio plays
a major role in ECT. The attenuation constant surrounding the objects
b
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the
original
phantom
and
its
reconstructed image. Based on the projections generated by
four
different
phantoms
(a,b,e,f),
four
reconstructed
images (c,d,g,h) were obtained.
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and c f b and d,..), most of the information including sizes
and location, but not the fi are preserved well.
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105
is not zero. From (3-70) and (3-73) » we observe that the projections of
the phantom (Figure 27) in TCT case and ECT case can be represented by
the following two equations:
P x (y d ,0) = m V^+M w CD-W t )
. (5.16)
P r (y d ,0) = -[1-exp (-2 m W)]x exp[~ m w(--W)]
M
2
(5-17)
where;
W, = 2W = 2 (ro~y^) p
For M w «y" the equations (5-16) and (5-17) reduce to
P x (y d ,0) » m W]
(5.18)
P r (y d ,0) a f[l-exp(-2,aW)] a 5C2 m W]
M
M
(5-19)
Again, let R =c/m , then (3-76) holds. If there are two or more objects
in
the
phantom,
then
according
to
the
rule
of
superposition,
the
projection P t (l,<£) will be
P t (l,4>) = P](1,0)+P 2 (1.^)
In
this
obtained,
situation
using
several
the
different conditions:
same
different
reconstructed
reconstruction
algorithm
(5-20)
images
can
(5*15)»
be
with
106
ECT
TCT
B
Mi
P*Oi,o)
PT<*,°)
Source
Figure 27*
Detectors
Source
Detectors
A comparison between ECT and TCT for a more complicated
case. When the
can not be ignored, the projections are
different from those in Figure 11.
Equations (5-16) and
(5.17) were used to modify equations (3«70 and (3-73) to
obtain the correct projections in both cases(ECT and TCT).
The superposition rule was used to modify cases in which
more than one object was passed through by a line.
107
Case
1:
let
Mi=nM2
and
ci=c2,
i.e.,
ci/Mi=C2/nM2,
then
from
(5.18), we knows'
P1
0 •*) = n
p 2( 1
(5-21)
and the reconstructed images based on (5-15) will be
/u 2 (r,0) = n Ml (r,0)
Comparing with
the original
(5-22)
cond i t i on (,ui=n,u2) , the opposite result
was obtained. Figure 28 shows
the reconstructed
images for
this
case.
Case
2:
let mi =M 2,
and
ci = n c 2 ,
i.e.,
ci/mi
=nc2/m2,
then
from
(5.18) we know
Pi (1,0) = nP 2(l,^)
(5.23)
and the reconstructed image will be
Ml (r ,0) = m 2 (r , 6 )
(5-2M
This image reproduces the emitter concentration information in the
orignal phantom very well. Figure 29 shows the reconstructed image
of this case.
Case 35 let mi = M2 and ci=c2, i.e., ci/mi =02/1x2, then from (5-18)
we know
Pi (1,0) = P 2 (1,0)
(5.25)
and the reconstructed image will be
Ml (r ,6) = M2 (r ,0)
(5-26)
108
A ECT PARALLEL BEAM RECONSTRUTION
RAOTUS OF A.B.C(O),C(I): T.9.I.9.7.9.•.0
ORIGINS OF A.B.C:(1.3.9).(-1.-1.9).(.9.0)
ATT. OF A,B,C.V:0.080.0. 160.0.340,0.00029
OISTRU. OF A.B.C: 1.0. 1.0. I.O
# OF VIEWS: 33
DISTANCE BETWEEN TWO DETECTORS: 0.9
RECON. REA: B1»PI
GRAY LEVEL
013349678
ECT PARALLEL BEAM RECONSTRUTION
RADIUS OF A.B.C(O),C(I): 1.5.1.5.7.9.6.0
ORIGINS OF A.B.C:(1.2.9).(-1.-I.5).(.5.0)
ATT. OF A.B.C.W:0.240.0.080.0.160.0.00025
OISTRU. OF A.B.C: 1.0. 1.0. 1.0
0 OF VIEWS: 32
DISTANCE BETWEEN TWO DETECTORS: 0.5
RECON. REA: 81*PI
GRAY LEVEL
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Reconstructed images for case 1. Reconstructed images with
the same c, but different n, for objects A, B and C are
shown.
a)Gray
levels 7 and 6 ' n objects
A
and
B
corresponded to m of 0.08 and 0.16, respectively, b) Gray
levels 5 and 7 in objects A and B corresponded to ti of 0 . 2 k
and 0.08, respectively.
109
The reconstructed images from a single object and a double object
are
the
same
under
this
condition.
Figure 30
shows
the
then
from
reconstructed image of this case.
Case
let
Mi=nM2
and
ci = n c 2 ,
i.e.,
ci/
mi=C2/m2
(5.18), we know
Pi (1,40 = nP 2 (],<£)
(5-27)
and the reconstructed images will be
M l ( r ,6 )
This
image
emitter
reproduces
concentration
=
either
in
n^2 (r,0)
the
the
(5'28)
attenuation
original
constant
phantom
or
very
the
well.
Figure 31 shows the reconstructed images of this case.
We conclude that reconstructed
images can not respond
to all
of
the information in-M and c in every case. However, from Figure 30 and
Figure 31 »
objects,
we
the
know
that
information
when
of
the
m or
ratio
c
( c/m )
could
be
is
eqi^al
reproduced
between
very
two
well.
Figure 29 and Figure 30 show that when the ratio ( c/m ) is not equal and
the attenuation constant
is equal between two objects, the information
of C could be reproduced very well.
Two factors may interfere in the accuracy of reconstructed images.
One is m w , the other
case in which m w = m .
is the number of gray
levels.
Figure 32 shows the reconstructed
We simulated the
images for case
2 and 3- It seems that the same conclusion applies as in the case where
H
W «M.
However, from the results of following simulation, the accuracy
of reconstructed images in this case (mw-m) is not as good as nV)«fx.
1 10
8 CCT PARALLEL BEAM RECONSTRUTION
RADIUS OF A,B.C(0).C(I): 1.3.1.S.T.3.«.0
ORIGINS OF A.B.C:(1.2.3).(-1.-1.3).(.3.0)
ATT. OF A,B,C,W:0.080.0.080,0.080.0.00023
DISTRU. OF A.B.C: 3.0. 1.0. 2.0
T OF VIEWS: 32
01 STANCE BETWEEN TWO DETECTORS: 0.3
RECON. REA.: 81«PI
GRAY LEVEL
CCT PARALLEL BEAM RECONSTRUTION
RADIUS OF A,B.C(0).C(L): 1.3.1.5.7.3.6.0
ORIGINS OF A.B.C:(1,2.3).(-1.-1.9).(.3.0)
ATT. OF A.B.C.W:0.080.0.080.0.080,0.00029
OISTRU. OF A.B.C: 1.0, 2.0. 3.0
0 OF VIEWS: 32
DISTANCE BETWEEN TWO DETECTORS: 0.9
RECON. REA: BI'PI
GRAY LEVEL
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Reconstructed images for case 2.
Reconstructed images with
the same n , b u t d i f f e r e n t c , f o r objects A, B and C are
shown.
a)
Gray
levels 7 and 3 in objects
A
and B
corresponded t o ft of 3-00 and 1.00, respectively, b) Gray
levels 2 and
i n objects A and B corresponded to m of 1.00
and 2.00, respectively.
111
ECT PARALLEL BEAM RECONSTRUTION
RAOIUS OF A,B,C(O ) , C ( I ) : 2 . 0 . 1 . 0 . 7 . 0 . 6 . 0
O R I G I N S O F A . B . C : ( 1 . 2 . 5 ) , ( - 1 1. 5 ) . ( . 5 . 1)
ATT. OF A.B.C.W:0.144.0.144.0.144.0.00025
OISTRU. OF A.B.C: I . O . 1 . 0 . I . O
0 OF VIEWS: 32
DISTANCE BETWEEN TWO DETECTORS: 0 . 5
RECON.
REA: 81-PI
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GRAY LEVEL
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Reconstructed image for case
Reconstructed image with
constant c and fx for objects A, B and C are shown. Gray
levels of 6 and 6 in objects A and B corresponded to the
ratio of c/yu both equal to 1.
112
To test the accuracy of this algorithm, the following simulation
has been done. Reconstructed
value of
images with
the same c/m and different
for objects A and B are shown in Figure 33* With a small
M w . the images of two objects do not vary much (Figure 33a) unless the
Mb is increased (Figure 33b). With a comparable m w . we can distinguish
the difference between
A
easier
two
to
distinguish
and
B very well
objects
by
rather than using vision (Figure 33)-
(Figure 33c,d).
using
the
We alsc
average
It
is much
gray
level
calculated the average
image value to test the accuracy of this algorithm (Figure 33) •
The
results show that the average image value method performs better than
the average gray
level method
an opposite result, the
in most case.
lower
Compared with Figure 33»
the higher accuracy, was obtained.
According to (5-16)-(5-18), the results in Figure 3k were expected.
seems that the gray
images, affects
It
levels, which were set by normalizing the data of
the accuracy of
the algorithm.
However, to avoid
a
misleading result, as in Figure 33» we can increase the number of gray
1evels.
Several different methods can be used to study the errors in image
reconstruction
algorithm.
Of
the
two
most
1985; Slaney et al., 198^4; Nahamoo et al.,
Soumekh
and
squared
error
function
Kaveh,
1986;
(MSE)
is 0(r) and
the relative MSE is:
Robinson and
analysis.
popular
(Slaney
198^; Pan and
Greenleaf,
Basically, assuming
and
Kak, 1983;
1985). one
the actual
the reconstructed object function
Kak,
is mean
object
is 0' (r) then
113
ECT PARALLEL BEAM RECONSTRUTION
RAOIUS OF A.B.C(O),C(I): I.3. I .S.7.5.6.0
ORIGINS OF A.B.C:( L.3.3).(-1,-L.3».(.3.0)
ATT. OF A,B,C,W:0.240.0.0E0.0.100.0.00033
OISTRU. OF A.B.C: 3.O. I.O. 2.0
0 OF VIEWS: 32
DISTANCE BETWEEN TWO DETECTORS: 0.3
RECON. AREA: B1*PI
JJ-ECT PAR»LLET BEAM OF CO'JS T PU T I ON
RAOIUS OF A.B.C(O).C(I ): 1.3. 1.3.7.5 6 O
ORIGINS OF A.B.C:(1,2 3).(-1.-1.3). (.5.0)
ATT. OF A.B.C.W :0.0n0.0. 1GD.0 2<1O.O 00035
OISTRU. OF A.B.C: 1.0. 3.O. 3 O
0 OF VIEWS: 32
DISTANCE BETWEEN TWO DETECTORS: 0.5
RECON. AREA: BI'PI
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Figure 31.
Reconstructed images for case U.
Reconstructed images with
same r a t i o (c/n) f o r objects A, B and C are shown, a) Gray
levels of 2, 3.5 and 5 i n objects A, B and C corresponded
to either c or n of objects of 1, 2 and 3. resp. b) Gray
levels of 7, h and 5.5 i n objects A, B and C corresponded
t o e i t h e r c o r ju o f o b j e c t s o f 3 . 1 a n d 2 , r e s p .
1
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Recontructed images with
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b , c and d with figure 30a, 30b, 31a and 31b respectively,
shows that the gray level i n each image varies equally.
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•• iUn:i.,..ti*UM,,..:iin.,
:XJSISI»:
115
I
.11..
•1. .•+*-»TTXjn?::..
.s..
:tt
...
:*nM*n«iHnu&*«n»
,1
...
..
1....
...
0123^567
.:+%;?*$
GRAY LEVEL
::::»—#•.
««i:,
:...:s:
in:
•
X:«»•:I::
J JIX:1:2zHfib
.im.
.
:•i*n<:2::.:jj
• i.
...
_
2:5:2:;.
:::::::1.1..
.:
:
::
•• •
:
GRAY LEVEL
..
,,{..••••V
:«#«•
..:!••»•
01234567
.:+%#*$
01234567
GRAY LEVEL
:sn*H.;
;:::
.:«!•
•»—*::
>•:
::::::
:::
•rt**..
1
til*:.
:«si:.
•«:.
•«:.
•
••:«522n,,,,,nn;!!.
::
U2fV
:
•r"t
••!•-»*:
•xin>
:
h*
:
••;:rT*
SJS**.
•:£"!
SSS*: .mi:
:»SUS«
nil
. .: u « .
. :ui .
..•tn:
....
.
..*98*.
HM;.
...:ussvn«s»^smii«::
Figure 33-
'"liZllU
....••Ksmanyfin*:,.
Reconstructed
images with comparable /u.
Reconstructed
images with comparable n and the same c/n for objects A and
B are shown. a) With M "0.00025. average gray levels 4.5
and 4.8 in objects A and B corresponded to M (c) of 0.1 (1)
and 0.12 (1.2), resp.
b) With /u =0.00025. average gray
levels 3*73 and 4.65 in objects A and B corresponded to n
(c) of 0.1 (1) and 0.15 (1-5) » resp.
c) With v =0.12,
average gray levels 4.95 and 5-46 in objects A and B
corresponded to fi (c) of 0.1 (1) and 0.12 (1.2), resp. d)
With M =0.15, average gray levels ^.6l and 5-46 in objects
A and B corresponded to ai (c ) of 0.1 (1) and 0.15 (1-5).
resp.
I1
01234567
GRAY LEVEL
.:+*#*$
111111....1
....I I I .Xl ...I
...
.—: S » — - » » . :
.;.taamom•«•»*::
•*.:X*"•r*— 1.
...........
Average Grajr level(AC) = TG*/N
ui 4
where G^ is the gray level of each
pixel corresponded to the object.
«*«?»».
• •. ••••<::j-_j ' •'a .... jp*
1«.%•-%•<
For example, the AG of object A is
:fr
(3x7+4xl9)/26 = 3.73,
jj
•
•
»
•
•
•
•
!
•
•
»«t«
m
- . . . . . . . a . . . ^0
• J.
r.
11!#•»•»!..I
I::X*~S:.t.
the AG of object B is
»«t»
s::. .11111
9
(3x2+4x5+5x19)/26 = 4.65
a
:X*"»
...1...
.
. —S:
Average Image value(AI) = .5LIi/N»
:**•»:'
where Ij is the real image value
j
without convert to gray level of
• - -s*2 • •.. •.•••••-•i.'x"!**,
1.
t..
,......;
.
.i.
nt
B/A
A.
AG
B.
B/A
A.
Al
B.
1.5 3 846. 4 .769. 1.240 0.505. 0.724.
•
(0. 1
ni
w
0.00025
Original setup
A ( u. c ) .
B(fi.c).
O
1
JU
each pixel corresponded to the object,
.:
...
...
0
or
B/A
1. 434
0 . 5*17, 0 . 6 4 2 .
1. 174
3
(0.11.1.1). 1. 1 4. 769. 5.231. 1.097
0.5G6. 0.612.
1 .08 1
5 . 1 5 4 . 5. 3 8 5 , 1 . 0 4 5
0.576, 0.596,
1.035
0.201. 0.255.
t .269
4
(0.105,1.05).
ni
( 0 . 1 2 . f . 2 ) . 1 . 2 4 5 3 8 . 4 . 8 0 8 . 1. 0 5 9
Z
2
1
1
5
O
ni
( 0 . 1. 1 ) . ( 0 . 1 5 . 1 . 3 ) . l . S 4 .692. 5 .615. 1. 196
6
0.12
( 0 . 1 2 . 1 . 2 ) . 1 . 2 4 . 9 2 3 . 5 . 6 1 5 . 1. 1 4 0
0.244, 0.272. 1.115
7
0.11
( 0 . 1 1 . 1 . 1 ) . 1 . 1 3 . 4 2 9 . 5 . 6 1 5 . 1. 0 3 5
0 . 2 6 3 . 0 . 2 7 6 , 1. 0 4 9
S
0.10s
Figure 3^.
( 0 . 1 0 S . 1 . 0 3 ) . 1.0S S.615. 3 .654, 1.007
0.275. 0.278.
1.013
Comparison of the accuracy of two methods.
Comparing the
ratio (B/A) between AG and Al with original, A! perform
better in most cases (in #1, 1. 1*36:1 -5 with 1.25:1.5). In
^w << ^ x c s a e (#1.2,3 and U) , the ratio (B/A) of Al is very
close to the ratio (B/A) of or igfnal (1 .l»36: 1 .5) . As M
start to increase, the ratio (B/A) of Al is not like the
o r igi n a l ( 1 . 2 6 1 : 1 . 5 ) .
117
JJ[0(r)-0'(r)]2dr
//[0(r)]2 dr
The other
method uses
a single
figure to
represent the
error.
The
figure represents a numerical comparison
of
values on a specific cut through the test
image. The f i r s t method seems
likely to
be much more
needed than the second.
accurate,
but
The second
much longer computing
method is clearer
t h i s reason many workers i n t h i s area
Kak, 1983; Soumekh and Kaveh,
the true and reconstructed
1983;
time is
and easier.
(Nahamoo e t a l . ,
198^t;
For
Pan and
Robinson and Greenleaf, 1986)
have
used the second method.
Emission CT
reasons.
is not
The most
as accurate
as transmission
attenuation constant.
Sorenson,
Several methods
suggested in the
literature
197^; Kay and Keyes,
1975;
from the distribution of
for approximation
(Budinger and
1979; Gorden and Herman, 197^; Alters et a l . ,
distinguish
here
postcorrection methods,
is applied
the
In
Gullberg, 1977;
Both pre-
first-order
Sorenson, 197^;
Jaszczak et a l . ,
1981;
Chang, 1978).
precorrection
.on whether
and post-correction
procedures
Kay and Keyes, 1975;
We
and
the correction
before reconstructing,
correction
1977;
1979; Budinger et a l . ,
so-called
generally depending
to raw data projections
reconstructed image.
complexity.
between
the
attenuation
Gullberg,
Gullberg, 1979;
1981; Tretiak and Delaney, 1978; B e l l i n i e t a l . ,
can
several
important reason is that the projection measurements
can not separate the emitter distribution
have been
CT f o r
or
to the
methods vary
(Budinger
in
and
Gullberg, 1979;
118
Jaszczak et al.,
1981) ,
factor.
for
Thus,
attenuating
many
materials,
quantitative results.
proposed
/u(instead of /i(x,y))is used as the correction
the
correction
Convolution
(Tretiak and
compensation,
distributions
Delaney,
in cases
of
emission
does
not
sources
produce
and Fourier methods also
1978;
Bellini et
al.,
where attenuation is constant.
and
accurate
have been
1979)
for
These seem to
work well for convex regions of uniform attenuation; however, more work
must be done to evaluate possible
frequencies
in
considerably
the
difficulties in handling low spatial
projection data.
slower
than
substantially improved
the
Iterative
analytic
methods,
accuracy (Budinger et
al.,
Herman, 197^; Alters et al., 1981; Chang, 1978).
(expressed in
1979;
capable
of
Gorden and
However, image noise
an iterative
attenuation correction than for a first-order correction.
Development
better method
further study.
is two-to-three times
are
although
higher for
of a
% RMS)
methods,
to compensate for
the attenuation
constant needs
Chapter 6
SYSTEM DESIGN
The
projections
phantom.
describes
a
system
from a phantom and display
A similar
Chan,
1984;
this
system
include
design1
system
Chan,
is
a
thermoelastic
1988)
shown
to
generate
reconstructed
image of
imaging system described in
in
receiving
F i g u r e 35pulse
Major
components
generator,
transducer
a
array,
a
of
signal
graphic display capability
(Figure 3&) •
intensity of
the
at
different
conversion system conditions and
numerically coded values
the
image
is
computer
a
and
with
The
transducers
The signal
translates
detect
the
conditioning and data
the signals
into a set of
in a format acceptable to the computer. After
recontructed,
processing and display.
phantom,
The rotating phantom generates
angles.
thermoelastic wave.
system
conditioning
and a processing and control
projections
this
(Lin and
this
rotating
data conversion interface,
the
the
i s used i n this experiment. A block diagram of
microwave
piezoelectric
a
used
Each
a
software
portion of
package
this
is
system
used
is
for
image
described
in
this chapter.
6.1
Microwave D e v i c e s
Coaxial
waveguide
applicators
can
be
designed
to
operate
throughout the microwave frequency range. They provide a greater degree
of
control
over
t h e SAR d i s t r i b u t i o n and a t
119
the same
time allow very
120
Pulsed Microwave Energy
Rotating Phantom
Transducer
Image Reconstruction
Display
Figure 35•
A block diagram of the thermoe1astic
imaging system.
121
I'ower
Meter
Directional
coupler
cm
Pulse Microwave
Generator
Stub
Tuner
ApplIcator
Hydrophone Output
(Project Ions)
Rotary Phantom
Pulse Generator
Function Generator
Ramp Generator
Ch.irge
A m pI i f i e r
Tank f i l l e d
with water
Hydrophone
Array
Bandpass
S/H
Computer
ADC
Bus
Main Frame
•Control
Ampl1f ier
Keyboard
Display
Figure 3 6 .
PrInter
Major components of imaging system.
Digltal
Scope
122
little
radiation
potential
leakage
outside
electro-magnetic
(Michaelson and
designed for
cylindrical
to
region,
other
instrumentation
that
of
less
than
1.25
permits
and
is
It consists of a concentric, circular
the
propagation
of
waveguide mode. This applicator operating at 2^50 MHz
VSWR
minimizing
A specific dimension of the applicator
use at 2^50 MHz.
waveguide
aperture
interference
1987)-
Lin,
the
a
radiation
pattern
TE11
coaxial
has a measured
that
is
circularly
symmetric with maximum radiation in the forward direction.
Microwave
energy
is
generated
by
a
microwave
pulse
generator
(Epsco PH^Ok) with plug-in at 2^50 MHz. The duration of pulse and
peak
power
of
microwave
during
experimentation
1985)*
This
RG501), a
control
could
by
adjusted
applying
system
pulse generator
be
an
consists
to
external
of
a
ramp
meet
this
control
generator
the
requirment
system
(Su,
(Tektronix
(Tektronix PG505) and a function generator
(Tektronix FG501) (see Figure 37) The operating ranges of this source
generator include: (1) maximun power output
is 40 KW; (2) pulse widths
range from O.k ms to 25 ms; and frequency ranges from k8^ MHz to 2757
MHz.
For all
the experiments reported
pulse width, and
peak
power
of
about
in this thesis, 2 microseconds
30
Kilowatts
are
used
unless
otherwise stated.
6•2
Rotating Phantom
In order to reconstruct the. image, the collection of
projections
in different angles is necessary. A projection generated as the source
and
detector
are
rotated
through
a
fixed
angle
4>
is
equal
to
a
123
INPUT
GATE IN
1
OUTPUT
PQ 600
hAMP
FQ 601
PH 40K
GATE
INPUT
QEN
IN
OUTPUT
0 Q
Figure 37«
Arrangement for
external pulse
projection which
To avoid
transducer
during
rotates at
a
the noise due
the
fixed
designed
the
microwave
source with
an
is generated by rotating the phantom through a fixed
angle -<t>.
special
controlling
experiment,
angle
holder
(for
with
to movement
the
with
example, ir/16)
each
time
used
to
screws
is
a
applicator
phantom
three
a
of
rotator
is
and
which
used.
hold
the
object, and then put into the desired hole, such as hole 1, 2, 3
A
test
9
on a plastic plate(see Figure 38), to keep the test object in position.
All
of
the holes are drilled
at specific
position
through
a
plastic
124
plate which is attached to a Delrin gears combination (PIC Design Co.,
Middlebury, CT).
the gears.
The object is rotated at a fixed angle by controlling
Using this setup, a complete projection (180 degree or 360
degree) can be obtained.
6.3
Hydrophone Transducer Array
A
prototype
20x20
element
piezoelectric
transducer
array
was
designed and fabricated by International Transducer Corporation (ITC).
The piezoelectric array element consists of a lead zirconate titanate.
All
transducers
have
a
free-field
(re: lV/^Pa) at 70 kHz, and
voltage
sensitivity
each device has a. coaxial
cable that exists at the bottom. The array
of
-220
dB
(RG-174) output
is sealed within a square
metal frame that measures 21 cm on a side. Each transducer is 8 mm in
diameter
and
2
mm
in
thickness. The
distance
between
two
adjacent
transducers is 8.25 mm. The area image sensors offer inherent geometric
stability essential for reliable measurement. One set (1x20X32) of data
allows
us
to
reconstruct
a
2-D
image,
according
to
the
projection
theorem. This setup could generate 20 slice images, meaning, that this
system can be extended to create a 3~D image.
6.4
Signal Conditioning and Storage
The output from each transducer element, containing
the desired
biomedical signal must be conditioned and converted to digital format
for
the
computer
to
analyze.
Piezoelectric
finite resistance. The output of
materials
the transducer
have
high
but
is small which would
not meet the requirment of resolution. In order to improve the dynamic
125
\b
Figure 38.
The setup of rotating phantom.
(a)The overall structure of
rotator and phantom. (b)The gears combination and plastic
plate with 9 holes.
(c)The plastic holder
with three
plastic screws used to hold different sizes of phantoms,
(d) k d i f f e r e n t si2es o f phantoms.
126
response of the transducer at low and medium frequencies, the output is
fed directly
into the negative
feedback resistor
input of
a charge amplifier. A
large
is added to the system to avoid saturation problems
which may be generated due to the op-amp bias current drift.
The dynamic response of the transducer at high frequency
simple
avoid
linear
function due to
aliasing
introduced.
which
is
according
overall
and
its
mechanical
high-frequency
noise,
resonance.
a
is not a
In order
to
filter
is
band-pass
The cutoff frequencies are selected with 25 and
the
to
linear
range
previous
of
the
experiments
bandwidth of the filter
transducer
(Olsen
and
frequency
Lin,1983) •
250 kHz
response,
Since
the
is so wide, an overlapping high-pass
and low-pass cascaded filter is desirable.
The
imposes
need
to
generate
requirements
on
images
the
data
of
good
resolution
conversion
and
low
devices. Two
noise
different
methods are used to convert the signal into a digital code (Figure 36):
(1)
a
serial
combination
converter (ADC)
is
of
be
used
image
is
processing.
The
corresponds
to 8 bits
sample-hold (S/H)
to
acquire
digitized
resolution
of
the
and
data
into
256
ADC.
A
analog-to-digital
set
gray
hybrid
for
further
levels,
which
para 11el/serial
design which considers the cost effect and time limitation is suggested
to convert the 400 data from transducers to computer serially and then
store the data in the computer. The entire operation is under control
by the INTEL
86380
system (Chan, 1 988) . (2) the signal is detected by a
digital storage scope, Nicolet 4094A
(Table XI), and
then stored onto
127
5-1A
inch
computer,
disks.
INTEL
Later,
86380, one
the
signal
by
one and
is
recalled
stored
as a
Figure 36, we know that compared with route (1)
and
resolution
could
be
obtained
by
using
then
data
keyed
into
file.
From
higher sampling rate
the
digital
scope.
A
disadvange of the second method would be the manual control. Thus, both
of the data acqusition time and
the errors rate, which
is caused
human error, will be increased. However, the complete signal
in the disk which may provide more
by
is stored
information for system development
in the early stages.
6.5
Microcomputer and Display
The INTEL 86380 microcomputer
central
processing unit
system
iRMX86 release 5-
(CPU).
Memory (RAM), peripheral
incorporates the Intel 8086 as the
It runs at 8 MHz under
It has 896
the operating
Kbyte of on board Random-Access
chassis-house the 20 Mbyte 8 inch Winchester
drive and 8 inch disk for 1 M-byte flexible disks, which is sufficient
for this particular application. Moreover, the system supports language
compilers and interpreters with Intel's Universal Development Interface
(UDI).
A
color
monitor
and
a
graphic
controller
are connected
to
the
Intel system to display the reconstructed images. Color II monitor is a
medium resolution Cathode Ray Tube (CRT) designed by Amdek
Ine. The RGB
(Red, Green, and Blue) input permit up to 16 computer controlled color
graphics. The control lines are interfaced to a graphic controller with
positive TTL signals. The graphic controller (RGB-Graph) is a multibus
128
TABLE XI
SPECIFICATION OF SCOPE
a
MAINFRAME
Memory Size:
Addressable Subgroups:
Storage Capacity:
Di splay:
Expans i on:
Numerics (YT Mode):
Numeric Displays:
(a) Normal:
(b) Grid:
Arithmetic Functions:
16K words, 16 bi ts.
Halves (8K), Quarters (i»K) .
Up to 32 Waveforms.
5-inch, high definition.
Up to X256, both axes, cursor- i nteract i ve.
Time and voltage plus channel i dent i f i er.
Absolute numerics.
Numeric scale per grid mark.
Data Move and variety of routines
available on diskettes.
i«562 PLUG-IN
Buffer Memory Size:
Overa11 Accuracy:
Vertical Resolution:
Aperture Uncertainty:
D i g i t i z i ng Rate:
a. Maximum
b. Minimun
Trigger Delay
to First Sample:
16K x 16-bits
0.2% F.S.
12-bits (0.025%)
50 pSec
500 nSec/pt.
200 Sec/pt.
210 nSec
DISK RECORDER
Disk Recorder Type:
5 ~ ' A " Floppy, DSDD, soft-sectors, SG TPI.
Storage Capacity/Diskette: Twenty 16K, forty 8 K or eighty 1»K records.
a
Borrowed from Nicolet company
129
compatible
color
video
(Montreal,
Canada).
It
board
has
from
16-bits
Matrox
Electronics
512x512
pixel
System
resolution
Ltd.
and
a
writing speed of 800 ns/pixel. The resolution of both is sufficient for
this system. However, the limited
16 display colors make images
look
more discrete than they actually are.
6.6
Image Process ing Software
The
ACQl»
is
a
software
package
developed
by
Chan(1988).
This
package consists of pascal-callable subroutines and utility program for
the
imaging
system
prototype.
PASC86 Pascal
language or
friendly
and
easily
overall
structure
specifically
to
All
routines
were
written
ASM86 assemble language. This
expandable
of
ACQ4
allow
data
software.
software
acquisition
Figure 39
package.
from
It
20x20
Intel
package is a
represents
was
an
designed
piezoelectric
transduces array, interactve manipulation of the acquired
and display pseudo-color images on the color display.
in
image data,
130
DATA
ACQUISITION
IMAGE
PROCESSING
SUBTRACTION /
NORMALIZATION
ENHANCEMENT
LINEAR IMAGE
MANIPULATION
SMOOTHING
FILTERING
EDGE
ENHANCEMENT
FILTERING
AVERAGE
THRESHOLDING
NON-LINEAR IMAGE
MANIPULATION
LINEAR
GREY SCALE
TRANSFORM­
ATION
PIECEVISE
LINEAR
GREY SCALE
TRANSFORM­
ATION
INTERACTIVE
MODE
Figure 39.
CHECK
DISPLAY
An overall structure of ACQU
HISTOGRAM
EQUALIZA­
TION
Chapter 7
METHOD o f EXPERIMENT
This chapter is divided
into two sections (1) a preliminary study
of the system and (2) experimental
protocol. The preliminary study of
the system describes some experimental
study
from
preliminary
the
system
study,
described
some
in
phantoms
results which related
chapter
were
6.
chosen
From
and
the
an
to this
results
of
experimental
protocol was designed to test the performance of the system.
7.1
Pre! iminary Study of System
Thermoelastic Wave of Single Transducer
7.1.1
The setup for this experiment is shown as Figure 36 with 2450 MHz
and 2 ms pulse width microwave source. The
induced thermoelastic wave
was received by the transducer. The signal was amplified and filtered
before feeding into a digitial scope. 1.024 ms of signal digitized at 2
MHz was
collected
and
analyzed
by
a
Nicolet
fast
Fourier
transform
program. Figure 40a shows a typical signal generated at the surface of
a water tank. The transducer was 14 cm underneath the microwave source.
By taking the speed of sound in water at about 1540 m/s, a propagation
delay of 90 ms from the triggering of the microwave pulse was detected.
According to (2.8), a computer program was
inplemented to compute the
magnitudes of the Fourier transforms of these pressure waves. Figure 41
is a plot of pressure intensities as a function of frequencies for a
131
132
2^50
MHz,
ms
2
component
and
pulse
microwave.
the pressure is
The
curve
is
a
broad
the highest a t about 30 KHz.
shows the accompanying frequency spectrum of
frequency
Figure 40b
Figure AOa. With a 25 KHz
and 250 KHz bandpass f i l t e r , the s p e c t r a l contents o f
the signal
from
agrees
2*t
KHz
to
86
KHz.
In
general,
this
result
spread
with
the
theoretical prediction.
The attenuation coefficient of
a function of
frequency
the acoustic wave varies rapidly as
(see Figure 3)*
velocity and attenuation constant of
water
in
the
been
done.
operating
The
setup
frequency
is
In order
the bulk
thermoe1astic wave propagating i n
region
identical
to estimate
to
the
the
following
experiment
one mentioned
earlier.
has
By
moving the detector away from the applicator along a s t r a i g h t l i n e w i t h
0 . 5 cm increment, namely, 6 . 0 , 6 . 5 , 7-0, 7 - 5 . 8 . 0 , 8 . 5 . 9 - 0 , 9»5»
10.5»
11.0,
the digital
and
11.5 cm,
a
sequence of
signals were recorded
through
scope. F i g u r e k2 shows the sequences o f these s i g n a l s . Time
delay and amplitude of each signal was measured, then f i t i n t o a
equation
linear
velocity
and/or
exponential
regression
of
wave
method.
equation
The
propagating
in
water
on scope display, both cases
fit
and
estimate
Moreover,
the
by
results
attenuation constant about 0.103 /cm.
98.^%).
10.0,
the
applying
the
( F i g u r e J»3)
is
about
least
show
1520
m/s
linear
squares
that
the
and
the
Although the signals had sampled
linear
of
equation very well
initial
magnitude
(100%
of
wave
Vo=6.360V. A similar experiment for cat's brain has been done by Lin et
al.
(1988).
They
report
attenuation constant
that
the bulk
is O.56 /cm.
velocity
is
1523 m/s
and
the
133
F i g u r e 1*0.
E x p e r i m e n t a l r e s u l t o f t h e r m o eI a s t i c w a v e .
Thermoe 1as t i c
wave induced a t the water tank i r r a d i a t e d w i t h 2^50 MHz, 2
Ats m i c r o w a v e p u l s e s . ( a ) T h e r e c e i v e d w a v e , (b) i t s f r e q u e n c y
spectrum (Borrowed from Chan,1988).
FREQUENCY SPECTRUM OF
A MICROWAVE-INDUCED ACOUSTIC SIGNAL
2450 MHz, 2.0 fxS
o
C
0
1
o
To
L
"itf
10
iff
FREQUENCY , Hz
Figure 41.
Frequency
spectrum
of
thermoelastic
wave.
Frequency
spectrum of wave induced by thermoe1astic mechanism i n
w a t e r i r r a d i a t e d w i t h 2 ^ 5 0 M H z , 2 ms m i c r o w a v e p u l s e .
135
£30.eb008aV
Figure h 2 .
A sequence of thermoelastic waves.
The waves were obtained
by moving
the detector
away
from applicator
along
a
s t r a i g h t l i n e w i t h 0 . 5 cm increment.
136
a
80r
u
QJ
</> 70
Lii
60
CD
t =1.41+d/0.15 2
§ 5 0
CL
o
£T
a.
R. 5 . : 100 %
40
7
8
9
10
11
12
DISTANCE d (cm)
Vj = 6.360exp(-0.]03-d)
R.S.J 98.4 %
•a
>
UJ
Q
D
h;
_J
CL
JL
8
9
DISTANCE
F igure
•
10
(cm)
11
12
The least square regression of previous figure.
(a) Shows
the wave w i t h a bulk v e l o c i t y o f 1520 m/sec. (b) Shows the
wave with an attenuation coefficient of 0.103 /cm and
Vo=6.360 V.
137
Estimation of Phantom Attenuation Coefficient
7-1.2
An experiment was designed t o estimate the a t t e n u a t i o n c o e f f i c i e n t
of phantom according to the following method.
and transducer
at
the amplitude of
the same p o s i t i o n and w i t h a distance of
vo
=
thickness of
Zj
(7.1)
water.
and an attenuation coefficient of a;
was placed between the transducer
of
then
exp(-awZ)
is attenuation coefficient of
A phantom with a
Z cm,
the detected signal Vn can be written as
vn
where
By f i x i n g the applicator
and applicator. Again,
the amplitude
the detected signal Vp w i l l be
Vp = VQ exp { - [ a , Z ,+<\, (Z-Z,) ]}
I f we divided
(7.1)
by
(7.2),
(7.2)
then take the natural
log of both sides,
and the following equation can be obtained:
Vp
l n - n - = Z : (C£j—gc:)
n
Thus the attenuation coefficient of phantom a j
(7-3)
can be written
as
Vp
a
In
and
(7.i») , the value of
Vn
are
available
i =
(7-M
avi-z\]n-V^
has been found out i n last section.
during
the
experiment.
Figure kk
Z j ,
shows
Vp,
the
sequence o f signals which was measured from d i f f e r e n t phantoms, namely,
0.9% saline, simulated muscle
(which was developed by
the University of
138
Washington;
composition
polyethylene power,
and
air.
signals
and
are
problem.
h
show
slightly
To
weight:
75-^% water
These phantoms
F i g u r e Uka
by
reduce
were
that
filled
even
estimate.
Then
calculating,
the
according
estimate
a
without
An
error
into
due
to
2.16
stuff,
to
cm wide
phantom
unstable
the
repeating the same procedure f i v e times,
an
super
15-20%
and 0.9068% s a l t . ) , glycol, glycerol,
different.
the
8.^5%
in
source
glass
both
case,
may
unstable
tube.
cause
source,
the
this
after
an average value was chosen as
(7-M.
attenuation
a,
can
be
coefficient
obtained.
of
After
phantoms
are
0.209, 0.225, 0.238, 0.280 and 0.5^5 cm"1, respectively.
From chapter
2,
different biological
(Figure 3) •
among
those
brain,
Compare
this
in a
muscle
2.18:
specific
for
muscle,
ratio
the attenuation of
and
with
tendon
the
2.72).
one phantom from another
frequency,
example
phantom,
2.^5:
that
tissues are different
tissues,
liver,
simulated
2.029:
i-e.
we know
at
are
ratio
glycol
1:
in
we
the
can
find out
the
ratio
among
blood,
2.258:
2.58:
5*16:
18.06.
phantom,
and
if
then i t
wave
frequency
water,
that
sonic
through the sonic
MHz,
among
I t seems
phantom,
1
a
0.9%
saline
glycerol
ratio
phantom,
phantom
(1:
a system can distinguish
should not have any problem
distinguishing one tissue from another.
7.1.3
Phantoms
Thin-Walled glass
solutions were put
biological
tissue
test
tubes of
into a water
in
tank
varing sizes
filled with various
and used as phantoms
these experiments.
These
tubes were
to simulate
filled with
139
Figure 44.
Thermoelastic wave measured for d i f f e r e n t phantoms.
The
signals
are
detected
by
the
same
transducer
where
(a)nothing, (b) simulated muscle, (c)0.g% saline, (d)glycol,
(e) vegetable o i l , (f)glycerol, ( g ) a i r , and (h)nothing, are
located between applicator and transducer.
no
0.9%
saline,
section).
muscle
All
the
phantom,
tubes
are
glycol
0.1
cm
thick
therefore, covering the whole length of
tubes of diameter sized, namely 0.9>
These
special
test
test
tubes
were
designed holder
tube
was
located
held
in
on
(see
tubes
were
between
Figure 45).
maintained
the
transducer
experiment.
nine holes
0cm) ,
Moreover,
are
(0cm,
With
in
array
the
(3cm, 0cm),
middle
2cm) ,
of
setting,
the
by
and
the
in
length;
the use of
holder,
then
a
the
(2cm,
(-2cm,
between
relative
were
reading of
-2cm),
~3cm),
the
separations
phantom
coordinate
(-2cm,
cm
the nine holes on the plastic
positions
(2cm,
24
previous
38). By adjusting the screws, the
the
and
are
stable positions
this
fixed
and
(see
the transducer array. Four test
holder was put i n t o p o s i t i o n which one of
plate
glycerol
1 . 2 , 1 . 5 . and 1 . 8 cm were chosen.
(see Figure
right
and
known
positions
during
the middle of
2cm),
and
the
these
(0cm, -2cm),
3cm),
test
(0cm,
(~3cm,
0cm),
respectively.
Array Signals and Projection
7.1.4
The
thermoelastic
transducer
into a
array.
digital
identical
which
waves
Signals
coming
scope Nicolet
is
are
detected
out
4094A.
demonstrated
in
from
The
through
the
S/H
gain of
Figure 46.
each
element
channels
were
each channel
These
two
is
waves
of
fed
not
were
coming from the same transducer, b u t going through d i f f e r e n t processing
channels. The amplitude of
the f i r s t wave i s about
the
2.48
second
wave
is
about
Vp-p-
This
applying a normalizing factor to each output.
3*25 ^p"p> but that
problem
can be
solved by
Ul
M
-4-
0
J
Figure US.
Relative location of
2
3
4
1
1
1
cm
the nine holes.
1^2
Figure i«6.
Non-uniform gain of processing channels.
same transducer through d i f f e r e n t channel
amplitude. (Borrowed from Chan,1988).
Two waves from
shows d i f f e r e n t
143
A
major
(Chan,1988)
problem
is
that system,
A
single
trace
All
caused by
data
a
acquisition
fixed triggering
transducer
the arrival
time.
of
located at
the sides of
these acoustic waves
the
the array
samples-and-holds would be placed i n t o the hold mode
for
Figure 47 shows the signals which were detected from the
in
digital
It
scope.
one
row
seems
(17
elements)
that
these
of
waves
transducers
do
not
array simultaneously. The reading error
Either
reading error
and
In
the
samp1e-and-ho1d
Figure 48.
system
signal
digitization.
transducer
automated
the data would represent the i n i t i a l peak responses of
to
computer.
the
a reading error
array elements.
was used
of
early
triggering or
using
arrive
(A)
in the automating data acquisition system.
i n Chan
the
is described in
late triggering will
attenuation correction procedure described
at
the
cause some
Even with an
(1988),
the problem
s t i l l exi sts.
Before solving this problem,
output
to
signal
onto
amplitude
peak,
a
of
digital
5~l/4"
the
storage
the
humans
error
error
by
error
to be very small
six
one
Later,
arrival
by
one,
the
acoustic
and
data
wave.
then
storing
were
read
Instead
of
from
the
the
the
first
the f i r s t peak-to-peak value was read as the data. By the use of
this method,
stored
scope
diskettes.
first
signals were obtained by feeding S/H
is
expected.
in the digital
times),
caused
from
DC
However,
drift
can avoided but
Table XII
shows
this
(0.435%)- To complete t h i s t a b l e ,
scope and displayed at different
then the reading by author
was
reading
reading
17 data were
times
taken. Moreover,
(totally
comparing
(Vuvw/vy i> v
VKVl
F i g u r e 1»7.
Two r o w s o f t h e r m o e 1a s t i c w a v e .
Signals detected from one
row
of
transducers
(a)without
object;
(b)with
object
between applicator and transducer. (Explain i n text).
Figure k S .
Reading error
from single
triggering
signal.
(a)Four
acoustic
waves
arriving
at
the
transducer
not
simultaneously, b u t (b)using a same t r i g g e r i n g which leads
t o e r r o r s (A) show i n wave 2 and 3»
11+6
the signals without object
the
same
channel,
amplitude
( F i g u r e U~]a)
the wave pattern
changed.
These
and with object
looks
changes
are
like
each
later
(Figure 47b)
other
defined
in
except
the
part
of
as
projections.
Total
l80° or
projections can be obtained by rotating the phantom through
360°. One row o f data were detected without the object between
the applicator
data
set
is
and
transducers
called
located between
"raw
through one
data"
applicator
and
(Figure 47a).
transducers,
can be obtained through the same row o f
set of
fixed
row
of
transducers.
Later,
another
transducers
This
the
object
was
row of
data
then
(Figure 47b).
This
data can be c a l l e d "object data". By r o t a t i n g the phantom i n a
angle
through
180°
or
360°,
complete
obtained. Based on the information of
projection for
each angle of
"object
data"
could
be
"raw data" and "object data", the
view could be
found.
Figure 49
shows
a
wave related t o the raw data and 16 waves related to object data which
were detected by the same transducer through l 8 0 ° .
Moreover,
Figure 5
the
Type
is difficult
2
thermoelastic
wave
which
was
t o detect when the phantom i s far
applicator.
this
situation can also be
found
t h a t t h e ECT
is d i f f i c u l t to obtain by using this system.
major result.
problem.
to irradiate a
Increasing the microwave energy
However,
the power
output
of
in
away from the
in Figure 49-
short microwave depth of penetration through water
the shortage of microwave power
described
It
seems
Because of a
(refer to Table I I ) ,
f a r away phantom i s the
i s the way t o solve
the microwave
generator
this
in our
11.7
TABLE XI I
READING ERROR TEST
Number of
Transducer
(#)
Average
Readi ng
Number o f Reading
( X )
1
2
3
5
6
Standard
Devi ation
( x )
S.D.
1
2
1.8i» l . 8 2 1 . 8 0 l . 8 2 1 . 8 0 1 . 8 2
1 - 7 2 1 . 7 2 1.7*4 1 . 7 2 1 . 7 1 1 . 7 0
1.8166
1.7183
0.01371*
0.01213
3
3.H» 3.12 3.11* 3.11* 3.13 3.13
3 . 2i« 3 . 2 i » 3 . 2 0 3 . 2 3 3 . 2 1 3 . 2 0
3.1333
3.2200
0.0071)5
0.01732
5
6
3-58 3-58 3.57 3.56 3.56 3.60
3.12 3.10 3.1* 3.12 3.12 3.12
3.5750
3.1200
0.013814
0.01155
7
8
* • • 2 5 1| . 2 6 14.26 * . 2 5 k . 2 k 1..26
3 . i » 8 3 . ^ 8 3 • i»6 3.1»8 3 . 5 0 3 . 5 0
*4 . 2 5 3 3
3.^833
0.0137^
9
10
3.66 3-65 3.6U 3.62 3-63 3.63
3 . i » 6 3 . ^ 3 3.1«l4 3 - ^ 2 3 - ^ 2 3 . H
3.63B3
3-^350
0.013143
O.OI38I4
11
12
3.68 3.70 3-67 3.70 3.68 3.69
3 - ^ 2 3 . W 3 .I4O 3 . i » 6 3 . 3 9 3 - ^ 2
3.6867
3.^217
0.01106
0.02339
13
lit
2.78 2.76 2.78 2.75 2.7I» 2.76
2.7*4 2 . 7 3 2 . 7 2 2 . 7 1 * 2.7*» 2 . 7 2
2.7617
2.7317
0.011»62
0.00898
15
16
3 - * 9 3 . 5 2 3 . 5 0 3.148 3 . 5 0 3 - 5 0
2 . H 2 . 11» 2 . 1 3 2 . I I 4 2 . H 2 . 1 2
3-^983
2.1350
0.01213
0.007614
17
2.58 2.59 2.56 2.60 2.57 2.57
2.5783
0 .013I4I4
Average of S.D./X :
0.00715
0.^35%
148
system can not meet t h i s r e q u i s i t i o n . Thus, from now on we only discuss
the TCT case.
There
are
ultrasound
several
imaging,
ways
to define
subtraction of
the projection.
a data
from those with objects present i s used
source
function
necessary
in
this
combination of
Chan
(1988).
(John
in
set with no object present
Additional
described
subtraction and normalization
For
most
of
CT
case,
the
the object data,
p
the
normalization
in previous
is
sections.
A
was one technique used i n
projection
can be
(3.3)• Here, we define the p r o j e c t i o n by taking the natural
raw data over
traditional
to remove the variation of
Chan,1988).
system which was
In
defined
log of
as
the
i.e.
R (x • y *)
(xi ,y i) = In 0 (x j ! y | J
(7.5)
where R is raw data, 0 is object data.
This
but
also
(3.3)
definition
the
and
form
(7.5)
not
looks much
is
a
constant of background,
as ajj is small,
7.1.5
take
like
constant,
Z
In
normalization
(3-3) o^Z,
The only
where
is the distance of
into
consideration,
difference between
is
the
attenuation
the wave t r a v e l .
As
long
through
the
then this term can be eliminated.
Image R e c o n s t r u c t i o n and P r o c e s s i n g
Images
can
be
reconstruction program.
case,
only
another
one
for
reconstructed
from
projections
Two d i f f e r e n t algorithms, one f o r p a r a l l e l ray
equally
spaced
fan beam case,
were written
in
H9
im
fSBSSBSSn
\J\j\A- v/NA/^V-v^r
Figure 1*9.
One row o f
signals detected by
same
transducer.
The f i r s t
signal i s raw data, another sixteen were detected through
16 different angle of views, with rotating the phantom
through l8o°
150
F0RTRAN77
then
implanted
into
INTEL8638O
microcomputer
(Appendix
B) .
After testing the performance of these two algorithms, the parallel ray
case
was
chosen
as
several problems for
CPU t i m e used
case.
a
standard
the use of fan beam algorithm.
the reconstruction region
case, for example, only 6 unit
of
the
(19)
simulation studies
(4-95 cm)
region.
All
First of
are
a l l , the
is
too small
to cover
every
region can be covered by using
these
problems
have
been
found
in
(refer to Chapter 4). The latter two problems can be
increasing the number of
transducers
There
of tranducers. Thirdly, the noise appeared around the edge
reconstruction
solved by
algorithm.
i s much l o n g e r t h a n t h e CPU t i m e used i n t h e p a r a l l e l r a y
Secondly,
one row
reconstruction
in this experiment
transducers. However,
is fixed,. Thus, only
the number of
the parallel
ray
reconstruction algoritm is used in this study.
The
concept
operations
of
infornmation,
been
image
which modify
visibility
has
of
or
an
useful
so
implanted
processing
image or
information
called "noise".
into
INTEL8638O
can be defined
groups
of
while
An
images
as
processes
to enhance
suppressing
The
the
non-useful
image processing package
system.
or
standard
ACQ4
smoothing
f i l t e r and thresholding techniques were applied in these experiments.
7.2
Experimental Protocol
For
four
the purpose of
different
combination
undertaken:
of
setups
several
testing the performance of
were
performed
different
setups,
(see
a
the system,
Table XI11).
specific
twenty
With
objective
a
was
(a) Image
with that
(b)Image
1,^,5>
and 6 - -
to compare
the actual
size of
phantoms
from the images i n single phantom case.
1,11,13»
and
15
~~
to
compare
different
biological
materials with that from the images i n single phantom case.
(c)Image
with that
(d)Image
10,11,
and
12 - -
to compare different phantom
locations
from the image.
7»8,
and
9
—
to
find
the minimum
spacing between
two
objects that can s t i l l be detected distinctively and the relative
posi t ion of
(e) Image
images.
16,17
—
to
compare
the
relative
size
of
two
to
compare
different
phantoms
with that from the images.
(f)Image
18,19.
and
20
—
biological
materials with that from the images i n two phantom case.
(g)Image H and 15 — t o check
the error due to rotating phantom.
(h)Image 1,2, and 3 — to check the error
due to unstable source.
(i)Image 21,22,23* and 2 b - - general case test.
i s s u m m a r i z e d a s f o l l o w s . A 2 ms p u l s e d
The experimental protocol
2^50
MHz
microwave
diameter of
of water.
3 0 mm)
energy
was
applicator
delivered
by
an
into the surface of
Elmed-107
(with
a 25x30x50 cm3
I t generated thermoelastic waves at the surface of
a
tank
the water
surface. The forward and reflected power was monitored so that the peak
power was about 30 KW(in most of
placed on a plastic
rack. After
the cases).
The hydrophone array was
the thermoelastic waves arrived a t the
array, the measuring of the i n i t i a l peak
to peak response of one row of
TABLE X I I I
IMAGE RECONSTRUCTION PROTOCOL
Pulsed Microwave
Phantom
Image No.
Nos .
1
2
3
i*
5
6
7
1
1
1
1
1
1
2
8
2
9
2
10
11
12
13
1<4
15
16
1
1
1
1
1
1
2
17
2
18
2
19
2
Medium8
Size''
0(cm)
Position 0
1
1
I
1
1
1
2
2
2
2
2
2
3
3
3
U
2
2
2
2
1
1
2
1
U
2
20
2
21
3
22
3
23
3
k
1
1
1
1
1
<•
2
2
k
2U
0
3
1
2
2
2
Medium 1: Simulate Muscle, 2: Glycerol,
bDiameter
Power (KW)
(x.y)
26.5
26.5
26.5
26.5
26.5
26.5
30.0
5
5
5
5
5
5
U
6
1
9
30.0
U
30.0
8
U
30.0
30.0
29-5
28.5
28.5
28.5
30.0
5
2
5
5
5
i.
6
30.0
6
1
9
I
9
1
9
2
3
9
2
3
9
2
3
9
2
3
9
29.3
30.0
30.0
30.0
30.0
30.0
30.0
Glycol, U: 0.9% Saline
of Test Tube Size(cm), I: 0.9, 2: 1.2,
1.5, *»: 2.0
Coordinate Reading of Pos i t ion(cm), 1:(0,3), 2:(2,-2), 3:(2,2),
h: (0,-2), 5:(0,0), 6:(0,2), 7:(-2.-3), 8:(-2,3), 3: ("3,0).
153
elements was made. The amplified, and f i l t e r e d signals were referred t o
as "raw data". Another
sixteen irradiations with the phantoms i n place
and rotated i n fixed angles produced sixteen sets of
data referred to
as
be
"object
data".
From
those
images were then reconstructed
data,
projections
can
from those projections.
setups, d i f f e r e n t images were generated.
defined.
The
With different
Chapter 8
RESULTS and DISCUSSIONS
The
images obtained from the experiment protocol described i n the
previous chapters are presented and disccussed i n this chapter.
8.1
Computer Simulation Model
Taking into consideration the limits of
views,
comparing
the
reconstructed images
difference
transducers and angles of
between
the
of
and
the
to test the performance is not appropriate. I t is
necessary to seek a computer simulation model
basis
phantom
comparison.
This
transducers
and
angles
of
projection,
and
then use
model
view
the
shall
as
same
to create an
use
the
experimental
image as the
same
case
to
amount
of
generate
the
reconstruction algorithm
to obtain
the comparable image.
Figure 50
shows
a
"time-of-f1ight"
image
(Chan,
1988)
which was
generated as follows. Signals coming out from the S/H channels were fed
into a digital
scope. The signals were sampled at
from the starting point
channel
computer
source
was
measured.
to the beginning of
The
for processing.
function
time-of-f1ight
in
image
the
400
This
data
domain.
the
time
the f i r s t response of each
were
is only a crude
time
concept,
points
2 MHz and t h e
then
keyed
into
the
image t o i l l u s t r a t e the
However,
irradiation
according
wave
pattern
to
the
of
the
system can be described i n Figure 51• I t i s clear that neither parallel
15*»
155
rays nor
seems
fan beam case can be used as a model
that
the reconstruction region
fan-beam case,
projections
but smaller
among
those
than
three
in our
the parallel
cases
also
to simulate our
case.
case
than
the
rays case. Moreover,
the
are
is
larger
different.
With
It
the
p r o j e c t i o n s from the wave p a t t e r n i n our case, some e r r o r s appeared i n
reconstructed
images which
image si2e
is
larger
off
phantom;
center
is
expected.
than actual
(3)
size;
These errors
(2)
the distance of
included
(1)
the
the distortion appeared for
two objects
in
image
is
large
than actual distance.
For
convenience,
two images which were constructed using
algorithm from two different projections
generate projections,
another
using
( one using parallel
fan-beam
the system.
to
fan-beam
more
detectors
However,
(29
instead of
to
to simulate
avoid
the
increasing the area of
8.2
19)
regions
in
to
the
In order
case,
more
were used in the simulated fan-beam case.
the method i s not very accurate.
a better model
system
reconstruction
rays
to generate projections)
were used as an index to check the performance of
cover
the same
this system or
rebinning
problem
Further study either to find
to modify
is
the setup of
necessary.
For
this
example,
the i n i t i a l wave may solve t h i s problem.
Experimental Results
To reduce the noise, a l l
the experimental
applying a smoothing f i l t e r . The operator of
results were obtained by
this filter
is
156
Figure 50.
A time-of-f1ight image.
The different colors represent
different propagating time of signals. At the middle the
propagating times i s shortest and the longest propagating
time appears a t the corners. (Borrowed from Chan,1988).
157
Thermoelastic wave source
Reconstruction r e g i o n of
( 1 ) parallel r a y s c a s e
( 3 ) experimental c a s e
(2) fan-beam case
Phantom
\
Detector plane
(1): Projection of phantom for case 1
(2): Projection of phantom for case 2
Figure 51«
A comparison of simulation models.
A model to describe the
wave propagating through the phantom based on information
from previous
figure.
Neither
parallel
rays
case
nor
fan-beam case only can be used to described the case of
this experiment.
158
A
thresholding
distinction.
processing
technique
Figure 52
procedure.
was
then
shows
a
applied
for
sequence
Comparing
of
Figure 52a
the
purpose
images
with
using
Figure 52b,
of
object
the
image
it
seems
that the image after smoothing f i l t e r processing, noise indeed has been
reduced,
but
the
edge
information
is
not
phenomena agrees w i t h the prediction which
After
the thresholding is applied,
ring
which
Table XIV
encloses
is
two
the gray
it
different
level
sharp
as
before.
is described
is clear
sizes
scale of
as
in chapter
the high pixel
value
(Pmax.)
decreased in each case after
of
circles
2k sets of
(Figure 52b).
images which includes
with a
the
are
the smoothing method is applied. Thus,
the
distinguish
from
results, only
8. 2 . 1
scale
(Pmin.)
is
increased ,
in the image. Thus,
background.
In
the
i.e., a
pixel
the phantom i s easy t o
following
experimental
the image after thresholding i s displayed.
S ! ng /e P h a n t o r n
Figures
tubes
the
level
i s reduced. When a low end threshold was
low minimum value
low value disappears
and the gray
I t is clear
(VGS)
noise with relative high value
applied,
3.
t h i s phantom composed a
the reconstructed images after smoothing and thresholding.
that
This
of
53
1.8
to 56
cm,
1.5
equivalent
materials,
that when
the phan.tom
area,
location
the
the reconstructed
are
four
cm,
1.2
processed
cm,
respectively.
is
located at
information of
images.
Also,
as
and
From
images
0.9
these
set
cm
the
size of
filled
figures,
the middle of
the phantom
for
single
with
it
is
test
muscle
clear
the reconstruction
is represented well
by
phantom i s decreased,
Figure 52-.
A sequence of images for image processing.
(a)The original
reconstructed image of phantom (b)the image after smoothing
the original
image
(c)the image after thresholding the
image b .
gure 52.
Continued
TABLE XIV
GRAY LEVEL SCALE OF
Image
Reconstructed Image
Pmin.aPmax.k
No.
VGSC
2
-0.063 0.1 8 0 0 . 0 1 5 1
-0.099 0.1 8 0 0.0171)
3
i»
1
5
6
7
8
9
IMAGES
Smoothing Image
Pmin. Pmax.
VGS
Thresholding Image
Pmin.
Pmax.
VGS
-0.072 0.171 0.0152
-0.063 0 . 1 5 1 0.01314
-0.099 0. 11+7 0.0151*
- 0 . 0 7 2 0.1 5 0 0 . 0 1 3 9
0 . 0 1 7 3 0. 151 0.00814
- 0 . 0 0 6 8 0. 11+7 0 . 0 0 9 6
0 . 0 1 1 3 0. 150 0 . 0 0 8 7
-0.01+5 0.1 2 6 0 . 0 1 0 7
-0. 0 3 6 0 . 0 8 1 0 . 0 0 7 3
-0.01.5 0. 112 O.OO98
-0.036 0 . 0 6 5 0.0063
-0.036 0.081 0.0073
- 0 . 0 3 6 0.01+8 0 . 0 0 5 2
0 . 0 1 3 9 0. 112 0 . 0 0 6 1
0 . 0 0 1 9 0 . 065 0.001+2
-0.001+5 0 . 01+8 0.0033
- 0 . 0 7 2 0 . 1 2 6 0.0121+
- 0 . 0 9 0 0 . 1 8 0 0.0169
-0.135 0 . 162 0 . 0 1 8 6
-0.072 0.082 0.0096
- 0 . 0 9 0 0.166 0.016
-0.135 0 . 1 2 5 0.0163
-0.01143 0 . 082 0.0060
0 . 0 0 6 0 0 . 166 0 . 0 1 0 0
- 0 . 0 3 7 5 0 . 125 0 . 0 1 0 2
-0. 011*6 0. 1 1 1
- 0 . 001*5 0. 123
- 0 . 0 0 8 3 0. 11)2
- 0 . 0 0 7 5 0. 100
0 . 0 0 6 0 0. 106
- 0 . 0 0 2 3 0.099
10
-0.090 0.171 0.0163
11
12
13
U
15
-0.081 0 . 1 3 5 0.0135
-0.099 0.1 8 0 0.017^
-0.072 0 . 1 3 5 0.0129
- 0 . 0 6 3 0 . 1 0 8 0.0107
-0.090 0 . 1 1 1
-0.081 0.1 2 3
-0.099 0.11+2
- 0 . 0 7 2 0.100
-0.051* 0.1 0 6
- 0 . 0 6 3 0.099
16
17
- 0 . 0 6 3 0.126 0.0118
- 0 . 0 6 3 0.1U 0 . 0 1 3 0
- 0 . 0 6 3 0.086 0 . 0 0 9 3
-0.0&3 0 .11 5 0.0111
- 0 . 0 0 7 1 0.086 0 . 0 0 5 8
0 . 0 0 3 8 0. 115 O.OO69
18
- 0 . 0 8 1 0.225 0.0190
- 0 . 0 9 9 0.153 0.0152
- 0 . 0 9 0 0.198 0 . 0 1 8
- 0 . 0 8 1 0.175 0. 0 1 6 0
-0.099 0.1 2 7 0.011 (1
- 0 . 0 9 0 0.161* 0 . 0 1 5 9
0 . 0 1 5 0 0. 175 0.01
-0.011*3 0. 127 0 . 0 0 8 8
0 . 0 0 5 3 0 . 161+ 0 . 0 0 9 9
-0.1 0 8 0.153 0.0163
-0 . 1 1 7 0 . 0 9 9 0.0135
- 0 . 0 9 9 0.1 2 6 0.011+1
- 0 . 0 9 9 0 . 1 7 1 0.0169
-0.1 0 8 0. 1 lfc 0 . 0 1 3 9
- 0 . 1 1 7 0 . 0 8 1 0.0121*
- 0 . 0 9 9 0.1 0 3 0.0126
- 0 . 0 9 9 0.1 2 6 0.011(1
-0.021+8
-0.01)28
-0.0233
-0.011)6
-0.05b 0.153 0.0129
19
20
21
22
23
2U
aPmin.:
0.0126
0.0128
0.0151
0.0108
0.0100
0.0101
Minimum value among the p i x e l s
i n image
kpmax.: Maximum value among the p i x e l s
i n image
CVGS
: The value of each gray scale.
0.0079
0.0080
0.00914
0.0068
0.0063
0.0063
0. 1 11+ 0 . 0 0 8 7
0.081 0.0078
0 . 103 0 . 0 0 7 9
0 . 126 0 . 0 0 8 8
162
the
gray
level
in
the
decreased. The sizes of
middle
of
image
and
the
l i m i t a t i o n from 0.825 cm transducer,
seems reasonable. When t h e o b j e c t s
of
of
image
are
the images do not correspond to the actual size
very well when using a non-parallel rays model
one unit
sizes
transducer
source.
However, with a
the relative size of
size goes down t o 0.9
size, more noise begins
the response
cm,
to appear.
close to
This
shows
that in order to detect a small object, the use of a smaller transducer
is necessary.
Figures 57
tubes
of
to 59 are
1.8
respectively.
cm
It
three processed
filled
seems
images are the same,
with
the
result
scale
reconstruction.
difficult
to
gray
With
set
a
CT,
limited
case during
the experiment.
the
Figure 60) . In each
1112)
gray
obtain
image,
different
corresponding
image,
the
same
This
of
stretching
glycerol,
these
increase
of
gray
four
straight
gray
level
level
lines
for
(12)
may
during
colors),
as
CT
the
it
is
number
to cover
scales
in
every
was
used
the
(see
and minimum
(mi and
the two ends,
then a
(G1
each
reasonable
the contrast of
(Mi and M2)
the
a
used
(15
such
images was used as
gray
is
image parameter
to
the maximum
was
display
level,
the
In order
linear
interpolation along
the
single test
and
the middle
well.
level
gray
to represent
value among the p i x e l s of
linear
in
for
saline
transformation
standard
x-ray
image,
levels
phantoms very
traditional
displayed
0.9%
sets
i . e . , the results do not response to the variety of
of
a
glycol,
gray
attenuation coefficient
since
images
and G2)was
pixel.
used
Thus,
represent
in
to
a
different
jimim
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Reconstructed images set 1.
With setup no. 1 (refer to
Table XIII)
the images reconstructed from
(a)simulated
parallel
rays
projections;
(b) simulated
fan-beam
projections;
(c) experimenta1 projections, by Shepp-Logan
f i1ter.
F i g u r e 5*»«
Reconstructed images set U.
With setup no. k (refer to
Table XIII)
the
images reconstructed from
(a)simulated
parallel
rays
projections;
(b)simulated
fan-beam
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(c) experimenta1 projections, by Shepp-Logan
f i1ter.
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Reconstructed images set 5With setup no. 5 (refer to
Table XIII)
the images reconstructed from
(a)simulated
parallel
rays
projections;
(b)simulated
fan-beam
projections;
( c ) e x p e r i m e n t aI p r o j e c t i o n s , b y S h e p p - L o g a n
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With setup no. 6 (refer to
Table XIII)
the
images reconstructed
from
(a)simulated
parallel
rays
projections;
(b)simulated
fan-beam
projections;
(c) experimental projections, by Shepp-Logan
f i1ter.
16?
values
(Vi and V2). This phenomenon appeared i n most of
experiment
more
(see Table XIV).
than one object
at
To avoid
the
same
However, by averaging the gray
i t can be seen that
this misleading result,
time was used
levels of
the higher
in
Figures
1.8
cm
61
the
glycol
at
images
different
the middle position
reconstructed
images
found i n a simulation case
are
(glycerol>
of phantom, the higher
for
(Figure 62b).
single test
locations.
The
tubes
of
locations
of
is different from the case of
(Figure 57) >
distorted.
there are no appropriate rebinning
This
This
because
situation
i s due
the
shape
also
the r e l a t i v e position showed that the
original
location of the phantom very well.
of
can
be
to the fact that
techniques used during this
However,
stage.
images responded to the
Double Phantoms
8. 2 . 2
Two
to find
identical
identical
2.2
phantoms with different
separations
were processed
the minimum spacing between objects that can s t i l l be detected
distinctively.
of
images,
i n the reconstructed image.
these two cases are not i n the middle. I t
the phantom at
experiments.
the attenuation coefficient
and 62 are processed
filled with
later
this
testing
the middle portions of
glycol> muscle equivalent material> 0.9% saline)
the average gray level
images of
Figures
phantoms
cm,
separation of
3-6
63
to
65
(1.8 cm s i z e ,
cm,
and
k.2
are
the reconstructed
filled with glycerol)
cm,
respectively.
It
images
for
two
with separation
is
clear
that
a
less than 2 . 2 cm i s not easy t o d e t e c t . I n an ideal case,
as long as the separation between the two objects i s greater than 0.825
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f i1ter.
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im«i j i:::
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11
Reconstructed images set 12.
With setup no. 12 (refer t o
Table XIII)
the
images reconstructed
from
(a)simulated
parallel
rays
projections;
(b)simulated
fan-beam
projections;
(c) experimenta1 projections, by Shepp-Logan
f i1ter.
cm
to
(1 u n i t ) ,
false
t h e separation should be d e t e c t a b l e . Some f a c t o r s may
results,
namely
shadowing,
diffraction
and
led
refraction
problems, wave scattered by two objects also interfere with each other
1982).
(Greenleaf,
wave equations.
the used of
This
interference
However,
it
problem
can be
solved
by
using
i s a very complicated problem, and so far
f i r s t order approximation of
wave equations does not solve
this problem. Further study in this area
i s necessary. From the figures
we can conclude that
tubes
must
be
Moreover, as
the phantom
the smallest
placed
and
still
separation a t which two 1.8 cm t e s t
be
detected
in the case of a single phantom,
in the
image
is
reasonable.
is
more
than
2.2
cm.
the relative position of
The distorted shapes
in the
reconstructed image i s due t o the rebinning problem.
Figure 66
glycerol,
one
Figure 46),
Figure
67
i s a processed image set for
that
and
is
a
is
1.8
another
cm
that
processed
and
is
image
located
0.9
set
two test tubes f i l l e d with
at
position
cm
and
located
for
two
test
k
at
tubes
(refer
to
position
6.
filled
with
simulated muscle materials, one that
i s 1.8 cm and located a t p o s i t i o n
h,
located
and
images,
another
that
is
1.2
cm
and
even with distorted shape of
the respective
sizes of
the phantoms
the
at
images,
are very
the minimum separation i n these two cases
position
it
is
6.
In both
apparent
clear. Moreover,
is 3-0
cm,
that
since
two phantoms can
be distinguished without difficulty.
Figures 68 t o 70 are three processed image sets for two test tubes
located
at
position
1
and 9-
Figure 68
represents
the
images
of
two
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Table XIII)
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(b)simulated
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pp-Logan
b
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the images
Table XIII)
parallel
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Droiections;
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by
Shepp-Logan
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Reconstructed images set 8.
(a)simulated
Table XIII)
the
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parallel
rays
projections;
(b)simulated
fan-beam
projections;
( c ) e x p e r i m e n t aI p r o j e c t i o n s , b y S h e p p - L o g a n
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Reconstructed images set 16.
With setup no. 16 (refer t o
Table XIII)
the
images reconstructed
from
(a)simulated
parallel
rays
projections;
(b)simulated
fan-beam
projections;
( c ) e x p e r i m e n t aI p r o j e c t i o n s , b y Shepp-Logan
f i1ter.
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Reconstructed images set 17With setup no. 17 (refer t o
Table XIII)
the
images reconstructed from
(a)simulated
parallel
rays
projections:
(b)simulated
fan-beam
projections;
(c) experimenta1 projections, by Shepp-Logan
f i1ter.
18.0
t e s t tubes, one t h a t i s 1.8 cm f i l l e d w i t h g l y c e r o l and another t h a t i s
1.2 cm f i l l e d w i t h simulated muscle m a t e r i a l . Figure 69 represents the
images of
two test
and another
the
that
images of
tubes,
is
one
that
is
1.8 cm f i l l e d w i t h 0.9%
1.2 cm f i l l e d w i t h g l y c e r o l .
two test
tubes,
one that
is
saline
Figure 70 represents
1.8 cm f i l l e d w i t h simulate
muscle material and another t h a t i s 0 . 9 cm f i l l e d w i t h 0.9% s a l i n e . The
shapes
of
all
three
correspondences
attenuation
processed
images
to the image size,
coefficient
are
very
are
distorted.
relative position,
consistent.
The
However,
the
and the relative
same
conclusion
has
been found i n processed images using three objects.
Figures 71 to l b are four processed image sets of
located
at
processed
positions
images
of
2,
3>
1 .5
three
muscle material. Figure 72
and
9>
cm
respectively.
test
tubes
three test tubes
Figure 71
filled
i s the processed images of
with
is
the
simulated
three 1 .5 cm t e s t
tubes f i l l e d with simulate muscle material, 0.9% saline, and glycerol,
respectively.
Figure 73
with different
1 .5
material,
respectively.
cm
test
The
tube
results
saline,
F i g u r e ~jb
images
and
is
0.9
the
1.5 cm, and 1.8 cm,
sizes,
images can represent
system.
processed
of
three
cm w i t h
processed
tubes f i l l e d with glycerol,
tubes are 1.2 cm,
various
these
0.9%
with
different size test
of
the
sizes and f i l l e d with different materials,
glycerol,
test
is
do
the performance of
not
significantly
1.2 cm w i t h
images
the size of
and
when
of
three
the three
With setups
relative
a general
differ
tubes
simulate muscle
respectively.
phantom materials
test
location,
case for
this
compared
with
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Reconstructed images set 18.
With setup no. 18 (refer to
Table XIII)
the
images reconstructed from
(a) simulated
parallel
rays
projections;
(b)simulated
fan-beam
projections;
(c) experimenta1 projections, by Shepp-Logan
fiIter.
182
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Figure 69.
j :•••••••
Reconstructed images set 19With setup no. 19 (refer to
Table XIII)
the images reconstructed from
(a)simulated
parallel
rays
projections;
(b)simulated
fan-beam
projections;
( c ) e x p e r i m e n t aI p r o j e c t i o n s , b y S h e p p - L o g a n
f i1ter.
8
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Reconstructed images set 20.
clu(j
mu.
Table XIII)
the
images reconstructed from
(a)simulated
parallel
rays
projections;
(b)simulated
fan-beam
projections;
(c) experimenta 1 projections, by Shepp-Logan
f i1ter.
181*
single
test
tube cases and double test
tubes cases,
except
that
there
i s an increase of wave scattering interference among objects due t o the
fact that more objects were involved i n this case.
I n summary,
and
three
location,
the results show that
objects
larger
shape,
than the actual
size.
several
factors
need
image one,
relative
size,
two
relative
As was expected, a d i s t o r t e d image
center phantom and the size of
by using a correction factor.
factor,
in
and attenuation constant.
results from off
is
consistently
this system can
All
of
i n the image
these problems may be recovered
In order
to be
the object
to investigate this correction
considered.
We
will
discuss
these
factors in the next section.
8.3
D iscussion
From previous
the reconstructed
rotating error,
sections,
images,
we
know
such as
rebinning model,
that
several
reading
error,
factors may
an unstable
diffraction and reflection of
and multiple scattering problems. These problems w i l l
affect
source,
waves,
be discussed as
follows.
The
reading
Although
the
error
error
problem
rate
has
(0.^35%)
been
is
discussed
very
low
in
in
section
this
7.1.^.
system,
an
accurate automated data acquisition system which also can save a l o t of
data acquisition time is preferred.
Error
due
experiment.
same
source
to
With
six
a
an
unstable
fixed
times.
source
phantom,
The
data
the
was
detected
phantom was
was
read
by
the
following
irradiated by
through
17
the
different
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Mil
Reconstructed images set 21.
With setup no. 21 (refer to
Table XIII)
the
images reconstructed from
(a)simulated
parallel
rays
projections;
(b)simulated
fan-beam
projections;
(c) experimenta1 projections, by Shepp-Logan
f i1ter.
b
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Figure 7 2 .
JL1 I
4
:1:»«:i•!Iii:!::I:!:1:::m<
»: 111. it 111 s 1::
111111:1:1n:
Um: t it 11..»u 1 n 1
Reconstructed images set 22.
With setup no. 22 (refer to
Table XIII)
the
images reconstructed from
(a)s imu1ated
parallel
rays
projections;
(b)simulated
fan-beam
projections;
(c) experimentaI projections, by Shepp-Logan
f iIter.
187
0123^567
GRAY L E V E L
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Figure 73.
Reconstructed images set 23.
With setup no. 23 (refer to
Table XIII)
the images reconstructed from
(a)simulated
parallel
rays
projections;
(b)simulated
fan-beam
projections;
(c) experimental projections, by Shepp-Logan
f i1ter.
t
i:: j •• :iti:•••
«-»«•»«•
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188
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Figure 1U.
i:
i !•-»•« t:
With setup no. 2U (refer to
Reconstructed images set 2 U .
the images reconstructed
from
(a)simulated
Table XIII)
parallel
rays
projections;
(b)simulated
fan-beam
projections;
(c)experimentaI projections, by Shepp-Logan
f i1ter.
189
transducers, then the mean
for each
individual
in this test
(50
and standard derivation
(SD)
transducer. The average error rate
(Table XV).
was counted
(SD/X)
i s 2.25%
There are two ways t o reduce the error caused
by the unstable source. One i s t o change the source, and another
use
the
The
averaging method.
first,
projections
second,
and
obtained
at
Figure 75
third
three
shows
images
four
were
different
reconstructed
reconstructed
times.
The
fourth
reconstructed from the projections obtained by averaging
three projections.
three original
In an
middle
of
I t seems the averaged image has
is to
images.
from
the
image
was
the previous
less noise than the
images.
ideal
the
case,
where a
reconstruction
symmetrical
area,
the
phantom
is
projections
located at
obtained
the
with
a
rotated phantom and without a rotated phantom w i l l have the same value.
In
a
practice
rotated.
case,
some
Table XV shows
noise may
pair
of
with
a
the
when
i s 3-09%.
reconstructed
fixed
phantom
images.
and
fact
that
phantom
Figure 76a
Figure 76b
is
is
the
the
not
centered
in
object
errors,
is
due to an
Figure 76
shows a
reconstructed
reconstructed
of Figure 76b i s larger
is
the
i.e., in our experiment
due to the rotating phantom s t i l l exists.
rotated phantom. The size
to
generated
that with a combination of
unstable source and rotating phantom,
the error
be
image
image
with
than Figure 76a due
the rotating case which
generated a wider projection than a fixed phantom.
As mentioned
has
been
found
in section 8.1,
for
our
so far
experiment.
no suitable rebinning method
The
distortion
of
shape
that
TABLE XV
SOURCE AND MECHANISM ERROR TEST
Number o f
Transducer
(#)
Numbers o f
R e a di n g
(N)
Average a
Readi ng
( XI )
S.D.
1
2
6
6
2.05
1.67167
3
4
6
6
3.27667
3.13833
5
6
6
6
7
8
0.06191
0.07081
Average
Reading
b
Standard
Devia t ion
( X2 )
S.D.
2. 165
1.63167
0.08401
0.03716
0.04308
0.03184
3-34167
3.23333
0.07625
0.09843
2.95
2.46
0.06137
0.09292
2.93166
2.585
0.05728
0.10420
6
6
3.3
2.1*2
0.05686
0.06128
3.43333
2.45333
0.06574
0.05153
9
10
6
6
2.51167
2.76333
0.02478
0.03197
2 .62666
2.66
O.O6968
0.19088
11
12
6
6
2.83838
0.09668
0.01155
2.77666
2.71833
0.11968
2.79
13
U
6
6
2.59833
2.66
0.05956
0.04123
2.5^
2.625
0.05956
0.05025
15
16
6
6
2.95
1.775
0.15716
0.02566
2.775
1.77333
0.12698
0.05706
17
6
2.30667
0.04460
2.18333
0.02427
Average of S.D./X :
3
Standard
Devi a t ion
2 . 25%
0.10335
3-09%
Reading i s taken from fixed phantom at different radiated times,
k Reading i s taken from rotated phantom at six different angles.
191
a
Figure 75*
Reconstructed images and averaged image.
(a), (b), and (c)
are the reconstructed images from the detected projections,
(d)
is
the
reconstructed
image
from
the
averaged
projections.
Figure 75-
Continued
193
a
Figure 76.
The reconstructed images with/without rotated.
The image
reconstructed from the projections which obtained by (a)
f i x e d the phantom; (b) r o t a t i n g the phantom.
19A
appeared
in
most
rebinning method
way
to generate
of
the
is still
reconstructed
was
expected.
The
under development, however, there is another
the projection without
The use of standard
images
horn
instead of
using the rebinning
applicator
technique.
in a microwave system
may generate the thermoelastic wave pattern close to the parallel
case (Olsen and
rays
Lin, 1983)• This technique has been tried during this
study. However, with our microwave generator (PH^OK), the power density
is too low to generate the thremoe1astic waves.
With
the
resolution of
limitation
of
size
and
gray
level,
the
images is lower than the traditional x-ray CT where more
transducers
and
gray
transducers
and
increasing
However,
transducer
another
levels
factor
were
the
that
operating frequency. With a
used.
gray
may
Decreasing
levels
also
low operating
may
limit
the
solve
the
size
this
of
the
problem.
resolution
is
the
frequency, the error due to
wave scattering may turn out to be very critical. There are two ways to
increase the operating frequency. One is the use of a narrow
beam with
a high frequency response transducer. In this case only high frequency
components
Another
of
way
signals
is
to
will
choose
be
the
According to (2.8), the maximum
function
summarizes
of
pulse
results
widths
for
5
detected
microwave
frequency
pulses
and
frequency
Conway,
and
pulse
1982).
width.
intensity of thermoelastic waves
and
microwave
different
different pulse widths. It is clear
microwave
(Casper
appear
frequencies.
microwave
frequencies
is a
Figure 77
with
10
that the shorter pulses and higher
to
stimulate
higher
frequency
195
components. Therefore,
using
a
high
a
frequency
high operating frequency
and
short
pulse width
can be achieved
by
microwave source and
a
high frequency response transducer.
The problem of wave scattering still
high
frequency
calculate
thermoelastic
wave
waves.
scattering,
then
exists even with the use of
Using
the
making
wave
the
equation
modification
to
of
reconstruction algorithm may be the right approach to obtain a better
result.
Several methods were proposed, such as first order diffraction
tomograghy with Born approximation and
al.,
1986),
inverse
method
Rytov approximation (Slaney et
(Boerner
amplitude conjugations (Guo et al.,
and
Chan,
1£}86) ,
phase
1986) . Most reports only provided
the results of computer simulation. The results from experimental
show that a higher order approximation
good
biological
tissue
imaging
cases, phase information
data
is necessary in order to obtain
(Ermert
is quite
and
and
Dohlus,
1986).
In
important to reconstruct the
these
image.
Therefore, further study on the development of reconstruction algorithm
and some modification of the hardware system
catch
enough
should be able
information
to record
sampling rate, such as
for
image
is necessary
reconstruction.
in order to
The
new
the signals through every channel
10 MHz, for 20 ms, 20
system
at a high
before and after the
/JLS
expected wave arriving time. Pattern recognition routine can be written
to track
the first
peak of
this wave and measure both
amplitude at that time. Fast
with
this
kind
of
ADC and
processing.
the phase and
large memories will
After
that,
algorithm can be used to generate a better image.
a
new
be required
reconstruction
196
Acoustic Frequency and Pulse Width
Acoustic
Freq., KHz
• *33 MHz
• 2450, MHz_
^ 9^5 MHz
O 5000 MHz
A 8000 MHz
1 00 -tAAJ
80 -
60 -
40 -
20 -
Pulse Width, microseconds
Figure 77*
Thermoelastic waves at frequency of maximum
intensity.
Five
frequencies
and
ten
pulse
widths
were
used
to
calculate the maximum intensity of generated thermoelastic
wave. With a high frequency and short pulse width microwave
source, a maximum intensity of thermoelastic wave appeared
at high frequency. (Borrowed from Chan, 1988).
Chapter 9
Conclusion
This thesis is concerned with the development of computer assisted
tomography
using
development
involves
receiving
microwave-induced
system,
the
the
design
display of two-dimensional
results
led
fabrication
implementation
modification of reconstruction
have been performed
and
thermoelastic
of
of
signal
algorithm, and
the
waves.
a
This
scanning
and
conversion,
the
image processing
and
thermoe1astic tissue tomography. Experiments
to demonstrate the performance of this system. The
us to new
approaches
for
further
study
and
the potential
use of this new modality.
Results from two preliminary studies
system.
In
thermoelastic wave study,
led us to develop this
results agree
both
theoretically
and empirically with findings previously reported by other
In computer simulation of
in shape, size and
well
to
the
reconstructed
images
researchers.
reconstruction algorithms, results show that
location, the reconstructed
original
image
simulation
correspond
well
phantoms.
to
the
images correspond
The
gray
attenuation
very
levels
of
constant
of
phantom in TCT case; in ECT case, however, they are proportional to the
ratio
of
the
emitter
concentration
Moreover, this system shows
with a separation of 1
and
that objects of
the
1
attenuation
unit of
constant.
transducer size
unit transducer from each other can be detected.
197
198
The
hardware,
used
to
generate
the
projections
of
test
object
consists of a microwave devices, rotating phantom, interface electronic
and
computer.
A
test object
is
immersed
in a
tank
of water
at whose
surface a single microwave pulse of 2 us with 30 KW peak power at 2^50
MHz
is
launched. Thermoe1astic waves
induced
at
the water
surface are
detected, on propagating through the objects, by a 20x20 piezoelectric
transducer array
are
then
repeated
amplified
by
the
and
rotating
projections
obtain
at the bottom of
were
the object
of
techniques, the processed
Results
from
microwave-induced
modality.
The
band-limited.
obtained.
tomograph
this
the water tank. The received signals
When
the
same
procedure,
through complete angles,
Reconstruction
object.
Through
algorithm
the
use of
was
the complete
was
image
applied
to
processing
images are displayed on a color monitor.
system
thermoelastic
experimental
demonstrate
waves
as
protocol
a
was
the
potential
possible
use
medical
designed
to
of
imaging
test
the
performance of this system. Several different setups with various size,
location,
and
test
objects
were
preliminary results show that
were
very
consistent
in
used
in
this
stage
the system corresponding
shape,
relative
size,
of
study.
The
to those setups
relative
location
and
attenuation constant. Currently, with the thermoelastic waves generated
from
this
system,
objects
of
0.9
cm
in
size
(about
the
size
of
transducer) with separation of 2.2 cm from each other can be detected.
Moreover,
biological
materials
with
contents can be differentiated from
high,
moderate,
the reconstructed
this system has its own limitation and drawbacks.
and
low
water
images. However,
199
Some
limitations
experiments.
and
drawbacks
There are three major
have
been
problems
found
during
the
in this system we cannot
eliminate. (1) Because of the limitation of microwave power output, the
thermoelastic signal
(refer to chapter 3) emitted from
be detected. Without this
provide
some
unique
information, the
tissue
characteristics
without enough power, instead of standard
during
the experiment.
In order
wave patterns generated
by
ECT of
was
phantoms cannot
phantoms which can
not
obtained.
Also,
horn, an applicator was used
to reduce the error due to the cone
the applicator, the use of
an appropriate
rebinning method is necessary. Without applying this correction factor,
the shape of reconstructed image is distorted.
(2) With a manual data
acquisition system, not only is there human error, but also it prolongs
the
data
acquisition
time.
(3)
A
low
operation
frequency,
not
only
reduces the resolution of the system, but also increases the error due
to
wave
scattering,
(such
as
diffraction,
refraction,
and
multiple
scattering). Due to these limitaions, the reconstructed image is not as
good as a computer simulated result.
This
system
microwave-induced
is
a
prototype
thermoelastic
wave
system
The
following
improvements
obtain
tomography. The
system is not complete. Many areas still
studies.
to
image
design
need to be improved
are
suggested
to
using
of
this
in future
conclude
this
thes i s.
(1)
The
use of
that can reduce
a
standard horn which
the distortion error
needs
a high
power
source
due t o the wave propagating
pattern and can avoid the rebinning problem.
200
(2) Investigation of an optimum combination of microwave frequency
and
pulse
widths
and
the design
frequency transducer that can
of
a
narrow
band-limited
high
increase the spatial resolution and
reduce the errors due to wave scattering.
(3)
Design
of
an
automated
data
acquisition
system
that
can
acquire both amplitude and phase information. This information is
required for the diffraction tomography system.
(4)
Modification
of
reconstruction
algorithm
to
include
diffraction and refraction effects based on the wave equation.
(5)
Design
phantoms.
a
system
This
information which
that
system
can
can
acquire .the
provide
us
signal
with
emitted
another
from
kind
is related to the characteristic of the tissue.
This is a very complicated study, thus theoretical work should
undertaken to build a mathematical
reconstruction.
of
be
model for a complete and exact
APPENDICES
201
202
Appendix A
SPECTRA OF THERMAL EXPANSION WAVE
This Appendix shows the normalized
magnitude plot of
the Fourier
transform of pressure for the thermal-expansion mechanism (free surface
case). The
and
curves
Figure 82
frequency and
in
show
the
the
Figure
78, Figure 79. Figure 80, Figure 8l,
frequency
pulse widths. All
spectra
for
different
microwave
the value were obtained from a program
which was based on equation (2.9).
203
C
C NAME: SPEC: CALCULATE THE FREQUENCY SPECTRUM FOR US WAVE.
C
DIMENSION FJ (200)
READ, IO,BETA,VEL,S,ARFA
C
C
C
C
C
C
C
TA: MW PULSE WIDTH
(US)
10: MW POWER DENSITY (W/CM**2)
W:
US FREQUENCY
(M HZ)
BETA: COEF. OF THERMAL EXPANSION *10 ***4 1/K)
VEL: SOUND VELOCITY
(KM/S)
S: SPECIFIC HEAT
(J/KG*K)
ARFA: ATTENUATION CONS.
(1/KM)
C FJ: FREQUENCY SPECTRUM.
DO 2 1=1,6
READ, TA
PRINT 1»,TA
FORMATC 1',3X,'FOR PULSE WlDTH=' ,F5•2, 1 USEC 1 ,/)
DO 2 J =2 ,7
1»
K =7-J
W=3-l 1 »l6*2/(10**K)
DO 2 N=1,9
FJ (N) =0.
PL"1-COS(W*TA)
DATA=8 .8 ^5*BETA*ARFA* 10*0.001*(VEL**2) /S
F J (N)=DATA*SQRT (PL)/(W**2+lf*(VEL**2)*(ARFA**2)
FR=W*(10**6)/(3-1^16*2)
Jl-J-1
PRINT 3,N,J1,W,PL,FR,FJ (N)
FORMAT (5X,IB,'*10**M1,2X,'W = I ,F8.6 ,5X,'PL =',F8.6,6X
3
*
')=' » F9.6)
W=W+3 - 1A 16*2/(10**K)
2
CONTINUE
STOP
END
SENTRY
1000 1.0 1.5 h.2 0.062
0.5
1.0
2.0
5.0
10.0
20.0
20/4
Appendix A
(continued)
FREQUENCY SPECTRUM OF
A MICROWAVE-INDUCED ACOUSTIC SIGNAL
433 MHz, 2.0 JLLS
'o .
>T'o
GQ
W
E-
o
_
10 4
FREQUENCY
Figure 78*
Frequency
spectrum 1 of
thermoelastic wave.
Frequency
spectrum of wave induced by thermoe1astic mechanism i n
water i r r a d i a t e d w i t h ^33 MHz, 2 ms microwave p u l s e .
205
Appendix A
(continued)
FREQUENCY SPECTRUM OF
A MICROWAVE-INDUCED ACOUSTIC SIGNAL
2450 MHz, 2.0 £iS
«
FREQUENCY , Hz
Figure 79»
Frequency spectrum 2 of
thermoe1astic wave.
Frequency
spectrum of wave induced by thermoelastic mechanism i n
water i r r a d i a t e d w i t h 2l»50 MHz, 2
microwave pulse.
US
206
Appendix A
(continued)
FREQUENCY SPECTRUM OF
A MICROWAVE-INDUCED ACOUSTIC SIGNAL
8000 MHz, 2.0 /LLS
'o
m
Z
£3
EZ
n
i
o
l
o
"itf
io'
icr
FREQUENCY , Hz
Figure 80.
F r e q u e n c y s p e c t r u m 3 of
thermoelastic wave.
Frequency
spectrum of wave induced by thermoelastic mechanism i n
w a t e r i r r a d i a t e d w i t h 8000 MHz, 2 y.s microwave p u l s e .
207
Appendix A
(continued)
FREQUENCY SPECTRUM OF
A MICROWAVE-INDUCED ACOUSTIC SIGNAL
2450 MHz, 0.5 /J.S
o
C
«
I
O
10
FREQUENCY , Hz
Figure 8l.
Frequency
s p e c t r u m >4 o f
thermoel ast i c wave.
Frequency
spectrum of wave induced by thermoe1astic mechanism i n
w a t e r i r r a d i a t e d w i t h 2 ^ 5 0 MHz, 0 . 5 MS m i c r o w a v e p u l s e .
208
Appendix A
(continued)
FREQUENCY SPECTRUM OF
A MICROWAVE-INDUCED ACOUSTIC SIGNAL
2450 MHz, 10.0 /J,S
>-T'o
cn
jz;
w
EZ
b
1C f
j
i
i »11111
10 4
j—» 11 u i
j
111
10
FREQUENCY
Figure 82.
Frequency
spectrum 5 of
thermoelastic wave.
Frequency
spectrum of wave induced by thermoelastic mechanism i n
w a t e r i r r a d i a t e d w i t h 2US0 M H z , 1 0 . 0 m s m i c r o w a v e p u l s e .
209
Appendix B
Program Listing .
Five computer
programs were written
in
Fortran
in
this
Three of those for the three different geometries(paral lei
equiangle
fan-beam,
and
equally
spaced
fan-beam
listing.
projection,
projection)
were
written to simulated TCT case.
In each program, two different filters
(refer
(3-36),
to
equations
(3.62b)) were used.
(3.23).
(3-51). C3-S^b) ,
(3-58),
and
The fourth one was written to simulate ECT case,
then the final one was used to reconstructed the experimental data,;
C THE FOLLOWING PROGRAM IS USED TO RECONSTRUCT THE IMAGE OF THREE OBJECT
C WHICH RADIATED BY A PARALLEL SOURCE.
C THERE ARE FIVE STEPS IN THIS PROGRAM:
C
1. MAIN PROGRAM: INITIALIZATION, A SET UP OF THE SYSTEM.
C 2. GEDA: ROTATED THE OBJECTS BY A FIXED ANGLE AND THE PROJECTIONS OF
C
OBJECT WAS OBTAINED.
C 3- RECON: RECONSTRUCT IMAGE FROM THOSE PROJECTIONS. EITHER SHEPP-LOGA
C
OR RAMACHANDRAN FILTER WAS USED.
C 4. COTRAN: TRANSFER THE IMAGE FROM POLAR SYSTEM TO RETANGULAR SYSTEM.
C 5- GRAY: SETTING THE GRAY LEVELS FOR IMAGE DISPLAY.
C
C PR: THE PROJECTIONS OF DIFFERENT ROTATED ANGLES.
C P: RECONSTRUCTED IMAGE.
C
C: THE IMAGE AFTER CHANGE THE COORDINATE SYSTEM.
C MP: THE IMAGE WITH GRAY LEVEL.
DIMENSION CA (-18:18,-18:18),PR (18,-18:18),IC (-18:18,-18:18)
DIMENSION CM (18,18,64) ,C(16,-18:18) ,P (18,64)
REAL*8 C,CM
C
C INITIAL I ED THE ARRAYS AND PARAMETERS
C INPUT THE PARAMETERS OF SYSTEM.
C
PI—3-14159
10
20
30
210
50
400
DO 400 INU=1,12
READ(8,*)N1,N2,N3,N4,DS,NG
N21=DS*N2
DO 10 1=1,N1
DO 10 J—N21.N21
PR (I,J)=0.0
CALL GEDA (N1,N2,N3,N4,DS,N21,PR,PI)
IR1-DS*N4
DO 20 1-1,IR1
DO 20 J=1,N3
P (I ,J)=0.0
DO 20 K-l.Nl
CM(K, I ,J)-0.0
DO 30 1=1,N1
DO 30 J—N21.N21
C(I,J)-0.0
CALL RECON(N1,N2,N3,N4,N21,IR1,PI,PR,P,C,CM)
PMAX=0.0
PMIN=10.0
DO 210 1=1,IR1
DO 210 J=1,N3
UT=P (I ,J)
PMAX=AMAX1(PMAX.UT)
PMIN=AMIN1(PMIN.UT)
CONTINUE
DO 50 I —IR1.IR1
DO 50 J—IR1.IR1
CA (I ,J)-0.0
I C(I ,J)-0
CALL COTRAN (P,IR1,N3,CA,PI,PMIN,PMAX)
CALL GRAY(IR1,IR1,CA,IC,PMAX,PMIN,NG)
CONTINUE
STOP
END
210
SUBROUTINE GEDA (N1 ,N2,N3,N4,DS,N21 ,PR,PI)
DIMENSION PR (N1,-N21:N21)
READ (8,*) A,B,RL,RS,RA,RB,RC,DA,DB,DD,W,ATW,ATA,ATB,ATC
WRITE(6,1) A,B,RL,RS
OA=DA-W/2
0B=DB-W/2
OD=DD-W/2
1
2
3
4
WRITE (6,2) OA,RA,OB,RB,OD,RC
WRITE (6,3) ATA,ATB,ATC,ATW
WRITE (6,4) N1.N2
FORMATC 1 ' ,2X,'RADIUS OF A,B,C (0) ,C(I) ;' ,1»F5-2)
F0RMAT(2X,'ORIGINS OF A,B,C:',6F7.3)
FORMAT(2X,'ATT. OF A.B,C.W:',4F8.5)
FORMAT(2X,'# OF VIEWS:',13.' # OF DET/UN IT:' , I 3)
C
C FIND THE ORIGINS AND LOCATIONS OF THREE OBJECTS WHICH WERE ROTATED
C
BY A FIXED ANGLE.
C
AL-SQRT(RA**2+(DA-W/2)**2)
BL-SQRT(RB**2+(DB-W/2)**2)
CL=SQRT(RC**2+(DD-W/2)**2)
THEA=ATAN2(RA,(DA-W/2))
THEB=ATAN2(RB,(DB-W/2))
IF( RC .EQ. 0.0) GO TO 5
THEC=ATAN2(RC,(DD-W/2))
5
6
7
GO TO 7
IF( DD .LT. W/2) GO TO 6
THEC>=0.0
GO TO 7
THEC=3-1^159
N21-DS*N2
DO 80 N-l.Nl
THE«=
N*PI/N1
YA=AL^eS I N (THEA-THE)
YB=BL*SIN(THEB-THE)
YC-CL*SIN(THEC-THE)
C
C CALCULATE THE ATTENUATIONS OF X-RAY THRU THE OBJECT, AND GENERATE
C THE DATA SETS WHICH BE USED TO RECONSTRUCT THE IMAGE.
C
YD=-DS
DO 70 I—N21.N21
QA=A**2-(YD-YA)**2
QB=B**2-(YD-YB)**2
QL=RL**2-(YD-YC)**2
QS=RS*A2- (YD-YC)**2
IF( QA .LE. 0.0001) GO TO 20
W0A=2*SQRT(QA)
20
30
40
41
GO TO 30
WOA-O.
IF( QB .LE. 0.0001) GO TO AO
W0B=2*SQRT(QB)
GO TO 41
W0B=0.
IF( QL .LE. 0.0001) GO TO 42
IF( QS .LE. 0.0001) GO TO 43
211
it3
k2
50
70
80
W0C=2ft(SQRT(QL)-SQRT(QS))
GO TO 50
W0C=2*SQRT(QL)
GO TO 50
WOC-O.
ATT" -(ATW*(W-WOA-WOB-WOC)+ATA*WOA+ATBftWOB+ATC*WOC)
PR (N, I)=-ATT/(ATW*W)
YD=YD+ 1.0/N2
CONTINUE
RETURN
END
SUBROUTINE RECON (N1,N2,N3,N't,N21,IR1,PI,PR,P,C,CM)
DIMENSION PR(Nl,-N21sN21) ,C(Nl,-N21sN21) ,CM(N1, IR1.N3) ,P(IR1.N3)
REAL*8 CH,C,CM
C
C FIND THE COORDINATE OF SOME POINTS WHICH WE ARE INTERSTING, AND DO
C THE LINEAR INTERPOLATION OF THOSE POINT.
C
DL=1.0/N2
DO 10 J=1,N1
DO 10 IR=1,IR1
DO 10 ITHER=1,N3
CH=IR*COS (JAp|/N1-PI *ITHER*2/N3)
IF (CH .LT. 0.) GO TO I*
K1 = 1DI NT(CH)
I F(K1 .EQ. IRl) GO TO 5
K2=K1+1
5
it
6
C
C
GO TO 6
K2=K1
GO TO 6
K2 a I D I NT(CH)
K1-K2-1
DO 15 IH-K1.K2
SUM3=0.0
DO 20 I 1— N21.N21
I H1«= IH— I 1
C
C
C
C
IF (IH1 .EQ. 0) GO TO U?
MR=MOD (I HI,2)
IF (MR .NE. 0) GO TO l» 1
CC=0
C
C4l
GO TO 20
CC=-l/((P|ftlHl)*ft2tDL)
C
Clt7
C20
GO TO 20
CC=l/(lt*DL)
SUM3=SUM3+PR(J, I D^CC
C
777777777777777777777777777
7 SHEPP-LOGAN FILTER
7
777777777777777777777777777
C
C
20
15
SUM3 =SUM3+PR(J,ll)/(2*(PI*DL)ft*2*(l-l»*(IH-ll)*A2))
C(J,IH)-SUM3
DH=CH-K1
IF( DH .LE. 0.0001) GO TO 25
DC=ABS(C(J,K1)-C(J,K2))
IF( DC .LE. 0.0001) GO TO 25
CM (J, IR, ITHER)=C (J,K1)+(C (J,K2)-C(J,K1))^DH
GO TO 10
25
10
CM(J, IR, I THER)-C(J,K1)
CONTINUE
C
C FOLLOW THE PROJECTION THEOREM TO FIND THE VALUES OF EACH POINT IN
C IMAGE PLANE.
C
C —
DO 35 IR-1,IR1
DO 35 ITHER=1,N3
SUM1=0.0
1»0
35
DO 1*0 J-l.Nl
SUM1 -SUM1+PI*CM(J, IR t I THER)/N1
P (IR, ITHER)- SUM1
RETURN
END
SUBROUTINE COTRAN (P,IR.NTH,C,PI.PMIN.PMAX)
DIMENSION P (IR,NTH),C(-1R:IR,- IR:IR),T(2)
C
C TRANSFERMATION OF TWO COORDINATE SYSTEM.
C
DO 10 IY°-IR,IR
DO 10 IX—IR, IR
IF ((I X.EQ.O) .AND. (IY.EQ.O) ) GOTO 10
IF ((IX.EQ.O) .OR. (IY.EQ.O)) GO TO 20
R1=(IX**2-HY**2)**0.5
1F (Rl .GT. IR) GO TO 30
GO TO 32
20
Rl-IX+IY
R1=ABS(Rl)
KR1 = INT(Rl)
32
KR2-KR1+1
!
DR-Rl-KRl
IF ((IY.EQ.O) .AND. (IX ;GT.O)) GO TO 8
IF ((IY.EQ.O) .AND. (IX.LT.O)) GO TO 2
IF ((IX.EQ.O) .AND. (IY.GT.O)) GO TO 1
IF ((IX.EQ.O) .AND. (IY.LT.O)) GO TO 3
GO TO 33
A=1.5*PI
GO TO 11
A=0.5*PI
3
1
GO TO 11
2
Ik
50
X
8
—->
ll —
— u
11
X —
->
33
A=ATAN2(YI,XI)
IF (A .GE. 0.) GO TO 11
A=A+2*PI
A1=A*NTH/(2*PI)
GO TO Ik
A1«=NTH
GO TO 3*»
A1-NTH/2
I Al°l NT(A 1)
IA2-IAl+1
DA=A1 — IA1
IF (IA1 .NE. 0) GO TO 50
IAl-NTH
IF((DR.LE.0.0001) .AND. (DA.LE.0.0001)) GO TO 5
14
5
12
30
10
DO li» I-KR1.KR2
I 1-1-KR1+ 1
T (I 1) «=P (I , IA1)+(P(I , I A2)-P (I , IA1)) ADA
C(IX, IY)-T(1)+(T(2)-T(1))*DR
I F (C (I X, I Y) .LE. PMAX) GO TO 12
C (I X, IY) =PMAX
GO TO 10
C(IX, IY)-P (KR1 , IA1)
GO TO 10
IF (C (IX,IY) .GE. PMIN) GOTO 10
C (I X. IY)-PMIN
CONTINUE
C (0,0)-(C(1,0)+C(0, 1)+C(-1,0)+C(0,-1)) A
RETURN
END
SUBROUTINE GRAY(I X,IY,C,IC,PMAX,PMIN,N)
DIMENSION C (-1X:IX.-IY:IY),IC(-1X:IX.-IY:IY)
CHARACTERS 1 ZE,ON,TW,TH,FO,FI,SI,SE,EI,MP (-18:18,-18:18)
DATA ZE,0N,TW,TH,FO,FI,S I,SE/' 1 ,'. 1 ,':','+','%','#','*',•$'/
q bbsbbsbsbtsbbbssisbbbbbbbbbbbbbbbbdbbbbbbbbbsbbbcbbbbscbbsbssssbbbsesss
C ASSIGN DIFFERENT CHARACTERS TO DIFFERENT SCALE TO REPRESENT THE GREY
C
LEVELS.
Q bsbsessssassscssasi&&acb:bbbbbesscbsssbbflbsobbbf:a:babssbb!sesssss===bs===e=
C
221
220
1
2
SCA=(PMAX-PMIN)/N
DO 220 I=-IY,IY
DO 220 J=-I X,I X
DO 221 L-l.N
LEVEL=PMIN+L*SCA
IF (C(I,J) .GT. LEVEL) GO TO 221
IC (I ,J)-L-1
GO TO 220
CONTINUE
CONTINUE
DO 230 I=-1X,IX
DO 230 J=-IY,IY
I F (I C (I ,J) .GT. 3) GO TO 1»
IF (I C (I ,J) .GT. 1) GO TO 2
IF (IC (I,J) .GT. 0) GO TO 1
MP (I,J)"ZE
GO TO 230
MP (I,J)-ON
GO TO 230
I F (I C(I,J) .GT. 2) GO TO 3
MP (I,J)=TW
5
GO TO 230
MP(I,J)=TH
GO TO 230
I F (I C(I,J) .GT. 5) GO TO 6
IF (I C (I ,J) .GT. 1») GO TO 5
MP (I ,J)-FO
GO TO 230
MP (I, J) =F I
6
IF (IC (I ,J) .GT. 6) GO TO 7
3
k
GO TO 230
MP (I ,J)=SI
GO TO 230
215
7
230
227
228
229
MP(I,J)-SE
CONTINUE
WRITE (6,227)
FORMAT(6X, 1
0123^567')
WRITE (6,228)
FORMAT(6X,'GRAY LEVEL
.:+%#*$',/)
WRITE (6,229) ((MP(I,J),J=IX,-IX,-1),l = IY,-IY,-l)
FORMAT(5X.37A1
RETURN
END
)
C THE FOLLOWING PROGRAM IS USED TO RECONSTRUCT THE IMAGE OF THREE
C OBJECTS WHICH RADIATED BY A FAN BEAM X-RAY(EQUI ANGULAR).
C THERE ARE FIVE STEPS IN THIS PROGRAM:
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
216
1. GENERATED THE GEOMETRY OF THE OBJECTS AND THE SET UP OF THE MODEL
2. ROTATED THE OBJECTS BY A FIXED ANGLE AND CALCULATED THE AMPLITUDE
OF EACH DETECTOR.
3. FOLLOW THE BASIC EQUATIONS TO RECONSTUCT THE IMAGE.
i». CHANGE THE COORDINATE SYSTEM OF IMAGE.
5- USING THE DIFFERENT CHARACTER TO REPRESENT THE GREY LEVELS.
THERE ARE TWO FILTERS IN THIS PROGRAM:
1. LAKES FILTER
2. MODIFIED LAKES (SMOOTHING) FILTER
AN: THE PROJECTIONS
C: THE IMAGE AFTER
RT: THE IMAGE AFTER
TRT: THE IMAGE WITH
OF DIFFERENT ANGLES.
RECONSTRUCTION.
CHANGE COORDINATE SYSTEM.
GREY LEVELS.
DIMENSION RT(-2l»:2l»,-2i4:2l») , C { S k , - 2 k i 2 k ) ,GH ( G k , 2 k , J 2 ) ,TRT(37.37)
DIMENSION CM (Gh,2k,J2) ,U(2l»,72).AN (6^.-56:56) ,T(2)
REAL*8 CON,C,CM
CHARACTERS ZE,ON,TW,TH,FO,F I ,SI ,SE. E I ,MP (37,37)
DATA ZE,ON,TW,TH,FO,FI,SI,SE/' ','. 1 ,': 1 ,'+','% 1 ,,'* 1 ,'$ 1 /
DATA EI/ 1 @ 1 /
C -
c 11111
C INPUT THE PARAMETERS OF SYSTEM.
C
DO 1»00 INU-1,1
READ (8,*) A,B,RL,RS,RA,RB,RC,DA,DB,DD,W,DS
READ (8,A) N1,N2,N3,Nl»,ATW, ATA,ATB,ATC
WRITE(6,1) A,B,RL,RS,RA,RB,RC,DA,DB,DD,W,DS
WRITE (6,2) N1,N2,N3« N't, ATW,ATA, ATB,ATC
1
FORMAT('1 1 ,3X,12F6.2)
2
FORMAT(AX,1» 12,1»F9«6)
C
C FIND THE ORIGINS AND LOCATIONS OF THREE OBJECTS WHICH WERE ROTATED
C
BY A FIXED ANGLE.
C
AL=SQRT(RA**2+(DA-W/2)**2)
BL«=SQRT(RB**2+(DB-W/2)**2)
CL«=SQRT(RC**2+(DD-W/2)**2)
THEA=ATAN2(RA,(DA-W/2))
THEB«=ATAN2(RB,(DB-W/2))
IF( CL .EQ. 0.0) GO TO 5
THEC=ATAN2(RC,(DD-W/2))
5
6
7
GO TO 7
IF( DD .LT. W/2) GO TO 6
THEC-O.O
GO TO 7
THEC«=3-1^159
PI-3.1*159
N11=16*N1
N21=DS*N2
21 '
ALFA=ATAN2(DS,W)
THERA-ALFA/N21
C
C
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
C 2222
C CALCULATE THE ATTENUATIONS OF X-RAY THRU THE OBJECT, AND GENERATE
C THE DATA SETS WHICH BE USED TO RECONSTRUCT THE IMAGE.
C
DO 80 N-1.N11
THE=
N *PI/(8.0*N1)
YA-ALASIN(THEA-THE)
ZA=W/2+AL*C0S(THEA-THE)
YB=BL*SIN(THEB-THE)
ZB-W/2+BL*C0S(THEB-THE)
YC-CL*SIN (THEC-THE)
ZC=W/2+CL*C0S(THEC-THE)
NC=0
20
30
1*0
i»l
1*3
1*2
50
DO 70 I— N21.N21
YD-W*TAN(NC*THERA-ALFA)
W0=1+(YD/W)**2
QA=A*A2A(1+YD**2/Wft*2)-(ZA*YD/W-YA)**2
QB=B**2*(1+YD**2/W**2)-(ZB*YD/W-YB)**2
QL=RL**2*(1+YD**2/W**2)-(ZC*YD/W-YC)**2
QS=RSAA2A(1+YD**2/WAA2)-(ZC*YD/W-YC)**2
IF( QA .LE. 0.0001) GO TO 20
W0A=2*SQRT(QA)/WO
GO TO 30
W0A=0.
IF( QB .LE. 0.0001) GO TO i»0
W0B«=2*SQRT(QB)/WO
GO TO It 1
W0B=0.
IF (QL .LE. 0.0001) GO TO U 2
IF (QS .LE. 0.0001) GO TO i»3
W0C=2*(SQRT(QL)-SQRT(QS))/WO
GO TO 50
W0C=2*SQRT(QL)/WO
GO TO 50
W0c«=0.
70
ATT= -(ATW*(W-WOA-WOB-WOC)+ATArtWOA+ATBftWOB+ATCftWOC)
ATT1=-(ATW*W)
AN (N, I) «=ATT/ATT1
NC=NC+1
80
CONTINUE
C 33333
C FIND THE COORDINATE OF SOME POINTS WHICH WE ARE
C THE LINEAR INTERPOLATION OF THOSE POINT.
C
C
N31=36*N3
DC=W/2.0
RI=DC*SIN (ALFA)
IR1-INT (RI)*N1»
C
DO 160 J=1,Nll
INTERSTING, AND DO
BETA= J *PI/(8.0*Nl)+PI/2
DO 160 IR°1,IR1
DO 160 ITHER=1,N31
TETA=PI*ITHER/(l8*N3)
Ql^DC+IR^SIN(BETA-TETA)/Nfc
R1 -1 RftCOS(BETA-TETA)/Nl»
UC-(Q1*A2+R1**2)**0.5
GH (J,IR,ITHER)=ATAN2(R1,Q1)/THERA
CON=GH (J,IR,ITHER)
Kl-1 D I NT(CON)
K2-KI+1
C
DO H5 IH-K1.K2
SUM3-0.0
DO HO 11— N21.N21
IHI-IH-I 1
HIA«IHI*THERA
IF (IHI .EQ. 0) GO TO 11*7
MR-MOD (IHI,2)
IF( MR .NE. 0) GO TO Hi
C 3.1
SMOOTH FILTER
ObtSSBSBSBBBSSBBSBBSatlttSBBBBaBBBBBBeBBBBBBBBBBISBBSSBSSBSSBSSSSSSSSBSS
C
K3=-l
C
CHI
CH2
C
C
C H7
C HO
C H5
GO TO H2
K3-1
UC1-2*UC*SIN(HIA)
CC=(K3*C0S(HIA*PI/UC1)+1)/(THERA**2-UC1**2)
GO TO HO
CC=( 1 -COS (PI /(2*UC)))/THERA**2
SUM3=SUM3+AN(J,I1)**CC*C0S(I1*THERA)
C (J, IH) =SUM3*DC*THERA/(PI **2)
^^^^^^^^
C
3-2
q / q / ^ q> q;
^^^^^
q« (v ft'q' q<qi&q<
^^^^ ^^^^
LAKES FILTER
&
^
%
CC=0.
Hi
H7
HO
H5
155
160
GO TO HO
CC=-1/(2*(PI*SIN(HIA))**2)
GO TO HO
CC=1/(8*THERA**2)
SUM3 =SUM3+AN(J, I 1)*CC*COS (I 1*THERA)
C(J, IH) =SUM3*DC*THERA/(UC**2)
SMA=C0N-K1
IF (SMA .LE. 0.01) GO TO 155
CM(J, IR, ITHER)=C(J,K1)+(C(J,K2) -C (J,K1))*SMA
GO TO 160
CM (J, IR, ITHER)=C(J,K1)
CONTINUE
C
Q
BBSBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBSBBBBBBBBBSBBBBBBBBBBBBBBSBSE
C FOLLOW THE PROJECTION THEOREM TO FIND THE VALUES OF EACH POINT IN
C IMAGE PLANE.
C
C
DO 185 IR=1,IR1
DO 185 ITHER=1,N31
180
185
SUM1-0.0
DO 180 J-l.Nll
SUM1 -SUM1+CM(J,IR,ITHER)
U (I R, ITHER)«=SUM1*PI/(8*N1)
C
UMAX-0.0
UMIN-10.0
DO 210 1-1,IR1
DO 210 J=1,N31
UT-U (I ,J)
IF(UT .LE. -10.0) GO TO 210
UMIN-AMIN1(UMIN.UT)
UMAX=AMAX1(UMAX,UT)
C
210
CONTINUE
DO 25 1 * IY— IR1.IR1
C kkkkk
C TRANSFERMATION OF TWO COORDINATE SYSTEM.
C
DO 25*» I X— I Rl, I Rl
IF ((I X. EQ.O) .AND. (IY.EQ.O) ) GO TO 25 1 *
IF ((I X.EQ.O) .OR. (IY.EQ.O)) GO TO 310
Rl1«IX**2+IYftft2
R1=SQRT (Rl1)
IF (Rl .GT. IRl) GO TO 253
310
312
370
360
372
GO TO 312
Rl-IX+IY
Rl-ABS (Rl)
KR1-INT(Rl)
KR2-KR1+1
SMA3=R1-KR1
IF ((IY.EQ.O)
IF ((IY.EQ.O)
IF ((IX.EQ.O)
IF ((IX.EQ.O)
.AND.
.AND.
.AND.
.AND.
(IX.GT.O))
(IX.LT.O))
(IY.GT.O))
(IY.LT.O))
GO
GO
GO
GO
TO
TO
TO
TO
311
313
360
370
GO TO 372
AM=1.5*PI
GO TO 231
AM=0.5*PI
GO TO 231
XI = IX
YI = 1Y
AM=ATAN2(YI,XI)
231
311
313
232
2hO
IF (AM .GE. 0.) GO TO 231
AM-AM+2*PI
AM1-AM*N31/(2*PI)
GO TO 232
AM1=N31
GO TO 232
AM1-N31/2
IAl-INT(AMI)
I A2=l Al+1SMA4=AM1-IA1
IF (IA1 .NE. 0) GO TO 2^0
IA1=N31
IF ((SMA3.LE.0.0001) .AND. (SMAi».LE.0.0001)) GO TO 252
220
21*5
252
253
25k
DO 2l»5 I =KR1,KR2
I 1 = 1 —KR1+1
T (I 1) =U (I , I Al)+(U (I , IA2)-U (I , IAl))*SMAi»
RT(IX, IY) =T(1)+(T(2)-T(1))*SMA3
GO TO 25b
RT(I X, I Y)-U(KR1, I Al)
GO TO 254
RT(I X,IY)-UMIN
CONTINUE
RT(0,0)-(RT(1,0)+RT(0,1)+RT(-1,0)+RT(0,- 1)) A
DO 215 I — IR1, IR1
DO 215 J— IR1, IR1
Jl=-J
I 1-1 +IR1+ 1
J2-J1+IR1+1
TRT(I 1,J2)=RT (I ,J)
215
CONTINUE
IR12-2*IR1+1
C
C 55555
C ASSIGN DIFFERENT CHARACTERS TO DIFFERENT SCALE TO REPRESENT THE GR
C
LEVELS.
C
SCALE®(UMAX-UMIN)/8.0
SC1=UMIN+SCALE
SC2-SC1+SCALE
SC3=SC2+SCALE
SCli=SC3+SCALE
221
SC5-SCl»+SCALE
SC6-SC5+SCALE
SC7=SC6+SCALE
SC8=SC7+SCALE
DO 220 1-1,IR12
DO 220 J-1,IR12
UT=TRT(I,J)
IF (UT .LE. SC8) GO TO 221
MP (I ,J)-EI
GO TO 220
IF (UT .LE. SC7) GO TO 222
MP(I,J)=SE
222
GO TO 220
IF (UT .LE. SC6) GO TO 223
MP(I,J)"=S I
223
GO TO 220
IF (UT .LE. SC5) GO TO 22l»
MP (I ,J)*=F I
22h
225
226
GO TO 220
IF (UT .LE. SCl») GO TO 225
MP (I ,J)-FO
GO TO 220
IF (UT .LE. SC3) GO TO 226
MP (I ,J)»TH
GO TO 220
IF (UT .LE. SC2) GO TO 227
MP (I ,J)=TW
221
227
228
220
229
1»00
GO TO 220
IF (UT. LE. SCI) GO TO 228
MP (I ,J)«ON
GO TO 220
MP(I,J)=ZE
CONTINUE
WRITE (6,229) ((MP (I,J),I-MR 12) ,J-l, I R12)
FORMAT(5X,37A1 )
CONTINUE
STOP
END
C T H E F O L L O W I N G PROGRAM I S U S E D T O R E C O N S T R U C T T H E IMAGE OF T H R E E 2 2 2
C O B J E C T S W H I C H R A D I A T E D BY A F A N BEAM X - R A Y ( E Q U L I S P A C E ) .
C T H E R E ARE F I V E S T E P S I N T H I S PROGRAM:
C
1 . GENERATED T H E GEOMETRY OF T H E O B J E C T S AND T H E S E T U P OF T H E M O D E L .
C 2 . R O T A T E D T H E O B J E C T S BY A F I X E D ANGLE AND C A L C U L A T E D T H E A M P L I T U D E
C
OF EACH D E T E C T O R .
C 3 . FOLLOW T H E B A S I C E Q U A T I O N S T O R E C O N S T U C T T H E I M A G E .
C k . CHANGE T H E C O O R D I N A T E S Y S T E M OF I M A G E .
C 5 - U S I N G T H E D I F F E R E N T CHARACTER T O R E P R E S E N T T H E GREY L E V E L S .
C
C
C
C
C
C
AN: THE PROJECTIONS
C:
T H E IMAGE A F T E R
R T : T H E IMAGE A F T E R
T R T : T H E IMAGE W I T H
OF DIFFERENT ANGLES.
RECONSTRUCTION.
CHANGE C O O R D I N A T E S Y S T E M .
GREY L E V E L S .
DIMENSION RT(-2l»:2i»,-2i»:2i») , C
, - 2 h : 2 k ) , G H ( 6 ^ , 2 ^ , 7 2 ) . T R T ( i » l ,1»1)
D I M E N S I O N C M ( 6 i » , 2 i » , 7 2 ) , U ( 2 l » , 7 2 ) , AN ( 6 ^ , - 5 6 : 5 6 ) , T ( 2 )
R E A L * 8 C O N , C , CM
CHARACTERS 1 Z E , ON , T W , T H , F O , F I , S I , SE , E I , M P ( 3 7 , 3 7 )
DATA Z E , O N , T W , T H , F O , F I , S I , S E / ' 1 , ' . 1 , ' : ' , ' + ' , ' % ' , , ' * 1 , ' £ ' /
DATA E I / ' @ ' /
C
C 11111
C I N P U T T H E PARAMETERS OF S Y S T E M .
C
DO i » 0 0 I N U ° 1 t h
READ ( 8 , * ) A , B , R L , R S , R A , R B , R C , D A , D B , D D , W , D S
READ ( 8 , * ) N l , N 2 , N 3 , N i » , A T W , A T A , A T B , A T C
WRITE (6,1) A,B,RL,RS,RA,RB,RC,DA,DB,DD,W,DS
WRITE (6,2) Nl,N2,N3,Nl4,ATW,ATA,ATB,ATC
1
FORMAT ( ' 1 1 , 3 X , 1 2 F 6 . 2 )
2
FORMAT ( i * X , i * l 2 , i * F 9 . 6 )
C
C F I N D T H E O R I G I N S AND L O C A T I O N S OF T H R E E O B J E C T S W H I C H WERE ROTA
C BY A F I X E D A N G L E .
C
AL«=SQRT ( R A * * 2 + ( D A - W / 2 ) * * 2 )
BL=SQRT(RB*A2+(DB-W/2)* * 2 )
CL=SQRT(RC**2+(DD-W/2)* * 2 )
THEA=ATAN2 (RA, (DA-W/2))
THEB=ATAN2 (RB,(DB-W/2))
I F ( C L . E Q . O . O ) GO T O 5
THEC=ATAN2 (RC, (DD-W/2))
GO T O 7
5
I F ( 0 0 . L T . W / 2 ) GO T O 6
THEC=0.0
GO T O 7
6
THEC=3.11* 159
7
PI-3.U159
N1 1-16*N1
N21=DS^N2
DBETA=2VkPI/Nll
C
C 22222
223
C C A L C U L A T E T H E A T T E N U A T I O N S OF X - R A Y T H R U T H E O B J E C T , AND GENERA
C T H E DATA S E T S W H I C H B E U S E D T O R E C O N S T R U C T T H E I M A G E .
C —
DO 7 0 N - l . N l l
THE= N *DBETA
YA=AL*SIN(THEA-THE)
XA-W/2+ALAC0S(THEA-THE)
YB=BL*SIN (THEB-THE)
XB=W/2+BL*C0S(THEB-THE)
YC-CL*SIN (THEC-THE)
XC-W/2+CL*C0S (THEC-THE)
DO 7 0 I — N 2 1 . N 2 1
YD-I/N2
WO" 1 + ( Y D / W ) f t * 2
QA=A**2*W0-(XAAYD/W-YA)**2
QB«B**2*W0- (XB*YD/W-YB) * * 2
QL=RL**2ftW0- (XC*YD/W-YC) * * 2
QS=RS**2*W0-(XC*YD/W-YC)**2
I F ( QA . L E . 0 . 0 0 0 1 ) GO T O 2 0
W0A-2ASQRT(QA)/WO
GO T O 3 0
20
W0A=0.
30
I F ( QB . L E . 0 . 0 0 0 1 ) GO T O k O
W0B=2*SQRT(QB)/WO
GO T O h 1
40
W0B=O.
41
I F ( Q L . L E . 0 . 0 0 0 1 ) GO T O U 2
I F (QS . L E . 0 . 0 0 0 1 ) GO T O 1»3
W0C=2* (SQRT (QL) -SQRT (QS)) /WO
GO T O 5 0
i»3
W0C-2*SQRT(QL)/W0
GO T O 5 0
42
W0C=0.
50
WD-(W**2+YD*ft2) * * 0 . 5
70
AN ( N , I ) - ( A T W * (WD-WOA-WOB-WOC) + A T A * W O A + A T B * W O B + A T C * W O C ) / (ATW*W)
C 33333
C F I N D T H E C O O R D I N A T E OF SOME P O I N T S W H I C H WE ARE I N T E R S T I N G , AND DO
C T H E L I N E A R I N T E R P O L A T I O N OF T H O S E P O I N T .
C
C
N31=16*N3
DC«=W/2.0
DL=0.5/N2
RI=DC*DS/((DS**2+W**2)**0.5)
IRl«=lNT (Rl) *Uh
DO 1 6 0 J = 1 , N 1 1
BETA= J *DBETA+PI/2
DO 1 6 0 I R = 1 , I R 1
DO 1 6 0 I T H E R - 1 . N 3 1
TETA=2*P I*ITHER/N31
UU=1.0+1R*SIN (BETA-TETA)/DC/N4
CON= I R*COS (BETA-TETA) / (UU*DL) /N4
I F (CON . L T . 0 ) GO T O 1 2 0
K l = l D I N T (CON)
IF ((N21-CON) .LE. 0.00001) GOTO 121
121
120
122
K2=K1+1
GO T O 1 2 2
K2-K1
GO T O 1 2 2
K 2 = l D I N T (CON)
K1=K2-1
DO l l » 5 I H - K 1 . K 2
SUM3=0.0
DO 1 A 0 I 1 — N 2 1 . N 2 1
IHI-IH-I1
0/ ^& q/
^
^ ^ & q/ & & & ^ ^ qf & ^^
^^ ^ &. & & & ^^ ^
^ <t) ^m ^m ^^ ^^^^^^^^ /n^^^^ ^^ ^^^^^^^ ^^^^^^^^^
C % 3.1
LAKES FILTER
C
C
C
C
C
C 1 i* 1
C
C ll»7
ClfcO
CI 45
I F ( I H I . E Q . 0 ) GO T O l < t 7
MR«=MOD ( I H I , 2 )
I F ( MR . N E . 0 ) GO T O 1*»1
CC«=0.
GO T O U O
CC=-1/((PI*IHI) **2*2)
GO T O 11*0
CC=l/8
SUM3 - S U M 3 + A N ( J , I l ) * C C / ( ( D C f t > * 2 + ( l l > f t D L ) * * 2 ) * * 0 . 5 )
C(J,IH)-SIM'3*DC/(UU**2*DL)
%
[eCS=SSCSSSBSSSBSESSSSSB8BS&SS&eSBSSSBBSBSBSeBBSSSSSS=SS=SSSS=5S=
C 3.2
SMOOTH F I L T E R
qbbbbsbbibkbibbbbhbbmbbiibbbabafllbaaababflibbaibiabbbbbkbbbsbbisbbebsbbbbk
1kO
SUM3=SUM3+AN ( J , I 1) / ( ( ( D C * * 2 + (11*DL) * * 2 ) * * 0 . 5 ) * (1- i i * I H I * * 2 ) )
1k5
155
160
C (J,IH)-SUM3*DC/ ((UU*PI)A*2*DL)
SMA=CON-K1
I F ( S M A . L E . 0 . 0 1 ) GO T O 1 5 5
CM ( J , I R , I T H E R ) = C ( J , K 1 ) + ( C ( J , K 2 ) - C ( J , K 1 ) ) * S M A
GO T O 1 6 0
CM ( J , I R , I T H E R ) - C ( J . K l )
CONTINUE
C
C FOLLOW T H E P R O J E C T I O N T H E O R E M T O F I N D T H E V A L U E S OF EACH P O I N T I N
C IMAGE P L A N E .
C
C
DO 1 8 5 I R = 1 » I R 1
DO 1 8 5 I T H E R - 1 . N 3 1
SUM1=0.0
DO 1 8 0 J = 1 , N 1 1
180
SUM1 = S U M 1 + C M ( J , I R , I T H E R )
185
U (IR,ITHER)=SUM1ADBETA
C
C
UMAX-0.0
UMIN-10.0
DO 2 1 0 1 = 1 , I R 1
DO 2 1 0 J = 1 , N 3 1
UT=U ( I , J)
UMIN=AMIN1(UMIN.UT)
UMAX=AMAX1(UMAX,UT)
210
CONTINUE
C 44444
C T R A N S F E R M A T I O N OF TWO C O O R D I N A T E S Y S T E M .
C
DO 2 5 4 I Y = - I R 1 , I R 1
DO 2 5 k I X — I R 1 ff I R 1
I F ( ( I X . E Q . O ) . A N D . ( I Y . E Q . O ) ) GO T O 2 5 4
I F ( ( I X . E Q . O ) . O R . ( I Y . E Q . O ) ) GO T O 3 1 0
R1® ( I X * * 2 + I Y * * 2 ) * * 0 . 5
I F ( R 1 . G T . I R 1 ) GO T O 2 5 3
GO T O 3 1 2
310
312
370
360
372
R1=IX+IY
R1=ABS (Rl)
KRl-INT(Rl)
KR2-KR1+1
SMA3=R1-KR1
I F ( ( I Y . E Q . O ) . A N D . ( I X . G T . O ) ) GO T O 3 1 1
IF ((IY.EQ.O) .AND. (IX.LT.O)) G0TO313
I F ( ( I X . E Q . O ) . A N D . ( I Y . G T . O ) ) GO T O 3 6 0
I F ( ( I X . E Q . O ) . A N D . ( I Y . L T . O ) ) GO T O 3 7 0
GO T O 3 7 2
AM»1.5ftPI
GO T O 2 3 1
AM=0.5*PI
GO T O 2 3 1
XI-IX
YI=IY
231
311
313
232
AM=ATAN2 ( Y I , X I )
I F ( A M . G E . 0 . ) GO T O 2 3 1
AM-AM+2*PI
AM1«=AM*N31/(2*PI)
GO T O 2 3 2
AM1-N31
GO T O 2 3 2
AM1=N31/2
I A 1= I N T ( A M I )
IA2°IAl+1
240
245
252
247
253
254
SMA4-AM1-IA1
I F ( I A 1 . N E . 0 ) GO T O 2 4 0
IA1=N31
I F ( ( S M A 3 . L E . 0 . 0 0 0 1 ) . A N D . ( S M A 4 . L E . 0 . 0 0 0 1 ) ) GO T O 2 5 2
DO 2 4 5 I = K R 1 , K R 2
I 1= 1 - K R 1 + 1
T ( I 1) =U ( I , IA1) + ( U ( I , IA2) -U ( I , IA1))*SMA4
RT ( I X, IY)-T(1) + (T (2)-T(l))*SMA3
I F ( R T ( I X , I Y ) . L E . UMAX) GO T O 21*7
R T ( I X , I Y ) «=UMAX
GO T O 2 5 ^
RT ( I X , I Y) "U (KR1, IA1)
GO T O 2 5 4
I F ( R T ( I X , I Y ) . G E . U M I N ) GO T O 2 5 4
RT (IX,IY)—UMIN
CONTINUE
R T ( 0 , 0 ) «= ( R T ( 1 , 0 ) + R T ( 0 , 1 ) + R T ( - 1 , 0 ) + R T ( 0 , - 1 ) ) / 4
DO 2 1 5 I — I R 1 . I R 1
215
DO 2 1 5 J — I R 1 , I R 1
Jl—J
I 1 = 1 + I R 1+ 1
J2=J1+1Rl+1
TRT (I 1 ,J2)=RT(I ,J)
CONTINUE
IR12-2*IR1+1
C 55555
C A S S I G N D I F F E R E N T CHARACTERS T O D I F F E R E N T SCALE T O R E P R E S E N T T H E
C LEVELS.
Q
BSBBBCBBBBCBBBBBBBBSSBHBCBBBBBBBBCBBBCBBCBSBBBBSBesSBBSBBBBSSSB
C
221
222
223
221*
225
226
227
228
220
229
1»00
SCALE-(UMAX-UMIN)/8.0
SC1-UMIN+SCALE
SC2=SC1+SCALE
SC3-SC2+SCALE
SCl«»SC3+SCALE
SC5"SC1»+SCALE
SC6=SC5+SCALE
SC7=SC6+SCALE
SC8=SC7+SCALE
DO 2 2 0 1 - 1 , I R 1 2
DO 2 2 0 J « 1 , I R 1 2
UT«=TRT(I,J)
I F ( U T . L E . S C 8 ) GO T O 2 2 1
MP ( I , J ) = E I
GO T O 2 2 0
I F ( U T . L E . S C 7 ) GO T O 2 2 2
MP ( I , J ) - S E
GO T O 2 2 0
I F ( U T . L E . S C 6 ) GO T O 2 2 3
MP ( I , J ) = S I
GO T O 2 2 0
I F ( U T . L E . S C 5 ) GO T O 2 2 k
MP ( I , J ) = F I
GO T O 2 2 0
I F ( U T . L E . SC*») GO T O 2 2 5
MP ( I ,J)»FO
GO T O 2 2 0
I F ( U T . L E . S C 3 ) GO T O 2 2 6
MP ( I , J) =TH
GO T O 2 2 0
I F ( U T . L E . S C 2 ) GO T O 2 2 7
M P ( I , J ) «=TW
GO T O 2 2 0
I F ( U T . L E . S C I ) GO T O 2 2 8
MP ( I , J ) = 0 N
GO T O 2 2 0
MP ( I , J ) «=ZE
CONTINUE
WRITE (6,229) ((MP ( I , J ) , 1 - 1 , I R 1 2 ) , J = 1 , I R 1 2 )
FORMAT ( 5 X . 3 7 A 1 )
CONTINUE
STOP
END
227
C T H E F O L L O W I N G PROGRAM I S U S E D T O RECONSTRUCT T H E IMAGE OF T H R E E O B J E C T
C W H I C H R A D I A T E D B Y PHOTONS W I T H D I F F E R E N T D I S T R U B U T I O N .
C T H E R E ARE F I V E S T E P S I N T H I S PROGRAM:
C
1 . GENERATED T H E GEOMETRY OF T H E O B J E C T S AND T H E S E T U P OF T H E M O D E L .
C 2 . R O T A T E D T H E O B J E C T S B Y A F I X E D ANGLE AND C A L C U L A T E D T H E A M P L I T U D E
C
OF EACH D E T E C T O R .
C 3 . FOLLOW T H E B A S I C E Q U A T I O N S T O RECONSTUCT T H E I M A G E .
C i » . CHANGE T H E C O O R D I N A T E S Y S T E M OF I M A G E .
C 5 . U S I N G T H E D I F F E R E N T CHARACTER T O R E P R E S E N T T H E GREY L E V E L S .
C
C
C
C
T H E R E ARE TWO D I F F E R E N T I N T H I S PROGRAM
C
C
1.
RAMACHANDRAN F I L T E R
C
2.
SHEPP-LOGAN FILTER
C
C
C
A N : T H E P R O J E C T I O N S OF D I F F E R E N T A N G L E S .
C C : T H E IMAGE A F T E R R E C O N S T R U C T I O N .
C R T : T H E IMAGE AFTER* CHANGE C O O R D I N A T E S Y S T E M .
C T R T : T H E IMAGE W I T H GREY L E V E L S .
C
DIMENSION RT(-18:18,-18:18),C (32,-18:18),GH(32, 18,72),TRT(37.37)
D I M E N S I O N CM ( 3 2 , 1 8 , 7 2 ) , U ( 1 8 , 7 2 ) , P R ( 3 2 , - 1 8 : 1 8 ) , T ( 2 )
R E A L * 8 CON
CHARACTERS ZE,ON,TW,TH,FO,FI,SI,SE,EI,MP(37.37)
DATA Z E , O N , T W , T H , F O , F I , S I , S E / '
'.V
' %' ,
'/
DATA E I / ' § ' /
PI-3.1A159265
c
c
C I N P U T T H E PARAMETERS OF S Y S T E M .
C
READ ( 8 , * ) N S E T
DO k O O I N U - l . N S E T
READ ( 8 , * ) A , B , C L , C S , R A , R B , R C , D A , D B , D C , D , D S
READ ( 8 , f t ) N 1 , N 2 , N 3 • N ' t , A T W , A T A , A T B , A T C , D I A , D I B , D I C
WRITE (6,1) A,B,CL,CS,RA,RB,RC,DA,DB,DC,D,DS
WRITE (6,2)
N1 ,N2,N3,N1»,ATW,ATA,ATB,ATC,DI A.DIB.DIC
1
FORMAT('11,3X»12F6.2)
2
FORMAT(4X,I 3
,312.7F8-5)
WRITE (6,3)
WRITE (6,4)
3
FORMAT(20X,' .:+%#*$§')
it
FORMAT(9X,'GRAY LEVEL 0123^5678',/)
C
C F I N D T H E O R I G I N S AND L O C A T I O N S OF T H R E E O B J E C T S W H I C H WERE R O T A T E D
C BY A F I X E D A N G L E .
C
CLA= (RAft*2+DAft*2)*ftO.5
CLB™ ( R B f t f t 2 + D B f t * 2 ) f t f t O . 5
CLC° (RC**2+DCftft2) ftftO.5
THEA=ATAN2(RA,DA)
THEB=ATAN2(RB,DB)
I F (RC . E Q . 0 . 0 ) GO T O 5
228
5
THEC=ATAN2 (RC.DC)
GO T O 7
IF ( DC . L T . O.O)
THEC=0.0
GO T O 6
GO TO 7
6
7
THEC=3.1A159265
PHI-2API/N1
N21=DSAN2
DO 1 0 0 N = 1 , N 1
YD—DS
XA-CLAACOS (THEA-N*PHI)
XB=CLBACOS ( T H E B - N A P H I )
XC=CLCACOS ( T H E C - N A P H I )
YA-CLAASIN(THEA-NAPHI)
YB-CLBASIN (THEB-NAPHI)
YC=»CLCAS I N ( T H E C - N A P H I )
C
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
C
C C A L C U L A T E T H E A T T E N U A T I O N S OF X - R A Y T H R U T H E O B J E C T , AND GENERATE
C T H E DATA S E T S W H I C H BE U S E D T O RECONSTRUCT T H E I M A G E .
C
DO 2 0 0 I = - N 2 1 , N 2 1
AA=AAA2- (YD-YA) AA2
AB=BAA2- (YD-YB) AA2
ACL=CLAA2-(YD-YC)AA2
ACS-CSAA2-(YD-YC)AA2
I F ( A C L . L E . 0 ) GO T O 1 0
I F ( A C S . L E . 0 ) GO T O 2 0
WWC1=D-XC-SQRT(ACL)
WWC2-D-XC-SQRT (ACL)+2ASQRT(ACS)
WlC-SQRT (ACL)-SQRT (ACS)
C21= (1-EXP (-ATCA (SQRT (ACL) -SQRT (ACS)))) /ATCAD I C
C22= (EXP (-ATCA (SQRT (ACL)+SQRT (ACS))) -EXP (-ATCA2*SQRT (ACL))) /ATC
&ADIC
CW21=C21AEXP(-ATWAWWC1)
C W 2 2 = C 2 2 A E X P (-ATWAWWC2)
IF (AA .LT. 0) GO TO 25
WWA =D -XA -SQRT (AA)
60
25
WIA=2ASQRT(AA)
Al=(1-EXP(-2AATAASQRT(AA)) ) /ATAADI A
AW1=A1AEXP(-ATWAWWA)
I F ( A B . L T . 0 ) GO T O k O
WWB=D-XB-SQRT(AB)
WIB=2ASQRT(AB)
B 1= ( 1 - E X P ( - 2 A A T B A S Q R T ( A B ) ) ) / A T B
BW1=B1AEXP(-ATWAWWB)
I F ( X A . G T . X B ) GO T O 6 0
PR(N,I) =AW1AEXP(-( (ATB-ATW)AWIB+(ATC-ATW)AWIC) )+BWlAEXP(-(ATC& A T W ) A W I C ) + C W 2 2 A E X P ( - ( ( A T B - A T W ) A W I B + ( A T A - A T W ) AWI A ) ) + C W 2 1
GO T O 2 0 0
P R ( N , I ) - B W 1 A E X P ( - ( ( A T A - A T W ) AWI A + ( A T C - A T W ) AW I C ) ) + A W 1 AEXP ( - ( A T C 6 ATW) AW I C ) + C W 2 2 A E X P ( - ( ( A T B - A T W ) A W I B + ( A T A - A T W ) A W I A ) ) + C W 2 1
GO T O 2 0 0
I F ( A B . L T . 0 ) GO T O 3 0
WWB-D-XB-SQRT(AB)
WIB=2ASQRT(AB)
AO
30
20
10
200
C
C51
C
C101
100
Bl=(1-EXP(-2AATB*SQRT (AB)))/ATB*DIB
B W 1 = B 1 * E X P (-ATWAWWB)
PR(N,I) =BW1AEXP(-(ATC-ATW)AW IC)+CW22AEXP (-(ATB-ATW) AWIB) +CW21
GO T O 2 0 0
P R ( N , I ) = A W J A E X P ( - ( A T C - A T W ) * W I C ) + C W 2 2 A E X P ( - ( A T A - A T W ) AWI A ) + C W 2 1
GO T O 2 0 0
PR(N,I)™CW21+CW22
GO T O 2 0 0
Cl>= ( 1 - E X P ( - 2 A S Q R T ( A C L ) A A T C ) ) / A T C A D I C
WWC-D-XC-SQRT(ACL)
P R ( N , I ) - C 1 A E X P (-ATWAWWC)
GO T O 2 0 0
P R ( N , I ) "=0
YD-YD+1.0/N2
W R I T E ( 6 , 5 1 ) ( P R (N , I ) . I — N 2 1 . N 2 1 )
FORMAT(5X.7F10.6)
W R I T E (6,101)XA,YA,XB,YB,XC,YC
FORMAT(4X.6F8.3)
CONTINUE
C
C F I N D T H E C O O R D I N A T E OF SOME P O I N T S W H I C H WE ARE I N T E R S T I N G , AND DO
C T H E L I N E A R I N T E R P O L A T I O N OF T H O S E P O I N T .
C
N31=36AN3
DL=1.O/NA
i r 1 = dsana
C
C141
C
C
C
C
C
C
C
C
C
ClUl
C
CU»7
C1A3
c
DO 1 6 0 J " 1 , N 1
DO 1 6 0 I R = 1 , I R 1
DO 1 6 0 I T H E R = 1 , N 3 1
GH(J,IR,ITHER)=IRACOS (JAP|/(Nl/2 )-PI*ITHER/(18AN3))/DL/NA
CON-GH (J,IR,ITHER)
WRITE (6, li»1)C0N
F ORMAT ( 3 X , F 1 0 . 6 )
K l - I D I N T (CON)
K2-K1+I
DO 1 A 5 I H = K 1 , K 2
SUM3=0.0
DO 1 A 0 I I — N 2 1 . N 2 1
888888888888888888888888888888
8 RAMACHANDRAN F I L T E R
8
888888888888888888888888888888
IHI=IH-11
I F ( I H I . E Q . 0 ) GO T O H » 7
MR=MOD ( I H I , 2 )
I F (MR . N E . 0 ) GO T O 1 4 1
CC=0
GO T O 1 U 3
C C = - 1 / ( ( P | A I H I ) AA2ADL)
GO T O 1 A 3
CC-I/(A ADL)
SUM3=SUM3+PR(J, 11)*CC
C
C
777777777777777777777777777
7 SHEPP-LOGAN FILTER
7
777777777777777777777777777-
140
CHO
SUM3 - S U M 3 + P R ( J . I l ) / ( 2 A ( P | A D L ) * * 2 * ( l - f c A ( | H - l l ) A A 2 ) )
CONTINUE
230
1A5
C
C1h2
150
155
C170
160
C(J,IH)-SUM3
WRITE (6, H»2)K1,K2,C(J,K1) ,C(J,K2)
FORMAT(UX,213» 2F9.6)
SMA1=C0N-K1
I F ( SMA1 . L E . 0 . 0 1 ) GO T O 1 5 5
SMA2=ABS (C ( J , K1) - C (J ,K2))
I F ( S M A 2 . L E . 0 . 0 1 ) GO T O 1 5 5
IF (C (J, Kl) .GT. C ( J ,K2) ) GOTO 150
CM ( J , I R , I T H E R ) = C ( J , K 1 ) + ( C ( J . K 2 ) - C ( J , K 1 ) ) * S M A 1
GO T O 1 6 0
CM ( J , I R , I T H E R ) - C ( J , K 1 ) - ( C ( J , K 1 ) - C ( J , K 2 ) ) * S M A 1
GO T O 1 6 0
CM ( J , I R , I T H E R ) - C ( J , K 1 )
WRITE (6, ] k ] ) CM(J, IR, ITHER)
CONTINUE
q rbhnbmbbabhaaabbbbmasbbmmbhnbutibabmabaebaascksssecesssasbaabsii
C FOLLOW T H E P R O J E C T I O N T H E O R E M T O F I N D T H E V A L U E S OF EACH P O I N T I N
C IMAGE P L A N E .
C
C
DO 1 8 5 I R = 1 , I R 1
DO 1 8 5 I T H E R - 1 . N 3 1
sum1=0.0
180
185
C
C200
DO 1 8 0 J = 1 , N 1
SUM1 «=SUM1+P I ACM ( J , I R , I T H E R ) / ( N l / 2 )
U ( I R , I T H E R ) «* SUM1
WRITE (6, 200) ( ( U ( I R , ITHER) ,ITHER-1,N31) • I R - 1 . I Rl)
FORMAT(8X.9F10.5)
UMAX=0.0
UMIN-10.0
DO 2 1 0 1 - 1 , I R 1
DO 2 1 0 J = 1 , N 3 1
UT=U ( I ,J)
UMAX=AMAX1(UMAX,UT)
UMIN-AMIN1(UMIN.UT)
CONTINUE
210
C
C T R A N S F E R M A T I O N OF TWO C O O R D I N A T E S Y S T E M .
C
DO 2 5 ^ I Y — I R 1 , I R 1
DO 2 5 ^ I X — I R 1 . I R 1
I F ( ( I X . E Q . O ) . A N D . ( I Y . E Q . O ) ) GO T O 2 5 1 *
I F ( ( I X . E Q . O ) . O R . ( I Y . E Q . O ) ) GO T O 3 1 0
Rl1=IX**2+IY**2
R1=SQRT (R11)
I F ( R l . G T . I R l ) GO T O 2 5 3
GO T O 3 1 2
310
Rl-IX+IY
R1=ABS (Rl)
312
K R 1 *= I N T ( R 1 )
KR2=KR1+1
SMA3-R1-KR1
I F ( ( I Y . E Q . O ) . A N D . ( I X . G T . 0 ) ) GO T O 3 1 1
I F ( ( I Y . E Q . O ) . A N D . ( I X . L T . 0 ) ) GO T O 3 1 3
I F ( ( I X . E Q . O ) . A N D . ( I Y . G T . O ) ) GO T O 3 6 0
I F ( ( I X . E Q . O ) . A N D . ( I Y . L T . 0 ) ) GO T O 3 7 0
231
231
311
313
232
GO T O 3 7 2
AM-1-5*PI
GO T O 2 3 1
AM=0.5*PI
GO T O 2 3 1
XI-IX
Y! = IY
AM=ATAN2 ( Y I , X I )
I F (AM . G E . 0 . ) GO T O 2 3 1
AM-AM+2*PI
AM1»AM*N31/ (2*P I )
GO T O 2 3 2
AM1«=N31
GO T O 2 3 2
AM1-N31/2
IA1= I NT (AMI)
IA2=IAl+1
240
245
246
247
248
250
251
252
253
254
215
SMA4=AM1-IAl
I F ( ( S M A 3 . L E . 0 . 0 0 0 1 ) . A N D . ( S M A 4 . L E . 0 . 0 0 0 1 ) ) GO T O 2 5 2
I F ( S M A 3 - L E . 0 . 0 0 0 1 ) GO T O 2 4 7
I F ( S M A 4 . L E . 0 . 0 0 0 1 ) GO T O 2 5 0
DO 2 4 5 I - K R 1 . K R 2
I1-I-KR1+1
I F ( U ( I , I A l ) . G T . U ( I , I A 2 ) ) GO T O 2 4 0
T ( I 1 ) " U ( I , I A 1 ) + ( U ( I , I A2) - U ( I , IAl))ftSMA4
GO T O 2 4 5
T (I 1)«U(I, IA1)-(U(I ,IA1)-U(I , IA2))*SMA4
CONTINUE
I F ( T ( 1 ) . G T . T ( 2 ) ) GO T O 2 4 6
RT ( I X, IY) - T (1) + (T (2) - T (1)) *SMA3
GO T O 2 5 4
RT (I X, IY) - T (1) - (T (1) - T (2))*SMA3
GO T O 2 5 4
I F ( U ( K R 1 , I A 1 ) . G T . U ( K R 1 , I A 2 ) ) GO T O 2 4 8
R T ( I X , I Y ) «*U ( K R I , I A 1 ) + ( U (KR I , I A 2 ) - U (KR 1 , I A 1 ) ) * S M A 4
GO T O 2 5 4
RT ( I X,IY)=U (KRl, IA1)-(U (KRl, I Al) - U (KRl, I A2)) *SMA4
GO T O 2 5 4
I F ( U ( K R l , I A l ) . G T . U ( K R 2 , I A I ) ) GO T O 2 5 1
R T ( I X , I Y ) «=U ( K R l , I A 1 ) + ( U ( K R 2 , I A l ) - U ( K R l , I A l ) ) * S M A 3
GO T O 2 5 4
RT ( I X, IY) =U (KRl, IA1) — (U (KRl, I Al) - U (KR2, IA1))*SMA3
GO T O 2 5 4
RT ( I X, IY) =U (KRl, I Al)
GO T O 2 5 4
RT ( I X,IY)=UMIN
CONTINUE
RT (0,0) = (RT (1,0) +RT (0,1)+RT (-1,0)+RT ( 0 , - 1 ) ) / 4
DO 2 1 5 I —IR1, I R 1
DO 215 J — I R 1 , IR1
J1--J
I 1 = 1 + I R 1+ 1
J2=J1+IR1+1
TRT ( I 1, J2) °RT ( I ,J)
CONTINUE
IR12=2*IR1+1
232
C
C175
WRITE ( 6 , 1 7 5 ) ((TRT ( I ,J ) , I =1 , IR12) , J = 1 , I R 1 2 )
FORMAT(3X,16F6.1)
C A S S I G N D I F F E R E N T CHARACTERS T O D I F F E R E N T SCALE T O R E P R E S E N T T H E G R E Y
C LEVELS.
221
222
223
22k
225
226
227
228
220
229
i»00
SCALE= (UMAX-UMIN) / 8 . 0
SC^UMIN+SCALE
SC2-SC1+SCALE
SC3=SC2+SCALE
SClf=SC3+SCALE •
SC5-SCU+SCALE
SC6-SC5+SCALE
SC7=SC6+SCALE
SC8=SC7+SCALE
DO 2 2 0 I - 1 . I R 1 2
DO 2 2 0 J = 1 , I R 1 2
UT=TRT ( I , J )
I F ( U T . L E . S C 8 ) GO T O 2 2 1
MP ( I , J ) " E I
GO T O 2 2 0
I F ( U T . L E . S C 7 ) GO T O 2 2 2
MP ( I , J ) « S E
GO T O 2 2 0
I F ( U T . L E . S C 6 ) GO T O 2 2 3
MP(I,J)«SI
GO T O 2 2 0
I F ( U T . L E . S C 5 ) GO T O 2 2 k
MP ( I , J ) - F I
GO T O 2 2 0
I F ( U T . L E . SCi») GO T O 2 2 5
MP ( I , J ) « F O
GO T O 2 2 0
I F ( U T . L E . S C 3 ) GO T O 2 2 6
MP ( I ,J)=TH
GO T O 2 2 0
I F ( U T . L E . S C 2 ) GO T O 2 2 7
MP ( I , J ) = T W
GO T O 2 2 0
I F ( U T . L E . S C I ) GO T O 2 2 8
MP ( I , J ) = O N
GO T O 2 2 0
MP ( I , J ) «"ZE
CONTINUE
WRITE (6,229) ((MP ( I , J ) . J - 1 . I R 1 2 ) , I - I R 1 2 , 1 , - 1 )
FORMAT(5X.37A1 )
CONTINUE
STOP
END
C T H E FOLLOWING PROGRAM I S USED T O RECONSTRUCT THE IMAGE OF PHANTOMS
C WHICH I R R A D I A T E D BY A T H E R M O E L A S T I C WAVES. T H I S PROGRAM I S I M P L A N T
C I N THE I N T E L 8 6 3 8 O M I N I C O M P U T E R .
PROGRAM REA
D I M E N S I O N CA ( - 1 0 : 1 0 , - 1 0 : 1 0 ) , P R ( T o , - 9 : 9 ) , I C ( 6 0 , 6 0 )
DIMENSION CM(16,9,32) ,C (32,-9:9) ,P(9,32)
R E A L 8 8 C , CM
PI=3-1^159265
OPEN ( 1 , F I L E = 1 S U P R / T E S T . I N I )
READ ( 1 , 2 ) N 1 , N 2 , N 3 , N A , N G , I E T . D S
2
FORMAT ( 2 X , 6 1 2 , F 5 . 2 )
N21=DS*N2
DO 1 0 1 = 1 , N l
DO 1 0 J — N 2 1 . N 2 1
10
PR(I,J)=0.0
DO 3 0 1 = 1 , N l
DO 3 0 J = - N 2 1 , N 2 1
30
C(l,J)=0.0
PMAX=0.0
PMIN=10.0
IR2=10
50
A5
11
31
AO
51
52
60
59
DO 5 0 I = - I R 2 , I R 2
DO 5 0 J = - I R 2 , I R 2
CA(I,J)=0.0
DO b 5 1 = 1 , I E T
DO A 5 J - l , I E T
I C (I , J) =0.0
CALL GEDA ( N 1 , N 2 1 , P R )
WRITE (6,11)
FORMAT ( 5 X , ' G E D A D O N E ' )
CALL RECON ( N 1 , N 3 , N A , N 2 1 , I R 1 , P I , P R , P , C , C M )
WRITE (6,31)
FORMAT ( 5 X , ' R E C O N D O N E ' )
DO AO 1 = 1 , I R 1
DO AO J = 1 , N 3
UT-P (I , J)
PMAX=MAX1(PMAX.UT)
PMIN=AMIN1(PMIN.UT)
CALL COTRAN ( P , I R 1 , I R 2 , N 3 , C A , P I , P M I N , P M A X )
WRITE (6,51)
FORMAT ( 5 X , ' C O N T R A N D O N E ' )
CALL GRAY ( I R 2 , I R 2 , I E T , C A , I C , P M A X , P M I N , N G )
WRITE (6,52)
FORMAT ( 5 X , ' G R A Y D O N E ' )
WRITE (6,60) ( ( I C ( I , J ) , J = I E T , 1 , - 1 ) , 1= 1ET,1,-1)
FORMAT('1',2X,60I2)
OPEN ( 2 , F I L E = ' S U P R / R E A . O U T ' , C A R R I A G E = ' F O R T R A N ' )
W R I T E ( 2 , 6 0 ) ( ( I C ( I , J ) , J = I E T , 1 , - 1 ) , 1= 1 E T , I , - 1 )
WRITE (2,59)
FORMAT ( 5 X , ' O U T P U T I N S U P R / R E A . O U T AND R E A . C O D ' )
STOP
END
23b
S U B R O U T I N E GEDA ( N 1 , N 2 1 , P R )
D I M E N S I O N PR ( N 1 , - N 2 1 : n 2 1 ) , D P ( - 8 : 8 ) , P I ( 1 6 , - 8 : 8 )
OPEN ( 3 , F I L E = ' S U P R / D E T . D A ' )
READ ( 3 , 2 0 ) ( D P ( I ) , I = - 8 , 8 )
READ ( 3 , 2 0 ) ( ( P I ( I , J ) , J = - 8 , 8 ) , 1 = 1 , 1 6 )
FORMAT(17F5.2)
DO 3 0 1 = 1 , 1 6
DO 3 0 J = - 8 , 8
PR ( I , J ) - L O G ( D P ( J ) / P I ( I , J ) )
RETURN
END
20
30
S U B R O U T I N E RECON ( N 1 , N 3 , N 4 , N 2 1 , I R 1 , P I , P R , P , C , C M )
D I M E N S I O N PR ( N 1 , - N 2 1 : N 2 1 ) , C ( N 1 , - N 2 1 : N 2 1 ) , C M ( N 1 , I R 1 , N 3 )
DIMENSION P (IR1,N30
REAL*8 CH,C,CM
n11=n1
5
1»
6
20
15
25
10
kO
35
DL=1.0/N4
DO 1 0 J = 1 , N 1
DO 1 0 I R = 1 , I R 1
DO 1 0 1TH E R = 1 , N 3
CH=IR*COS (J*PI/N11-PI * ITHER*2/N3)
I F (CH . L T . 0 . 0 ) GO T O k
Kl = lDI NT (CH)
I F ( ( I R 1 - C H ) . L E . 0 . 0 0 0 0 1 ) GO T O 5
K2=K1+1
GO T O 6
K2=K1
GO T O 6
K 2 = I D I N T (CH)
K1=K2 — 1
DO 1 5 I H - K 1 . K 2
SUM=0.0
DO 2 0 I 1 = - N 2 1 , N 2 1
SUM=SUM+PR ( J , I l ) / ( 2 * ( P I * D L ) * * 2 * ( l - i » * ( I H - l 1 ) * * 2 ) )
C (J, I H) =SUM3
DH=CH-K1
I F (DH . L E . 0 . 0 1 ) GO T O 2 5
CM ( J , I R , I T H E R ) = C ( J , K l ) + ( C ( J , k 2 ) - C ( J , K l ) ) * D H
GO T O 1 0
CM ( J , I R , I T H E R ) = C ( J , K 1 )
CONTINUE
DO 3 5 I R = 1 , I R 1
DO 3 5 I T H E R = 1 , N 3
SUM1=0.0
DO 4 0 J = 1 , N 1
SUM1=SUM1+P h'«CM ( J , I R , I T H E R ) / N 1 1
P (IR,ITHER)=SUM1
RETURN
END
235
20
21
3
1
33
11
i»
2
3*»
S U B R O T I N E COTRAN ( P , I R , I R 2 , N T H , C , P I , P M I N , P M A X )
DIMENSION P (IR.NTH) , C ( - I R2: I R2, - I R2: IR2) ,T(2)
DO 1 0 I Y = - I R 2 , I R 2
DO 1 0 I X = - I R 2 , I R 2
IF ((IX.EQ.O) .AND. (IY.EQ.O)) GOTO 10
I F ( ( I X . E Q . O ) . O R . ( I Y . E Q . O ) ) GO TO 2 0
R1=(!X**2+IY**2)**0.5
GO TO 2 1
Rl-IX+IY
R1=ABS (Rl)
I F ( R l . G T . I R ) GO T O 3 0
KR1 = I N T ( R l )
KR2=KR1+1
DR=R1-KR1
I F ( ( I Y . E Q . O ) . A N D . ( I X . G T . O ) ) GO T O h
I F ( ( I Y . E Q . O ) . A N D . ( I X . L T . O ) ) GO T O 2
IF ( ( I X.EQ.O) .AND. (IY.GT.O)) GOTO 1
I F ( ( I X . E Q . 0 ) . A N D . ( I Y . L T . O ) ) GO T O 3
GO T O 3 3
A = 1 .5>'<P I
GO T O 1 1
A=0.5'VPI
GO T O 1 1
X I= I X
Y I= I Y
A=ATAN2 ( Y I , X I )
I F (A . G E . 0 . ) GO T O 1 1
A+A+2*PI
A1=A*NTH/ (2*P I)
GO T O 3 ^
A1=NTH
GO T O $ k
Al=NTH/2
I A 1= 1 N T ( A l )
IA2-IAl+1
15
1^
5
12
DA=A1 - I A 1
I F ( I A 1 . N E . 0 ) GO T O 1 5
IA1=NTH
I F ( ( D R . L E . 0 . 0 0 0 1 ) . A N D . ( D A . L E . 0 . 0 0 0 1 ) ) GO T O 5
DO 1 4 I = K R 1 , K R 2
I 1 = 1 - K R 1+ 1
T (I 1) =P (I , I A1) + (P ( I , I A2) - P ( I , I A1)) *DA
C (I X, IY) =T(1) + (T(2)-T(1))*DR
I F ( C ( I X , I Y ) . L E . PMAX) GO T O 1 2
C (IX, IY)=PMAX
GO T O 1 0
C(I X.IY)=P (KR1,I A1)
GO T O 1 0
I F (C ( I X , I Y ) . G E . P M I N ) GO T O 1 0
C (IX, IY)=PMIN
CONTINUE
C ( 0 , 0 ) = (C ( 1 , 0 ) + C ( 0 , 1 ) + C ( - 1 , 0 ) + C ( 0 , - 1 ) ) A
RETURN
END
SUBROUTINE GRAY ( I X , I Y , I E T , C , I C , P M A X , P M I N , N )
D I M E N S I O N C ( - I X : I X , - I Y : I Y ) , I C ( I E T , I E T ) , RC ( 6 0 , 6 0 ) , P 0 ( 4 0 0 )
IXO=I X-1
IYO=IY-1
DO 3 0 I = - 1 X 0 , I X o
C(I ,- IY) = C ( I , - IYO)
C (I , I Y)=C (I , IYO)
C (-IX,I)=C (-1X0,I)
C (IX, l)=C (1X0, I)
C ( - I X , - I Y ) = 0 . 5 * (C ( - I X , - I YO) + C ( - I X O , - I Y ) )
C ( - I X , I Y ) = 0 . 5 * ( C ( - I X , I YO) + C ( - I X O , I Y ) )
C ( I X, - IY) =0. 5 * (C ( I XO, - I Y) +C ( I X, - I YO))
C(I X,IY) =0.5*(C(IXO,IY)+C (I X,IYO))
DO 4 1 I = - 9 , 1 0
DO 4 1 J = - 9 , 1 0
I N=(1+9)*20+ (J+10)
PO ( I N ) = C ( I , J )
OPEN ( 1 0 , F I L E = ' S U P R / R E 4 . C 0 D ' )
W R I T E ( 1 0 , 4 2 ) (PO ( I ) , 1 = 1 , 4 0 0 )
FORMAT ( 4 0 0 F 6 . 3 )
DO 4 0 1 = 1 , 5 8 , 3
I I= (I -1)/3"10
30
41
42
10=11+1
RC
RC
RC
4 0 RC
20
10
DO 4 0 J = 1 , 5 8 , 3
JJ=(J-l)/3-10
J0=JJ+1
RC ( I , J ) = C ( I I , J J )
RC ( I , J + l ) = 0 . 6 7 * 0 ( I I , J J ) + 0 . 3 3 * 0 ( I I , J O )
RC(I, J+2) =0-33*C(I I,JJ)+0.67*0 (II,JO)
R C ( 1+ 1 , J ) = 0 . 6 7 * 0 ( I I , J J ) + 0 . 3 3 * 0 ( 1 0 , J J )
( 1+ 1 , J + l ) = 0 . 3 6 * 0 ( I I , J J ) + 0 . 2 3 * ( 0 ( I I , J O ) + C ( 1 0 , J J ) ) + 0 . 1 8 * 0 ( 1 0 , J O )
( 1 + 1 , J + 2 ) = 0 . 3 6 * 0 ( I I , J O ) + 0 . 2 3 * (C ( I I , J J ) + C ( I 0 , J O ) ) + 0 . 1 8 * C ( I 0 , J J )
RC(1+2, J) =0.33*C(I I,JJ)+0.67*0 (10,JJ)
( 1 + 2 ,J + l ) = 0 . 3 6 * 0 ( I 0 , JJ ) + 0 . 2 3 * ( 0 ( I I , J J ) + C ( l 0 , J O ) ) + O . l 8 * C ( l I , J 0 )
(I+2,J+2)=0.36*0(I 0,JO)+0.23*(C(I 0,JJ)+C(I I,JO))+0.18*C(II,JJ)
SCA= ( P M A X - P M I N ) / N
DO 1 0 1 = 1 , I E T
DO 1 0 J = l , I E T
DO 2 0 L = 1 , N
GL=PMIN+L*SCA
I F ( R C ( I , J ) . G T . G L ) GO T O 2 0
I C (I ,J)=L-1
GO T O 1 0
CONTINUE
CONTINUE
RETURN
END
CITED LITERATURE
Abragram, A.:
Principles of
London, 1961.
Nuclear Magnetism. Oxford University Press
Alters, T.E., Simon, W., Chesler ,D.A., and Correia, J.A.:
A t t e n u a t i t i o n c o r r e c t i o n i n gamma emission computed tomography.
J. Comput. Assist. Tomogr., 5189"9^» 1981.
Barret, H. and Swindell, W.:
Academic Press, 1981.
Radiological
Imaging. V . 2 , New York,
Bates, R.H.T., Garden, K.L., and Peters, T.M.: Overview of computerized
tomography with emphasis on future developments.
Proc. IEEE,
71:356-372, 1983.
Bellini, S., Piacentini, M., Cafforio, C., and Rocca, F . : Compensation
of tissue absorption in emission tomography.
IEEE Trans. Acous.
Speech, S i g. Proc.. 27:213-218, 1979Bocage, E.M.:
French Patent: 536, 464,(1921). Quoted in Massiot.J.:
History of Tomography. In Medical Mundi . 19:106, 1974.
Boerner, W.M. and Chan, C.Y.: Inverse in electromagnetic imaging. In
Medical Applications of Microwave Imaging . eds. Larsen.L.E., and
J a c o b i . J . H . , I E E E P r e s s , N . J . ,1986.
Borth, D.E. and Cain, C.A.: Theoretical analysis of acoustic signal
generation in materials irradiated with microwave energy.
IEEE
Trans. Microwave Theory Tech., 25:944-954, 1977Bowen, T . , Nason, R . L . , and P i f e r , A.E.: Thermoacoustic imaging induced
by deeply penetrating radiation. In Acoustic Imaging, eds. Kaveh,
M . , Muler, R . K . , and Greenleaf, J . F . , New York,Plenum Press,
13:409-427, 1984a.
Bowen, T . , Connor, W.G., Nasoni, R . L . , P i f e r , A.E., B e l l , R . , Cooper,
D.H., and Sembroski, G.H.: Observation of Acoustic singles from a
phantom in an 18 Mev electron beam for cancer therapy. In Acoustic
Imaging , eds. Kaveh, M . , Muler, R . K . , and Greenleaf, J . F . , New
York,Plenum Press, 13:429-434, 1984b.
Bowen, T . : Radiation-induced thermoacoustic soft tissue imaging.
Ultrasonics Symposium , 1:817—822, 1981.
Bowen, T . , Nasoni, R . L . P i f e r , A . E . , and Sembroski, G . H . : Some
experimental results on the thermoacoustic imaging of tissue
equivalent phantom materials. In Ultrasonics Symposium ,
1:823-827, 1981.
237
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I
VITA
Personal data
Jenn-Lung Su
Born 2/15/55. Peikang, Taiwan, R.O.C.
Educat ion
1985-1988
University of Illinois at Chicago
Ph.D. in Bioengineering, August 1988
1982-1985
University of Illinois at Chicago
MS, Bioengineering, August 1985
1973-1977
Chung-Yuan University
Chung-Li, Taiwan, R.O.C.
BS w i t h Honor, Biomedical Engineering, June 1977
Honors and Awards
Chung Yuan University Assistantship, August 1983
College of Engineering of Chung Yuan U. Assistantship, August 1982
Member of Chung Yuan Honors Society, February 1977
Chung Yuan University scholarship, March 1977
Chung Yuan University scholarship, October 1976
Research and Teaching Experience
1986-1988
Research Assistant (Dept. of BioE, U of I at Chicago)
Research Project: Medical Imaging System.
1987 Fall
Teaching Assistant (Dept. of BioE, U of
Teaching Course: Nonionizing Radiation.
1982-1985
Research Assistant (Dept. of BioE, U of I at Chicago)
Research Project: Bioelectromagnetics, Thermoelastic
Pressure Waves.
I98O-I982
Research Assistant (Dept. of BioE, Chung-Yuan Univ.)
Research Project: The Market Survey Study of Medical
Instruments in Taiwan.
1979-1982
Administration 6 Teaching Assistant (Dept. of BioE )
Assistant Course: Engineering Mathematics, Electronic
Circuits, Bioinstrumentations, and Digital Circuits.
Teaching Course: Analog and Digital Circuits Lab.
247
I at Chicago)
21*8
Act i vi t ies
Student Member of
Student Member of
Student Member of
Soc i ety
IEEE Engr. in Medicine and Biology Society
IEEE Microwave Theory and Techniques Society
IEEE Acoustics, Speech, and Signal Processing
Adviser of Chinese Student Society in UIC,
1/1987—Present
President of Chinese Student Society in UIC, 1/1986— 1/1987
Vicepresident of Biomed. Engr. Society in Chung Yuan U. 9/1976-6/1977
Faculty Team of Tennis in Chung Yuan U. 8/1979—7/1982
Soft tennis team of Chung Yuan University, 5/1974—6/1977
Theses
Su, Jenn-Lung (1985)* "Experimental Study of Acoustic Imaging of
Induced Thermal Expansion of Simulated Biological Tissue" M.S. in
B i.oengineer ing, University of Illinois at Chicago.
Su, Jenn-Lung (1988)"Computer Assisted Tomography Using MicrowaveInduced Thermoelastic Waves" Ph.D. in Bioengineering, University
of Illinois at Chicago.
Presentat i ons
J.C. L i n , J . L . Su and Y. Wang "Measurement o f Microwave-induced
Thermoelastic Propagation i n Cat Brains" 8th Annual BEMS, Madison,
23,1986.
J . L . Su and J.C. Lin "Acoustic Imaging of Induced Thermal Expan­
sion of Biological Tissue" IEEE/7th Annual Conference of EMBS,
Chicago, 628-631, 1985-
Pub!i cat i ons
J.C. Lin, J . L . Su and Y. Wang,"Microwave-Induced Thermoelastic
Pressure Wave Propagation i n The Cat Brain", Bioelectromagnetics
9:141-147.1988.
J.L. Su and J.C. Lin,"Thermoelastic Signatures of Tissue Phantom
Absorption and Thermal Expansion" IEEE Trans. Bimed. Eng. 34:
No.2, 179-182,1987.
W.H. Chang, W.C. Hu and J.L. Su,"The Market Survey Study of Medi­
cal Instruments in Taiwan" (in Chinese) Tech. Report, National Sci­
ence Counci1,1981.
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