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Imaging geophysical data В Эtaking the viewer into account.

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Archaeological Prospection
Archaeol. Prospect. 11, 35–48 (2004)
Published online 5 February 2004 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/arp.221
Imaging Geophysical DataTaking the
Viewerinto Account
T. J. DENNIS*
Department of Electronic Systems Engineering, University of Essex, Colchester CO4 3SQ,
UK
ABSTRACT
Acommonwayofpresentinggeophysicaldatafromtwo-dimensionalsourcesisasagrey-scaleimage.
Some theoretical background to discrete image representation is described, and the deleterious effects of inappropriate (too sparse) sampling and display of such images discussed in an archaeological context. In high-quality images, such as magazine illustrations or digital television, the sampling
densities can be sufficiently high to avoid the appearance ofartefacts.Geophysicalimagesin contrast
are often sampled at very low densities; if the effective area of each sample is significantly less than
the sample spacing, then the classic effect called ‘aliasing’ in communication engineering, caused by
the violation of Nyquist’s criterion, will be seen.Knowledge of the sensor’s footprint can be used to select an appropriate sample density, and so minimize this source of distortion.To maximize the visibility
of what may be low-contrast structures immersed in a high level of background noise, it is helpful also
to consider the bandpass nature of the spatial frequency response of the human visual system. The
non-linear phenomenon of visual masking is shown to influence the choice on presentation methods.
Copyright 2004 JohnWiley & Sons,Ltd.
Key words: geophysics; image; alias; sample; visualmasking; spatial frequency; scanning
Introduction
Sampling
In geophysical imaging processes, like resistivity
or magnetometry, where the aim is to get an
accurate two-dimensional view of the pattern
of variation, the ideal would be to cover the areas
of interest with a dense or near-continuous array
of sample points. In practice, this is rarely, if ever
possible, and the imperfections can have serious
consequences on the visibility of features of interest, and the introduction of unwanted artefacts.
Guidance on how the data should be treated
can be obtained by consideration of two-dimensional sampling theory, and the related spatial
frequency (SF) response properties of the
intended final recipient of the data, the human
visual system.
In digital electronic signal representation,
whether temporal such as speech, or spatial
such as an image, the signal is discrete in two
ways: time and amplitude. The number of amplitude levels required is dictated by the need to
render any distortion that results from the quantization process imperceptible to the recipient in
each circumstance; for example, telephone
speech is of acceptable quality if non-linearly
quantized to a resolution of 256 levels, hence
needing log2256 ¼ 8 bits per sample. High-quality audio systems require 65 536 levels or more,
distributed linearly over the dynamic range. The
wider dynamic range of such signals compared
with telephone speech is the principal reason for
the difference. Digital image signals are less
demanding on amplitude, 256 levels being adequate for most purposes to render quantization
effects invisible. (A caveat here is that more than
* Correspondence to: T. J. Dennis, Department of Electronic
Systems Engineering, University of Essex, Colchester CO4
3SQ, UK. E-mail: tim@essex.ac.uk
Copyright # 2004 John Wiley & Sons, Ltd.
Received 6 May 2003
Accepted 20 October 2003
T. J. Dennis
36
Figure 1. Idealized amplitude spectrum of a bandlimited time
waveform sampled at frequency Fs.
8 bits, or floating point representation, should be
used for signal processing operations such as
filtering, in order to avoid propagation of rounding errors. Only at the final output should scaling
back to 8 bits/256 levels be performed.)
For a time-varying signal the second choice of
the system designer concerns an appropriate
sample rate. The relevant theorem was proposed
by Nyquist (1928) and can be paraphrased as, ‘A
signal that is strictly bandlimited to W Hz can be
recovered without distortion from instantaneous
samples taken at a uniform period T, where
T 1/2W’. This can be understood by reference
to a spectral description, in which it can be
shown that the Fourier domain spectrum of an
ideally sampled signal consists of an infinite
series of replications of the baseband signal at
intervals of Fs, as in Figure 1. Recovery of the
signal from its samples requires isolation of
the baseband replication (the one centred at the
origin) by means of an appropriate electrical
(low-pass) filter. It is easy to see that the baseband replication can be recovered without interference from the nearest replications (centred at
Fs) only if Fs 2W.
In practice, the ideals are difficult to achieve on
several counts, the most serious being an inability to ensure strict bandlimitation of the source
signal, because it is impossible to construct the
perfect ‘brickwall’ low-pass filter that would be
required. As a result, to cope with the imperfections of realizable filters, the sample rate has to
be somewhat higher than Nyquist would imply.
For example, in telephone speech, the nominal
bandwidth of the audio signal is 3.4 kHz, but the
sample rate used in the telephone system is
8 kHz, not 6.8 kHz.
The effect of interference owing to imperfect
filtering on the time-domain or waveform
description of the signal is known as ‘aliasing’.
In the telephone example, consider a single sinewave component at 4.7 kHz. This is above the
range that can be handled by the 8 kHz sample
rate, and it will appear in the first replication
spectrum at 8 þ 4.7 ¼ 12.7 and more significantly
also at 8 4.7 ¼ 3.3 kHz, which is within the baseband signal range and cannot then be removed. It
is this effect that accounts for some of the ‘squeaks
and whistles’ on the telephone system.
Two-dimensional sampling
For static images, the sampling process has to be
extended to two dimensions, and the process can
be better understood by reference to the twodimensional Fourier transform. Figure 2, left
half, is a gradiometer image of part of an iron
Figure 2. A gradiometer image (left) and its two-dimensional Fourier spectrum.The log amplitude of the spectrum is displayed to
accommodate its wide dynamic range.
Copyright # 2004 John Wiley & Sons, Ltd.
Archaeol. Prospect. 11, 35–48 (2004)
Imaging Geophysical Data
37
age site near Gosbecks Archaeological Park in
Colchester, Essex. It represents an area of 64 by
64 m. The central section uses two transects per
metre, the lines of which run up and down the
image. North is at the top. The periphery is
‘faded out’ to avoid Fourier edge-effects.
Along-transect the sample density is four per
metre. The right half of Figure 2 represents the
log magnitude two-dimensional discrete Fourier
transform of the image. The U and V axes are
horizontal and vertical spatial frequency respectively, with zero frequency (which represents the
mean grey level of the image) in the centre. It is
computed using the two-dimensional discrete
Fourier transform
Gðk; lÞ ¼
M1
X N1
X
gðm; nÞ ej2ðkm=Mþln =NÞ
m¼0 n¼0
here G(k, l) are the complex transformed coefficients of the input function g(m, n), sampled on a
grid size M bypN.
ffiffiffiffiffiffiffiffiElectronic engineers prefer the
symbol ‘j’ for 1. Fast computation algorithms
are used in practice (for examples see Press et al.,
1992).
As images are ‘real’ in a mathematical sense,
the full transform has complex conjugate symmetry, any effect of which disappears on an
amplitude-only spectrum, so each feature
appears twice on radially opposite sides of the
plot. The spectrum illustrates a number of points.
Given that the image is scaled to four pixels per
metre in both directions, the extremes of the axes
represent SFs of two cycles per metre. The range
of significant energy on the U axis is one half that
in V because the original sample density in the X
direction was two per metre.
A particularly powerful aspect of two-dimensional spectra is the way periodic or near-periodic
structure in the image appears as discrete energy
concentrations in the Fourier domain; hence, the
stride pattern of the machine operator is responsible for the horizontal line of components on the
V axis (labelled (1) on Figure 2), and the regular
(modern) cultivation patterns in the image running at a site bearing of approximately N16 E
emerge as a line of energy (2) inclined at 16
clockwise from the þU axis. The largest amplitude features are a result of the paired ditches in
the image, and it is interesting to note from the
Copyright # 2004 John Wiley & Sons, Ltd.
Figure 3. Spectral replications in two-dimensional sampling.
Ideal pre- and post-filters should have a frequency response
corresponding to the highlighted area.
presence of faint energy concentrations on the
line of (3), that other more detailed periodic
features also must be present on the same or close
alignment. These are not obvious in the image.
The two-dimensional analogy of the time
domain sampling process is typically a regular
grid of sample points. The pattern is most conveniently orthogonal, but samples do not have to
be equally spaced in the X and Y directions.
Figure 3 is a picture of the central area of the
two-dimensional spectrum that might result
from ideal sampling of a band-limited version
of an image: it comprises an infinite two-dimensional array of replications of the baseband spectrum, which is highlighted in the centre; the
replications’ spacings in U and V are proportional to the reciprocal of the spatial sample
spacing in X and Y respectively. A result of this
is that as sample spacing increases (i.e. the
samples become more sparse) there is a danger
that the spectral replications will overlap, generating alias components as in the time domain
case. This may have serious consequences on the
quality and reliability of the image sampled.
Returning to Figure 2, the spectral feature
labelled (2a) is likely to be an aliased continuation of (2). Had the sample density been four per
metre in both directions, then the spectrum
would have been square, and feature (2) is likely
Archaeol. Prospect. 11, 35–48 (2004)
38
T. J. Dennis
Figure 4. Effect of downsampling with and without a pre-filter: (1) original; (2) pre-filtered to halve bandwidth in U and V; (3) subpicture1after one in four downsampling, showing severe aliasing distortion; (4) same with sub-picture 2.
to have extended further from the origin. By
reducing the sample density in X to two per
metre, we are halving the spectral replication
separation on the U axis, and the continuation
of (2) as a result protrudes into the baseband: (2a)
in quadrant (U, V) comes from the one on the
opposite side, in (þU, V).
Figure 4 is an illustration of the visual effects of
inappropriate sampling densities. Sub-image (1)
is the upper left 256 by 256 pixel section of a 512
by 512 full-sized original; (2) is the same, but
with the spatial bandwidth halved vertically and
horizontally by a sharp-cut two-dimensional
spatial lowpass filter. (Spatial filtering can be
done by direct manipulation of the complex
two-dimensional discrete Fourier transform
(DFT) spectrum, or by spatial convolution using
a mask that is derived from the inverse DFT of
the impulse response. In this case it was done in
two passes by the same convolution mask operating horizontally then vertically.) Sub-image (3)
Copyright # 2004 John Wiley & Sons, Ltd.
is constructed by extracting alternate samples
from sub-image (1) in both directions, so that
the total number of samples is 25% of the original, and (4) is the same process done to
sub-image (2). The source comes from a highresolution scanner working from a good quality
photographic print, so the amount of detail
( ¼ high spatial frequency energy) in the original
picture is large. As a result, (3) is severely
affected by aliasing distortion, which shows itself
as spurious patterns on high-contrast detail,
whereas (4) is not distorted.
Consequences for geophysical sampling
As has been noted, geophysical scanning processes have to be performed at sample densities
that are practical on-site. According to Nyquist,
the aliasing effects discussed above will occur if
the area being sampled contains detail with
spatial frequency components that exceed a
Archaeol. Prospect. 11, 35–48 (2004)
Imaging Geophysical Data
frequency with a physical wavelength that is
equivalent to two sample spacings. To put this
into context, suppose the operator’s walking
speed and the setting of a gradiometer’s datalogger result in four samples per metre in the direction of traverse, and traverses are spaced at 1 m,
then in the walk direction the highest SF that can
be reliably reproduced will have a wavelength of
0.5 m (SF ¼ 2 cycles m1). Across-line the frequency limit will be 0.5 cycle m1. The result of
this is that any (typically sharply defined) feature
containing spectral energy above the Nyquist
frequencies will have it aliased to a lower frequency, as was shown by Figure 2, where it may
appear as an artefact, in the worst cases like
Figure 4, sub-image (3). Figure 5 is a qualitative
illustration of these effects.
Figure 5a is a synthesized ‘ground truth’ scene.
It contains a 128 by 128 pixel zone plate, which is a
two-dimensional radial linear frequency sweep
and has the useful property that the horizontal
and vertical components of the spatial frequency are proportional to position with respect to
the centre of the pattern. When acted on by some
filtering or sampling process it gives a direct view
of the frequency domain effects. The other features are two copies of a typical set of what might
be expected on a site, with (right) and without the
addition of some background granularity, generated by low-pass filtered uncorrelated Gaussian
noise. Such noise is likely to arise from random,
but fixed, features within the field pattern being
examined, and from electronic effects. Only the
latter could be reduced by time-averaging the
raw signals, rarely an option in practice. Not
considered in the example are correlated features
owing to structure in the underlying geology, or
the regular patterns that can arise from recent or
former cultivation activity.
Figure 5b has been subjected to a simulated
sampling process, such that it is scaled down by a
factor of six in the horizontal direction and 1.5 in
the vertical. This is the same 1:4 sample density
ratio as the commonly used ‘1 transect m1, 4
samples m1, in magnetometry. The picture is
then rescaled back to the original, using simple
pixel replication, sometimes called ‘nearest neighbour’ interpolation. This is equivalent to a sampling process lacking a pre-filter and hence
violates Nyquist’s criterion, with the effects
Copyright # 2004 John Wiley & Sons, Ltd.
39
showing most clearly on the zone plate. The
pattern in converted to a series of aliased repeats
in the horizontal direction, which is the most
severely affected because of its lower sample
density. The same effect occurs vertically, but is
less extreme.
The features show characteristic distortions
that will be recognized from practical experience:
the circle has a poorly defined outline; features
almost parallel to the traverse direction are partially broken up; the cross-traverse feature is
moderately well represented because of the high
along-traverse sample density; the ‘post holes’ are
distorted, poorly defined or absent altogether.
The situation of these examples is unduly
pessimistic, however, owing to the effective point
sampling process. In practical situations, for
example, a single-sensor total field magnetometer, the field strength is measured in a particular direction over a small volume of space. That
field is, however, modified to a greater or lesser
extent by variations in the magnetic properties of
the materials in its vicinity; it is the vector sum of
these effects along the sensor axis that the sensor
returns. Looking at it another way, Earth’s field is
distorted over a volume of space that is much
greater than the volume of a single near-point
anomaly. Therefore, rather than ‘seeing’ the
anomaly as a point, the sensor will behave as if
it has a spatial impulse response (or point spread
function) of finite area (or volume if sampling in
three dimensions is allowed). The difference
between total field magnetic instruments and
the more common gradiometer configuration is
that the latter performs a fixed spatial high-pass
filtering operation on one axis of the field, usually
the vertical, resulting in zero gain at zero spatial
frequency. Results from the total field instrument
are usually processed for display in much the
same way, however, as it is probably the local
variations of field strength that are of interest. Its
advantage is that the cut-off frequency of the
spatial filter is now a controllable parameter,
and if made sufficiently low—for example, by
simply subtracting the signal level averaged over
the whole of a scanned area—can render visible
large-scale features that are undetectable by the
gradiometer configuration.
Results of practical measurements of the
impulse response on a Geoscan Research FM18
Archaeol. Prospect. 11, 35–48 (2004)
40
T. J. Dennis
Figure 5. Synthesized test image, incorporating zone plate and typical archaeological features. Noise is added to the rightmost
section. The image size is 528 by 138 pixels. (b) As (a) with downsizing to 88 by 92 pixels, then resizing. No pre-filter. Note the alias
components on the zone plate. (c) As (b) but using Gaussian pre-filter. (d) As (c) but using bicubic interpolation at the upsizing stage.
gradiometer are shown in Figure 6. The instrument was fixed in position with its sensor tube
vertical, while a small (relative to vertical separation) ferrite toroid was moved in 5-cm steps on a
horizontal line directly beneath it parallel to the
local S–N magnetic axis. The value taken at each
Copyright # 2004 John Wiley & Sons, Ltd.
step was the average of 40 samples as logged by
the instrument at a rate of 4 Hz. Drift errors,
assumed linear over the measurement period,
have been compensated by reference to dummy
averages taken at the start and end of each run,
with the sample not present. These do not appear
Archaeol. Prospect. 11, 35–48 (2004)
Imaging Geophysical Data
41
Figure 6. Gradiometer responses with horizontal displacement and vertical separation to a ferrite toroid, 2 cm diameter and1cm
thick. As a precaution, the toroid was degaussed prior to the measurement process.
in the plot. No other processing, such as smoothing, has been applied. Runs were taken for vertical separations of the FM18’s lower sensor to the
centre of the sample of 0.3, 0.4, 0.5 and 0.6 m.
As expected, peak signal amplitude is strongly
affected by vertical separation, but the width of
the response, arbitrarily taken as the point where
its amplitude has halved, varies relatively little
from approximately 0.5 m, certainly between 30
and 40 cm vertical separation. It is not possible to
estimate properly for the greater separations
owing to the low response relative to noise: these
are difficult measurements to take because they
have to be done out of doors away from any
ferrous materials, and the extended time interval
required means that extraneous factors such as
gradiometer thermal drift and possibly even
‘glitches’ in Earth’s field cannot be fully compensated by the drift correction. Another point of
interest to note from these curves is the
well-known asymmetry caused by the local dip
angle of Earth’s field, which in mid-northern
latitudes results in a peak response to the south
of zero displacement, and a small negative peak
on the north side (Scollar et al., 1990).
The effect of the finite-sized sensitivity footprint, which is two dimensional in practice, will
Copyright # 2004 John Wiley & Sons, Ltd.
be to act predominantly as a low-pass filter, but
with enhanced sensitivity to spatial frequencies
having wavelengths 0.4–0.8 m in the north–south
direction, the approximate range of distance
between the positive and negative peaks, and
dependent on vertical spacing. Compared with
the sample spacing of 0.25 m commonly used in
the direction of transit with gradiometers, the
positive response width of 0.5 m is probably
sufficient to avoid aliasing. This is not the case
in the other direction at one or even two traverses
per metre, as was demonstrated in the discussion
on Figure 2.
Any advantage of the filtering effect is negated
if an anomaly of sufficient strength to exceed the
dynamic range of the instrument is encountered,
resulting in hard limiting.
Returning to the synthesized examples, in
Figure 5c a Gaussian pre-filter has been applied,
using a two-dimensional Gaussian smoothing
kernel of diameter (measured as two standard
deviations) 3.6 pixels. Gaussian smoothing produces a Gaussian roll-off in the frequency
domain, which is insufficient to prevent completely the alias components appearing on the zone
plate. It improves the continuity of the other
features, however, and there is significant
Archaeol. Prospect. 11, 35–48 (2004)
42
reduction in the granularity. The reduction in
noise level is an important and valuable byproduct of the use of a pre-filter. In the absence
of a pre-filter, noise energy that is above the
Nyquist limit appears in the baseband signal
because of spectral folding; furthermore it is at
lower frequencies than before, and hence potentially more visually intrusive, as in Figure 5b.
Figure 5d has the same processing as 5c, but
uses the bicubic interpolation method to construct
the output picture; this removes much of the
blockiness. It is discussed further in a later section.
Resistivity example
The spatial response of resistivity systems is
more complex than for magnetometry, and
is strongly affected by probe configuration. This
is discussed in detail by Clark (1996). It has
similarities in that measurements again do not
relate to a single point, but are the result of
contributions from a hemisphere of soil comparable in diameter to the probe spacings. It is
subject to some of the same sampling artefacts,
particularly aliasing, as this example shows.
Gosbecks Archaeological Park, Colchester, is a
large and important site to the southwest of the
town, containing a variety of Roman and Iron
T. J. Dennis
Age structures, including the remains of a Roman
Theatre (Hawkes and Crummy, 1995) at OS grid
coordinates (596846, 222309). The Park is owned
by Colchester Borough Council, and is the subject
of a long-term geophysical scanning programme
conducted on behalf of Colchester Museum that
aims eventually to cover the whole area.
Figure 7 is a twin-probe resistivity plot taken
in June 2002 on the Gosbecks Theatre site using a
Geoscan Research RM4 instrument. The graticule
lines are spaced at 20 m intervals, and sampling
was on a 1 m square grid. A full description of
the site based on partial excavation in 1967 can be
found in Dunnett (1971). In its present state it
takes the form of a low mound rising about 1 m
above the surrounding ground level, and has
always produced poor results on geophysical
investigation, probably because of the depth of
any physical remains below the cut turfs that
comprise the bulk of the mound. In the mid1990s, a presumed ground plan (Crummy, 1997),
based on Dunnett (1971), was traced on the surface for the benefit of visitors to the Park using a
playing-field line marker and powerful weedkiller. This ground plan was still visible in 2003; the
theatre area is mown regularly, whereas the
surround is cut for hay every 3 years or so.
To enhance smaller features, Figure 7 has been
processed by a Gaussian high-pass filter that
Figure 7. Gosbecks, Colchester, RomanTheatre resistivity plot. Data collected by P. J. Cott, A. Dyson and T. J. Dennis; reproduced
by courtesy of P. J. Cott.
Copyright # 2004 John Wiley & Sons, Ltd.
Archaeol. Prospect. 11, 35–48 (2004)
Imaging Geophysical Data
increases the gain on high spatial frequencies by
a factor of about 10:1 relative to near-zero frequency. Recognizable structure is visible as a
result. Of particular immediate interest was the
white (higher resistance) semicircular feature,
labelled A. Figure 8, top image, shows the original ground pattern design at the same scale and
orientation. Surveying on site showed that A
corresponded exactly to the gap in the modern
seating plan. Given that the precise location of
the Theatre cannot be inferred from Dunnett
(1971), it is likely that this is not a coincidence,
and that the resistivity meter is detecting a difference resulting from the absence of vegetation
in the ground marks. Dark marks (reduced resistance) labelled C do appear to be from the theatre
itself, and do not align exactly with A. Feature B
also appears to be an artefact of modern origin, as
further surveying checks showed that it aligned
exactly with the then boundary between mown
and rough vegetation.
43
Patterned features labelled D initially were
thought to be machine operator faults, but given
the suspicion about A, a to-scale sampling simulation was carried out on the ground plan design,
with the result shown in the lower section of
Figure 8. This is a classic example of an alias, or
interference pattern, and suggests that features D
are the result of another interaction between the
coarse sample grid common in resistivity work
and the ground markings.
Image reproduction issues
The design of an electronic image display system, principally for economic reasons, has to take
account of the properties of the ultimate receptor,
the human eye. For example, in the domestic
television system, very approximately the number of scan lines is chosen such that adding more
lines (that is, improving the resolution) will not
significantly improve the picture quality for a
viewer sitting at eight to ten times picture height
from the screen. Cost of transmission in terms of
analogue bandwidth, or bits per second in the
digital case, is proportional to the square of
the number of lines, so it is in the interests of
the broadcasters to use the minimum possible.
The human visual system
Figure 8. Upper: ground plan of surface marks applied to theatre site for visitors, kindly supplied by Philip Crummy of Colchester Archaeological Trust Ltd. Lower: simulated sampling
process, showing alias patterns. Compare with patterning D
on Figure 7.
Copyright # 2004 John Wiley & Sons, Ltd.
A simple and highly revealing model for human
vision is that the eye–brain system acts as a twodimensional spatial filter (there is a temporal
component as well, but this only affects moving
video sequences). Experiments by Campbell and
Robson (1968) on foveal (fine detail vision),
results summarized by Pearson (1975), measured
the spatial frequency response in detail. Subjects
are presented with small visual fields (corresponding to the foveal region) containing a spatial sinusoid. The frequency is under control of
the experimenter, and the subject is asked to vary
the amplitude (contrast) of the pattern until it is
just perceptible. That there is a roll-off (loss in
sensitivity) at high frequencies would be
expected, and is due to imperfections in the
visual optics and the finite density and size of
receptor cells in the retina. What is possibly
unexpected is that there is also a significant
Archaeol. Prospect. 11, 35–48 (2004)
T. J. Dennis
44
Figure 9. Logarithmic spatialfrequency sweep. View from1-2 m, and observe where the patternsjust disappear towardsthe top of
the picture as a function of horizontal position.
Figure10. Simultaneous contrast illusion.
roll-off at low spatial frequencies, with a peak
response (at high light levels) at about 3.5 cycles
per degree. This is equivalent to a wavelength of
approximately 10 mm at a distance of 2 m. The
effect can be seen by observing Figure 9 from a
distance of 1–2 m. It shows a logarithmic spatial
frequency sweep, frequency increasing left to
right, and contrast decreasing from a maximum
Copyright # 2004 John Wiley & Sons, Ltd.
at the bottom to zero at the top. The test is to
observe how the upper limit of visibility of the
pattern varies across the image: there should be a
broad peak somewhere near the centre. It is this
bandpass filter effect that is thought to explain
the well-known simultaneous contrast illusions,
of the kind shown in Figure 10. The grey level at
the centre of the circle is actually the same as at
the edge of the picture.
The implication of this result for the reproduction of geophysical data—which as we have seen
is not always collected in the most ideal ways—
is that to maximize the chance of spotting what
may be faint or elusive features, the size of the
display should aim to ensure that these features’
spatial frequency lies near the peak of the sensitivity curve. The temptation is sometimes to
reproduce images constructed from sparse data
sets at a larger size than they deserve. The eye’s
loss of sensitivity at the low spatial frequencies
that would then reflect scene structure suggests
the converse: using a small size display should
improve perceived quality.
Masking
The eye’s spatial frequency response measurement was done at low contrast, at the threshold
of vision, and in common with most systems fed
Archaeol. Prospect. 11, 35–48 (2004)
Imaging Geophysical Data
45
Figure 11. Visual masking of low contrast features overlaid by high contrast detail. Observe from a distance, or defocus vision to
reveal text in the stripe region.
with small input stimuli, can be treated as a
linear effect, that is the response to the sum of
inputs A and B is the same as the sum of the
responses to A and B presented separately.
For large input stimuli, the situation is somewhat different, with the appearance in vision, as
in other perceptual systems, of a powerful masking effect: the response to a large stimulus presented at the same time/place as a small one is
that the small one is not perceived (Pearson,
1975). Figure 11 shows the effect. Text of the
same low video contrast (4.3% of the dynamic
range) has been added to both halves of the
picture, in which mean grey levels are the
same. The high-contrast stripes fill approximately 70% of the video dynamic range. Viewed
from nearby the stripes completely mask the text;
view from further away, or defocus the vision,
Copyright # 2004 John Wiley & Sons, Ltd.
and the characters are revealed. Both of these
actions are equivalent to a spatial low-pass filter,
which reduces the stripes’ effective amplitude.
Masking is an important aspect of vision that is
usefully exploited in the lossy ‘psychovisual’
image compression techniques like JPEG and
MPEG (Ghanbari, 1999): these introduce
significant error into the decoded image, but it
is confined to areas of high spatial frequency
detail and/or contrast, where it will not be
noticed.
Presentation of results
Combined effects of spatial frequency response
and masking can be seen in Figure 12, which
shows versions of an image of a common household object reconstructed from an original 32 by
Archaeol. Prospect. 11, 35–48 (2004)
46
T. J. Dennis
Figure12. (a) Low-resolution image upsized from 32 24 to 640 480 using pixelreplication. (b) As (a) but using bicubic interpolation.Inset: upsized to 96 72 pixels.
Copyright # 2004 John Wiley & Sons, Ltd.
Archaeol. Prospect. 11, 35–48 (2004)
Imaging Geophysical Data
47
the original has been enlarged by a factor of three
up to 96 72 pixels: the assertion is that at this
scale, and viewed at normal reading distance, the
spatial frequencies comprising the important
detail of the image lie nearer the peak of the
eye’s sensitivity curve.
The same effect can be seen in Figure 13, which
shows a 40 40 m area from a gradiometer scan
of a site in Norfolk. Transect spacing was 1 m,
with four samples per metre along transect. The
zig-zag method was used; transects are aligned
north–south. Figure 13a is upscaled to eight
pixels per metre in both directions, using pixel
duplication as in Figure 12a. Figure 13b is scaled
to four pixels per metre, using bicubic interpolation, which affects only the horizontal direction
on the image. There is evidence of two rows of
four post holes to the right of centre in the upper
half of the picture; they are most clearly visible in
the smaller properly interpolated image.
Finally Figure 14 shows an extended region of
the site, incorporating areas rescanned at two
transects per metre, which greatly improves quality. The picture resolution is four pixels per metre.
The original post holes are now clear, and there is
possibly another set aligned at right angles to the
first near the upper left edge of Figure 14.
Figure 13. (a) Excessive upsizing of fluxgate gradiometer image using pixel replication. Scan density one track per metre,
four samples per metre. (b) Resizing to reasonable level, bicubic interpolation.
24 pixels in size. Figure 12a enlarges the original
by a factor of 20 in each direction, and uses no
interpolation at all—each pixel is simply replicated so that it fills an area 20 20 pixels in the
output image. In 12b, bicubic interpolation has
been used, and generates a smoothly varying
pattern of contrast; in 12a, the severely blocky
structure, with its high-contrast edges, masks
much of the underlying pattern making the
object difficult to recognize. In 12b, it is seen as
a 13 amp plug, and the apparent quality is
further improved in the small inset in which
Copyright # 2004 John Wiley & Sons, Ltd.
Figure 14. More of Figure 13b, incorporating regions rescanned at two transects per metre.
Archaeol. Prospect. 11, 35–48 (2004)
T. J. Dennis
48
Conclusions
The concept of spatial frequency as a means of
interpreting the processes involved in sampling
in two dimensions is well-known in digital
image processing for video applications, but
probably is not taken into account in the context
of geophysical image acquisition systems. Video
sampling rates and densities can be designed so
that distortions owing to the sampling/reconstruction process are effectively eliminated.
This luxury is not available in geophysical imaging, where sample densities are dictated by
practical on-site considerations rather than theoretical rigour. Knowledge of the effects of suboptimum sampling, in particular aliasing owing
to the absence of an appropriate pre-filter, can
lead to an understanding of artefacts that may
appear.
The bandpass nature of human vision, where
there is a definite peak in sensitivity at mid-range
spatial frequencies, has implications for the display of sparsely sampled data, especially in the
situation where a feature of interest is poorly
defined owing to low contrast or high levels of
background noise. The recommendation is that
reproduction size—on a computer display, for
example—should be such that original sample
sizes are not enlarged by more than a factor of
two or at most four. Hence the common sample
density combination of four samples per metre
with one transect per metre is best viewed on
screen (assuming is has the usual pixel density of
72 per inch) at four samples per metre, that is 1:1
enlargement in the scan direction, 4:1 across
scan. Any greater enlargement tends to make
the image appear too ‘soft’, and larger scale
Copyright # 2004 John Wiley & Sons, Ltd.
fainter features actually become less visible. To
avoid blocky artefacts, with their damaging
masking effect, the enlargement process should
use a higher order interpolation method, bilinear
or bicubic.
On printed documents intended for close
viewing, depending on the printer’s half-toning
resolution, 72 pixels per inch is too low as it can
render the sample pattern visible. To avoid this
problem a density of 108 per inch is generally
satisfactory.
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Archaeol. Prospect. 11, 35–48 (2004)
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