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# Multiple View Geometry in Computer Vision

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Geometric
Computer Vision
Marc Pollefeys
Fall 2009
http://www.inf.ethz.ch/personal/pomarc/courses/gcv/
Geometric Computer Vision course schedule
(tentative)
Lecture
Exercise
Sept 16
Introduction
-
Sept 23
Geometry & Camera model
Camera calibration
Sept 30
Single View Metrology
Measuring in images
(Changchang Wu)
Oct. 7
Feature Tracking/Matching
Correspondence computation
Oct. 14
Epipolar Geometry
F-matrix computation
Oct. 21
Shape-from-Silhouettes
Visual-hull computation
Oct. 28
Multi-view stereo matching
Project proposals
Nov. 4
Structure from motion and visual SLAM
Papers
Nov. 11
Multi-view geometry and
self-calibration
Papers
Nov. 18
Shape-from-X
Papers
Nov. 25
Structured light and active range
sensing
Papers
Dec. 2
3D modeling, registration
and range/depth fusion
Papers
(Christopher Zach?)
Dec. 9
Appearance modeling and imagebased rendering
Papers
Dec. 16
Final project presentations
Final project presentations
Projective Geometry and
Camera model
Class 2
points, lines, planes
transformations
camera model
Read tutorial chapter 2 and 3.1
http://www.cs.unc.edu/~marc/tutorial/
Chapter 1, 2 and 5 in Hartley and Zisserman
Homogeneous coordinates
Homogeneous representation of 2D points and lines
ax + by + c = 0
(a,b,c ) T ( x,y,1) = 0
The point x lies on the line l if and only if
T
l x =0
Note that scale is unimportant for incidence relation
(a,b,c ) T ~ k (a,b,c ) T , в€Ђk в‰ 0
(x , y ,1)T
~ k ( x , y ,1) , в€Ђk в‰ 0
T
equivalence class of vectors, any vector is representative
Set of all equivalence classes in R3пЂ­(0,0,0)T forms P2
Homogeneous coordinates ( x1 , x 2 , x 3 )
T
Inhomogeneous coordinates ( x, y )
T
but only 2DOF
Points from lines and vice-versa
Intersections of lines
The intersection of two lines l and l' is x = lГ— l'
Line joining two points
The line through two points x and x' is l = x Г— x'
Example
Note:
x Г— x ' = [ x ]Г— x '
пѓ¦xпѓ¶
пѓ§ пѓ·
( 0 ,1, - 1) пѓ§ y пѓ·
пѓ§1пѓ·
пѓЁ пѓё
with пЃ›x пЃќп‚ґ
y =1
x =1
пѓ¦xпѓ¶
пѓ§ пѓ·
(1, 0 , - 1) пѓ§ y пѓ·
пѓ§1пѓ·
пѓЁ пѓё
пѓЄ
пЂЅ -z
пѓЄ
пѓЄпѓ« y
z
0
-x
- yпѓ№
пѓє
x
пѓє
0 пѓєпѓ»
Ideal points and the line at infinity
Intersections of parallel lines
T
l = (a , b , c ) and l' = (a , b , c ')
Example
T
lп‚ґ l' пЂЅ пЂЁb , пЂ­ a , 0 пЂ©
(a , b )
T
tangent vector
normal direction
x =1 x = 2
(x , x
Ideal points
1
l в€ћ = (0 , 0 ,1)
Line at infinity
2
,0 )
T
2
2
P = R в€Єl в€ћ
T
Note that in P2 there is no distinction
between ideal points and others
3D points and planes
Homogeneous representation of 3D points and planes
ПЂ1 X 1 + ПЂ 2 X 2 + ПЂ 3 X 3 + ПЂ 4 X 4 = 0
The point X lies on the plane ПЂ if and only if
T
ПЂ X =0
The plane ПЂ goes through the point X if and only if
T
ПЂ X =0
Planes from points
T
T
T
Solve ПЂ from X 1 ПЂ = 0, X 2 ПЂ = 0 and X 3 ПЂ = 0
пѓЄ Tпѓє
пѓЄX 2 пѓє ПЂ пЂЅ 0
пѓЄX T пѓє
пѓ« 3пѓ»
пѓЄ
пѓє
(solve ПЂ as right nullspace of пѓЄ X T2 пѓє )
пѓЄX T пѓє
пѓ« 3пѓ»
Points from planes
T
T
T
Solve X from ПЂ 1 X = 0, ПЂ 2 X = 0 and ПЂ 3 X = 0
пѓЄ Tпѓє
пѓЄПЂ 2 пѓє X пЂЅ 0
пѓЄПЂ T пѓє
пѓ« 3пѓ»
(solve X as right nullspace of пѓЄ ПЂ T пѓє )
пѓЄ 2пѓє
пѓЄПЂ T пѓє
пѓ« 3пѓ»
Representing a plane by its span
X =Mx
T
ПЂ M =0
M пЂЅ пЃ›X 1 X 2 X 3 пЃќ
Lines
Representing a line by its span
(4dof)
W пЂЅ пѓЄ Tпѓє
О»A пЂ« ОјB
пѓ«B пѓ»
Dual representation
W пЂЅ пѓЄ Tпѓє
пѓ«Q пѓ»
*
*
W W
T
= WW
О»P пЂ« ОјQ
*T
= 0 2Г— 2
Example: X-axis
W пЂЅпѓЄ
пѓ«1
0
0
0
0
1пѓ№
пѓє
0пѓ»
W пЂЅпѓЄ
пѓ«0
*
0
1
1
0
(Alternative: PlГјcker representation, details see e.g. H&Z)
0пѓ№
пѓє
0пѓ»
Points, lines and planes
M пЂЅ пѓЄ Tпѓє
пѓ«X пѓ»
M пЂЅ пѓЄ T пѓє
пѓ«ПЂ пѓ»
W
M ПЂ = 0
X
W
M X пЂЅ0
ПЂ
*
Conics
Curve described by 2nd-degree equation in the plane
2
2
ax + bxy + cy + dx + ey + f = 0
or homogenized
2
xпЃЎ
x1
x3
,yпЃЎ
x2
x3
2
2
ax 1 + bx 1 x 2 + cx 2 + dx 1 x 3 + ex 2 x 3 + fx 3 = 0
or in matrix form
пѓЄ
T
x C x = 0 with C пЂЅ пѓЄ b / 2
пѓЄпѓ« d / 2
5DOF:
{a : b : c : d : e : f }
b/2
c
e/2
d / 2пѓ№
пѓє
e/2
пѓє
f пѓєпѓ»
Five points define a conic
For each point the conic passes through
2
2
ax i + bx i y i + cy i + dx i + ey i + f = 0
or
(x
2
i
c = (a , b , c , d , e , f
, x i y i , y i , x i , y i ,1).c = 0
2
stacking constraints yields
пѓЄ 2
пѓЄ x2
пѓЄx2
3
пѓЄ 2
пѓЄ x4
пѓЄx2
пѓ« 5
x1 y 1
y1
2
x1
y1
x2 y2
y2
2
x2
y2
x3 y3
y3
2
x3
y3
x4 y4
y4
2
x4
y4
x5 y5
y5
2
x5
y5
1пѓ№
пѓє
1пѓє
1пѓє c пЂЅ 0
пѓє
1пѓє
1пѓєпѓ»
)T
Tangent lines to conics
The line l tangent to C at point x on C is given by l=Cx
x
l
C
Dual conics
A line tangent to the conic C satisfies l T C * l = 0
In general (C full rank):
C пЂЅC
*
-1
Dual conics = line conics = conic envelopes
Degenerate conics
A conic is degenerate if matrix C is not of full rank
m
e.g. two lines (rank 2)
T
C = lm + ml
l
T
e.g. repeated line (rank 1)
C = ll
T
l
Degenerate line conics: 2 points (rank 2), double point (rank1)
*
Note that for degenerate conics (C * ) в‰ C
T
X QX = 0
вЂў
вЂў
вЂў
вЂў
(Q : 4x4 symmetric matrix)
9 d.o.f.
in general 9 points define quadric
tangent plane ПЂ = QX
T
Q
пѓЄ
пЃЇ
пЂЅпѓЄ
пѓЄпЃЇ
пѓЄ
пѓ«пЃЇ
п‚·
п‚·
п‚·
п‚·
пЃЇ
п‚·
пЃЇ
пЃЇ
п‚·пѓ№
пѓє
п‚·
пѓє
п‚·пѓє
пѓє
п‚·пѓ»
*
ПЂ Q ПЂ=0
вЂў
relation to quadric Q * = Q -1
(non-degenerate)
2D projective transformations
Definition:
A projectivity is an invertible mapping h from P2 to itself
such that three points x1,x2,x3 lie on the same line if and
only if h(x1),h(x2),h(x3) do.
Theorem:
A mapping h:P2п‚®P2 is a projectivity if and only if there
exist a non-singular 3x3 matrix H such that for any point
in P2 reprented by a vector x it is true that h(x)=Hx
Definition: Projective transformation
пѓ¦ x '1 пѓ¶ пѓ© h11 h12 h13 пѓ№ пѓ¦ x1 пѓ¶
пѓ§
пѓ· пѓЄ
пѓєпѓ§ пѓ·
x' = H x
or
пѓ§ x ' 2 пѓ· пЂЅ пѓЄ h 21 h 22 h 23 пѓє пѓ§ x 2 пѓ·
пѓ§ x' пѓ· пѓЄh
пѓ§ пѓ·
8DOF
пѓЁ 3 пѓё пѓ« 31 h 32 h 33 пѓєпѓ» пѓЁ x 3 пѓё
projectivity=collineation=projective transformation=homography
Transformation of 2D points, lines
and conics
For a point transformation
x' = H x
Transformation for lines
l' = H
-T
l
Transformation for conics
-T
C ' = H CH
-1
Transformation for dual conics
*
*
C ' = HC H
T
Fixed points and lines
H e = О»e
H
T
(eigenvectors H =fixed points)
(пЃ¬1=пЃ¬2 пѓћ pointwise fixed line)
l = О» l (eigenvectors H-T =fixed lines)
Hierarchy of 2D transformations
Projective
8dof
Affine
6dof
Similarity
4dof
Euclidean
3dof
пѓЄ
h
пѓЄ 21
пѓЄпѓ« h 31
пѓЄ
a
пѓЄ 21
пѓЄпѓ« 0
пѓЄ
sr
пѓЄ 21
пѓЄпѓ« 0
пѓЄ
r
пѓЄ 21
пѓЄпѓ« 0
h12
h 22
h 32
a 12
a 22
0
sr12
sr 22
0
r12
r22
0
h13 пѓ№
пѓє
h 23
пѓє
h 33 пѓєпѓ»
tx пѓ№
пѓє
ty
пѓє
1 пѓєпѓ»
tx пѓ№
пѓє
ty
пѓє
1 пѓєпѓ»
tx пѓ№
пѓє
ty
пѓє
1 пѓєпѓ»
transformed
squares
invariants
Concurrency, collinearity,
order of contact (intersection,
tangency, inflection, etc.),
cross ratio
Parallellism, ratio of areas,
ratio of lengths on parallel
lines (e.g midpoints), linear
combinations of vectors
(centroids).
The line at infinity lв€ћ
Ratios of lengths, angles.
The circular points I,J
lengths, areas.
The line at infinity
пЂ­T
пЂ­T
l п‚ўп‚Ґ пЂЅ H A l п‚Ґ пЂЅ пѓЄ T пЂ­ T
пѓ«пЂ­ t A
пѓ¦0пѓ¶
0пѓ№ пѓ§ пѓ·
пѓє пѓ§ 0 пѓ· пЂЅ lп‚Ґ
1пѓ» пѓ§ пѓ·
пѓЁ1пѓё
The line at infinity lп‚Ґ is a fixed line under a projective
transformation H if and only if H is an affinity
Note: not fixed pointwise
Affine properties from images
projection
H PA
пѓЄ
пЂЅ 0
пѓЄ
пѓЄпѓ« l1
0
1
l2
0пѓ№
пѓє
0 HA
пѓє
l 3 пѓєпѓ»
rectification
l в€ћ пЂЅ пЃ›l1
l2
l 3 пЃќ , l 3 в‰ 0
T
Affine rectification
lв€ћ
v1
l1
l2
l3
v2
v 2 = l3 Г— l4
l4
v 1 пЂЅ l1 п‚ґ l 2
lв€ћ пЂЅ v1 п‚ґ v 2
The circular points
пѓ¦1пѓ¶
пѓ§ пѓ·
IпЂЅпѓ§iпѓ·
пѓ§0пѓ·
пѓЁ пѓё
пѓЄ
I п‚ў пЂЅ H S I пЂЅ пЂ­ s sin пЃ±
пѓЄ
пѓЄпѓ«
0
пѓ¦ 1 пѓ¶
пѓ§
пѓ·
J пЂЅ пѓ§пЂ­ iпѓ·
пѓ§ 0 пѓ·
пѓЁ
пѓё
s sin пЃ±
s cos пЃ±
0
tx пѓ№пѓ¦ 1 пѓ¶
пѓ¦1пѓ¶
пѓ§ пѓ·
пѓєпѓ§ пѓ·
iпЃ±
t y пѓ§ i пѓ· пЂЅ se пѓ§ i пѓ· пЂЅ I
пѓє
пѓ§0пѓ·
1 пѓєпѓ» пѓ§пѓЁ 0 пѓ·пѓё
пѓЁ пѓё
The circular points I, J are fixed points under the
projective transformation H iff H is a similarity
The circular points
вЂњcircular pointsвЂќ
2
2
lв€ћ
2
x1 + x 2 + dx 1 x 3 + ex 2 x 3 + fx 3 = 0
x3 пЂЅ 0
2
2
x1 + x 2 = 0
I пЂЅ пЂЁ1, i , 0 пЂ©
T
J пЂЅ пЂЁ1, - i , 0 пЂ©
Algebraically, encodes orthogonal directions
I=
(1, 0 , 0 )T
T
(
)
+ i 0 ,1, 0
T
Conic dual to the circular points
C
*
в€ћ
пЂЅ IJ пЂ« JI
T
T
C
*
в€ћ
пѓЄ
пЂЅ 0
пѓЄ
пѓЄпѓ« 0
0
1
0
0пѓ№
пѓє
0
пѓє
0 пѓєпѓ»
lв€ћ
I
J
C в€ћ пЂЅ H S C в€ћH S
*
*
T
The dual conic C *в€ћ is fixed conic under the
projective transformation H iff H is a similarity
Note: C *в€ћ has 4DOF
lв€ћ is the nullvector
Angles
Euclidean: l = (l1 , l 2 , l 3 )
m = (m 1 , m 2 , m 3 )
T
cos Оё =
T
l1 m 1 + l 2 m 2
(l
2
1
+ l 2 )(m 1 + m 2 )
2
2
T
Projective: cos пЃ± пЂЅ
пЂЁl
2
*
l Cп‚Ґ m
T
*
T
*
Cп‚Ґ l m Cп‚Ґ m
T
*
l C в€ћ m = 0 (orthogonal)
Transformation of 3D points,
For a point transformation
X' = H X
(cfr. 2D equivalent)
( x' = H x )
Transformation for lines
ПЂ' = H
-T
(l' =
ПЂ
H
-T
l)
Transformation for conics
-T
Q' = H QH
-1
(C ' =
H CH
(C '* =
HC H
-T
-1
)
Transformation for dual conics
*
*
Q' = HQ H
T
*
T
)
Hierarchy of 3D transformations
Projective
15dof
пѓЄ T
пѓ«v
tпѓ№
пѓє
vпѓ»
Affine
12dof
пѓЄ T
пѓ«0
tпѓ№
пѓє
1пѓ»
Similarity
7dof
пѓЄ T
пѓ«0
tпѓ№
пѓє
1пѓ»
Angles, ratios of length
Euclidean
6dof
пѓЄ T
пѓ«0
tпѓ№
пѓє
1пѓ»
Volume
Intersection and tangency
Parallellism of planes,
Volume ratios, centroids,
The plane at infinity ПЂв€ћ
The plane at infinity
ПЂ п‚ўп‚Ґ пЂЅ H
пЂ­T
A
ПЂп‚Ґ
пЂЅ пѓЄ T пЂ­T
пѓ«- t A
пѓ¦0пѓ¶
пѓ§ пѓ·
0пѓ№ пѓ§ 0 пѓ·
пЂЅ ПЂп‚Ґ
пѓє
пѓ§
пѓ·
1пѓ» 0
пѓ§ пѓ·
пѓЁ1пѓё
The plane at infinity ПЂп‚Ґ is a fixed plane under a
projective transformation H iff H is an affinity
1.
2.
3.
4.
T
canonical position ПЂ в€ћ = (0 , 0 , 0 ,1)
contains directions D = ( X 1 , X 2 , X 3 , 0 )T
two planes are parallel пѓ› line of intersection in ПЂв€ћ
line // line (or plane) пѓ› point of intersection in ПЂв€ћ
The absolute conic
The absolute conic О©в€ћ is a (point) conic on ПЂп‚Ґ.
In a metric frame:
2
2
2
X1 пЂ« X 2 пЂ« X3 пѓј
пѓЅпЂЅ0
X4
пѓѕ
or conic for directions:
(with no real points)
пЂЁ X 1 , X 2 , X 3 пЂ©I пЂЁ X 1 , X 2 , X 3 пЂ© T
The absolute conic О©в€ћ is a fixed conic under the
projective transformation H iff H is a similarity
1. О©в€ћ is only fixed as a set
2. Circle intersect О©в€ћ in two circular points
3. Spheres intersect ПЂв€ћ in О©в€ћ
пЃ—
*
в€ћ
пЂЅпѓЄ T
пѓ«0
0пѓ№
пѓє
0пѓ»
the projective transformation H iff H is a similarity
1. 8 dof
2. plane at infinity ПЂв€ћ is the nullvector of О©в€ћ
T
*
3. Angles:
cos Оё =
Camera model
Relation between pixels and rays in space
?
Pinhole camera
Pinhole camera model
T
( X , Y , Z ) пЃЎ ( fX / Z , fY / Z )
T
пѓ¦X
пѓ§
пѓ§Y
пѓ§Z
пѓ§
пѓ§ 1
пѓЁ
пѓ¶
пѓ·
пѓ·
пѓ·
пѓ·
пѓ·
пѓё
пѓ¶
пѓ·
пѓ§
пѓ· пѓЄ
пѓ·
пѓ· пЃЎ пѓ§ fY пѓ· пЂЅ пѓЄ
пѓ§ Z пѓ· пѓЄ
пѓ·
пѓЁ
пѓё пѓ«
пѓ·
пѓё
f
1
пѓ¦X
0пѓ№пѓ§
пѓєпѓ§ Y
0 пѓ§
пѓє Z
0 пѓєпѓ» пѓ§пѓ§
пѓЁ 1
linear projection in homogeneous coordinates!
Pinhole camera model
пѓ¦X
пѓ§
пѓ· пѓЄ
пѓє пѓЄ пѓєпѓ§ Y
f
0 пѓ§1
пѓ§ fY xпѓ· пЂЅ PX
пѓЄ
пѓєпѓЄ пѓє Z
пѓ§ Z пѓ· пѓЄ
1 пѓєпѓ» пѓЄпѓ« 0 пѓєпѓ» пѓ§пѓ§
пѓЁ
пѓё пѓ«
пѓЁ 1
P пЂЅ diag ( f , f ,1) пЃ›I | 0 пЃќ
пѓ¶
пѓ·
пѓ·
пѓ·
пѓ·1
пѓ·
пѓё
пѓ¦X
0пѓ№пѓ§
пѓєпѓ§ Y
0 пѓ§
пѓє Z
0 пѓєпѓ» пѓ§пѓ§
пѓЁ 1
пѓ¶
пѓ·
пѓ·
пѓ·
пѓ·
пѓ·
пѓё
Principal point offset
T
( X , Y , Z ) пЃЎ ( fX / Z + p x , fY / Z + p y )
( px, py )
пѓ¦X
пѓ§
пѓ§Y
пѓ§Z
пѓ§
пѓ§ 1
пѓЁ
T
principal point
пѓ¶
пѓ·
пѓ¦ fX пЂ« Zp x пѓ¶ пѓ© f
пѓ§
пѓ· пѓЄ
пѓ·
пѓ· пЃЎ пѓ§ fY пЂ« Zp x пѓ· пЂЅ пѓЄ
пѓ§
пѓ· пѓЄ
пѓ·
Z
пѓЁ
пѓё пѓ«
пѓ·
пѓё
px
f
py
1
пѓ¦X
0пѓ№пѓ§
пѓєпѓ§ Y
0 пѓ§
пѓє Z
0 пѓєпѓ» пѓ§пѓ§
пѓЁ 1
пѓ¶
пѓ·
пѓ·
пѓ·
пѓ·
пѓ·
пѓё
T
Principal point offset
px
пѓ¦ fX пЂ« Zp x пѓ¶ пѓ© f
пѓ§
пѓ· пѓЄ
py
пѓ§ fY пЂ« Zp x xпѓ· пЂЅ пѓЄK пЃ›I | 0fпЃќX cam
пѓ§
пѓ· пѓЄ
Z
1
пѓЁ
пѓё пѓ«
пѓЄ
K пЂЅ
пѓЄ
пѓЄпѓ«
f
пѓ¦X
0пѓ№пѓ§
пѓєпѓ§ Y
0 пѓ§
пѓє Z
0 пѓєпѓ» пѓ§пѓ§
пѓЁ 1
пѓ¶
пѓ·
пѓ·
пѓ·
пѓ·
пѓ·
пѓё
px пѓ№
пѓє calibration matrix
py
пѓє
1 пѓєпѓ»
Camera rotation and translation
~
~ ~
X cam = R (X - C )
пѓ¦X
~ пѓ§
пѓ©R - R C пѓ№пѓ§ Y
X cam пЂЅ пѓЄ
пѓєпѓ§
1 пѓ» Z
пѓ«0
пѓ§
пѓ§ 1
~
пѓЁ
x пЂЅ KR
C cam
X
K пЃ›I |I0| пЃќ-X
пЃ›
x = PX
пЃќ
пѓ¶
пѓ·
пѓ· пЂЅ пѓЄ0
пѓ· пѓ«
пѓ·
пѓё
P пЂЅ K пЃ›R | t пЃќ
~
- RC пѓ№
пѓєX
1 пѓ»
~
t пЂЅ -R C
CCD camera
пѓЄпѓЄ
пѓєпѓєпѓЄ
KK пЂЅпЂЅ
пЃЎmy y p y
пѓєпѓєпѓЄ
пѓЄпѓЄ
пѓЄпѓ«пѓЄпѓ«
11 пѓєпѓ»пѓєпѓ»пѓЄпѓ«
f
px пѓ№
пѓє
py
пѓє
1 пѓєпѓ»
General projective camera
пѓЄ
K пЂЅ
пѓЄ
пѓЄпѓ«
px пѓ№
пѓє
py
пѓє
1 пѓєпѓ»
s
пЃЎx
пЃ›
~
P пЂЅ KR I | C
пЃќ
11 dof (5+3+3)
non-singular
P пЂЅ K пЃ›R | t пЃќ
intrinsic camera parameters
extrinsic camera parameters
вЂў Due to spherical lenses (cheap)
вЂў Model:
R
R
2
2
4
4
( x , y ) пЂЅ (1 пЂ« K 1 ( x пЂ« y ) пЂ« K 2 ( x пЂ« y ) пЂ« ...) пѓЄ пѓє
пѓ« yпѓ»
straight lines are not straight anymore
Camera model
Relation between pixels and rays in space
?
Projector model
Relation between pixels and rays in space
(dual of camera)
?
(main geometric difference is vertical principal point offset
to reduce keystone effect)
Meydenbauer camera
vertical lens shift
to allow direct
ortho-photographs
Affine cameras
Action of projective camera on points
and lines
projection of point
x = PX
forward projection of line
X (Ој ) = P(A + ОјB) = PA + ОјPB = a + Ојb
back-projection of line
T
О =P l
пЃђ X пЂЅ l PX
T
T
(l T x = 0; x
= PX )
Action of projective camera on conics
back-projection to cone
T
T T
T
x
Cx
=
X
P CPX = 0
Q co = P CP
( x = PX )
C пЂЅ PQ P
*
*
T
пЃђ Q пЃђ пЂЅ l PQ P l пЂЅ 0
T
*
T
*
T
(О = P l)
T
Image of absolute conic
A simple calibration device
(i)
compute H for each square
(corners @ (0,0),(1,0),(0,1),(1,1))
(ii) compute the imaged circular points H(1,В±i,0)T
(iii) fit a conic to 6 circular points
(iv) compute K from w through cholesky factorization
(в‰€ ZhangвЂ™s calibration method)
Exercises: Camera calibration
Next class:
Single View Metrology
Antonio Criminisi
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