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![Page 1: Projective Geometry and Geometric Invariance in Computer Vision Babak N. Araabi Electrical and Computer Eng. Dept. University of Tehran Workshop on image.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649e315503460f94b22504/html5/thumbnails/1.jpg)
Projective Geometry and Geometric Invariance in
Computer Vision
Babak N. Araabi
Electrical and Computer Eng. Dept.
University of Tehran
Workshop on image and video processingMordad 13, 1382
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2
Do parallel lines intersect?
Perhaps!
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Fifth Euclid's postulate
For any line L and a point P not on L, there exists a unique line that is parallel to L (never meets L) and passes through P.
At first glance it would seem that the parallel postulate ought to be a theorem deducible from the other more basic postulates.
For centuries mathematicians tried to prove it. Eventually it was discovered that the parallel
postulate is logically independent of the other postulates, and you get a perfectly consistent system even if you assume that parallel postulate is false.
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Alternative postulates
Non-Euclidean geometry; though not a typical one The projective axiom:
Any two lines intersect (in exactly one point). More intuitive approach:
Take each line of ordinary Euclidean geometry and add to it one extra object called a point at infinity. In addition, the collection of all the extra objects together is also called a line in projective plane (called the line at infinity).
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Why Projective Geometry?
A more appropriate framework when dealing with projection related issues.
Perspective projection: Photophraphy Human vision
Perspective projection is a non-linear mapping with Euclidean coordinates, but a linear mapping with homogeneous coordinates
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Overview
2D Projective Geometry 3D Projective GeometryApplication: Invariant matchingFrom 2D images to 3D space
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Projective 2D Geometry
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Homogeneous coordinates
0 cbyax Ta,b,c
0,0)()( kkcykbxka TT a,b,cka,b,c ~
Homogeneous representation of lines
equivalence class of vectors, any vector is representative
Set of all equivalence classes in R3(0,0,0)T forms P2
Homogeneous representation of points
0 cbyax Ta,b,cl Tyx,x on if and only if
0l 11 x,y,a,b,cx,y, T 0,1,,~1,, kyxkyx TT
The point x lies on the line l if and only if xTl=lTx=0
Homogeneous coordinates
Inhomogeneous coordinates Tyx,
T321 ,, xxx but only 2DOF
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Points from lines and vice-versa
l'lx
Intersections of lines
The intersection of two lines and is l l'
Line joining two points
The line through two points and is x'xl x x'
Example
1x
1y
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Ideal points and the line at infinity
T0,,l'l ab
Intersections of parallel lines
TTand ',,l' ,,l cbacba
Example
1x 2x
Ideal points T0,, 21 xx
Line at infinity T1,0,0l
l22 RP
tangent vector
normal direction
ab , ba,
Note that in P2 there is no distinction
between ideal points and others
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A model for the projective plane
exactly one line through two points
exaclty one point at intersection of two lines
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Duality
x l
0xl T0lx T
l'lx x'xl
Duality principle:
To any theorem of 2-dimensional projective geometry there corresponds a dual theorem, which may be derived by interchanging the role of points and lines in the original theorem.
For instance, the basic axiom that "for any two points, there is a unique line that intersects both those points", when turned around, becomes "for any two lines, there is a unique point that intersects (i.e., lies on) both those lines"
In projective geometry, points and lines are completely interchangeable!
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Conics
Curve described by 2nd-degree equation in the plane
022 feydxcybxyax
0233231
2221
21 fxxexxdxcxxbxax
3
2
3
1 , xxyx
xx or homogenized
0xx CT
or in matrix form
fed
ecb
dba
2/2/
2/2/
2/2/
Cwith
fedcba :::::5DOF:
6 parameters
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Five points define a conic
For each point the conic passes through
022 feydxcyybxax iiiiii
or
0,,,,, 22 cfyxyyxx iiiiii Tfedcba ,,,,,c
0
1
1
1
1
1
552555
25
442444
24
332333
23
222222
22
112111
21
c
yxyyxx
yxyyxx
yxyyxx
yxyyxx
yxyyxx
stacking constraints yields
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Projective transformations
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.
Definition:
A mapping h:P2P2 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
Theorem:
Definition: Projective transformation
3
2
1
333231
232221
131211
3
2
1
'
'
'
x
x
x
hhh
hhh
hhh
x
x
x
xx' Hor
8DOF
projectivity=collineation=projective transformation=homography
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Mapping between planes
central projection may be expressed by x’=Hx(application of theorem)
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Removing projective distortion
333231
131211
3
1
'
''
hyhxh
hyhxh
x
xx
333231
232221
3
2
'
''
hyhxh
hyhxh
x
xy
131211333231' hyhxhhyhxhx 232221333231' hyhxhhyhxhy
select four points in a plane with know coordinates
(linear in hij)
(2 constraints/point, 8DOF 4 points needed)
Remark: no calibration at all necessary, better ways to compute
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More examples
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Transformation of lines and conics
Transformation for lines
ll' -TH
Transformation for conics-1-TCHHC '
xx' HFor a point transformation
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A hierarchy of transformations
Projective linear group
Affine group (last row (0,0,1))
Euclidean group (upper left 2x2 orthogonal)
Oriented Euclidean group (upper left 2x2 det 1)
Alternative, characterize transformation in terms of elements or quantities that are preserved or invariant
e.g. Euclidean transformations leave distances unchanged
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Class I: Isometries
(iso=same, metric=measure)
1100
cossin
sincos
1
'
'
y
x
t
t
y
x
y
x
1
11
orientation preserving:orientation reversing:
x0
xx'
1
tT
RHE IRR T
special cases: pure rotation, pure translation
3DOF (1 rotation, 2 translation)
Invariants: length, angle, area
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Class II: Similarities
(isometry + scale)
1100
cossin
sincos
1
'
'
y
x
tss
tss
y
x
y
x
x0
xx'
1
tT
RH
sS IRR T
also know as equi-form (shape preserving)
metric structure = structure up to similarity (in literature)
4DOF (1 scale, 1 rotation, 2 translation)
Invariants: ratios of length, angle, ratios of areas, parallel lines
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Class III: Affine transformations
11001
'
'
2221
1211
y
x
taa
taa
y
x
y
x
x0
xx'
1
tT
AH A
non-isotropic scaling! (2DOF: scale ratio and orientation)
6DOF (2 scale, 2 rotation, 2 translation)
Invariants: parallel lines, ratios of parallel lengths,
ratios of areas
DRRRA
2
1
0
0
D
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Class VI: Projective transformations
xv
xx'
vP T
tAH
Action non-homogeneous over the plane
8DOF (2 scale, 2 rotation, 2 translation, 2 line at infinity)
Invariants: cross-ratio of four points on a line
ratio of ratio of area
T21,v vv
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Action of affinities and projectivitieson line at infinity
2211
2
1
2
1
0v
xvxvx
xx
x
vAA
T
t
000 2
1
2
1
x
xx
x
vAA
T
t
Line at infinity becomes finite,
allows to observe vanishing points, horizon,
Line at infinity stays at infinity,
but points move along line
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Decomposition of projective transformations
vv
sPAS TTTT v
t
v
0
10
0
10
t AIKRHHHH
Ttv RKA s
K 1det Kupper-triangular,decomposition unique (if chosen s>0)
0.10.20.1
0.2242.8707.2
0.1586.0707.1
H
121
010
001
100
020
015.0
100
0.245cos245sin2
0.145sin245cos2
H
Example:
![Page 27: Projective Geometry and Geometric Invariance in Computer Vision Babak N. Araabi Electrical and Computer Eng. Dept. University of Tehran Workshop on image.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649e315503460f94b22504/html5/thumbnails/27.jpg)
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Overview transformations
1002221
1211
y
x
taa
taa
1002221
1211
y
x
tsrsr
tsrsr
333231
232221
131211
hhh
hhh
hhh
1002221
1211
y
x
trr
trr
Projective8dof
Affine6dof
Similarity4dof
Euclidean3dof
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 l∞
Ratios of lengths, angles.The circular points I,J
lengths, areas.
![Page 28: Projective Geometry and Geometric Invariance in Computer Vision Babak N. Araabi Electrical and Computer Eng. Dept. University of Tehran Workshop on image.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649e315503460f94b22504/html5/thumbnails/28.jpg)
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Number of invariants?
The number of functional invariants is equal to, or greater than, the number of degrees of freedom of the configuration less the number of degrees of freedom of the transformation
e.g. configuration of 4 points in general position has 8 dof (2/pt) and so 4 similarity, 2 affinity and zero projective invariants
![Page 29: Projective Geometry and Geometric Invariance in Computer Vision Babak N. Araabi Electrical and Computer Eng. Dept. University of Tehran Workshop on image.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649e315503460f94b22504/html5/thumbnails/29.jpg)
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Projective geometry of 1D
x'x 22H
The cross ratio
Invariant under projective transformations
T21, xx
3DOF (2x2-1)
02 x
4231
43214321 x,xx,x
x,xx,xx,x,x,x Cross
22
11detx,x
ji
ji
ji xx
xx
![Page 30: Projective Geometry and Geometric Invariance in Computer Vision Babak N. Araabi Electrical and Computer Eng. Dept. University of Tehran Workshop on image.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649e315503460f94b22504/html5/thumbnails/30.jpg)
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Projective 3D Geometry
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Projective 3D Geometry
Points, lines, planes and quadrics
Transformations
П∞, ω∞ and Ω ∞
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3D points
TT
1 ,,,1,,,X4
3
4
2
4
1 ZYXX
X
X
X
X
X
in R3
04 X
TZYX ,,
in P3
XX' H (4x4-1=15 dof)
projective transformation
3D point
T4321 ,,,X XXXX
![Page 33: Projective Geometry and Geometric Invariance in Computer Vision Babak N. Araabi Electrical and Computer Eng. Dept. University of Tehran Workshop on image.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649e315503460f94b22504/html5/thumbnails/33.jpg)
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Planes
0ππππ 4321 ZYX
0ππππ 44332211 XXXX
0Xπ T
Dual: points ↔ planes, lines ↔ lines
3D plane
0X~
.n d T321 π,π,πn TZYX ,,X~
14 Xd4π
Euclidean representation
n/d
XX' Hππ' -TH
Transformation
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Planes from points
0π
X
X
X
3
2
1
T
T
T
2341DX T123124134234 ,,,π DDDD
0det
4342414
3332313
2322212
1312111
XXXX
XXXX
XXXX
XXXX
0πX 0πX 0,πX π 321 TTT andfromSolve
(solve as right nullspace of )π
T
T
T
3
2
1
X
X
X
0XXX Xdet 321
Or implicitly from coplanarity condition
124313422341 DXDXDX 01234124313422341 DXDXDXDX 13422341 DXDX
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Points from planes
0X
π
π
π
3
2
1
T
T
T
xX M 321 XXXM
0π MT
I
pM
Tdcba ,,,π T
a
d
a
c
a
bp ,,
0Xπ 0Xπ 0,Xπ X 321 TTT andfromSolve
(solve as right nullspace of )X
T
T
T
3
2
1
π
π
π
Representing a plane by its span
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Lines
T
T
B
AW μBλA
T
T
Q
PW* μQλP
22** 0WWWW TT
0001
1000W
0010
0100W*
Example: X-axis
(4dof)
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Points, lines and planes
TX
WM 0π M
Tπ
W*
M 0X M
W
X
*W
π
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Hierarchy of transformations
vTv
tAProjective15dof
Affine12dof
Similarity7dof
Euclidean6dof
Intersection and tangency
Parallellism of planes,Volume ratios, centroids,The plane at infinity π∞
The absolute conic Ω∞
Volume
10
tAT
10
tRT
s
10
tRT
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The plane at infinity
π
1
0
0
0
1t
0ππ
A
AH
TT
A
The plane at infinity π is a fixed plane under a projective transformation H iff H is an affinity
1. canonical position2. contains directions 3. two planes are parallel line of intersection in π∞
4. line // line (or plane) point of intersection in π∞
T1,0,0,0π
T0,,,D 321 XXX
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Application inInvariant Matching
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How to define an invariant 2D identifier patch on an image?
Consider a multilateral identifier patch.
Lines are invariant under the perspective projection.
• 3 (4) reference points are required to obtain an affine (projective) invariant representation of the multilateral identifier patch.
• Weak-perspective/perspective camera models.
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Affine invariant representation of a planar point Three reference points. Ratio of areas are invariant. X at intersection of
R1R2 and R3A.
(r1,r2) invariants defined by:
R3
R1R2 X
A
)()(
)()(
32232
321321
RARRXR
RXRRRR
areararea
areararea
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Projective invariant representation of a planar point Four reference points. Ratio of ratio of areas
are invariant. X at intersection of
R1R4 and R3A.
(r1,r2) invariants defined by:
R3
R1R2
X
A
R4
)(
)(
)(
)(
)(
)(
)(
)(
42
322
42
32
42
321
421
321
RAR
RAR
RXR
RXR
RXR
RXR
RRR
RRR
area
arear
area
area
area
arear
area
area
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Affine invariant parallelogram:Three reference points
Select A to make a parallelogram with three reference points R1, R2, and R3
R3
R2 R1
A
)()( 32121
321 ARRRRRR areaarea
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Affine invariant parallelogram:
Construction of identifier patch (1)
B2
B1
R3
R2 R1
A
Select B1 on R2R3
)()( 32121
1211 ARRRRRB areararea 1.01 r
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Affine invariant parallelogram:
Construction of identifier patch (2)
C2
C1
B2
B1
R3
R1
Select C1 on B1R1
)()( 123121
2311 RBRBRBC areararea 25.01 r
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Affine invariant parallelogram:
Construction of identifier patch (3)
D2
D1
C2
C1
R3
R1
Select D1 on C1R1
)()( 123121
3121 RCRCRCD areararea 1.03 r
),,(
),,(
321
321
rrr
RRR
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From 2D to 3D
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Perspective projection
Perspective projection of 6 (feature) points from 3D world (dorsal fin) to 2D image:
pi=(xi,yi,zi) qi=(ui,vi) i=1,…,6
p1
p2p3
p4
p5
p6
Focal point
Image plane
q1
q2q3q4
q5q6
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3D and 2D Projective Invariants
I1, I2, and I3 are 3D projective invariants (under PGL(4)).
i1, i2, i3, and i4 are 2D projective invariants (under PGL(3)).
There is no general case view-invariant. 2D invariants impose constraints (in the form of polynomial relations)
on 3D invariants.* PGL(n): Projective General Linear group of order n.
111
1
1
1
0
0
0
1
0
0
0
1
111111
??projectioneperspectiv
11
1
1
1
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
111111
4
3
2
1
5
5
4
4
3
3
2
2
1
1
0
0
3
2
1
5
5
5
4
4
4
3
3
3
2
2
2
1
1
1
0
0
0
33
44
i
i
i
i
v
u
v
u
v
u
v
u
v
u
v
u
I
I
I
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
T
T
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Invariants of point sets:
Algebraic invariants
Transfer images to a reference frame.
A unique projective/affine transformation between any two sets of 4/3 corresponding points.
Coordinate of any other point is invariant in the reference frame.
00.2
0.40.6
0.81
0
0.5
10
0.2
0.4
0.6
0.8
1
P1 P2P4
P3
00.2
0.40.6
0.81
0
0.5
10
0.2
0.4
0.6
0.8
1
P4
P3
P2P1
00.2
0.40.6
0.8
0.4
0.6
0.8
10
0.2
0.4
0.6
0.8
1
P1 P3P2
P4
Reference frame
Globalinvariant
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Invariant Relations
Constraints (here for 6 points) typically have the following form
This is a fundamental object-image relation for six arbitrary 3D points and their 2D perspective projections.
Therefore having at least 6 points in correspondence over a number of images, 3D invariants (Ik’s) can be estimated and invariant relations can be used as geometric invariant models.
4322
4222
3311
3111 11
iIIi
iIIi
iIIi
iIIi
0)())1()1(())1()1()(( 343323232414131221 iiIiiiiIIiiiiIiiII
n 6 points3n-15 Ik’s2n-8 ik’s
2n-11 relationsor
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Where are we going?
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Thanks for yourAttention!
Questions?