computer vision cmput 499/615 lecture 8: 3d projective geometry and it’s applications 3d...
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Computer Visioncmput 499/615
Lecture 8:Lecture 8:
3D projective geometry and 3D projective geometry and it’s applicationsit’s applications
Martin JagersandMartin Jagersand
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First 1D Projective lineProjective Coordinates
• Basic projective invariant in Basic projective invariant in PP11: the : the cross ratiocross ratio
a
a b
c
d
acbd
b cd
acbd=
Requires 4 points:3 to create a coordinate systemThe fourth is positioned within that system
inhomogeneous
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Cross ratio
• Basic projective invariant in Basic projective invariant in PP11: the : the cross ratiocross ratio
•Properties:Properties:– Defines coordinates along a 1d projective lineDefines coordinates along a 1d projective line– Independent of the homogeneous representation of Independent of the homogeneous representation of xx– Valid for ideal points Valid for ideal points – Invariant under homographiesInvariant under homographies
22
11
4231
43214321 det},;,{
ji
jiji xx
xxxx
xxxx
xxxxxxxx
homogeneous
HZ CH 2.5
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Projective 3D space
0),,,,( 44321 xxxxxX
),,()1,,,(
)/,/,/(),,,( 4342414321
ZYXZYX
xxxxxxxxxx
)0,,,( 321 xxx
•Projective transformation:Projective transformation: –4x4 nonsingular matrix 4x4 nonsingular matrix HH–Point transformationPoint transformation–Plane transformationPlane transformation
•Quadrics:Quadrics: QQ Dual: Dual: Q*Q*–4x4 symmetric matrix Q4x4 symmetric matrix Q
–9 DOF (defined by 9 points in general pose)9 DOF (defined by 9 points in general pose)
•PointsPoints
•PlanesPlanes0
),,,( 4321
Xπ
πT
Points and planes are dual in 3d projective
space.
ππ
XXTH
H
'
'
0XX QT 0* ππ QT
•Lines: 5DOF, various parameterizationsLines: 5DOF, various parameterizations
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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
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Planes
0ππππ 4321 ZYX
0ππππ 44332211 XXXX
0Xπ T
Dual: points ↔ planes, lines ↔ lines
3D plane
0X~
n dT 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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Representing a plane byby lin comb of 3 points
xX M 321 XXXM
0π MT
I
pM
Tdcba ,,,π T
a
d
a
c
a
bp ,,
Representing a plane by its nullspace span M
All points lin comb of basis
Canonical form:Given a plane nullspace span M is
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Points from planes
0X
π
π
π
3
2
1
T
T
T
0Xπ 0Xπ 0,Xπ X 321 TTT andfromSolve
(solve as right nullspace of )X
T
T
T
3
2
1
π
π
π
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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)
Join of two points: A, B
Intersection of two planes: P, Q
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Points, lines and planes
TX
WM 0π M
Tπ
W*
M 0X M
W
X
*W
π
Join of point X and line W is plane
Intersection of line W with plane is point X
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Affine space
Difficulties with a projective space:Difficulties with a projective space:– Nonintuitive notion of direction:Nonintuitive notion of direction:
– Parallelism is not represented– Infinity not distinguishedInfinity not distinguished– No notion of “inbetweenness”: No notion of “inbetweenness”:
– Projective lines are topologically circular– Only cross-ratios are availableOnly cross-ratios are available
– Ratios are required for many practical tasks
A
B +/-
Solution: find the plane at infinity!
•Transform the model to give ∞ its canonical coordinates
•2D analogy: fix the horizon line
Determining the plane at infinity upgrades the geometry from projective to affine
)1,0,0,0(π
)1,0,0(l
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From projective to affine
• Finding the plane (line, point) at infinity Finding the plane (line, point) at infinity – 2 or 3 sets of parallel lines 2 or 3 sets of parallel lines (meeting at “infinite” points)(meeting at “infinite” points)
– a known ratio can also determine infinite pointsa known ratio can also determine infinite points
l
2D
3D
1Dp
21
11
x
21
23
1x2(1+x)
= cross ratio
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From projective to affine
HZ
2D
3D
Projective Affine
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Affine space
•Affine transformationAffine transformation– 12 DOF12 DOF
– Leaves Leaves unchangedunchanged
• InvariantsInvariants– Ratio of lengths on a lineRatio of lengths on a line– Ratios of anglesRatios of angles– ParallelismParallelism
100034333231
24232221
14131211
aaaa
aaaa
aaaa
H A
Kutulakos and Vallino, Calibration-Free Augmented
Reality, 1998
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Metric space
•Metric transformation (similarity)Metric transformation (similarity)– 7 DOF7 DOF– Maps absolute conic to itself Maps absolute conic to itself
• InvariantsInvariants– Length ratiosLength ratios– AnglesAngles– The absolute conicThe absolute conic
1TS
sRH
O
t
Without a yard stick, this is the highest level of geometric structure that can be retrieved from images
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The absolute conic
• Absolute conic is an imaginary circle onAbsolute conic is an imaginary circle on• It is the intersection of every sphere with It is the intersection of every sphere with
• In a metric frameIn a metric frame
π
0),,(),,(
0
)1,0,0,0(
321321
4
23
22
21
TxxxIxxx
x
xxx
π
0
0*
*
T
T
I
0
0
On : π
*
absolute dual quadric
π
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From affine to metric
• IdentifyIdentify on on OR identify OR identify– via angles, ratios of lengths – e.g. perpendicular lines
•Upgrade the geometry by bringingUpgrade the geometry by bringing to its to its canonical form via an affine transformation canonical form via an affine transformation
2D3D
π *
021 dd T
1D ?
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Examples
2D
3D
Affine Metric
Two pairs of perpendicular lines
5 known points
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What good is a projective model?
Represents fundamental feature interactions
Used in rendering with an unconventional engine:
Achieving 3d tasksvia
2d image control
Visual Servoing
Physical Objects!
f1
f2
f3
Tcol(f1,f2,f3) = f1 - f2f3
collinearity task
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What good is a projective model?
Represents fundamental feature interactions
Used in rendering with an unconventional engine:
Achieving 3d tasksvia
2d image control
Visual Servoing
Physical Objects!
Y12
y22
y32
y31
y21
y11
image collinearity constraint
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A point meets a line in each of two images...
Collinearity?
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Collinearity?
But it doesn’t guarantee task achievement !
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All achievable tasks...
are ALL projectively distinguishable
coordinate systems
One point
Coincidence
Noncoincidence
1000
1000
1000
1000
0100
1000
1000
1000
1000
1000
0100
1000
0100
0010
Collinearity
General Position
2pt. coincidence
3pt. coincidence
1000
0100
1100
1000
1000
1000
1000
1000
1000
0100
0100
1000
0100
1100
100
1000
0100
0010
1110
1000
0100
0010
0001
4pt. coincidence
2 2pt. coincidence
Cross Ratio
General Position
Coplanarity
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Tpp
Tcol
Tcopl
Tcr
point-coincidence
collinearitycoplanarity
cross ratios ()
Cinj
Cwk
Task primitives
Tpp
Injective
Projective
Composing possible tasks
Task operators
T1 T2 = T1 T2 = T1•T2 T = 0 if T = 0
1 otherwise
T1
T2
AND OR NOT
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wrench: view 1 wrench: view 2
Twrench(x1..8) =
Tcol(x2,x5,x6)Tcol(x4,x7,x8)
Tpp(x1,x5) Tpp(x3,x7) 1
2
34
56
78
Result: Task toolkit
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GroupGroup TransformationTransformation InvariantsInvariants DistortionDistortion
ProjectiveProjective
15 DOF15 DOF
• Cross ratioCross ratio
• IntersectionIntersection
• TangencyTangency
AffineAffine
12 DOF12 DOF
• ParallelismParallelism
• Relative dist in 1dRelative dist in 1d
• Plane at infinityPlane at infinity
MetricMetric
7 DOF7 DOF
• Relative distancesRelative distances
• AnglesAngles
• Absolute conicAbsolute conic
EuclideanEuclidean
6 DOF6 DOF
• LengthsLengths
• AreasAreas
• VolumesVolumes
Geometric strata: 3d overview
1TS
sRH
O
t
1TA
AH
O
t
1TE
RH
O
t
v
AH TP v
t
π
3DOFπ
5DOF
Faugeras ‘95
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Perspective and projection
• Euclidean geometry is not the only representation- for building models from images- for building images from models
representations ?
• But when 3d realism is the goal, how can we effectively build and use
projectiveaffinemetric