more on single-view geometry class 10
DESCRIPTION
More on single-view geometry class 10. Multiple View Geometry Comp 290-089 Marc Pollefeys. Multiple View Geometry course schedule (subject to change). Single view geometry. Camera model Camera calibration Single view geom. Gold Standard algorithm. Objective - PowerPoint PPT PresentationTRANSCRIPT
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More on single-view geometry
class 10
Multiple View GeometryComp 290-089Marc Pollefeys
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Multiple View Geometry course schedule(subject to change)
Jan. 7, 9 Intro & motivation Projective 2D Geometry
Jan. 14, 16
(no class) Projective 2D Geometry
Jan. 21, 23
Projective 3D Geometry (no class)
Jan. 28, 30
Parameter Estimation Parameter Estimation
Feb. 4, 6 Algorithm Evaluation Camera ModelsFeb. 11, 13
Camera Calibration Single View Geometry
Feb. 18, 20
Epipolar Geometry 3D reconstruction
Feb. 25, 27
Fund. Matrix Comp. Structure Comp.
Mar. 4, 6 Planes & Homographies Trifocal TensorMar. 18, 20
Three View Reconstruction
Multiple View Geometry
Mar. 25, 27
MultipleView Reconstruction
Bundle adjustment
Apr. 1, 3 Auto-Calibration PapersApr. 8, 10 Dynamic SfM PapersApr. 15, 17
Cheirality Papers
Apr. 22, 24
Duality Project Demos
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Single view geometry
Camera model
Camera calibration
Single view geom.
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Gold Standard algorithmObjective
Given n≥6 2D to 2D point correspondences {Xi↔xi’}, determine the Maximum Likelyhood Estimation of P
Algorithm(i) Linear solution:
(a) Normalization: (b) DLT
(ii) Minimization of geometric error: using the linear estimate as a starting point minimize the geometric error:
(iii) Denormalization:
ii UXX~ ii Txx~
UP~TP -1
~ ~~
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More Single-View Geometry
• Projective cameras and planes, lines, conics and quadrics.
• Camera calibration and vanishing points, calibrating conic and the IAC
** CPPQ T
coneQCPP T
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Action of projective camera on planes
1ppp
10ppppPXx 4214321 Y
XYX
The most general transformation that can occur between a scene plane and an image plane under perspective imaging is a plane projective transformation
(affine camera-affine transformation)
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Action of projective camera on lines
forward projection
μbaμPBPAμB)P(AμX
back-projection
lPT
PXlX TT
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Action of projective camera on conics
back-projection to cone
CPPQ Tco
00
0CKK0|KC0KQ
TT
T
co
example:
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Images of smooth surfaces
The contour generator G is the set of points X on S at which rays are tangent to the surface. The corresponding apparent contour g is the set of points x which are the image of X, i.e. g is the image of G
The contour generator G depends only on position of projection center, g depends also on rest of P
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Action of projective camera on quadrics
back-projection to cone
TPPQC ** 0lPPQlQ T*T*T
The plane of G for a quadric Q is camera center C is given by =QC (follows from pole-polar relation)
The cone with vertex V and tangent to the quadric Q is the degenerate Quadric: TT
CO (QV)(QV)-QV)QV(Q 0VQCO
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The importance of the camera center
]C~|[IR'K'P'],C~|KR[IP
PKRR'K'P' -1
xKRR'K'PXKRR'K'XP'x' -1-1
-1KRR'K'HHx with x'
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Moving the image plane (zooming)
xKK'0]X|[IK'x'0]X|K[Ix
1-
10
x~k)(1kIKK'H T01-
100kIK
10x~kA
10x~A
10x~k)(1kIK
10x~k)(1kIK'
TT0
T0
T0
T0
'/ ffk
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Camera rotation
xKRK0]X|K[Rx'0]X|K[Ix
1-
-1KRKH
conjugate rotation
ii ee μ,μμ,
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Synthetic view
(i) Compute the homography that warps some a rectangle to the correct aspect ratio
(ii) warp the image
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Planar homography mosaicing
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close-up: interlacingcan be important problem!
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Planar homography mosaicingmore examples
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Projective (reduced) notation
T4
T3
T2
T1 )1,0,0,0(X,)0,1,0,0(X,)0,0,1,0(X,)0,0,0,1(X
T4
T3
T2
T1 )1,1,1(x,)1,0,0(x,)0,1,0(x,)0,0,1(x
dcdbda
000000
P
Tdcba ),,,(C 1111
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Moving the camera center
motion parallax
epipolar line
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What does calibration give?
xKd 1
0d0]|K[Ix
21-T-T
211-T-T
1
2-1-TT
1
2T
21T
1
2T
1
)xK(Kx)xK(Kx
)xK(Kx
dddd
ddcos
An image l defines a plane through the camera center with normal n=KTl measured in the camera’s Euclidean frame
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The image of the absolute conic
KRd0d]C~|KR[IPXx
mapping between p∞ to an image is given by the planar homogaphy x=Hd, with H=KR
image of the absolute conic (IAC)
1-T-1T KKKKω 1TCHHC
(i) IAC depends only on intrinsics(ii) angle between two rays(iii) DIAC=w*=KKT
(iv) w K (cholesky factorisation)(v) image of circular points
2T
21T
1
2T
1
ωxxωxx
ωxxcos
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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,±i,0)T
(iii) fit a conic to 6 circular points(iv) compute K from w through cholesky factorization
(= Zhang’s calibration method)
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Orthogonality = pole-polar w.r.t. IAC
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The calibrating conic
1T K1
11
KC
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Vanishing points λKdaλPDPAλPXλx
KdλKda limλ xlimvλλ
KdPXv
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ML estimate of a vanishing point from imaged parallel
scene lines
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Vanishing lines
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Orthogonality relation
2T
21T
1
2T
1
ωvvωvv
ωvvcos
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Five constraints gives us five equations and can determine w
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Calibration from vanishing points and lines
Assumes zero skew, square pixels and 3 orthogonal
vanishing points
Principal point is the orthocenter of the trinagle made of 3 orthogonol vanishing lines
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Assume zero skew, square pixels, calibrating conic is a circle;How to find it, so that we can get K?
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Assume zero skew , square pixels, and principal point is at the image centerThen IAC is diagonal{1/f^2, 1/f^2,1) i.e. one degree of freedom need one more Constraint to determine f, the focal length two vanishing points corresponding To orthogonal directions.
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Next class: Two-view geometryEpipolar geometry