Post on 19-Dec-2015
transcript
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 Models
Feb. 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 Tensor
Mar. 18, 20
Three View Reconstruction
Multiple View Geometry
Mar. 25, 27
MultipleView Reconstruction
Bundle adjustment
Apr. 1, 3 Auto-Calibration Papers
Apr. 8, 10
Dynamic SfM Papers
Apr. 15, 17
Cheirality Papers
Apr. 22, 24
Duality Project Demos
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
~ ~~
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
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)
Action of projective camera on lines
forward projection
μbaμPBPAμB)P(AμX
back-projection
lPT
PXlX TT
Action of projective camera on conics
back-projection to cone
CPPQ Tco 0CPXPXCxx TTT
000CKK0|KC
0KQ
TT
T
co
example:
Images of smooth surfaces
The contour generator is the set of points X on S at which rays are tangent to the surface. The corresponding apparent contour is the set of points x which are the image of X, i.e. is the image of
The contour generator depends only on position of projection center, depends also on rest of P
Action of projective camera on quadrics
back-projection to cone
TPPQC ** 0lPPQlQ T*T*T
The plane of 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 isTT
CO (QV)(QV)-QV)QV(Q 0VQCO
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'
Moving the image plane (zooming)
xKK'0]X|[IK'x'0]X|K[Ix
1-
10
x~k)(1kIKK'H T
01-
100kIK
10x~kA
10x~A
10x~k)(1kI
K10
x~k)(1kIK'
TT0
T0
T0
T0
'/ ffk
Synthetic view
(i) Compute the homography that warps some a rectangle to the correct aspect ratio
(ii) warp the image
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
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
The image of the absolute conic
KRd0d]C
~|KR[IPXx
mapping between ∞ 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=*=KKT
(iv) K (cholesky factorisation)(v) image of circular points
2T
21T
1
2T
1
ωxxωxx
ωxxcos
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 through cholesky factorization
(= Zhang’s calibration method)