More on single-view geometry
class 10
Multiple View Geometry
Comp 290-089
Marc Pollefeys
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.
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
Mar. 4, 6
Single view geometry
Camera model
Camera calibration
Single view geom.
Gold Standard algorithm
Objective
Given n≥6 2D to 2D point correspondences {Xi↔xi’},
determine the Maximum Likelyhood Estimation of P
Algorithm
(i) Linear solution:
~
(a) Normalization: Xi  UX i ~x i  Tx i
(b) DLT
(ii) Minimization of geometric error: using the linear
estimate as a starting point minimize the geometric error:
~ ~~
(iii) Denormalization: P  T-1~
PU
More Single-View Geometry
P T CP  Q cone
• Projective cameras and
planes, lines, conics and quadrics.
PQ *PT  C*
• Camera calibration and vanishing
points, calibrating conic and the IAC
Action of projective camera on planes
X 
X

Y


x  PX  p1p 2 p 3p 4 
 p1p 2 p 4  Y 
0
 1 
 1 
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
Xμ   P(A  μB)  PA  μPB  a  μb
back-projection
  PTl
 T X  lT PX
Action of projective camera on conics
back-projection to cone
Q co  P CP
T
x Cx  X P CPX  0
T
T
T
example:
T
T



T
K
K
Qco   C K | 0   CK 0
0
0 
 0
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
Action of projective camera on quadrics
back-projection to cone
C  PQ P
*
*
T
 T Q*  lT PQ*P T l  0
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
Q CO  (V T QV)Q - (QV)(QV) T
QCO V  0
The importance of the camera center
~
P  KR[I | C], P'  K' R' [I | C]
P'  K' R' KR  P
-1
x'  P' X  K' R' KR  PX  K' R' KR  x
-1
-1
x'  Hx with H  K' R' KR 
-1
Moving the image plane (zooming)
x  K[I | 0]X
-1
x'  K'[I | 0]X  K' K  x
 kI
-1
H  K' K    T
0
(1  k)~
x0 
1 
~
kI
(1

k)
x0 

 kI
K'   T
K T

1 
0
0
x0 
kA ~
 kI 0
 T

K
T


0
1
1

0
k f / f'
(1  k)~
x0  A ~
x0 
1  0T 1 
Camera rotation
x  K[I | 0]X
x'  K[R | 0]X  KRK -1x
H  KRK -1
conjugate rotation
μ, μe
i
, μe i

Synthetic view
(i) Compute the homography
that warps some a rectangle
to the correct aspect ratio
(ii) warp the image
Planar homography mosaicing
close-up:
interlacing
can be important problem!
Planar homography mosaicing
more examples
Projective (reduced) notation
X1  (1,0,0,0) T , X 2  (0,1,0,0) T , X 3  (0,0,1,0) T , X 4  (0,0,0,1) T
x1  (1,0,0) T , x 2  (0,1,0) T , x 3  (0,0,1) T , x 4  (1,1,1) T
a 0 0  d 
P  0 b 0  d 
0 0 c  d 
C  (a 1 , b 1 , c 1 , d 1 )T
Moving the camera center
motion parallax
epipolar line
What does calibration give?
x  K[I | 0]d 
0 
d  K 1x
cos  
T
T
d1 d 2
d d d
T
1
1
T
2
d2


x
x1 (K -T K -1 )x 2
T
1

T
(K -T K -1 )x1 x 2 (K -T K -1 )x 2

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
~ d
x  PX   KR[I | C]   KRd
0
mapping between p∞ to an image is given by the planar
homogaphy x=Hd, with H=KR
image of the absolute conic (IAC)

ω  KK
(i)
(ii)
(iii)
(iv)
(v)

T 1
 K -T K -1
C  H
IAC depends only on intrinsics
angle between two rays cos  
DIAC=w*=KKT
w  K (cholesky factorisation)
image of circular points
T
CH1

T
x
x1 ωx 2
T
1

T
ωx1 x 2 ωx 2

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)
Orthogonality = pole-polar w.r.t. IAC
The calibrating conic
1

 K 1
C  K T  1



1


Vanishing points
xλ  PXλ  PA  λPD  a  λKd
v  lim x λ   lim a  λKd  Kd
λ 
v  PX  Kd
λ 
Vanishing lines
Orthogonality relation
cos  
v
v1 ωv2  0
T
l1 ω*l 2  0
T
T
1
T
1
v ωv2

T
ωv1 v 2 ωv2

Calibration from vanishing points and
lines
Calibration from vanishing points and
lines
Next class: Two-view geometry
Epipolar geometry