Camera Model & Camera Calibration Slides are from Marc Pollefeys @ETH.
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Transcript of Camera Model & Camera Calibration Slides are from Marc Pollefeys @ETH.
Principal point offset
Tyx
T pZfYpZfXZYX )/,/(),,(
principal pointT
yx pp ),(
101
0
0
1
Z
Y
X
pf
pf
Z
ZpfY
ZpfX
Z
Y
X
y
x
x
x
Principal point offset
101
0
0
Z
Y
X
pf
pf
Z
ZpfY
ZpfX
y
x
x
x
camX0|IKx
1y
x
pf
pf
K calibration matrix
Camera rotation and translation
C~
-X~
RX~
cam
X10
RCR
1
10
C~
RRXcam
Z
Y
X
camX0|IKx XC~
|IKRx
t|RKP C~
Rt PXx
Finite projective camera
1yx
xx
p
ps
K
1yx
xx
p
p
K
C~
|IKRP
non-singular
11 dof (5+3+3)
decompose P in K,R,C?
4p|MP 41pMC
~ MRK, RQ
{finite cameras}={P4x3 | det M≠0}
If rank P=3, but rank M<3, then cam at infinity
Camera matrix decomposition
Finding the camera center
0PC (use SVD to find null-space)
432 p,p,pdetX 431 p,p,pdetY
421 p,p,pdetZ 321 p,p,pdetTFinding the camera orientation and internal parameters
KRM (use RQ decomposition ~QR)
Q R=( )-1= -1 -1QR
(if only QR, invert)
Cameras at infinity
00
dP
Camera center at infinity
0Mdet
Affine and non-affine cameras
Definition: affine camera has P3T=(0,0,0,1)
Basic equations
0Ap
minimal solution
Over-determined solution
5½ correspondences needed (say 6)
P has 11 dof, 2 independent eq./points
n 6 points
Ap
1p
1p̂3 3p̂P
minimize subject to constraint
Degenerate configurations
More complicate than 2D case
(i) Camera and points on a twisted cubic
(ii) Points lie on plane or single line passing through projection center
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
~ ~~
Calibration example
(i) Canny edge detection(ii) Straight line fitting to the detected edges(iii) Intersecting the lines to obtain the images corners
typically precision <1/10 (HZ rule of thumb: 5n constraints for n unknowns