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Cameras and Projections
Dan Witzner Hansen
Course web page:www.itu.dk/courses/MCV
Email:[email protected]
Previously in Computer Vision….
• Homographies• Estimating homographies• Applications (Image rectification)
Outline• Projections• Pinhole cameras• Perspective projection
– Camera matrix– Camera calibration matrix
• Affine Camera Models
Single view geometry
Camera model
Camera calibration
Single view geom.
Pinhole camera model
TT ZfYZfXZYX )/,/(),,(
101
0
0
1
Z
Y
X
f
f
Z
fY
fX
Z
Y
X
Pinhole camera model
101
0
0
Z
Y
X
f
f
Z
fY
fX
101
01
01
1Z
Y
X
f
f
Z
fY
fX
PXx
0|I)1,,(diagP ff
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
CCD camera
1yx
xx
p
p
K
11y
x
x
x
pf
pf
m
m
K
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 anatomy
Camera centerColumn pointsPrincipal planeAxis planePrincipal pointPrincipal ray
Camera center
0PC
null-space camera projection matrix
λ)C(1λAX
λ)PC(1λPAPXx
For all A all points on AC project on image of A,
therefore C is camera center
Image of camera center is (0,0,0)T, i.e. undefined
Finite cameras:
1
pM 41
C
Infinite cameras: 0Md,0
d
C
Column vectors
0
0
1
0
ppppp 43212
Image points corresponding to X,Y,Z directions and origin
Row vectors
1p
p
p
0 3
2
1
Z
Y
X
y
x
T
T
T
1p
p
p0
3
2
1
Z
Y
X
w
yT
T
T
note: p1,p2 dependent on image reparametrization
The principal point
principal point
0,,,p̂ 3332313 ppp
330 Mmp̂Px
Action of projective camera on point
PXx
MdDp|MPDx 4
Forward projection
Back-projection
xPX 1PPPP
TT IPP
(pseudo-inverse)
0PC
λCxPλX
1
p-μxM
1
pM-
0
xMμλX 4
-14
-1-1
xMd -1
CD
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)
Euclidean vs. projective
homography 44
0100
0010
0001
homography 33P
general projective interpretation
Meaningful decomposition in K,R,t requires Euclidean image and space
Camera center is still valid in projective space
Principal plane requires affine image and space
Principal ray requires affine image and Euclidean space
Cameras at infinity
00
dP
Camera center at infinity
0Mdet
Affine and non-affine cameras
Definition: affine camera has P3T=(0,0,0,1)
Affine cameras
Summary parallel projection
1000
0010
0001
P canonical representation
10
0KK 22 calibration matrix
principal point is not defined
A hierarchy of affine cameras
Orthographic projection
Scaled orthographic projection
1000
0010
0001
P
10
tRH
10rr
P 21T
11T
tt
ktt
/10rr
P 21T
11T
(5dof)
(6dof)
A hierarchy of affine cameras
Weak perspective projection
ktt
y
x
/10rr
1α
αP 2
1T1
1T
(7dof)
1. Affine camera= proj camera with principal plane coinciding with P∞
2. Affine camera maps parallel lines to parallel lines3. No center of projection, but direction of
projection PAD=0(point on P∞)
A hierarchy of affine camerasAffine camera
ktts
y
x
A
/10rr
1α
αP 2
1T1
1T
(8dof)
1000P 2232221
1131211
tmmmtmmm
A
affine 44100000100001
affine 33P
A
Next: Camera calibration
The principal axis vector
3m
camcamcam X0|IKXPx T1,0,0mMdetv 3
camcam PP k vv 4k
4p|MC~
|IKRP k
0)Rdet(
vector defining front side of camera
(direction unaffected)
vmMdetv 43 kkk camcam PP k
because
Depth of points
C~
X~
mCXPXPT3T3T3 w
(dot product)(PC=0)
1m;0det 3 MIf , then m3 unit vector in positive direction
3m
)sign(detMPX;depth
T
w
TX X,Y,Z,T
When is skew non-zero?
1yx
xx
p
ps
K
1 g
arctan(1/s)
for CCD/CMOS, always s=0
Image from image, s≠0 possible(non coinciding principal axis)
HPresulting camera:
