camera model
DESCRIPTION
Camera model. Relation between pixels and rays in space. ?. Pinhole camera. Pinhole camera model. Camera is a mapping between 3D world (object space) and 2D image; Camera model: matrix representing camera mapping; interested in central projection. P = Intersection of principle axis - PowerPoint PPT PresentationTRANSCRIPT
Camera model
Relation between pixels and rays in space
?
Pinhole camera
Pinhole camera model
TT ZfYZfXZYX )/,/(),,(
101
0
0
1
Z
Y
X
f
f
Z
fY
fX
Z
Y
X
linear projection in homogeneous coordinates!
Camera is a mapping between 3D world (object space) and 2D image;Camera model: matrix representing camera mapping; interested in central projection
P = Intersection of principle axisWith image plane
Line from cameraCenter perp. To imageplane
Focal plane Z=f
Plane through cameraCenter C, parallel to imagePlane is called principlePlane of the camera
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
So far, assumed the origin of the coordinate in the image plane is the Same as the principal point
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~
RXcam
X10
RC-R
1
10
C~
R-RXcam
Z
Y
X
camX0|IKx XC~
-|IKRx t|RKP C
~R-t PX=x
~
Points in space are usually expressed in terms of world coordinate frame, i.e. Euclidean
X~
Inhomogeneous vector in world coordinateCoordinates of camera center in world coordInhomogeneous 3 vector in camera coord.
C~
camX
General mapping for pinhole camera
9 deg of freedom: f, px,py,3 rot, 3 trans
CCD camera
1yy
xx
p
p
K
11y
x
y
x
pf
pf
m
m
K
Pinhole camera model assumes image coordinates are Euclidean coordinates with equal scales in both axial directions; CCD cameras does not have square pixels
mx is the number of pixels per unit distance in image coordinates in the x directionmy is the number of pixels per unit distance in image coordinates in the y directionαx = f mx and αy = f my : focal length of the camera in terms of the pixel dimensions in x and y directions10 degrees of freedom for CCD camera
When is skew non-zero?
1yx
xx
p
ps
K
1 g
arctan(1/s)
for CCD/CMOS, always s=0Example of skew: negative enlarging
General projective camera
1yx
xx
p
ps
K
1yx
xx
p
p
K
C~
|IKRP
non-singular
11 dof (5+3+3)
t|RKP
intrinsic camera parametersextrinsic camera parameters
Projection equation
• The projection matrix models the cumulative effect of all parameters• Useful to decompose into a series of operations
ΠXx
1****
****
****
Z
Y
X
s
sy
sx
fsx 0 x 'c
0 fsy y 'c0 0 1
1 0 0 0
0 1 0 0
0 0 1 0
R3x3
03x1
01x3 1
I
3x3T3x1
01x3 1
projectionintrinsics rotation translation
identity matrix
Camera parametersA camera is described by several parameters
• Translation T of the optical center from the origin of world coords• Rotation R of the image plane• focal length f, principle point (x’c, y’c), pixel size (sx, sy)
• blue parameters are called “extrinsics,” red are “intrinsics”
• The definitions of these parameters are not completely standardized– especially intrinsics—varies from one book to another
Projection matrices for Orhographic and scaled orthographic projections
Orthographic projection
Scaled orthographic projection
10rr
P 21T
11T
tt
ktt
/10rr
P 21T
11T
(5dof)
(6dof)
Radial distortion
• Due to spherical lenses (cheap)• Model:
R
y
xyxKyxKyx ...))()(1(),( 44
222
1R
http://foto.hut.fi/opetus/260/luennot/11/atkinson_6-11_radial_distortion_zoom_lenses.jpgstraight lines are not straight anymore
Camera model
Relation between pixels and rays in space
?
Projector model
Relation between pixels and rays in space
(dual of camera)
?
Affine cameras
Track back while zooming inKeep object of interest same size