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EECS 274 Computer Vision
Geometric Camera Models
Geometric Camera Models
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•
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Elements of Euclidean geometry
Intrinsic camera parameters
Extrinsic camera parameters
General form of perspective projection
• Reading: Chapter 1 of FP, Chapter 2 of S
Geometric camera calibration
Euclidean Geometry
Euclidean coordinate system
x OP.i
x
y
y
OP
.
j
OP
x
i
y
j
z
k
P
z OP.k
z
Planes
AP n 0 OP n OA n 0
P [ x, y, z ]T , n [a, b, c]T , OA n d
ax by cz d 0 Π P 0
where
a
b
Π
c
d
and
x
y
P
z
1
homogenous coordinate
Pure translation
OBP = OBOA + OAP , BP = BOA+ AP
AP: point P in frame A
Pure rotation
1st column:
iA in the basis of (iB, jB, kB)
3rd row:
kB in the basis of (iA, jA, kA)
i A .i B
i .j
B
R
A
A B
i A .k B
j A .i B
j A .jB
j A .k B
k A .i B A i TB
k A .jB A jTB
k A .k B A k TB
B
iA
B
jA
B
kA
Rotation about z axis
cos
sin
B
R
A
0
sin
cos
0
0
0
1
Rotation matrix
Elementary rotation
R=R x R y R z , described by three angles
Properties of rotation matrix
• Its inverse is equal to its transpose, R-1=RT , and
• Its determinant is equal to 1.
Or equivalently:
• Its rows (or columns) form a right-handed
orthonormal coordinate system.
Rotation group and SO(3)
• Rotation group: the set of rotation
matrices, with matrix product
– Closure, associativity, identity, invertibility
• SO(3): the rotation group in Euclidean
space R3 whose determinant is 1
– Preserve length of vectors
– Preserve angles between two vectors
– Preserve orientation of space
Pure rotations
OP i A
jA
P R P
B
B
A
A
A x
A
k A y i B
Az
jB
B x
B
k B y
Bz
Rigid transformation
B
P R P OA
B
A
A
B
Block matrix manipulation
A11
A
A21
A12
A22
B11
B
B21
B12
B22
What is AB ?
A11B11 A12 B21
AB
A21B11 A22 B21
A11B12 A12 B22
A21B12 A22 B22
Homogeneous Representation of Rigid Transformations
B P AB R
T
1 0
B
O A A P AB R AP BO A B A P
AT
1 1
1
1
Rigid transformations as mappings
Rotation about the k Axis
Affine transformation
• Images are subject to geometric distortion
introduced by perspective projection
• Alter the apparent dimensions of the
scene geometry
Affine transformation
• In Euclidean space, preserve
– Collinearity relation between points
• 3 points lie on a line continue to be collinear
– Ratio of distance along a line
• |p2-p1|/|p3-p2| is preserved
Shear matrix
Horizontal shear
Vertical shear
2D planar transformations
See Szeliski Chapter 2
2D planar transformations
2D planar transformations
3D transformation
Pinhole Perspective Equation
x
x
'
f
'
z
y' f ' y
z
Idealized coordinate system
Camera parameters
• Intrinsic: relate camera’s coordinate
system to the idealized coordinated
system
• Extrinsic: relate the camera’s coordinate
system to a fix world coordinate system
• Ignore the lens and nonlinear aberrations
for the moment
Intrinsic camera parameters
Units:
k,l : pixel/m
f :m
(See EXIF tags)
a,b: pixel
Physical Image Coordinates (f ≠1)
Normalized Image
Coordinates
Scale parameters: k, l (image sensor may not be square)
Offset: u0, v0
Manufacturing error: θ
Intrinsic camera parameters
Calibration matrix κ
P ( x, y , z ,1)T
The perspective
projection Equation
In reality
• Physical size of pixel and skew are always fixed
for a given camera, and in principal known
during manufacturing
• Some parameters often available in EXIF tag
• Focal length may vary for zoom lenses when
optical axis is not perpendicular to image plane
• Change focus affects the magnification factor
• From now on, assume camera is focused at
infinity
Extrinsic camera parameters
Explicit form of projection Matrix
riT denotes the i-th row of R, t , t , t , are the coordinates of t
x y z
T
ri can be written in terms of the corresponding angles
R can be written as a product of three elementary rotations,
and described by three angles
M is 3 × 4 matrix with 11 parameters
5 intrinsic parameters: α, β, u0, v0, θ
6 extrinsic parameters: 3 angles defining R and 3 for t
Explicit form of
projection Matrix
Note:
M is only defined up to scale in this setting!!
riT : i-th row of R
Theorem (Faugeras, 1993)
Camera parameters
A camera is described by several parameters
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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”
Projection equation
sx * * * *
x sy * * * *
s * * * *
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The projection matrix models the cumulative effect of all parameters
Useful to decompose into a series of operations
identity matrix
fs x
Π 0
0
0
fs y
0
intrinsics
•
X
Y
ΠX
Z
1
x'c 1 0 0 0
R
y 'c 0 1 0 0 3 x 3
0
1 0 0 1 0 1x 3
projection
rotation
03 x1 I 3 x 3
1 01x 3
1
T
3 x1
translation
Definitions are not completely standardized
– especially intrinsics—varies from one book to another
Camera calibration toolbox
• Matlab toolbox by Jean-Yves Bouguet
http://www.vision.caltech.edu/bouguetj/calib_doc/
• Extract corner points from checkerboard