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EECS 274 Computer Vision
Geometric Camera Models
Geometric Camera Models
•
•
•
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Elements of Euclidean geometry
Intrinsic camera parameters
Extrinsic camera parameters
General Form of the Perspective
projection equation
• Reading: Chapter 2 of FP, Chapter 2 of S
Quantitative Measurements and Calibration
Euclidean Geometry
Euclidean Coordinate Systems
 x  OP.i
 x



 y  OP.j  OP  xi  yj  zk  P   y 
 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
Coordinate Changes: Pure Translations
OBP = OBOA + OAP , BP = BOA+ AP
Coordinate Changes: Pure Rotations
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 


i
B
B
A
jA
B
kA

Coordinate Changes: Rotations about the 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
A rotation matrix is characterized by the following properties:
• 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
Coordinate Changes: 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
 
Coordinate Changes: Rigid Transformations
B
P  R P  OA
B
A
A
B
Block Matrix Multiplication
 A11
A
 A21
A12 
A22 
 B11 B12 
B

B
B
22 
 21
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
OA   A P   AB R AP  BOA  B  A P 
   
  AT  
1  1  
1

1
Rigid Transformations as Mappings
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
– Ratios of distance along a line
• |p2-p1|/|p3-p2| is preserved
Shear matrix
Horizontal shear
Vertical shear
2D planar transformations
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
The Intrinsic Parameters of a Camera
Units:
k,l : pixel/m
f :m
a,b : pixel
Physical Image Coordinates (f ≠1)
Normalized Image
Coordinates
The Intrinsic Parameters of a Camera
Calibration Matrix
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
• Focal length may vary for zoom lenses
• Optical axis may not be perpendicular to image
plane
• Change focus affects the magnification factor
• From now on, assume camera is focused at
infinity
Extrinsic Parameters
Explicit Form of the Projection Matrix
riT denotes the i-th row of R, tx, ty, tz, are the coordinates of t
riT 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 x 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 the 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  * * * *
•
•
The projection matrix models the cumulative effect of all parameters
Useful to decompose into a series of operations
identity matrix
 fsx
Π   0
 0
0
 fsy
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