3
CSE190A, Fall 06
Biometrics
Homogenous coordinates
A way to represent points in a projective space
1. Add an extra coordinate
e.g., (x,y) -> (x,y,1)=(u,v,w)
2. Impose equivalence relation
such that (
λ
not 0)
(u,v,w)
≈ λ
*(u,v,w)
i.e., (x,y,1)
≈
(
λ
x,
λ
y,
λ
)
3. “Point at infinity” – zero for
last coordinate
e.g., (x,y,0)
• Why do this?
– Possible to represent
points “at infinity”
• Where parallel lines
intersect
• Where parallel planes
intersect
– Possible to write the
action of a perspective
camera as a matrix
CSE190A, Fall 06
Biometrics
Euclidean -> Homogenous-> Euclidean
In 2-D
• Euclidean -> Homogenous: (x, y) -> k (x,y,1)
• Homogenous -> Euclidean: (u,v,w) -> (u/w, v/w)
In 3-D
• Euclidean -> Homogenous: (x, y, z) -> k (x,y,z,1)
• Homogenous -> Euclidean: (x, y, z, w) -> (x/w, y/w, z/w)
CSE190A, Fall 06
Biometrics
The camera matrix
Turn this expression into
homogenous coordinates
– HC’s for 3D point are
(X,Y,Z,T)
– HC’s for point in image
are (U,V,W)
U
V
W
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