# Invariants (continued) Summary

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• ## Solution: Homogenous coordinates.

• Represent points in plane as (x,y,w)
• (x,y,w), (kx, ky, kw), (x/w, y/w, 1) represent same point.
• If we think of these as points in 3D, they lie on a line through origin. Set of 3D points that project to same 2D point.

• ## Form a group.

• They can be composed
• They have inverses.
• Projective transformations equivalent to set of images of images.

• ## Strategy.

• Suppose P represents five points. V1 transforms P so that first 4 to canonical position, and fifth to (a,b,c).
• Next, suppose we are given TP, with T unknown. Find V2 to transform first 4 points of TP to canonical position.
• V2 = V1*T-1. V2P has fifth point = (a,b,c).
• For this to work, V1, V2 must be uniquely determined.

• ## Invariance isn’t captured by mathematical definition of invariance because 3D to 2D transformations don’t form a group.

• You can’t compose or invert them.

• ## If two objects are identical except for one point, they produce the same image when viewed along a line joining those two points.

• Along that line, those two points look the same.
• The remaining points always look the same.

• ## 3-D objects have no invariants.

• We can deal with this by focusing on planar portions of objects.
• Or special restricted classes of objects.
• Or by relaxing notion of invariants.
• ## However, invariants have become less popular in computer vision due to these limitations.

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