## Invariants
## Summary ## Rigid rotation is 3x3 orthonormal matrix. ## 3-D Translation is 3x4 matrix. ## 3-D Translation + Rotation ## Scaled Orthographic Projection: Remove row three and allow scaling. ## Planar Object, remove column 3.
## Planar Invariants
## Perspective Projection ## Problem: perspective is non-linear. ## 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.
## Perspective motion and projection
## Projective Transformations ## Mapping from plane to plane. ## Form a group. - They can be composed
- They have inverses.
- Projective transformations equivalent to set of images of images.
## Planar Projective Invariants ## 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.
## Transform to Canonical Position
## Affine ## Note that this is equivalent to what we did in the affine case. Affine coordinates are coordinates of 4th point after first three are transformed to (0,0), (1,0), (0,1).
## Cross Ratio ## Let p1, p2, p3, p4 be collinear points. ## Let (xj,yj) denote the coordinates of pj. ## Let |xj xk| denote the determinate of a matrix whose first column is xj1, xj2, and whose second column is xk1, xk2. ## Cross(p1,p2,p3,p4) = ## (|x1 x2| |x3 x4|)/(|x1 x3| |x2 x4|) ## This cross-ratio is invariant to projective transformations.
## Lines: Parameterization ## Equation for line: ax+by+c=0. ## Parameterize line as l = (a,b,c)T. ## p=(x,y,1)T is on line if
=0.
## The intersection of l and l’ is l x l’ (where x denotes the cross product). ## This follows from the fact that the cross product is orthogonal to both lines.
## Intersection of Parallel Lines ## Suppose l and l’ are parallel. We can write l=(a,b,c), l’ = (a,b,c’). l x l’ = (c’-c)(b,-a,0). This equivalent to (b,-a,0). ## This point corresponds to a line through the focal point that doesn’t intersect the image plane. ## We can think of the real plane as points (a,b,c) where c isn’t equal to 0. When c = 0, we say these points lie on the ideal line at infinity. ## Note that a projective transformation can map this to another line, the horizon, which we see.
## Invariants of Lines ## Notice that affine transformations are the subgroup of projective transformations in which the last row is (0, 0, 1). ## These map the line at infinity to itself. ## So parallel lines are affine invariants, since they continue to intersect at infinity.
## Invariance in 3D to 2D ## 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.
## Let f be a function on images. f is an invariant iff for every Object O, if I1 and I2 are images of O, f(I1)=f(I2). ## f is a non-trivial invariant if there exist two image I1 and I2 such that f(I1)~=f(I2).
## Non-Invariance in 3D to 2D ## Theorem: Valid objects are any 3D point sets of size k, for some k. There are no non-trivial invariants of the images of these objects under perspective projection.
## Proof Strategy ## Let f be an invariant. ## Suppose two objects, A and B have a common image. Then f(I)=f(J) if I and J are images of either A or B. ## Given any O0, Ok, we construct a series of objects, O1, …, Ok, so that Oi and O(i+1) have a common image for all i, and Ok and j have a common image. ## So for any pair of images, I, J, from any two objects, f(I) = f(J).
## Constructing O1 … Ok-1 ## Oi has its first i points identical to the first i points of Ok, and the remaining points identical to the remaining points of O0. ## 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.
## Invariants of Lighting
## How do we represent light? ## Ideal distant point source: ## - No cast shadows ## - Light distant ## - Three parameters ## - Example: lab with controlled ## light
## Planar Objects ## If n is surface normal of plane, l is light, a(x,y) is albedo at (x,y), intensity i(x,y) is: i(x,y) = a(x,y)n.l ## Then for any two points (x,y), (x’,y’): ## i(x,y)/i(x’,y’)= a(x,y)/a(x’,y’) is invariant to lighting changes.
## Non-Planar Objects ## For any I and J, there is an object, O, and two lights, l1 and l2, such that I is an image of O with light l1, and J is an image of O with light l2. ## Scale images to have intensity less than 1. ## Pick l1 = (1,0,0), l2=(0,1,0). ## Note surface normal n(x,y) is a unit vector. a(x,y)n(x,y) is an arbitrary vector of magnitude less than 1. ## a(x,y)nx(x,y) = I(x,y). (nx is x component of n). ## a(x,y)ny(x,y) = J(x,y).
## Summary ## Planar objects give rise to rich set of invariants. ## 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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