# UMD CMSC 828 - Invariants (concluded); Lowe and Biederman (28 pages)

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## Invariants (concluded); Lowe and Biederman

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- Pages:
- 28
- School:
- University of Maryland, College Park
- Course:
- Cmsc 828 - Advanced Topics in Information Processing

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Invariants concluded Lowe and Biederman Announcements No class Thursday Attend Rao lecture Double check your paper assignments Key Points Rigid rotation is 3x3 orthonormal matrix 3 D Translation is 3x4 matrix 3 D Translation Rotation is 3x4 matrix Scaled Orthographic Projection Remove row three and allow scaling Planar Object remove column 3 Projective Transformations Rigid Rotation of Planar Object Represented by 3x3 matrix When we write in homogeneous coordinates projection implicit When we drop rigidity 3x3 matrix is arbitrary Projective x u a b c x u r r r t r r t x 1 w y r r r t r r t y d e f y1 v v w r r r t 0 r r t 1 g h 1 1 w 1 1 Rigid rotation and translation Projective Transformation 1 1 1 2 1 3 x 1 1 1 2 x 2 1 2 2 2 3 y 2 1 2 2 y 3 1 3 2 3 3 z 3 1 3 2 z Notation suggests that first two columns are orthonormal and transformation has 6 degrees of freedom Notation suggests that transformation is unconstrained linear transformation Points in homogenous coordinates are equivalent Transformation has 8 degrees of freedom because its scale is arbitrary 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 p l 0 Line Intersection 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

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