# Lecture10 - EECS 442 Computer vision Multiple view geometry Affine structure from Motion Affine structure from motion problem Algebraic methods

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EECS 442 – Computer vision Multiple view geometry Affine structure from Motion Reading: [HZ] Chapters: 6,14,18 [FP] Chapter: 12 Some slides of this lectures are courtesy of prof. J. Ponce, prof FF Li, prof S. Lazebnik & prof. M. Hebert - Affine structure from motion problem - Algebraic methods - Factorization methods

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Structure from motion problem x 1 j x 2 j x 3 j X j M 1 M 2 M 3 Given m images of n fixed 3D points x ij = M i X j , i = 1 , … , m, j = 1 , … , n
From the m x n correspondences x ij , estimate: •m projection matrices M i •n 3D points X j x 1 j x 2 j x 3 j X j motion structure M 1 M 2 M 3 Structure from motion problem

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Affine structure from motion (simpler problem) Image World Image From the m x n correspondences x ij , estimate: •m projection matrices M i (affine cameras) •n 3D points X j
Finite cameras p q r R Q O P [ ] X T R K x = = 1 0 0 y 0 x s K o y o x α = 1 0 T R 0 1 0 0 0 0 1 0 0 0 0 1 K M 3 x 3 M Perspective projection matrix Homography (in 3D) Homography (in 2D)

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Affine cameras [ ] X T R K x = = 1 0 0 0 0 0 s K y x α = 1 0 T R 1 0 0 0 0 0 1 0 0 0 0 1 K M = 1 0 T R 0 1 0 0 0 0 1 0 0 0 0 1 K M = 1 0 0 y 0 x s K o y o x Projective case Affine case Parallel projection matrix (points at infinity are mapped as points at infinity)
Orthographic Projection Weak-Perspective Projection = 1 0 0 0 1 0 0 0 1 K = 1 0 0 0 0 0 0 K y x α Scaling function of the distance

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Orthographic Projection Weak-Perspective Projection = 1 0 0 0 1 0 0 0 1 K = 1 0 0 0 0 0 0 K y x α
Affine cameras [] X T R K x = = 1 0 0 0 0 0 s K y x α = 1 0 T R 1 0 0 0 0 0 1 0 0 0 0 1 K M = = × × = 1

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## This note was uploaded on 10/26/2010 for the course EECS 442 taught by Professor Savarese during the Fall '09 term at University of Michigan.

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Lecture10 - EECS 442 Computer vision Multiple view geometry Affine structure from Motion Affine structure from motion problem Algebraic methods

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