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Lecture-19-h

# Lecture-19-h - Lecture-19 Structure from Motion Problem...

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1 Lecture-19 Structure from Motion Problem • Given optical flow or point correspondences, compute 3-D motion (translation and rotation) and depth.

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2 Main Points • What projection? • How many points? • How many frames? • Single Vs multiple motions. • Mostly non-linear problem . Displacement Model Point Correspondences
3 3-D Rigid Motion (displacement ) Z Y X T Z Y X Z T Z Y X Y T Z Y X X + + + - = ¢ + - + = ¢ + + - = ¢ g b g a b a + - - - = + = ¢ ¢ ¢ Z Y X T T T Z Y X T Z Y X R Z Y X 1 1 1 g b g a b a Orthographic Projection (displacement model) Z Y X T Z Y X Z T Z Y X Y T Z Y X X + + + - = ¢ + - + = ¢ + + - = ¢ g b g a b a Y X T Z y x y T Z y x x + - + = ¢ + + - = ¢ g a b a

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4 Perspective Projection (displacement) Z T y x Z T y x y Z T y x Z T y x x Z Y Z X + + + - + - + = ¢ + + + - + + - = ¢ 1 1 g b g a g b b a Z Y X T Z Y X Z T Z Y X Y T Z Y X X + + + - = ¢ + - + = ¢ + + - = ¢ g b g a b a Instantaneous Velocity Model Optical Flow
5 3-D Rigid Motion + - - - = ¢ ¢ ¢ Z Y X T T T Z Y X Z Y X 1 1 1 g b g a b a + - - - = - ¢ - ¢ - ¢ Z Y X T T T Z Y X Z Z Y Y X X 0 0 0 g b g a b a + + - - - = ¢ ¢ ¢ Z Y X T T T Z Y X Z Y X 1 0 0 0 1 0 0 0 1 0 0 0 g b g a b a + W W - W - W W W - = 3 2 1 1 2 1 3 2 3 0 0 0 V V V Z Y X Z Y X & & & 3 2 1 2 1 3 1 3 2 V X Y Z V Z X Y V Y Z X + W - W = + W - W = + W - W = & & & 3-D Rigid Motion 3 2 1 2 1 3 1 3 2 V X Y Z V Z X Y V Y Z X + W - W = + W - W = + W - W = & & & V X X + W = & Cross Product W W W = W = 3 2 1 , Z Y X X

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6 Orthographic Projection (u,v) is optical flow 2 1 3 1 3 2 3 2 1 2 1
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Lecture-19-h - Lecture-19 Structure from Motion Problem...

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