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# Lecture-8 - Lecture-8 Structure from Motion Problem Given...

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1 Lecture-8 Structure from Motion Problem • Given optical flow or point correspondences, compute 3-D motion (translation and rotation) and shape (depth).

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2 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
3 Perspective Projection (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 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 ¢ x = X - a Y + b Z + T X - b X + g Y + Z + T Z ¢ y = a X + Y - g Z + T Y - b X + g Y + Z + T Z Instantaneous Velocity Model Optical Flow

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4 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
5 Orthographic Projection (u,v) is optical flow 2 1 3 1 3 2 3 2 1 2 1 3 1 3 2 V Z x y v V y Z x u V X Y Z V Z X Y V Y Z X + W - W = = + W - W = = + W - W = + W - W = + W - W = & & & & & X x Y y = = Perspective Projection (arbitrary flow) Z fY y Z fX x = = Z Z y Z Y f Z Z fY Y fZ y v Z Z x Z X f Z Z fX X fZ x u & & & & & & & & & & - = - = = - = - = = 2 2 2 1 2 3 3 1 2 2 2 1 3 3 2 1 ) ( ) ( y f xy f y Z V x Z V f v x f xy f y x Z V Z V f u W - W + - W + W - = W + W - W - - W + = 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 = & & & Z V X Y y Z V Z X f Z Z y Z Y f y v Z V X Y x Z V Y Z f Z Z x Z X f x u 3 2 1 2 1 3 3 2 1 1 3 2 + W - W - + W - W = - = = + W - W - + W - W = - = = & & & & & &

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