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CG - N u m e r i c a l S o l u t i o n o f O p t i c a l F...

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Unformatted text preview: N u m e r i c a l S o l u t i o n o f O p t i c a l - F l o w P r o b l e m s Example Example Computing Optical Flow t t t + δ t t + 2 δ ( , ) u v I x y t ( , , ) I x u t y v t t t ( , , ) + + + δ δ δ I x u t y v t t t ( , , ) + + + 2 2 2 δ δ δ = = I x u t y v t t t I x y t I x u t I y v t I t t high order terms I x u t I y v t I t t I x u I y v I t I x I y u v I t ( , , ) ( , , ) ( , ) ( , ) + + + = + + + +-- + + = + + = ⋅ = - δ δ δ ∂ ∂ δ ∂ ∂ δ ∂ ∂ δ ∂ ∂ δ ∂ ∂ δ ∂ ∂ δ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ intensity stays constant intensity stays constant E I x u I y v I t u x u y v x v y dxdy = + + + + + + ∫ ∫ ( ) [( ) ( ) ( ) ( ) ] ∂ ∂ ∂ ∂ ∂ ∂ λ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 2 2 2 2 2 Optimization Formulation x Discretize the governing equation, at ( i,j ): ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ u x u u u y u u v x v v v y v v i j i j i j i j i j i j i j i j =- =- =- =- + + + + 1 1 1 1 , , , , , , , , x Discretized expression : E I x u I y v I t u u u u v v v v i j i j i j i j i j j i i j i j i j i j i j i j i j i j = + + +- +- +- +- ∑ ∑ + + + + ( ) [( ) ( ) ( ) ( ) ] , , , , , , , , , , , , , ∂ ∂ ∂ ∂ ∂ ∂ λ 2 1 2 1 2 1 2 1 2 • At a pixel location ( k,l ): ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ λ E u I x u I y v I t I x u u u u u u u u k l k l k l k l k l k l k l k l k l k l k l k l k l k l k l , , , , , , , , , , , , , , , ( ) [( ) ( ) ( ) ( )] = + +-- +- +- +- =-- + + 2 2 1 1 1 1 ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ λ E v I x u I y v I t I y v v v v v v v v k l k l k l k l k l k l k l k l k l k l k l k l k l k l k l , , , , , , , , , , , , , , , ( ) [( ) ( ) ( ) ( )] = + +-- +- +- +- =-- + + 2 2 1 1 1 1 ∂ ∂ λ E = L Differentiation (k,l) (k,l+1) (k+1,l) (k,l-1) (k-1,l) • Putting it all together: 4 / ) ( 4 / ) ( ) ( 4 ) ( ) ( 4 ) ( 1 , , 1 1 , , 1 1 , , 1 1 , , 1 , , , , 2 , , , , , , , , , , , 2 , + +-- + +-- + + + = + + + = =-- + + =-- + + l k l k l k l k l k l k l k l k l k l k l k l k l k l k l k l k l k l k l k l k l k l k l k l k v v v v v u u u u u v v t I y I v y I u y I x I u u t I x I v y I x I u x I λ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ λ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ Differentiation (cont.) (k,l) (k,l+1) (k+1,l) (k,l-1) (k-1,l) [ ( ) ] [ ( ) ] , , , , , , , , , , , , , , 4 4 4 4 2 2 λ ∂ ∂ ∂ ∂ ∂ ∂ λ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ λ ∂ ∂ λ ∂ ∂ ∂ ∂ + + =- + + =- I x u I x I y v u I x I t I x I y u I y v v I y I t k l k l k l k l k l k l k l k l k l k l k l k l k l k l u u I x u I y v I t I x I y I x v v I x u I y v I t I x I y I y k l k l k l k l k l k l k l k l k l k l k l k l k l k l , , , , , , , , , , , , , , ( ) ( ) ( ) ( ) =- + + + + =- + + + + ∂ ∂ ∂ ∂ ∂ ∂ λ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ λ ∂ ∂ ∂ ∂ ∂ ∂ 4 4 2 2 2 2 u u I x u I y v I t I x I y I x v v I x u I y v I t I x I y I y k l k l k l k l k l k l k l k l k l k l k l k l k l k l , , , , , , , , , ,...
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