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cap6412 regularization with pdes yilmaz

# cap6412 regularization with pdes yilmaz - Vector Valued...

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Vector Valued Image Regularization with PDE’s: A Common Framework for Different Applications CVPR 2003 Best Student Paper Award

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Definitions Scalar images: Intensity images Vector valued images: RGB, HSV, YIQ… Regularization: Finding approximate solution for ill-posed problems. Let G be 2x2 matrix = d c b a G = = 2 1 , ) ( i i i g trace G
Divergence and Curl Vectors are obtained from image  gradients  I y A x A A A div y x + = = ) ( Divergence defines expansion or  contraction per unit volume ) , ( y x A vector function yy xx y x I I I I div I div I + = = = ) ( Curl of a vector field is defined by cross product between vectors

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Structure Tensor [ ] = = = = n i y x y x n i T i i I I I I I I 1 1 G
Structure Tensor [ ] = = = = n i y x y x n i T i i I I I I I I 1 1 G One dimensional image (gray level) = y y y x y x x x gray I I I I I I I I G ) 1 , ( ) , ( ) , 1 ( ) , ( - - = - - = y x I y x I I y x I y x I I y x

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Structure Tensor + + = + + = y y y x y x x x y y y x y x x x y y y x y x x x color T T T color B B B B B B B B G G G G G G G G R R R R R R R R B B G G R R G G r eigenvecto eigenvalue color = + + + + + 2 1 2 1 φ φ λ φ φ G Eigenvalue and eigenvectors of  G (spectral elements) Can be computed using Matlab functions
x derivative y derivative

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More Insight (hessian&tensor) Hessian of image I Laplacian of I Tensor: Matrix of matrices. They abuse the notation: Symmetric semi-positive definite matrix = yy yx xy xx I I I I H yy xx I I trace I I + = = = ) ( ) div( H
xx derivative yy derivative

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trace xx derivative yy derivative
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cap6412 regularization with pdes yilmaz - Vector Valued...

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