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Vector Valued Image Regularization with PDE’s: A Common Framework for Different Applications. CVPR 2003 Best Student Paper Award. Definitions. Scalar images: Intensity images Vector valued images: RGB, HSV, YIQ… Regularization: Finding approximate solution for ill-posed problems.
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Vector Valued Image Regularization with PDE’s: A Common Framework for Different Applications CVPR 2003 Best Student Paper Award
Definitions • Scalar images: Intensity images • Vector valued images: RGB, HSV, YIQ… • Regularization: Finding approximate solution for ill-posed problems. • Let G be 2x2 matrix
Divergence and Curl vector function Divergence defines expansion or contraction per unit volume Vectors are obtained from image gradients I Curl of a vector field is defined by cross product between vectors
Structure Tensor One dimensional image (gray level)
Structure Tensor Eigenvalue and eigenvectors of G (spectral elements) Can be computed using Matlab functions
y derivative x derivative
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 derivative xx derivative
yy derivative trace xx derivative
Image Regularization • Functional minimization: Euler-Lagrange equations • Divergence expression: Diffusion of pixel values from high to low concentration • Oriented Laplacians: Image smoothing in eigenvector directions weighted by corresponding eigenvalue.
Variational Problem • Define variational problem: Euler-Lagrange equation • Solution using classic iterative method:
Defining Energy Functional • Functional: (minimization based) increasing function • Euler-Lagrange is given by (relation to divergence based methods)
D Matrix is 2x2. Eigenvalues and eigenvectors of D are
Old School Laplacian Approach Second order image derivatives in directions of eigenvectors of G at that point (structure tensor) (Edge preserving smoothing) Solution of this PDE can be given by:
Laplacian Example (constant T) Constant T, such that direction (eigenvectors are same)
Laplacian Example (varying T) T is not constant and but independent of image content. There are similarities with bilateral filtering.
Relation Between T and D Homework Due 19 January 2005 Show the following: (Page 3 of the paper):
Relation Between T and D Let’s just solve for div(G). Rest can be solved similarly:
Relation Between T and D Using trace property: Hessian Kronecker func.
Rewriting the Formula super matrix notation
Numerical Implementation • Conventional approach • Compute image Hessians and Gradients • Proposed method • Use local filtering by Gaussians (2nd page) Similar to bilateral filtering
Numerical Implementation • Compute local convolution mask defining local geometry by T. • Estimate trace(THi) in local neighborhood of x. • Apply filtering for each trace(AijHj) in the vector trace(AH)
