Vector Valued Image Regularization with PDEs A Common

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

Definitions n n Scalar images: Intensity images Vector valued images: RGB, HSV, YIQ… Regularization: Finding approximate solution for ill-posed problems. Let G be 2 x 2 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

Structure Tensor One dimensional image (gray level)

Structure Tensor Eigenvalue and eigenvectors of G (spectral elements) Can be computed using Matlab functions

x derivative y derivative

More Insight (hessian&tensor) n Hessian of image I n Laplacian of I n Tensor: Matrix of matrices. They abuse the notation: Symmetric semi-positive definite matrix

xx derivative yy derivative

xx derivative yy derivative trace

Image Regularization n 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 n Define variational problem: Euler-Lagrange equation n Solution using classic iterative method:

Variational Problem Horn&Schunck Example

Defining Energy Functional n Functional: (minimization based) increasing function n Euler-Lagrange is given by (relation to divergence based methods)

D Matrix is 2 x 2. 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

Laplacian Example (varying T) T is not constant and but independent of image content. There are similarities with bilateral filtering.

Laplacian Example (varying T)

Relation Between T and D Homework Due 19 January 2005 Show the following: (Page 3 of the paper):

Relation Between T and D

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

Unified Expression

Numerical Implementation n Conventional approach n n Compute image Hessians and Gradients Proposed method n Use local filtering by Gaussians (2 nd page) Similar to bilateral filtering

Numerical Implementation 1. 2. 3. Compute local convolution mask defining local geometry by T. Estimate trace(THi) in local neighborhood of x. Apply filtering for each trace(Aij. Hj) in the vector trace(AH)

Comparison Between Two Implementations

Noise Removal and Image impainting

Reconstruction, Visualization Superscale Images