$\ell_0$ Minimization for Wavelet Frame Based Image Restoration

TL;DR

This paper introduces an $ ext{l}_0$ minimization model using wavelet frames for image restoration, which improves the preservation of sharp features and smoothness over existing methods, solved efficiently by the penalty decomposition method.

Contribution

It proposes a novel $ ext{l}_0$ minimization model based on wavelet frames for image restoration, along with an adapted PD algorithm and convergence analysis.

Findings

The model outperforms analysis and balanced approaches in restoring sharp features.

The PD method effectively solves the $ ext{l}_0$ minimization problem.

Numerical results demonstrate improved image quality with better feature preservation.

Abstract

The theory of (tight) wavelet frames has been extensively studied in the past twenty years and they are currently widely used for image restoration and other image processing and analysis problems. The success of wavelet frame based models, including balanced approach and analysis based approach, is due to their capability of sparsely approximating piecewise smooth functions like images. Motivated by the balanced approach and analysis based approach, we shall propose a wavelet frame based $\ell_0$ minimization model, where the $\ell_0$ "norm" of the frame coefficients is penalized. We adapt the penalty decomposition (PD) method to solve the proposed optimization problem. Numerical results showed that the proposed model solved by the PD method can generate images with better quality than those obtained by either analysis based approach or balanced approach in terms of restoring sharp features as well as maintaining smoothness of the recovered images. Some convergence analysis of the PD method will also be provided.