# A Greedy Approach to $\ell_{0,\infty}$ Based Convolutional Sparse Coding

## TL;DR

This paper introduces a fast greedy algorithm for convolutional sparse coding using the $oldsymbol{ ext{ell}_{0, ext{infty}}}$ norm, improving image processing tasks like noise removal and inpainting by maintaining local sparsity.

## Contribution

It develops an efficient greedy matching pursuit method tailored to the $oldsymbol{ ext{ell}_{0, ext{infty}}}$ norm for convolutional sparse coding, addressing non-convexity without relaxation.

## Key findings

- Effective salt-and-pepper noise removal
- Successful image inpainting results
- Code implementation available online

## Abstract

Sparse coding techniques for image processing traditionally rely on a processing of small overlapping patches separately followed by averaging. This has the disadvantage that the reconstructed image no longer obeys the sparsity prior used in the processing. For this purpose convolutional sparse coding has been introduced, where a shift-invariant dictionary is used and the sparsity of the recovered image is maintained. Most such strategies target the $\ell_0$ "norm" or the $\ell_1$ norm of the whole image, which may create an imbalanced sparsity across various regions in the image. In order to face this challenge, the $\ell_{0,\infty}$ "norm" has been proposed as an alternative that "operates locally while thinking globally". The approaches taken for tackling the non-convexity of these optimization problems have been either using a convex relaxation or local pursuit algorithms. In this paper, we present an efficient greedy method for sparse coding and dictionary learning, which is specifically tailored to $\ell_{0,\infty}$, and is based on matching pursuit. We demonstrate the usage of our approach in salt-and-pepper noise removal and image inpainting. A code package which reproduces the experiments presented in this work is available at https://web.eng.tau.ac.il/~raja