Analysis Operator Learning and Its Application to Image Reconstruction

TL;DR

This paper introduces a novel algorithm for learning analysis operators from training images, which improves image reconstruction tasks by optimizing the operator to enhance sparsity and reconstruction quality.

Contribution

The work presents a new analysis operator learning algorithm based on $\,\ell_p$-norm minimization and conjugate gradient on manifolds, advancing the analysis model for image reconstruction.

Findings

Competitive performance in denoising, inpainting, and super-resolution

Effective analysis operator learning improves reconstruction quality

Method outperforms some existing techniques in experiments

Abstract

Exploiting a priori known structural information lies at the core of many image reconstruction methods that can be stated as inverse problems. The synthesis model, which assumes that images can be decomposed into a linear combination of very few atoms of some dictionary, is now a well established tool for the design of image reconstruction algorithms. An interesting alternative is the analysis model, where the signal is multiplied by an analysis operator and the outcome is assumed to be the sparse. This approach has only recently gained increasing interest. The quality of reconstruction methods based on an analysis model severely depends on the right choice of the suitable operator. In this work, we present an algorithm for learning an analysis operator from training images. Our method is based on an $\ell_p$-norm minimization on the set of full rank matrices with normalized columns. We carefully introduce the employed conjugate gradient method on manifolds, and explain the underlying geometry of the constraints. Moreover, we compare our approach to state-of-the-art methods for image denoising, inpainting, and single image super-resolution. Our numerical results show competitive performance of our general approach in all presented applications compared to the specialized state-of-the-art techniques.