$\ell_1$-K-SVD: A Robust Dictionary Learning Algorithm With Simultaneous Update

TL;DR

The paper introduces $ abla$-K-SVD, a dictionary learning algorithm that minimizes the $ abla$-norm for robustness against non-Gaussian noise, updating dictionary atoms and sparse coefficients simultaneously.

Contribution

It proposes a novel $ abla$-K-SVD algorithm that enhances noise robustness and improves atom recovery in small training sets compared to existing methods.

Findings

Higher atom recovery rate than K-SVD and RDL in Gaussian and non-Gaussian noise.

Outperforms K-SVD and RDL with small training datasets.

Achieves better denoising performance, especially in structural similarity for Laplacian noise.

Abstract

We develop a dictionary learning algorithm by minimizing the $\ell_1$ distortion metric on the data term, which is known to be robust for non-Gaussian noise contamination. The proposed algorithm exploits the idea of iterative minimization of weighted $\ell_2$ error. We refer to this algorithm as $\ell_1$-K-SVD, where the dictionary atoms and the corresponding sparse coefficients are simultaneously updated to minimize the $\ell_1$ objective, resulting in noise-robustness. We demonstrate through experiments that the $\ell_1$-K-SVD algorithm results in higher atom recovery rate compared with the K-SVD and the robust dictionary learning (RDL) algorithm proposed by Lu et al., both in Gaussian and non-Gaussian noise conditions. We also show that, for fixed values of sparsity, number of dictionary atoms, and data-dimension, the $\ell_1$-K-SVD algorithm outperforms the K-SVD and RDL algorithms when the training set available is small. We apply the proposed algorithm for denoising natural images corrupted by additive Gaussian and Laplacian noise. The images denoised using $\ell_1$-K-SVD are observed to have slightly higher peak signal-to-noise ratio (PSNR) over K-SVD for Laplacian noise, but the improvement in structural similarity index (SSIM) is significant (approximately $0.1$) for lower values of input PSNR, indicating the efficacy of the $\ell_1$ metric.