$\ell_1$-K-SVD: A Robust Dictionary Learning Algorithm With Simultaneous Update
This work addresses robust dictionary learning for signal processing applications like image denoising, but it is incremental as it builds on existing K-SVD methods with a modified objective.
The paper tackles the problem of dictionary learning under non-Gaussian noise by proposing $\ell_1$-K-SVD, a robust algorithm that simultaneously updates dictionary atoms and sparse coefficients. The result shows higher atom recovery rates compared to K-SVD and RDL, with significant improvements in structural similarity (e.g., approximately 0.1 in SSIM) for image denoising under Laplacian noise.
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.