ℓ_1-K-SVD: A Robust Dictionary Learning Algorithm With Simultaneous Update
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We develop a dictionary learning algorithm by minimizing the ℓ_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 ℓ_2 error. We refer to this algorithm as ℓ_1-K-SVD, where the dictionary atoms and the corresponding sparse coefficients are simultaneously updated to minimize the ℓ_1 objective, resulting in noise-robustness. We demonstrate through experiments that the ℓ_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 ℓ_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 ℓ_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 ℓ_1 metric.
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