ℓ1-K-SVD: A robust dictionary learning algorithm with simultaneous update

• We propose an algorithm, which we refer to as ℓ1-K-SVD, to learn data-adaptive dictionaries in the presence of non-Gaussian noise. The fundamental idea behind the algorithm is to replace the usual ℓ2-norm-based data-fidelity metric with ℓ2-norm, and minimize it using iteratively reweighted least-squares (IRLS).
• In the dictionary update stage of ℓ1-K-SVD, we adopt a simultaneous updating strategy similar to K-SVD, that is found to result in faster convergence.
• We elucidate how the proposed idea can be extended to minimize the ℓp data error, where 0• We demonstrate experimentally that the ℓ1-K-SVD algorithm results in faster convergence and more accurate atom detection performance compared with the state-of-the-art algorithms. It is also shown that ℓ1-K-SVD is more suitable than the competing algorithms, when the training dataset contains fewer examples.
• As an application, we deploy the algorithm for image denoising. It is found that ℓ1-K-SVD results in peak signal-to-noise ratio (PSNR) values that are on par with the K-SVD algorithm, but the improvement in structural similarity index (SSIM) over K-SVD is approximately 0:08–0:10, indicating its efficacy in preserving the structural content of images.

We develop a new dictionary learning algorithm called the ℓ1-K-SVD, by minimizing the ℓ1 distortion on the data term. The proposed formulation corresponds to maximum a posteriori estimation assuming a Laplacian prior on the coefficient matrix and additive noise, and is, in general, robust to non-Gaussian noise. The ℓ1 distortion is minimized by employing the iteratively reweighted least-squares algorithm. The dictionary atoms and the corresponding sparse coefficients are simultaneously estimated in the dictionary update step. Experimental results show that ℓ1-K-SVD results in noise-robustness, faster convergence, and higher atom recovery rate than the method of optimal directions, K-SVD, and the robust dictionary learning algorithm (RDL), in Gaussian as well as non-Gaussian noise. For a fixed value of sparsity, number of dictionary atoms, and data dimension, ℓ1-K-SVD outperforms K-SVD and RDL on small training sets. We also consider the generalized ℓp,0