Image Denoising via Correlation Based Sparse Representation and Dictionary Learning

Error-based Orthogonal Matching Pursuit (OMPe) employed in many image denoising algorithms (e.g., K-means Singular Value Decomposition (K-SVD) algorithm) tries to reconstruct the clean image patch by projecting the observed noisy patch onto a dictionary and picking the atom with maximum orthogonal projection. This approach does indeed minimize the power in the residual. However minimizing the power in the residual does not guarantee that selected atoms will match the clean image patch. This leaves behind a residual that contains structures from the clean image patch. This problem becomes more pronounced at high noise levels. Firstly, we develop a simple method to prove that autocorrelation of residual does not match that of the contaminating noise. Then we propose a correlation-based sparse coding algorithm that is better able to pick the atom that matches the clean patch. This is achieved by picking atoms that force the residual patch to have autocorrelation similar to the autocorrelation of contaminating noise. Autocorrelation-based sparse coding and dictionary update stages are iterated and dictionaries are learned from noisy image patches. Also, a new residual correlation based regularization for image denoising is developed. The regularization can effectively render residual patches as uncorrelated as possible. It allows us to derive analytical solution for sparse coding (atom selection and coefficient calculation). It also leads to a new online dictionary learning update. The clean image is obtained by alternating between the two stages of sparse coding and dictionary updating. Experimental results of peak signal-to noise ratio (PSNR) and structural similarity index (SSIM) show that the proposed algorithm can significantly outperform the K-SVD denoising algorithm, especially at high noise levels.The proposed algorithm is compared with the K-SVD denoising algorithm, BM3D, NCSR and EPLL algorithms. Our results indicate that the proposed algorithm is better than K-SVD and EPLL denoising. The proposed algorithm gives visual results that are comparable or better than BM3D and NCSR algorithms. Keywords: Correlation regularization, dictionary learning, image denoising, residual correlation, sparse representation.