Learned-MAP-OMP: An unrolled neural network for signal and image denoising

Abstract

Learned Orthogonal Matching Pursuit (L-OMP) has been applied to signal and image denoising tasks. However, under high-noise scenarios, dictionaries generated by L-OMP networks often exhibit high coherence and poor convergence to the true dictionary, as they mimic OMP and fail to select optimal atoms from the learned dictionary. This results in error propagation, degrading L-OMP’s performance in signal denoising tasks. To address this, we propose an unrolled network based on Maximum a posteriori OMP (MAP-OMP), termed Learned-MAP-OMP (L-MAP-OMP). It learns the atoms of the dictionary with the highest MAP likelihood ratios by leveraging the statistical distributions of the measurement matrix, sparse signal, and noise vector. Numerical results demonstrate that dictionaries learned by L-MAP-OMP exhibit improved convergence to the true dictionary, lower coherence, and reduced test Mean Squared Error (MSE) in signal denoising tasks. In particularly, at a noise level of $0.1$, the coherence of the dictionary learned by L-MAP-OMP is $0.29$, while those learned by L-OMP and Learned Iterative Soft Thresholding Algorithm (LISTA) are $0.98$ and $0.48$, respectively. Consequently, we observe that L-MAP-OMP achieves a test MSE of approximately $-27$ dB, outperforming L-OMP and LISTA, which attain test MSE around $-22$ dB and $-20$ dB, respectively. Furthermore, in image denoising tasks, L-MAP-OMP showed statistically significant difference ($p<0.05$) in PSNR and SSIM compared to L-OMP, LISTA, DnCNN, and BM3D. Model selection based on Cohen’s d, mean, and variance further confirmed its superiority$-$outperforming LISTA and BM3D , surpassing L-OMP in several noise scenarios, and remaining competitive with DnCNN.