Expected Patch Log Likelihood with a Sparse Prior

Jeremias Sulam, Michael Elad

Abstract: Image priors are of great importance in image restoration tasks. These problems can be addressed by decomposing the degraded image into overlapping patches, treating the patches individually and averaging them back together. Recently, the Expected Patch Log Likelihood (EPLL) method has been introduced, arguing that the chosen model should be enforced on the final reconstructed image patches. In the context of a Gaussian Mixture Model (GMM), this idea has been shown to lead to state-of-the-art results in image denoising and debluring. In this paper we combine the EPLL with a sparse-representation prior. Our derivation leads to a close yet extended variant of the popular K-SVD image denoising algorithm, where in order to effectively maximize the EPLL the denoising process should be iterated. This concept lies at the core of the K-SVD formulation, but has not been addressed before due the need to set different denoising thresholds in the successive sparse coding stages. We present a method that intrinsically determines these thresholds in order to improve the image estimate. Our results show a notable improvement over K-SVD in image denoising and inpainting, achieving comparable performance to that of EPLL with GMM in denoising.