Algorithms for signal separation exploiting sparse representations, with application to texture image separation

We consider the problem of signal separation, where the observation y = x₁ +x₂ +v is composed of the signals x₁ and x₂ to be separated, along with additive noise v. We further assume that x₁ and x₂ have sparse representations with respect to two different dictionaries D₁ and D₂, respectively. Exploiting sparse representations for signal separation has been introduced recently under the name Morphological Component Analysis (MCA). In this work we consider several greedy MCA algorithms. First, we show that the alternated denoising, practiced extensively in past work on signal separation, is inferior to a far simpler direct approach. Secondly, we introduce the Minimum-Mean-Squared-Error (MMSE) estimator into the separation algorithm, in the form of a randomized average of several representations, and show the benefit it provides. Turning to the task of separating a mixture of texture images, we fuse the above-mentioned direct greedy algorithm as a local operation in a global system, and demonstrate the successful separation it leads to.