NMF with Sparse Regularizations in Transformed Domains

Non-negative blind source separation (non-negative BSS), which is also referred to as non-negative matrix factorization (NMF), is a very active field in domains as different as astrophysics, audio processing or biomedical signal processing. In this context, the efficient retrieval of the sources requires the use of signal priors such as sparsity. If NMF has now been well studied with sparse constraints in the direct domain, only very few algorithms can encompass non-negativity together with sparsity in a transformed domain since simultaneously dealing with two priors in two different domains is challenging. In this article, we show how a sparse NMF algorithm coined non-negative generalized morphological component analysis (nGMCA) can be extended to impose non-negativity in the direct domain along with sparsity in a transformed domain, with both analysis and synthesis formulations. To our knowledge, this work presents the first comparison of analysis and synthesis priors —as well as their reweighted versions— in the context of blind source separation. Comparisons with state-of-the-art NMF algorithms on realistic data show the efficiency as well as the robustness of the proposed algorithms.