Blind Source Separation Based on Sparsity

TL;DR

This work surveys sparsity-based approaches to blind source separation (BSS), contrasting them with classical ICA and highlighting how sparse representations and morphological diversity enable separation in underdetermined and noisy scenarios. It reviews MCA, MMCA, GMCA, and fast variants, and discusses adaptive dictionary learning via KSVD and its integration with MMCA (K-SVD+MMCA). The authors propose a Block Sparse KSVD framework (SAC+BK-SVD) to learn block-structured dictionaries during BSS and demonstrate via image separation experiments that adaptive dictionaries can improve separation quality and robustness, though at a higher computational cost. The work suggests practical implications for image and multichannel data, and outlines future directions such as faster dictionary learning (SimCo), single-dictionary strategies, and extensions to underdetermined BSS.

Abstract

Blind source separation (BSS) is a key technique in array processing and data analysis, aiming to recover unknown sources from observed mixtures without knowledge of the mixing matrix. Classical independent component analysis (ICA) methods rely on the assumption that sources are mutually independent. To address limitations of ICA, sparsity-based methods have been introduced, which decompose source signals sparsely in a predefined dictionary. Morphological Component Analysis (MCA), based on sparse representation theory, assumes that a signal is a linear combination of components with distinct geometries, each sparsely representable in one dictionary and not in others. This approach has recently been applied to BSS with promising results. This report reviews key approaches derived from classical ICA and explores sparsity-based methods for BSS. It introduces the theory of sparse representation and decomposition, followed by a block coordinate relaxation MCA algorithm, whose variants are used in Multichannel MCA (MMCA) and Generalized MCA (GMCA). A local dictionary learning method using K-SVD is then presented. Finally, we propose an improved algorithm, SAC+BK-SVD, which enhances K-SVD by learning a block-sparsifying dictionary that clusters and updates similar atoms in blocks. The implementation includes experiments on image segmentation and blind image source separation using the discussed techniques. We also compare the proposed block-sparse dictionary learning algorithm with K-SVD. Simulation results demonstrate that our method yields improved blind image separation quality.

Figures (20)

• Figure 1: Instantaneous linear mixture model (m=n)
• Figure 2: Barbara image with $80\%$ missing pixels (right). The result of the MCA impainting is given on the left.
• Figure 3: Illustration of sparse representation using overcomplete dictionary
• Figure 4: Illustration of the image decomposition(top) using MCA and the blind source separation (bottom) using MMCA. For the bottom part, each source is itself a mixture of morphological components(texture and cartoon)
• Figure 5: Two equivalent representations with different block structure.