Technical Report CIS-2008-07

Title: Using Joint Sparsity for Blind Separation of Noisy Multichannel Signals
Authors: Grigory Begelman, Ehud Rivlin and Michael Zibulevsky
Abstract: We call a set of vectors $z[k]\in R^m$ jointly sparse, when for the most of them all $m$ components are simultaneously [close to] zero. When recovering this set from [indirect] noisy observations using variational approach, joint sparsity prior can be expressed via convex penalty term $\sum_k \|z[k]\|_2$. In this work we explore joint sparsity in the context of blind source separation problem $X=AS+\xi$, where mixing matrix $A$ is and sources $S$ are unknown. We suppose $S$ to have sparse representation coefficients $C$ in some given signal frame (dictionary) $\Phi$: $S=C\Phi$. In this case, the mixtures' coefficients $AC$ are jointly sparse, therefore we can recover them robustly without knowledge of mixing matrix $A$, just using joint sparsity prior and solving a convex optimization problem. Another use of joint sparsity comes when the sources themselves are not scalars, but vectors (for example, color images: three RGB color layers usually have spatial similarities, therefore their [wavelet-type] representations are jointly sparse.) Our simulations show efficiency of the presented approach.

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