Generalized MCA
for blind source separation
GMCA (Multichannel Morphological Component Analysis) provides a general multichannel extension of MCA.
In the blind source separation framework (BSS), several observations (X) are recorded. Each observation is the linear combination of original sources (S). Formally, the BSS model is such that:
X = AS + N
The way the original sources are mixed is stored in the mixing matrix A. Noise is added (N) to account for instrumental noise/model imperfections. Nonetheless, both the sources (S) and the mixing matrix (A) are unknown. BSS then aims at estimating these parameters.
Concretely, the GMCA model states that the sources are linear combinations of morphological components and emphasizes on sparsity to estimate the sources and the mixing matrix.
A noiseless example :
These two pictures depict two different mixtures of several original sources.
A first example : Cameraman + Lena + noise
The original sources (left). Gaussian noise is added to each mixture (SNR = 20dB). The GMCA estimates (right) were computed from two mixtures (middle).
A second example : Harbors + noise
The original sources (left). Gaussian noise is added to each mixture (SNR = 14dB). The GMCA estimates (right) were computed from two random mixtures (middle).
In those examples, the redundant dictionary we used is composed of the union of Curvelets and DCT.
References :
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❖J.Bobin, J.-L. Starck, J. Fadili, Y.Moudden,"Sparsity and Morphological Diversity in Blind Source Separation", IEEE Transactions on Image Processing, Vol.16, N°11, p. 2662 - 2674, November 2007.
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❖J. Bobin, Y. Moudden, J.-L. Starck, "Enhanced Source Separation by Morphological Component Analysis", IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP 2006), May 2006.