The blind source separation problem is to extract the underlying source signals from a set of their linear mixtures, where the mixing matrix is unknown.
Abstract
The blind source separation problem is to extract the underlying source signals from a set of their linear mixtures, where the mixing matrix is unknown. It was discovered recently, that exploiting the sparsity of sources in an appropriate representation according to some signal dictionary, dramatically improves the quality of separation. The sparsity of the sources in a specific representation indicates the usefulness of the representation for purposes of separation. Multi scale transforms, such as wavelet or wavelet packets decompose signals into sets of local features with various degrees of sparsity.
Introduction
Detection of high-energy photons emitted as the result of positron decay is one of the most important low-level stages in PET imaging. A typical detector used in PET is based on the so-called Anger scintillation camera. Incident high-energy gamma quanta, generated due to positron decay, produce scintillation effect in the crystal. As the result, a shower of low energy photons in the visible and UV spectra is emitted. These photons are collected by an array of photo-multipliers (PMTs), optically coupled to the scintillation crystal, and invoke electric impulses in them. The PMT responses are utilized in estimation of the scintillation point coordinates.
In the blind source separation problem an N-channel sensor x(ζ) is generated by M unknown scalar source signals s(ζ), linearly mixed together by an unknown NxM mixing matrix A, and possibly corrupted by additive noise n(ζ):
x(ζ) = As(ζ) + n(ζ),
where the independent variable ζ is either time parameter t or spatial coordinates (x1,x2) in the context of the problems under consideration in this study. The problem at hand is to estimate the mixing matrix A and the M-dimensional source signal s(ζ). Sparse sources can be separated by each one of several techniques, such as BS InfoMax, ICA, histogram, Fuzzy Clustering. The basic idea of the solution is presented in the flow scheme:
After the mixing of independent income signals by using linear matrix, we have got the mixtures to try all algorithms on:
As the first step of the solution, we represent the mixture signals by using a dictionary. The meaning of it is that now our information is represented using fewer coefficients, which give us better sparsity. In this case, scatter plots of these coefficients show distinct orientations each of which specified a column of the mixing matrix:
In order to get better results, we can use different kinds of preprocessing, such as trash-off, transformation of coefficients onto the surface of a unit sphere, choosing “good subsets”. After this step, we can use one of the standard BSS algorithms for restoration of mixing matrix.
BSS Applications:
The famous applications of BSS are Functional MRI and unmixing of hyper spectral data.
The Software
The software was developed and coded following the MatLab design concept and methods. The project was written under the Win2000/NT operating systems.
Conclusion
Experiments with two-dimension simulated and natural signals demonstrate that sparse representations improve the efficiency of blind source separation. Our methods improve the separation quality by utilizing the structure of signals, wherein several subsets of coefficients have significantly better sparsity and separability than others. In this case, scatter plots of these coefficients show distinct orientations each of which specified a column of the mixing matrix. The “good subsets” we choose according to the global distortion as a measure of cluster quality. Finally, we restore the mixing matrix by using of these subsets of coefficients. This yields significantly better experimental results than those obtained by using standard algorithm.
Acknowledgment
We are especially grateful to our project supervisor Pavel Kissilev.
We would also like to express our gratitude to the Ollendorff Minerva Center Fund for supporting this project.