Ph.D Student | Kaftory Ran |
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Subject | Blind Separation of Time/Position Varying Mixtures |

Department | Department of Electrical Engineering |

Supervisor | Professor Emeritus Yehoshua Zeevi |

Full Thesis text |

We address the fascinating open engineering problem of blindly separating time/position varying mixtures, and attempt to separate the sources from such mixtures without having prior information about the sources or the mixing system. Unlike studies concerning instantaneous or convolutive mixtures, we assume that the mixing system (medium) is changing in time/position.

Attempts to solve this problem have utilized, so far, online algorithms based on tracking the mixing system by methods previously developed for the instantaneous or convolutive mixtures. In contrast with these attempts, we develop a batch algorithm in the form of Staged Sparse Component Analysis (SSCA). Accordingly, we assume that the sources are either sparse or can be 'sparsified'. In the first stage we estimate the mixing system filters, based on the scatter plot of the sparse mixtures' data using a proper grouping and curve/surface fitting. In the second stage, the mixing system is inverted yielding the estimated sources.

We study the structure of a time/position varying system and find that it constitutes a non-commutative ring. This implies that, in general, an inverse filter does not exist and filters do not commute. We use Zadeh's incompatible transform in order to find conditions for a filter to be commutative and an inverse filter to exist.

Applying these results, we study the conditions which enable the application of the SSCA. It is shown that 'sparsification' can be accomplished only by a transformation which is invariant to the mixing system. Sparse signal sources can be used for estimating the mixing system as long as a proper threshold is chosen and the above-threshold instances are frequent enough. It is shown that estimation of the mixing system is feasible only if some of the filters can be inverted, and that the inverse of the matrix exists only if the filters commute and the inverse filter of the determinant of the matrix exists. We study the conditions where the inverse of the mixing matrix is ill-conditioned and therefore amplifies the noise. Finally we prove that the sources can estimated up to some unknown time/position varying filters applied to the sources.

The SSCA approach is used for solving three types of mixtures: time/position varying instantaneous mixtures, single-path mixtures and multi-path mixtures. Methods for the 'sparsification' of the mixtures, estimation of the mixing system and separation of the sources are developed. Real image mixtures and simulated mixtures are used to test our approach yielding good results.