Blind source separation based on simultaneous learning of the sparse frame representations for multi sources from their mixtures

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Just as that any sentence can be constructed by several words in a dictionary, any signal or image can be either represented by several "words" in a "dictionary". Comparing with the large number of words in the dictionary, a sentence is usually be constructed by only very few words, so that these words are mapped into the dictionary sparsely. Then constructed sentence may be called as sparse coding with the dictionary. Similarly, with a dictionary for signal or image, one can represent any signal or image, and this can also be termed as sparse coding. Usually, there more words in the dictionary than the length of the words, i.e., the dictionary is over-complete, the dictionary has a structure of frame (a mathematical concept). The motivation of this research is to find more effective methods for finding this sparse representation, and then apply them to blind source separation (BSS).BSS by using the sparse representation usually includes repeated two steps, once it is given an arbitrar … More y initialization of the estimated dictionary. In the first step, for a given dictionary the sparse representation by the dictionary is conducted. In the second step, the dictionary is learned in part and the corresponding sparse representation is modified, while the other parts of sparse representation are kept. In these two steps, the dictionary learning is more important and there still very few effective methods for it. For this purpose, we worked out a method that is termed as adaptive non-orthogonal sparsifying transform. In this method, we take the multiplication of the frame and a sparse matrix as the source signal estimation. Though there may be many possible solutions, we choose the sparsest one as our result. As a feature of the method, the words in the dictionary are ordered by their energy, rather than randomly ordered as in the usual dictionary.In the above method, the dictionary learning and source estimation may converge very slowly and the computation is also very consuming. For solving these problems, we also worked out a method in which the dictionary and the sources are simultaneously estimated, by invoking the nonnegative matrix factorization (NMF). There are many nonnegative signals, such as image, in applications. However, usually, the result of NMF is not unique and not all of them are sparse. For solving this problem, we worked out a sparse NMF method, in which we select a sparse solution from the non unique solutions by a constraint. We propose a measure for measuring the sparsity of source signals and use its minimization as the constraint. As an alternative method, we also proposed to use a constraint on the dictionary, rather than on the sources, since the sources are usually very long and a constraint on them will be computation consuming. We found that the maximization of space spanned by the words in the dictionary is a good constraint.Evaluations showed that our methods are efficient. Then we applied them to blind spectral unmixing, BSS or denoising of images, beamforming and direction of arrival estimation. Less