Compressive Sensing and Combinatorial algorithms for image compression

Abstract

The initial motivation of this Masters Thesis is the design and analysis of an image compression method based on Compressive Sensing. Compressigve Sensing is a technique which allows coding sparse signals by projecting the signal onto random vectors. When signals are sparse, it is possible to encode the signal with a much smaller number of measurements than the length of the original signal. Two methods based on Compressive Sensing are proposed. Both of them initially apply a wavelet transform to obtain the signal in a convenient domain in which is supposed to be sparse. At the same time, in the wavelet transform domain some sub-blocks are generated, which are useful in order not to process the whole image, but sub-block per sub-block. The first method uses compressive sensing onto binary signals and the second one onto integers. By studying and testing the proposed methods, a new one not based on compressive sensing emerged which provided significant improvements. It is a combinatorial method which orders uniquely the possible combinations of a binary vector of length N with S nonzero coefficients. It is supported by a fast algorithm to cycle through the combinations. Results provided in this report allow having an idea of the efficiency and advantages and disadvantages of each one of the methods proposed.