On Optimization Of The Measurement Matrix For Compressed Sensing Systems

Compressed sensing (CS), is a great paradigm that goes against the traditional wisdom,the so-called Nyquist Sampling, in the signal processing field. It was proposed in2006byCande`s、Tao、Romberg and Donoho. By exploring the compressibility of signals, CS em-ploys non-adaptive linear projections that nearly preserve all the information of the signals,and then such signals can be exactly reconstructed under some certain conditions. CS not onlyofers a new way to lower the sample frequency, but also stimulates the work in other areas.CS still has many unsolved problems which need further study, such as the issues of develop-ing signal reconstruction algorithms and optimizing measurement matrices. There are manyresults available related with the former one, while less with the later.This thesis introduces the theoretical framework of CS, analyze two algorithms on opti-mizing the measurement matrices, and propose a new algorithm. The main contributions aregiven as follows:1. Introduce the theory of CS in terms of the sparse representation of signals, measure-ment matrices, signal reconstruction algorithms. The key-points of measurement matrices,such as the principles, the common matrices and the coherence of optimization and analyzed.2. Analyze two methods for optimizing the measurement matrices. In the first one,Michael Elad aims to minimize the mutual coherence of sensing matrix, and the second one,Vo Dinh Minh Nhat aims to let the measurement matrix have the structure as much similar asthe original dictionary.3. Propose a new method. The theories of tight frame and equiangular tight frame areintroduced. Meanwhile, the methods mentioned above have been fully tested and evaluated insome terms of compression ratio, the sparsity of signals and the reconstruction error. Numeri-cal experiments have shown our proposed methods yields a better performance.4. Based on the work mentioned above, analyze the relation between mutual coherenceand the optimization of the measurement matrix, and thus prepare for further work.