Reconstruction Method For Sparse Signals Based On Compressed Sensing

The traditional signal measurement and processing are consisted of sampling，compressing, transmission and decompressing. According to the Nyquist sampling theorem:the sampling frequency of signal is not less than two times of its own highest frequency. So,this way of sampling and then compressing will waste a lot of time，sensors, and memory ofdata storage. Aiming at the sparse or compressible signal,Compressed sensing （CS）is a newtheory which can compress the signal during the process of sampling. The advantage of thisapproach is that the measurement data of the signal is far less than traditional sampling. CS isnot constainted by the sampling theorem,so it is possible that we can acquire the highresolution signal. At present,the research for CS is mainly concentrated on the sensor matrixstructure and the optimization of the reconstruction algorithm. Basing on compressedSensing，this Master’s thesis stduies systematacially the theories and numerical methods ofthe leastl₂norm minimization problem with quadratic inequality constraint，the leastl₁normminimization problem with linear equality constraint and the leastl₀norm minimizationproblem with linear equality constraint,the main contributions are summarized as follows.1.Basing on the singular value decomposition of matrix, Lagrange multiplierformulation, the property of orthogonal matrix andl₂norm, the unique of solution for theleastl₂norm minimization problem with quadratic inequality constraint is proved,and thesufficient and necessary condition for the existence of solution is drived. Furthermore,we givethe numerical algorithms and examples to solve it.2.The leastl₁norm minimization problem with linear equality constraint can betransformed into a linear programming equivalently is proved,and the structure of the optimalsolution is derived. For nonsmooth inl₁norm, smooth function is constructed,discreteoptimal solution sequences are used to approximating the global optimal solution.Theproperty of smooth approximation function and the convergence of optimal solutionsequences guarantee the algorithm is feasible. The numerical results show that smoothapproximation is an effective technique.3.Basing on the projection theorem in the finite-dimensional inner product space andM-P inverse theorem,we stduy the numerical method of the leastl₀norm minimizationproblem with linear equality constraint. We also analyze the core thought of many greedyiterative algorithms,such as matching pursuit, orthogonal matching pursuit, subspace pursuit,regularized orthogonal matching pursuit and sparse adaptive matching pursuit. Finally, wecompare the advantages and disadvantages of this algorithm.