A Class Of Signal Reconstruction Algorithms Based On Compressed Sensing

Compressed sensing is a new theory frame of signal acquisition and processing that is based on optimization, operations research and matrix analysis, it broke through bottlenecks of the traditional signal sampling rate, and greatly promoted the combination of mathematical theory and engineering application. Moreover, it also has been a very active field of recent research with a wide range of applications, including nuclear magnetic resonance, medical imaging et al. Compressed sensing theory mainly has three parts, including signal sparse representation, the design of the measurement matrix and signal reconstruction algorithms. Specially, signal reconstruction algorithms that determine the signal reconstruction results are the core of compressed sensing theory. This paper mainly studies the solution of l₀ minimization problem and the related problem, and iterative hard thresholding algorithm in compressed sensing. The upper bound of the number of minimization problem solution is given, and several improved algorithms such as fast iterative hard thresholding algorithm are proposed in this paper. Main research results are listed as follows.?1? Signal reconstruction algorithms in compressed sensing are aimed to solve the l₀ minimization problem, so exploring the number of the solution for this problem has practical research significance. This paper proves that the range of the number of solutions on l₀ minimization problem is min{,[ /2]}[1, ]M NNC, and constructing specific measurement matrix?37? and observation signal b can achieve the upper bound that this bound is the best. Thus, the related l₀ minimization problem is also discussed in this paper, when spark????s+1, the number of solutions has boundary that is [1,C_N^s]. To construct?37? and b can also achieve the upper bound.?2? The fast iterative hard thresholding algorithm based on steepest descent method is proposed. From the numerical experiment we see that the improved algorithm is better than the original algorithm in the signal-noise ratio, reconstruction probability and running time. Meanwhile, the normalized iterative hard thresholding algorithm and Quasi-Newton iterative projection algorithm take the corresponding improvement, and put forward the fast normalized iterative hard thresholding algorithm and fast Quasi-Newton iterative projection, these improved algorithms can reduce the computing time and improve the efficiency of reconstruction.