压缩感知重构问题的凸松弛算法研究

发布时间：2018-05-31 19:31

本文选题：压缩感知 + 重构算法　；参考：《西安电子科技大学》2014年硕士论文

【摘要】：全新的信号处理理论-压缩感知理论是根据信号的稀疏特性或可压缩性质提出来的。它打破了限定采样速率的Nyquist采样定理,摒弃了先进行采样再实施压缩的信号处理模式,使得信号采样过程与压缩过程同时进行。通过求解优化问题就可以重构信号,从而有效的避免了大量采样数据的需求,同时解决了采样所得数据的存储、传输等高成本问题,为实现高效的信号处理带来了巨大的进展。该理论包含三个基本内容:稀疏表示信号、选取观测矩阵以及构造重构算法。其中所选算法的效果及收敛速度直接决定着该理论是否切实可行。因此压缩感知理论的核心内容是设计高效的重构算法。本文是在对压缩感知理论的基本知识及现有重构算法系统学习的前提下,研究了以下几方面的内容:首先,简要阐述了研究压缩感知的背景及意义,目前的研究现状以及典型的应用领域,并详细介绍了压缩感知理论的基础知识。其次,在各种重构算法中,深入研究了几种常用的凸松弛算法,同时研究了几种新颖的非单调线搜索方法,并在此基础上给出了一个改进的非单调线搜索Barzilai-Borwein梯度法。通过大量的仿真实验,发现在达到相同的相对误差时,改进的信号重构算法需要较少的迭代次数,但是其运行时间却比非单调Barzilai-Borwein梯度法有所增加。最后,针对上述算法存在的问题,提出了新非单调线搜索Barzilai-Borwein梯度算法。该算法在充分利用新非单调线搜索方法的收敛特性的基础上,通过目标函数的近似函数来搜寻最优解,从而获得迭代方向的取值,再利用新非单调线搜索方法求得步长。仿真结果表明,新非单调线搜索Barzilai-Borwein梯度算法不仅可以降低算法的运行时间,而且明显减少了算法的迭代次数,从而使得算法的收敛速度大大提高,算法的重构性能大大增强。
[Abstract]:A new signal processing theory, compression sensing theory, is proposed based on the sparse or compressible properties of signals. It breaks the Nyquist sampling theorem which limits the sampling rate and abandons the signal processing mode of sampling first and then compressing so that the signal sampling process and the compression process are carried out simultaneously. By solving the optimization problem, we can reconstruct the signal, thus effectively avoid the need of a large number of sampled data, at the same time, solve the problem of storage and transmission of the sampled data, and bring great progress for the realization of efficient signal processing. The theory includes three basic contents: sparse representation signal, selection of observation matrix and reconstruction algorithm. The effect and convergence speed of the selected algorithm directly determine the feasibility of the theory. Therefore, the core of compressed perception theory is to design efficient reconstruction algorithm. Based on the basic knowledge of compression perception theory and the learning of existing reconstruction algorithms, this paper studies the following aspects: firstly, the background and significance of the research on compression perception are briefly described. The present research situation and typical application field are introduced, and the basic knowledge of compressed perception theory is introduced in detail. Secondly, among all kinds of reconstruction algorithms, several commonly used convex relaxation algorithms are deeply studied, and several novel non-monotone line search methods are studied, and an improved non-monotone line search Barzilai-Borwein gradient method is presented. Through a large number of simulation experiments, it is found that the improved signal reconstruction algorithm needs less iterations when the relative error is the same, but its running time is longer than that of the non-monotone Barzilai-Borwein gradient method. Finally, a new nonmonotone line search Barzilai-Borwein gradient algorithm is proposed to solve the problem. On the basis of making full use of the convergence of the new nonmonotone line search method, the algorithm searches for the optimal solution by the approximate function of the objective function, and then obtains the value of the iteration direction, and then uses the new nonmonotone line search method to obtain the step size. Simulation results show that the new non-monotone line search Barzilai-Borwein gradient algorithm can not only reduce the running time of the algorithm, but also obviously reduce the number of iterations of the algorithm, so that the convergence speed of the algorithm is greatly improved, and the reconstruction performance of the algorithm is greatly enhanced.
【学位授予单位】：西安电子科技大学
【学位级别】：硕士
【学位授予年份】：2014
【分类号】：TN911.7

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