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信号重构作为压缩感知理论的核心之一,是指从长度为m的测量向量Y重构长度为n(m n)的稀疏信号Θ的过程。由于测量次数远小于原始信号维度,信号重构成为欠定方程求解问题,一般没有确定解。然而,若Θ满足一定的稀疏性条件,问题有确定解。文章首先从解析几何角度出发,分析了压缩感知中稀疏信号重构的原理,并对已有的两大类重构算法分别进行介绍:一类是针对l0范数最小化的一系列贪婪算法,一类是针对l1范数最小化的凸优化算法。对前一类算法,选取了代表性的OMP、ROMP、CoSaMP和SAMP算法进行研究,并分析了它们的优缺点;对后一类算法,着重阐述了将BP问题转换为LP问题的推导过程,并介绍了两类经典的凸优化算法:BP-Simplex和BP-Interior。最后,展望了信号重构算法的研究前景。
As one of the core of compression sensing theory, signal reconstruction refers to the process of reconstructing the sparse signal Θ of length n (m n) from the measurement vector Y of length m. Since the number of measurements is much smaller than the original signal dimension, the signal reconstruction becomes an undetermined equation solving problem, and generally no solution is determined. However, if Θ satisfies a certain sparsity condition, the problem has a definite solution. First of all, from the perspective of analytic geometry, the principle of sparse signal reconstruction in compressed sensing is analyzed, and two existing refactoring algorithms are introduced separately. One is a series of greedy algorithms aimed at the minimization of l0 norm, One is the convex optimization algorithm for l1 norm minimization. For the former algorithm, the representative OMP, ROMP, CoSaMP and SAMP algorithms were selected and analyzed, and their advantages and disadvantages were analyzed. For the latter algorithm, the derivation process of converting the BP problem into the LP problem was emphatically described. Two kinds of classical convex optimization algorithms are introduced: BP-Simplex and BP-Interior. Finally, the prospect of signal reconstruction algorithm is prospected.