Improvement On Reconstruction Algorithm Of Compressed Sensing Based On RIP Condition With Its Applications

Compressed sensing?CS?is a new sampling theorem after the Nyquist sampling theorem,which has attracted much attention in recent years.Its basic idea is to combine the sampling and compression process together.By a linear transformation under the condition,the original sparse signal is then measured by a sensing matrix to obtain a low-dimensional data,which is convenient for transmission.A minimal optimization model is solved to reconstruct the given original.Compressed sensing is a tool to reconstruct the sparse signals.To obtain a sparse solution,a direct idea is to minimize the number of nonzero elements in the signal.Unfortunately,minimizing the number of nonzero elements is a nonpolynomial problem.To address this problem,various methods are proposed,including under certain conditions,the convex relaxation method has the same solution as that minimization of the number of nonzero elements in the original signal.Moreover,sensing matrix satisfies restrict isometric property?RIP?,then the convex relaxation method can accurately recover the sparse signal.This paper thus focuses on the weighted l₁ optimization model to quickly solve the sparse solution of compressed sensing.For a weighted l₁ optimization model,a key point is how to determine the optimal weight.In general,the determination of the optimal weights is based on prior knowledge,such as the sensing matrix A,or the measurements y,or some prior knowledge about sparse signals.Here,we assume that we do not have any information on sparse signals in advance.So we then determine the support of sparse signals by measuring the relationship between the measurements and the measurement matrix so as to determine the weights in the weighted l₁ optimization model.Specifically,given the RIP condition,we first give the necessary condition for the exact solution of the compressed sensing,trying to give preliminary estimation of sparsity of the original signals.Based on that,we then also give the range of sparsity,that is,the upper and lower bounds of the interval where the sparsity is,and the support set of sparse signals,thereby determining the model parameter related to the following signal reconstruction.The objective of solving the weighted l₁ optimization problem is to solve the exact solution of the squeezed problem,in which the solution of the compression perception is nonzero in the support set and zero in the non-support set.We,therefore give a systematic discussion on the selection of weights of support sets.That is,the weight of the position corresponding to the support set is smaller,rather than supporting the corresponding position on the weight of the relative larger.As a result,we give the strategy of how to determine weights for the set of support and the relationship between the sensing matrix and measurement signal.Regarding to practical image application,we study the signal sparsity after the wavelet conversion with the given four basis functions,and then compare the reconstruction results corresponding to these basis function.We also analyze the results of these imagines with different level noises.It is then conclusive that our theoretical conclusion is valid for the sparse signal or the signal is sparse under some good transformation.We further discuss how to determine the position of supporting sets and assign weights to the effective supporting sets for weighted l₁ optimization model,thereby verifying the performance of signal reconstruction with such weights.Numerical results show that the improved algorithm can give better results.