Research On The Performance Of Compressed Sensing Reconstruction Algorithm Under Restricted Isometry Property

Signal sampling is a necessary means for analog physical world to dig-ital information world.For many years,the theoretical basis of signal sam-pling has been the Nyquist sampling theorem.The theorem states that the sampling rate of a signal must be more than twice its bandwidth to ensure that the original signal can be reconstructed.With the advent of the era of mobile Internet and 5G,great data processing pressure will be generated,and the traditional Nyquist sampling theorem can no longer meet such da-ta processing requirements.In recent years,compressed sensing(CS)has attracted wide attention in the fields of applied mathematics,computer sci-ence and electrical engineering due to its potential to surpass the limitations of traditional sampling theory.Compressed sensing is based on the basic fact that many signals can be represented by several non-zero coefficients in an appropriate basis or dictionary.Nonlinear optimization methods can then recover these signals from very few measurements.In order to recover the accurate original signal from the low-dimensional observation,the compressed sensing reconstruction algorithm is very important.At present,many reconstruction algorithms have been proposed and their performances have been studied.Studies show that the performance of the reconstruction algorithm depends on such parameter-s as the dimension,sparsity degree,observation noise power,observation times and the statistical distribution of non-zero elements of the sparse sig-nal.Although good progress has been made in the research of reconstruc-tion algorithm performance,there are still many problems to be solved.In the standard compressed sensing,it is usually assumed that the observation matrix is known a priori,but various errors and fluctuations in the actual system will lead to the deviation of the observation matrix,so it is neces-sary to study the reconstruction under such disturbance.In addition,how to make good use of signal's sparse structure(such as joint sparse structure,block sparse structure,etc.)to effectively improve the performance of the algorithm is also an important research topic in compressed sensing.In view of the above two points,the main work of this paper focuses on three aspects:first,it studies the influence of perturbation on the recon-struction performance of the algorithm,providing theoretical support for the application of compressive sensing in the case of complete perturba-tion;Secondly,the convergence performance of joint sparse reconstruction algorithm in different scenarios is studied.Finally,signal with special s-parse structure is combined with traditional sparse reconstruction algorith-m to design a reconstruction algorithm with high gain and low complex-ity.The restricted isometry property(RIP)of measurement matrix is an effective tool for analyzing algorithm performance.Therefore,this paper applies finite isometry property to conduct research on the performance of compressed sensing reconstruction algorithm.The main innovative work of this paper is as follows:First,the reconstruction performance of matching pursuit algorithms and threshold algorithms under complete perturbation are studied.Based on the restricted isometry property of the measurement matrix,sufficien-t conditions and reconstruction errors are obtained to ensure the conver-gence of the orthogonal matching pursuit algorithm and subspace pursuit algorithms,and the sufficient condition that ensure the convergence of the orthogonal matching pursuit algorithm is more relaxed than the existing results.Because it is the reconstruction error of the subspace pursuit algo-rithm derived by the method of strict convergence,the upper bound of the error is smaller than the existing upper bound.Similarly,based on the re-stricted isometry property of the measurement matrix,sufficient conditions are obtained to ensure the convergence of the iterative threshold algorith-m and the hard threshold algorithm in the case of complete perturbation,the reconstruction error and the number of iterations required to achieve the reconstruction error accuracy.In this paper,it is pointed out that the performance of the threshold algorithms are affected not only by the con-stant noise of the equidistant constraints of the observation matrix and the perturbation parameters of the matrix,but also by the iteration step ?.Secondly,on the basis of the first innovative work,this paper extends the research work from one-dimensional sparse signal reconstruction al-gorithm to multi-dimensional joint sparse signal reconstruction algorith-m,focusing on the reconstruction performance of synchronous orthogonal matching pursuit algorithm.Firstly,in the ideal undisturbed case,this pa-per improves the upper bound of restricted isometry constant,which guar-antees the convergence of simultaneous orthogonal matching pursuit algo-rithm in finite iterations.Then,based on the upper bound of the constrained constant,the reconstruction performance of the simultaneous orthogonal matching pursuit algorithm in noisy situations is studied,and the sufficient conditions and reconstruction errors to ensure the convergence of the al-gorithm are derived.Finally,based on the convergence condition in the case of noise,the convergence condition and reconstruction error of si-multaneous orthogonal matching pursuit algorithm in the case of complete perturbation are obtained.Thirdly,based on the above two innovations,this paper also studies the block sparse recovery model.Based on the characteristics of block sparsity,this paper proposes block compression sampling matching pur-suit and block normalized iterative threshold pursuit algorithms.Compared with the traditional compression sampling matching pursuit algorithm,the block compression sampling matching pursuit algorithm has stronger per-formance gain and lower computational complexity in reconstructing block sparse signal.When the iteration step is ?<1,the performance of block normalized iteration threshold tracking algorithm is better than that of ex-isting block iteration threshold pursuit algorithm.In addition,based on the block restricted isometry property of the measurement matrix,the suf-ficient conditions for guaranteeing the convergence of block compression sampling matching pursuit and block normalized iteration threshold pursuit algorithms and the number of iterations required for accurate reconstruc-tion of sparse signals are derived respectively,which provides an effective theoretical support for the reconstruction performance of these two algo-rithms.