Sparse Recovery Model And Applications Based On Analysis Operator

In the past ten years,compressed sensing has been one of the most popular research direc-tion in applied mathematics.As a new sampling theory,it exploits the sparsity of signals and use random sampling to perfectly reconstruct discrete signals by nonlinear reconstruction algorithms.The sampling rate is much less than the traditional Nyquist sampling rate.Once the theory of com-pressed sensing was proposed,it attracted widespread attention from the academic community and industrial community.It has wide applications ranging from information theory,image processing,earth science,optics to pattern recognition and wireless communications,and was rated as the top ten scientific and technological process in 2007 by the American Science and Technology Review.Besides,Donoho in Notices of the AMS has mentioned that the FDA approved two new MRI de-vices,which dramatically speed up important MRI applications using Compressed Sensing theory.The techniques of standard compressed sensing holds for signals which are sparse in standard basis or orthogonal basis.However,in many real applications,the signal may be sparse under a redun-dant dictionary or some transforms.For example,the radar signal is sparse in the Gabor frame,and the natural image is approximately sparse under the differential operator or discrete wavelet frame.Based on l1 analysis model,this dissertation includes several components as follows.Firstly,we introduce the Restricted Eigenvalue condition adapted to frame D(abbreviated as D-RE condition).It is a natural extension to the standard Restricted Eigenvalue condition,and is one of the weakest condition that yields guarantees on the l2-error of the analysis LASSO model(ALASSO)and analysis Dantzig Selector model(ADS).We establish the D-RE condition for several classes of correlated measurement matrices,such as correlated subGaussian random matrices and subsampled basis system.When referring to non-sparse signals,we consider the robust l2 D-nullspace property of correlated Gaussian matrices.Similarly,applying the robust l2 D-nullspace property we get robust l2 error estimations in the ALASSO and the ADS in non-sparse scenario.Secondly,we focus on recovering block signals represented by fusion frame.fusion frame system is a collection of subspaces,and is more flexible in real applications such as distributed processing,parallel processing,and data packet coding.Here we consider the sensing mechanism satisfying a simple incoherence property and an isometry property,including subGaussian random matrices and subsampled partial Fourier matrix.Based on such kind of sensing mechanism,we can faithfully recovery approximately block sparse signals in fusion frame by FLASSO model(4-8)and FDS model(4-9).The number of measurements is significantly reduced by a priori knowledge of incoherence parameter A,which is associated with the angles between the fusion frame subspaces.Thirdly,we find that k-means algorithm is equivalent to l0 minimization problem and its con-vex relaxation can be modeled as l1 optimization problem based on TV analysis operator.However,there are few existing results on provably giving correct identification of all cluster memberships,and the condition is too limited compared with some other methods.In Chapter 5,we propose a weighted sum-of-l1-norm relaxation model.The sufficient condition for exact recovery of clus-ters is better than existing results in literature.In addition,the model can be well applied in both simulated data by the probabilistic model and real data.