As a special shift-invariant spaces,spline subspaces yield many advantages so that there are many practical applications for signal or image processing.In this paper,we pay attention to the sampling and reconstruction problem in spline subspaces.We improve lower bound of sampling set conditions in spline subspaces.Based on the improved explicit lower bound,a improved explicit convergence ratio of reconstruction algorithm is obtained.The improved convergence ratio occupies faster convergence rate than old one.At the end,some numerical examples are shown to validate our results.
In this paper, we shall study the solutions of functional equations of the form Φ =∑α∈Zsa(α)Φ(M·-α), where Φ = (φ1, . . . , φr)T is an r×1 column vector of functions on the s-dimensional Euclidean space, a:=(a(α))α∈Zs is an exponentially decaying sequence of r×r complex matrices called refinement mask and M is an s×s integer matrix such that limn→∞M-n=0. We are interested in the question, for a mask a with exponential decay, if there exists a solution Φ to the functional equation with each function φj,j=1, . . . ,r, belonging to L2(Rs) and having exponential decay in some sense? Our approach will be to consider the convergence of vector cascade algorithms in weighted L2 spaces. The vector cascade operator Qa,M associated with mask a and matrix M is defined by Qa,Mf:=∑α∈Zsa(α)f (M·-α),f= (f1, . . . , fr)T∈(L2,μ(Rs))r.The iterative scheme (Qan,Mf)n=1,2,... is called a vector cascade algorithm or a vector subdivision scheme. The purpose of this paper is to provide some conditions for the vector cascade algorithm to converge in (L2,μ(Rs))r , the weighted L2 space. Inspired by some ideas in [Jia,R.Q.,Li,S.: Refinable functions with exponential decay: An approach via cascade algorithms. J. Fourier Anal. Appl., 17, 1008-1034 (2011)], we prove that if the vector cascade algorithm associated with a and M converges in (L2(Rs))r, then its limit function belongs to (L2,μ(Rs))r for some μ>0.
This paper discusses conditions under which the solution of linear system with minimal Schatten-p norm, 0 < p 1, is also the lowest-rank solution of this linear system. To study this problem, an important tool is the restricted isometry constant (RIC). Some papers provided the upper bounds of RIC to guarantee that the nuclear-norm minimization stably recovers a low-rank matrix. For example, Fazel improved the upper bounds to δA4r < 0.558 and δA3r < 0.4721, respectively. Recently, the upper bounds of RIC can be improved to δA2r < 0.307. In fact, by using some methods, the upper bounds of RIC can be improved to δA2r < 0.4931 and δ A r < 0.309. In this paper, we focus on the lower bounds of RIC, we show that there exists linear maps A with δA2r > 1/ 2^(1/2) or δAr > 1/3 for which nuclear norm recovery fail on some matrix with rank at most r. These results indicate that there is only a little limited room for improving the upper bounds for δA2r and δAr . Furthermore, we also discuss the upper bound of restricted isometry constant associated with linear maps A for Schatten p (0 < p < 1) quasi norm minimization problem.
The matrix rank minimization problem arises in many engineering applications. As this problem is NP-hard, a nonconvex relaxation of matrix rank minimization, called the Schatten-p quasi-norm minimization(0 < p < 1), has been developed to approximate the rank function closely. We study the performance of projected gradient descent algorithm for solving the Schatten-p quasi-norm minimization(0 < p < 1) problem.Based on the matrix restricted isometry property(M-RIP), we give the convergence guarantee and error bound for this algorithm and show that the algorithm is robust to noise with an exponential convergence rate.
This paper establishes new bounds on the restricted isometry constants with coherent tight frames in compressed sensing. It is shown that if the sensing matrix A satisfies the D-RIP condition δk< 1/3 or δ2k<2^(1/2)/2, then all signals f with D*f are k-sparse can be recovered exactly via the constrained 1 minimization based on y = Af, where D*is the conjugate transpose of a tight frame D. These bounds are sharp when D is an identity matrix, see Cai and Zhang's work. These bounds are greatly improved comparing to the condition δk< 0.307 or δ2k< 0.4931. Besides, if δk< 1/3 or δ2k<2^(1/2)/2, the signals can also be stably reconstructed in the noisy cases.
In this paper,we consider data separation problem,where the original signal is composed of two distinct subcomponents,via dual frames based Split-analysis approach.We show that the two distinct subcomponents,which are sparse in two diferent general frames respectively,can be exactly recovered with high probability,when the measurement matrix is a Weibull random matrix(not Gaussian)and the two frames satisfy a mutual coherence property.Our result may be signifcant for analysing Split-analysis model for data separation.
Low rank matrix recovery is a new topic drawing the attention of many researchers which addresses the problem of recovering an unknown low rank matrix from few linear measurements. The matrix Dantzig selector and the matrix Lasso are two important algorithms based on nuclear norm minimization. In this paper,we first prove some decay properties of restricted isometry constants, then we discuss the recovery errors of these two algorithms and give a new bound of restricted isometry constant to guarantee stable recovery, which improves the results of [11].
