Refinable subspaces of a refinable space

Local refinable finitely generated shift-invariant spaces play a significant role in many areas of approximation theory and geometric design. In this paper we present a new approach to the construction of such spaces. We begin with a refinable function which is supported on . We are interested in spaces generated by a function built from the shifts of .

     

Generalized Strang–Fix condition for scattered data quasi-interpolation

Quasi-interpolation is very useful in the study of the approximation theory and its applications, since the method can yield solutions directly and does not require solving any linear system of equations. However, quasi-interpolation is usually discussed only for gridded data in the literature. In this paper we shall introduce a generalized Strang–Fix condition, which is related to nonstationary quasi-interpolation. Based on the discussion of the generalized Strang–Fix condition we shall generalize our quasi-interpolation scheme for multivariate scattered data, too. AMS subject classification 41A63, 41A25, 65D10Zong Min Wu: Supported by NSFC No. 19971017 and NOYG No. 10125102.     

Frames of Periodic Shift-Invariant Spaces

This note studies Bessel sequences and frames of shift-invariant spaces generated by a countable set of periodic functions. We give characterizations under which the set of translations of the countable set is a Bessel sequence or a frame in terms of spectral decompositions of some self-adjoint operators.     

Periodic Nonuniform Dynamical Sampling in ℓ2(ℤ) and Shift-Invariant Spaces

In this article, we mainly study the periodic nonuniform dynamical sampling in ?²(?) and shift-invariant spaces. We first provide a su?cient and necessary condition for c∈?²(?) which can be reconstructed by its spatial and temporal samples. Then we give a concrete example to show that the su?cient and necessary condition is feasible. Finally, we discuss the periodic nonuniform dynamic sampling problem in shift-invariant spaces.     

Local sampling set conditions in weighted shift-invariant signal spaces

How to represent a continuous signal in terms of a discrete sequence is a fundamental problem in sampling theory. Most of the known results concern global sampling in shift-invariant signal spaces. But in fact, the local reconstruction from local samples is one of the most desirable properties for many applications in signal processing, e.g. for implementing real-time reconstruction numerically. However, the local reconstruction problem has not been given much attention. In this article, we find conditions on a finite sampling set X such that at least in principle a continuous signal on a finite interval is uniquely and stably determined by their sampling value on the finite sampling set X in shift-invariant signal spaces.     

System bandwidth and the existence of generalized shift-invariant frames

We consider the question whether, given a countable family of lattices ${({\mathrm{\Gamma }}_j)}_{j\in J}$ in a locally compact abelian group G, there exist functions ${(g_j)}_{j\in J}$ such that the resulting generalized shift-invariant system ${(g_j(??\gamma ))}_{j\in J,\gamma \in {\mathrm{\Gamma }}_j}$ is a tight frame of $L^2(G)$. This paper develops a new approach to the study of generalized shift-invariant system via almost periodic functions, based on a novel unconditional convergence property. From this theory, we derive characterizing relations for tight and dual frame generators, we introduce the system bandwidth as a measure of the total bandwidth a generalized shift-invariant system can carry, and we show that the so-called Calderón sum is uniformly bounded from below for generalized shift-invariant frames. Without the unconditional convergence property, we show, counter intuitively, that even orthonormal bases can have arbitrary small system bandwidth. Our results show that the question of existence of frame generators for a general lattice system is rather subtle and depends on analytical and algebraic properties of the lattice system.     

Exact iterative reconstruction algorithm for multivariate irregularly sampled functions in spline-like spaces: The -Theory

We prove that the exact reconstruction of a function from its samples on any ``sufficiently dense" sampling set can be obtained, as long as is known to belong to a large class of spline-like spaces in . Moreover, the reconstruction can be implemented using fast algorithms. Since a limiting case is the space of bandlimited functions, our result generalizes the classical Shannon-Whittaker sampling theorem on regular sampling and the Paley-Wiener theorem on non-uniform sampling.

     

基于平稳Contourlet变换的图像去噪方法

多尺度几何分析中的Contourlet变换可以实现灵活的多分辨、多方向图像表示,但是由于不具有平移不变性,在图像去噪中容易产生伪吉布斯现象,本文应用具有平移不变性且能有效表示图像纹理信息的平稳Contourlet变换,提出了软硬阈值结合的去噪法.试验结果表明该方法有效提高去噪声后图像的PSNR,有效保存图像纹理信息以及更好的视觉效果.