Local Sampling and Reconstruction in Shift-Invariant Spaces and Their Applications in Spline Subspaces

The local reconstruction from samples is one of the most desirable properties for many applications in signal processing. Local sampling is practically useful since we need only to consider a signal on a bounded interval and computer can only process finite samples. However, the local sampling and reconstruction problem has not been given as much attention. Most of known results concern global sampling and reconstruction. There are only a few results about local sampling and reconstruction in spline subspaces. In this article, we study local sampling and reconstruction in general shift-invariant spaces and multiple generated shift-invariant spaces with compactly supported generators. Then we give several applications in spline subspaces and multiple generated spline subspaces.     

Phaseless Sampling and Reconstruction of Real-Valued Signals in Shift-Invariant Spaces

Sampling in shift-invariant spaces is a realistic model for signals with smooth spectrum. In this paper, we consider phaseless sampling and reconstruction of real-valued signals in a high-dimensional shift-invariant space from their magnitude measurements on the whole Euclidean space and from their phaseless samples taken on a discrete set with finite sampling density. The determination of a signal in a shift-invariant space, up to a sign, by its magnitude measurements on the whole Euclidean space has been shown in the literature to be equivalent to its nonseparability. In this paper, we introduce an undirected graph associated with the signal in a shift-invariant space and use connectivity of the graph to characterize nonseparability of the signal. Under the local complement property assumption on a shift-invariant space, we find a discrete set with finite sampling density such that nonseparable signals in the shift-invariant space can be reconstructed in a stable way from their phaseless samples taken on that set. In this paper, we also propose a reconstruction algorithm which provides an approximation to the original signal when its noisy phaseless samples are available only. Finally, numerical simulations are performed to demonstrate the robustness of the proposed algorithm to reconstruct box spline signals from their noisy phaseless samples.

     

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 reconstruction for sampling in shift-invariant spaces

The local reconstruction from samples is one of most desirable properties for many applications in signal processing, but it has not been given as much attention. In this paper, we will consider the local reconstruction problem for signals in a shift-invariant space. In particular, we consider finding sampling sets X such that signals in a shift-invariant space can be locally reconstructed from their samples on X. For a locally finite-dimensional shift-invariant space V we show that signals in V can be locally reconstructed from its samples on any sampling set with sufficiently large density. For a shift-invariant space V(? ₁, ..., ? _N ) generated by finitely many compactly supported functions ? ₁, ..., ? _N , we characterize all periodic nonuniform sampling sets X such that signals in that shift-invariant space V(? ₁, ..., ? _N ) can be locally reconstructed from the samples taken from X. For a refinable shift-invariant space V(?) generated by a compactly supported refinable function ?, we prove that for almost all \((x_0, x_1)\in [0,1]^2\), any signal in V(?) can be locally reconstructed from its samples from \(\{x_0, x_1\}+{\mathbb Z}\) with oversampling rate 2. The proofs of our results on the local sampling and reconstruction in the refinable shift-invariant space V(?) depend heavily on the linear independent shifts of a refinable function on measurable sets with positive Lebesgue measure and the almost ripplet property for a refinable function, which are new and interesting by themselves.     

Multivariate generalized sampling in shift-invariant spaces and its approximation properties

Nowadays the topic of sampling in a shift-invariant space is having a significant impact: it avoids most of the problems associated with classical Shannon's theory. Under appropriate hypotheses, any multivariate function in a shift-invariant space can be recovered from its samples at Zd. However, in many common situations the available data are samples of some convolution operators acting on the function itself: this leads to the problem of multivariate generalized sampling in shift-invariant spaces. This extra information on the functions in the shift-invariant space will allow to sample in an appropriate sub-lattice of Zd. In this paper an L²(Rd) theory involving the frame theory is exhibited. Sampling formulas which are frame expansions for the shift-invariant space are obtained. In the case of overcomplete frame formulas, the search of reconstruction functions with prescribed good properties is allowed. Finally, approximation schemes using these generalized sampling formulas are included.     

Improved sampling and reconstruction in spline subspaces

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.     

Invariant Cocycles Have Abelian Ranges

 Let R be a discrete nonsingular equivalence relation on a standard probability space , and let V be an ergodic strongly asymptotically central automorphism of R. We prove that every V-invariant cocycle with values in a Polish group G takes values in an abelian subgroup of G. The hypotheses of this result are satisfied, for example, if A is a finite set, a closed, shift-invariant subset, V is the shift, μ a shift-invariant and ergodic probability measure on X, the two-sided tail-equivalence relation on X, a shift-invariant subrelation which is μ-nonsingular, and a shift-invariant cocycle. (Received 15 September 2001)     

Invariant Cocycles Have Abelian Ranges

 Let R be a discrete nonsingular equivalence relation on a standard probability space , and let V be an ergodic strongly asymptotically central automorphism of R. We prove that every V-invariant cocycle with values in a Polish group G takes values in an abelian subgroup of G. The hypotheses of this result are satisfied, for example, if A is a finite set, a closed, shift-invariant subset, V is the shift, μ a shift-invariant and ergodic probability measure on X, the two-sided tail-equivalence relation on X, a shift-invariant subrelation which is μ-nonsingular, and a shift-invariant cocycle.     

Local connectedness and extension of uniformly continuous functions

A metric space X is straight if for each finite cover of X by closed sets, and for each real valued function f on X, if f is uniformly continuous on each set of the cover, then f is uniformly continuous on the whole of X. The straight spaces have been studied in [A. Berarducci, D. Dikranjan, J. Pelant, An additivity theorem for uniformly continuous functions, Topology and its Applications 146-147 (2005) 339-352], which contains characterization of the straight spaces within the class of the locally connected spaces (they are the uniformly locally connected ones) and the class of the totally disconnected spaces (they coincide with the totally disconnected Atsuji spaces). We show that the completion of a straight space is straight and we characterize the dense straight subspaces of a straight space. In order to clarify further the relation between straightness and the level of local connectedness of the space we introduce two more intermediate properties between straightness and uniform local connectedness and we give various examples to distinguish them. One of these properties coincides with straightness for complete spaces and provides in this way a useful characterization of complete straight spaces in terms of the behaviour of the quasi-components of the space.     

On functions with given spherical means on symmetric spaces

The inversion problem for a local Pompeiu transformation of rank one on sym- metric spaces X of the noncompact type is studied. The reconstruction of a function defined in the ball B _R ⊂ X by its averages on balls of two fixed radii lying in B _R is obtained.     

Sampling and reconstruction for shift-invariant stochastic processes

In this paper, combining stochastic processes with shift-invariant spaces, we introduce shift-invariant stochastic processes. It is a general case of the classical band-limited stochastic processes and a kind of non-band-limited stochastic processes. Two sampling theorems are obtained for the shift-invariant stochastic processes. The results for band-limited stochastic processes and shift-invariant spaces are generalized by our new results.     

混合范数条件下平移不变信号的非均匀平均采样

L~p平移不变子空间中的采样研究通常要求生成函数属于一个不依赖于p的Wiener amalgam空间,此条件因不能控制p而显得太强.本文主要讨论生成函数属于混合范数空间时,非衰减平移不变空间中的非均匀平均采样与重构.生成函数属于混合范数空间的条件弱于Wiener amalgam空间且依赖于参数p.基于混合范数空间中的一些引理,针对两种平均采样泛函建立了采样稳定性,并给出了对应的具有指数收敛的迭代重构算法.     

Geometric Models for Quasicrystals II. Local Rules Under Isometries

The atomic structures of quasicrystalline materials exhibit long range order under translations. It is believed that such materials have atomic structures which approximately obey local rules restricting the location of nearby atoms. These local constraints are typically invariant under rotations, and it is of interest to establish conditions under which such local rules can nevertheless enforce order under translations in any structure that satisfies them. A set of local rules in is a finite collection of discrete sets {Y _i } containing 0, each of which is contained in the ball of radius ρ around 0 in . A set X satisfies the local rules under isometries if the ρ -neighborhood of each is isometric to an element of . This paper gives sufficient conditions on a set of local rules such that if X satisfies under isometries, then X has a weak long-range order under translations, in the sense that X is a Delone set of finite type. A set X is a Delone set of finite type if it is a Delone set whose interpoint distance set X-X is a discrete closed set. We show for each minimal Delone set of finite type X that there exists a set of local rules such that X satisfies under isometries and all other Y that satisfy under isometries are Delone sets of finite type. A set of perfect local rules (under isometries or under translations, respectively) is a set of local rules such that all structures X that satisfy are in the same local isomorphism class (under isometries or under translations, respectively). If a Delone set of finite type has a set of perfect local rules under translations, then it has a set of perfect local rules under isometries, and conversely. Received February 14, 1997, and in revised form February 14, 1998, February 19, 1998, and March 5, 1998.     

Dual frames in connected with generalized sampling in shift-invariant spaces

The aim of this article is to derive stable generalized sampling in a shift-invariant space by using some special dual frames in L²(0,1). These sampling formulas involve samples of filtered versions of the functions in the shift-invariant space. The involved samples are expressed as the frame coefficients of an appropriate function in L²(0,1) with respect to some particular frame in L²(0,1). Since any shift-invariant space with stable generator is the image of L²(0,1) by means of a bounded invertible operator, our generalized sampling is derived from some dual frame expansions in L²(0,1).     

Robustness of sampling and reconstruction and Beurling–Landau-type theorems for shift-invariant spaces

Beurling–Landau-type results are known for a rather small class of functions limited to the Paley–Wiener space and certain spline spaces. Here, we show that the sampling and reconstruction problem in shift-invariant spaces is robust with respect to two classes of probing measures as well as to the underlying shift-invariant space. As an application we enlarge the class of functions for which Beurling–Landau-type results hold.     

Tilings,packings, coverings,and the approximation of functions

A packing (resp. covering) ? of a normed space X consisting of unit balls is called completely saturated (resp. completely reduced) if no finite set of its members can be replaced by a more numerous (resp. less numerous) set of unit balls of X without losing the packing property (resp. covering property) of ?. We show that a normed space X admits completely saturated packings with disjoint closed unit balls as well as completely reduced coverings with open unit balls, provided that there exists a tiling of X with unit balls. Completely reduced coverings by open balls are of interest in the context of an approximation theory for continuous real‐valued functions that rests on so‐called controllable coverings of compact metric spaces. The close relation between controllable coverings and completely reduced coverings allows an extension of the approximation theory to non‐compact spaces. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rings of Continuous Functions on Spaces of Finite Rank and the SV Property

Let X be a compact topological space and let C(X) denote the f-ring of all continuous real-valued functions defined on X. A point x in X is said to have rank n if, in C(X), there are n minimal prime ?-ideals contained in the maximal ?-ideal M _x  = {f ? C(X):f(x) = 0}. The space X has finite rank if there is an n ? N such that every point x ? X has rank at most n. We call X an SV space (for survaluation space) if C(X)/P is a valuation domain for each minimal prime ideal P of C(X). Every compact SV space has finite rank. For a bounded continuous function h defined on a cozeroset U of X, we say there is an h-rift at the point z if h cannot be extended continuously to U ∪ {z}. We use sets of points with h-rift to investigate spaces of finite rank and SV spaces. We show that the set of points with h-rift is a subset of the set of points of rank greater than 1 and that whether or not a compact space of finite rank is SV depends on a characteristic of the closure of the set of points with h-rift for each such h. If X has finite rank and the set of points with h-rift is an F-space for each h, then X is an SV space. Moreover, if every x ? X has rank at most 2, then X is an SV space if and only if for each h, the set of points with h-rift is an F-space.     

Products of straight spaces

A metric space X is straight if for each finite cover of X by closed sets, and for each real valued function f on X, if f is uniformly continuous on each set of the cover, then f is uniformly continuous on the whole of X. A locally connected space is straight iff it is uniformly locally connected (ULC). It is easily seen that ULC spaces are stable under finite products. On the other hand the product of two straight spaces is not necessarily straight. We prove that the product X×Y of two metric spaces is straight if and only if both X and Y are straight and one of the following conditions holds:

(a)

both X and Y are precompact;

(b)

both X and Y are locally connected;

(c)

one of the spaces is both precompact and locally connected.

In particular, when X satisfies (c), the product X×Z is straight for every straight space Z.Finally, we characterize when infinite products of metric spaces are ULC and we completely solve the problem of straightness of infinite products of ULC spaces.     

Characterization of local sampling sequences for spline subspaces

The sampling theorem is one of the most powerful tools in signal analysis. It says that to recover a function in certain function spaces, it suffices to know the values of the function on a sequence of points. Most of known results, e.g., regular and irregular sampling theorems for band-limited functions, concern global sampling. That is, to recover a function at a point or on an interval, we have to know all the samples which are usually infinitely many. On the other hand, local sampling, which invokes only finite samples to reconstruct a function on a bounded interval, is practically useful since we need only to consider a function on a bounded interval in many cases and computers can process only finite samples. In this paper, we give a characterization of local sampling sequences for spline subspaces, which is equivalent to the celebrated Schönberg-Whitney Theorem and is easy to verify. As applications, we give several local sampling theorems on spline subspaces, which generalize and improve some known results.