This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbert-space formulation, we reinterpret Shannon’s sampling procedure as an orthogonal projection onto the subspace of band-limited functions. We then extend the standard sampling paradigm for a representation of functions in the more general class of “shift-invariant” functions spaces, including splines and wavelets. Practically, this allows for simpler—and possibly more realistic—interpolation models, which can be used in conjunction with a much wider class of (anti-aliasing) prefilters that are not necessarily ideal low-pass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned. Keywords—Band-limited functions, Hilbert spaces, interpolation, least squares approximation, projection operators, sampling,

We address the problem of representing and recognizing 3D free-form objects when (a) the object viewpoint is arbitrary, (b) the objects may vary in shape and complexity, and (c) no restrictive assumptions are made about the types of surfaces on the object. We assume that a range image of a scene is available, containing a view of a rigid 3D object without occlusion. We propose a new and general surface representation scheme for recognizing objects with freeform (sculpted) surfaces. In this scheme, an object is described concisely in terms of maximal surface patches of constant shape index. The maximal patches that represent the object are mapped onto the unit sphere via their orientations, and aggregated via shape spectral functions. Properties such as surface area, curvedness and connectivity which are required to capture local and global information are also built into the representation. The scheme yields a meaningful and rich description useful for object recognition. A novel conce...

We construct a new orthonormal basis for L 2 (R 2), whose elements are angularly integrated ridge functions — orthonormal ridgelets. The basis elements are smooth and of rapid decay in the spatial domain, and in the frequency domain are localized near angular wedges which, at radius r =2 j,haveradialextent∆r≈2 j and angular extent ∆θ ≈ 2 −j. Orthonormal ridgelet expansions expose an interesting phenomenon in nonlinear approxi-mation: they give very efficient approximations to objects such as 1 {x1 cos θ+x2 sin θ>a} e−x2 1−x2 2 which are smooth away from a discontinuity along a line. The orthonormal ridgelet coefficients of such objects are sparse: they belong to every ℓp, p>0. This implies that simple thresholding in the ridgelet orthobasis is, in a certain sense, a near-ideal nonlinear approximation scheme for such objects. Orthonormal ridgelets may be viewed as L2 substitutes for approximation by sums of ridge functions, and so can perform many of the same tasks as the ridgelets systems constructed by Candès (1997, 1998). Orthonormal ridgelets make available the machinery of orthogonal decompositions, which is not availble for ridge functions as they are not in L2 (R2). The ridgelet orthobasis is constructed as the isometric image of a special wavelet basis for

A Hilbert space embedding for probability measures has recently been proposed, with applications including dimensionality reduction, homogeneity testing and independence testing. This embedding represents any probability measure as a mean element in a reproducing kernel Hilbert space (RKHS). The embedding function has been proven to be injective when the reproducing kernel is universal. In this case, the embedding induces a metric on the space of probability distributions defined on compact metric spaces. In the present work, we consider more broadly the problem of specifying characteristic kernels, defined as kernels for which the RKHS embedding of probability measures is injective. In particular, characteristic kernels can include non-universal kernels. We restrict ourselves to translation-invariant kernels on Euclidean space, and define the associated metric on probability measures in terms of the Fourier spectrum of the kernel and characteristic functions of these measures. The support of the kernel spectrum is important in finding whether a kernel is characteristic: in particular, the embedding is injective if and only if the kernel spectrum has the entire domain as its support. Characteristic kernels may nonetheless have difficulty in distinguishing certain distributions on the basis of finite samples, again due to the interaction of the kernel spectrum and the characteristic functions of the measures. 1

Abstract. We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points). Call a configuration sharp if there are m distances between distinct points in it and it is a spherical (2m −1)-design. We prove that every sharp configuration minimizes potential energy for all completely monotonic potential functions. Examples include the minimal vectors of the E8 and Leech lattices. We also prove the same result for the vertices of the 600-cell, which do not form a sharp configuration. For most known cases, we prove that they are the unique global minima for energy, as long as the potential function is strictly completely monotonic. For certain potential functions, some of these configurations were previously analyzed by Yudin, Kolushov, and Andreev; we build on their techniques. We also generalize our results to other compact two-point homogeneous spaces, and we

To recover a sharp image from its blurry observation is the problem known as image deblurring. It frequently arises in imaging sciences and technologies, including optical, medical, and astronomical applications, and is crucial for allowing to detect important features and patterns such as those of a distant planet or some microscopic tissue. Mathematically, image deblurring is intimately connected to backward diffusion processes (e.g., inverting the heat equation), which are notoriously unstable. As inverse problem solvers, deblurring models therefore crucially depend upon proper regularizers or conditioners that help secure stability, often at the necessary cost of losing certain highfrequency details in the original images. Such regularization techniques can ensure the existence, uniqueness, or stability of deblurred images. The present work follows closely the general framework described in our recent monograph [18], but also contains more updated views and approaches to image deblurring, including, e.g., more discussion on stochastic signals, the Bayesian/Tikhonov approach to Wiener filtering, and the iterated-shrinkage algorithm of Daubechies et al. [30,31] for wavelet-based deblurring. The work thus contributes to the development of generic, systematic, and unified frameworks in contemporary image processing.

The most popular "fractal-based" algorithms for both the representation as well as compression of computer images have involved some implementation of the method of Iterated Function Systems (IFS) on complete metric spaces, e.g. IFS with probabilities (IFSP), Iterated Fuzzy Set Systems (IFZS), Fractal Transforms (FT), the Bath Fractal Transform (BFT) and IFS with grey-level maps (IFSM). (FT and BFT are special cases of IFSM.) The "IFS component" of these methods is a set of N contraction maps w = fw 1 ; w 2 ; : : : ; wN g, w i : X ! X over a complete metric space (X; d), the "base space" representing the computer screen. Most discussions of these methods, both practical as well as theoretical in nature, assume that the sets w i (X) are nonoverlapping (or at least ignore any overlapping), i.e. that w \Gamma1 i (x) exists for only one value i 2 f1; 2; : : : Ng. As such, given an image function u, its so-called fractal transform (Tu)(x) at any point x 2 X is given by OE i (u(w \Ga...

We discuss some problems of lattice tiling via Harmonic Analysis methods. We consider lattice tilings of R d by the unit cube in relation to the Minkowski Conjecture (now a theorem of Hajos) and give a new equivalent form of Hajos's theorem. We also consider "notched cubes" (a cube from which a reactangle has been removed from one of the corners) and show that they admit lattice tilings. This has also been been proved by S. Stein by a direct geometric method. Finally, we exhibit a new class of simple shapes that admit lattice tilings, the "extended cubes", which are unions of two axis-aligned rectangles that share a vertex and have intersection of odd codimension. In our approach we consider the Fourier Transform of the indicator function of the tile and try to exhibit a lattice of appropriate volume in its zero-set. 1991 Mathematics Subject Classification. Primary 52C22; Secondary 42. the electronic journal of combinatorics 5 (1998), #R14 2 0. Introduction 0.1 Results. We obtain...

Introduction This work has been done in collaboration with Bruno Forte, University of Verona, Italy. Much of the work to be outlined below was presented at the NATO ASI on Fractal Image Coding and Analysis held in Trondheim, Norway, July 8-17, 1995 and will appear as two papers in the Proceedings, edited by Yuval Fisher. Preliminary versions of these papers are available by anonymous ftp from links.uwaterloo.ca:/pub/Fractals/Papers/Waterloo/. The most popular fractal-based algorithms for both the representation as well as compression of computer images have involved some implementation of the method of Iterated Function Systems (IFS) on complete metric spaces, e.g. IFS with probabilities (IFSP), Iterated Fuzzy Set Systems (IFZS), Fractal Transforms (FT), the Bath Fractal Transform (BFT) and IFS with grey-level maps (IFSM). (FT and BFT are special cases of IFSM.) There are two major aspects of our work which are summarized below. First is the effort to formulat

Generalized eigenfunctions of the 3-dimensional relativistic Schrödinger operator √ ∆ + V (x) with |V (x) | ≤ C〈x 〉 −σ, σ> 1, are considered. We construct the generalized eigenfunctions by exploiting results on the limiting absorption principle. We compute explicitly the integral kernel of ( √ − ∆ − z) −1, z ∈ C \ [0, +∞), which has nothing in common with the integral kernel of (− ∆ − z) −1, but the leading term of the integral kernels of the boundary values ( √ −∆−λ∓i0) −1, λ> 0, turn out to be the same, up to a constant, as the integral kernels of the boundary values (−∆−λ∓i0) −1. This fact enables us to show that the asymptotic behavior, as |x | → +∞, of the generalized eigenfunction of ∆ + V (x) is equal to the sum of a plane wave and a spherical wave