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47
Ridgelets: A key to higherdimensional intermittency?
, 1999
"... In dimensions two and higher, wavelets can efficiently represent only a small range of the full diversity of interesting behavior. In effect, wavelets are welladapted for pointlike phenomena, whereas in dimensions greater than one, interesting phenomena can be organized along lines, hyperplanes, and ..."
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Cited by 116 (10 self)
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In dimensions two and higher, wavelets can efficiently represent only a small range of the full diversity of interesting behavior. In effect, wavelets are welladapted for pointlike phenomena, whereas in dimensions greater than one, interesting phenomena can be organized along lines, hyperplanes, and other nonpointlike structures, for which wavelets are poorly adapted. We discuss in this paper a new subject, ridgelet analysis, which can effectively deal with linelike phenomena in dimension 2, planelike phenomena in dimension 3 and so on. It encompasses a collection of tools which all begin from the idea of analysis by ridge functions ψ(u1x1+...+unxn) whose ridge profiles ψ are wavelets, or alternatively from performing a wavelet analysis in the Radon domain. The paper reviews recent work on the continuous ridgelet transform (CRT), ridgelet frames, ridgelet orthonormal bases, ridgelets and edges and describes a new notion of smoothness naturally attached to this new representation.
Orthonormal Ridgelets and Linear Singularities
, 1998
"... We construct a new orthonormal basis for L2 (R2), whose elements are angularly integrated ridge functions — orthonormal ridgelets. The new basis functions are in L2 (R2) and so are to be distinguished from the ridge function approximation system called ridgelets by Candès (1997, 1998), as ridge func ..."
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Cited by 57 (16 self)
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We construct a new orthonormal basis for L2 (R2), whose elements are angularly integrated ridge functions — orthonormal ridgelets. The new basis functions are in L2 (R2) and so are to be distinguished from the ridge function approximation system called ridgelets by Candès (1997, 1998), as ridge functions are not in L2 (R2). Orthonormal ridgelet expansions have an interesting application in nonlinear approximation: the problem of 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 an object are sparse: they belong to every ℓp, p>0. This implies that simple thresholding in the ridgelet orthobasis is, in a certain sense, a nearideal nonlinear approximation scheme. The ridgelet orthobasis is the isometric image of a special wavelet basis for Radon space; as a consequence, ridgelet analysis is equivalent to a special wavelet analysis in the Radon domain. This means that questions of ridgelet analysis of linear singularities can be answered by wavelet analysis of point singularities. At the heart of our nonlinear approximation result is the study of a certain tempered distribution on R2 defined formally by S(u, v) =v  −1/2σ(u/v) with σ a certain smooth bounded function; this is singular at (u, v) =(0,0) and C ∞ elsewhere. The key point is that the analysis of this point singularity by tensor Meyer wavelets yields sparse coefficients at high frequencies; this is reflected in the sparsity of the ridgelet coefficients and the good nonlinear approximation properties of the ridgelet basis.
Approximating by Ridge Functions
 J. Approx. Theory
, 1997
"... . This paper surveys certain aspects of the study of ridge functions. We hope it will also encourage some readers to consider researching problems in this area. After first explaining what ridge functions are and giving various motivations for their study, we turn to the problem of presenting algori ..."
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Cited by 43 (7 self)
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. This paper surveys certain aspects of the study of ridge functions. We hope it will also encourage some readers to consider researching problems in this area. After first explaining what ridge functions are and giving various motivations for their study, we turn to the problem of presenting algorithms for approximating by ridge functions. We then touch upon the topic of determining the degree of approximation by ridge functions, and that of recognizing functions which are linear combinations of ridge functions. x1. Introduction This short paper is an introduction to certain aspects of the study of ridge functions. After first explaining what ridge functions are and giving various motivations for their study, we turn to the problem of presenting algorithms for approximating by ridge functions. We then touch upon the topic of determining the degree of approximation by ridge functions, and that of recognizing when we have a function which is a linear combination of ridge functions. Thi...
Overview of methods for image reconstruction from projections in emission computed tomography
 PROC. IEEE
, 2003
"... Emission computed tomography (ECT) is a technology for medical imaging whose importance is increasing rapidly. There is a growing appreciation for the value of the functional (as opposed to anatomical) information that is provided by ECT and there are significant advancements taking place, both in t ..."
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Cited by 18 (1 self)
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Emission computed tomography (ECT) is a technology for medical imaging whose importance is increasing rapidly. There is a growing appreciation for the value of the functional (as opposed to anatomical) information that is provided by ECT and there are significant advancements taking place, both in the instrumentation for data collection, and in the computer methods for generating images from the measured data. These computer methods are designed to solve the inverse problem known as “image reconstruction from projections.” This paper uses the various models of the data collection process as the framework for presenting an overview of the wide variety of methods that have been developed for image reconstruction in the major subfields of ECT, which are positron emission tomography (PET) and singlephoton emission computed tomography (SPECT). The overall sequence of the major sections in the paper, and the presentation within each major section, both proceed from the more realistic and general models to those that are idealized and application specific. For most of the topics, the description proceeds from the threedimensional case to the twodimensional case. The paper presents a broad overview of algorithms for PET and SPECT, giving references to the literature where these algorithms and their applications are described in more detail.
Approximation of Multivariate Functions
 in Advances in Computational Mathematics
, 1994
"... . We discuss one approach to the problem of approximating functions of many variables which is truly multivariate in character. This approach is based on superpositions of functions with infinite families of smooth simple functions. x1. Introduction and Motivation There are numerous methods of appr ..."
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Cited by 17 (1 self)
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. We discuss one approach to the problem of approximating functions of many variables which is truly multivariate in character. This approach is based on superpositions of functions with infinite families of smooth simple functions. x1. Introduction and Motivation There are numerous methods of approximating functions of many variables. For example, we have the more classic methods using Polynomials, Fourier Series, or Tensor Products, and more modern methods using Wavelets, Radial Basis Functions, Multivariate Splines, or Ridge Functions. Many of these are natural generalizations of methods developed for approximating univariate functions. However functions of many variables are fundamentally different from functions of one variable, and approximation techniques for such problems are much less developed and understood. We will discuss in these few pages one approach to this problem which is truly multivariate in character. Hilbert's 13th problem, although not formulated in the followi...
Geometric Weakly Admissible Meshes, Discrete Least Squares Approximations and Approximate Fekete Points
, 2009
"... Using the concept of Geometric Weakly Admissible Meshes (see §2 below) together with an algorithm based on the classical QR factorization of matrices, we compute efficient points for discrete multivariate least squares approximation and Lagrange interpolation. ..."
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Cited by 16 (12 self)
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Using the concept of Geometric Weakly Admissible Meshes (see §2 below) together with an algorithm based on the classical QR factorization of matrices, we compute efficient points for discrete multivariate least squares approximation and Lagrange interpolation.
Ridge functions and orthonormal ridgelets
 J. Approx. Theory
"... Orthonormal ridgelets are a specialized set of angularlyintegrated ridge functions which make up an orthonormal basis for L 2 (R 2). In this paper we explore the relationship between orthonormal ridgelets and true ridge functions r(x1 cos θ + x2 sin θ). We derive a formula giving the ridgelet coeff ..."
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Cited by 14 (2 self)
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Orthonormal ridgelets are a specialized set of angularlyintegrated ridge functions which make up an orthonormal basis for L 2 (R 2). In this paper we explore the relationship between orthonormal ridgelets and true ridge functions r(x1 cos θ + x2 sin θ). We derive a formula giving the ridgelet coefficients of a ridge function in terms of the 1D wavelet coefficients of the ridge profile r(t), and we study the properties of the linear approximation operator which ‘kills ’ coefficients at high angular scale or high ridge scale. We also show that partial orthonormal ridgelet expansions can give efficient nonlinear approximations to pure ridge functions. In effect, the rearranged weighted ridgelet coefficients of a ridge function decay at essentially the same rate as the rearranged weighted 1D wavelet coefficients of the 1D ridge profile r(t). This shows that simple thresholding in the ridgelet basis is, for certain purposes, equally as good as ideal nonlinear ridge approximation. Key Words and Phrases. Wavelets. Ridge function. Ridgelet. Radon transform. Best mterm approximation. Thresholding of wavelet coefficients.
Statistical inversion for medical Xray tomography with few radiographs II: Application to dental radiology,” Phys
 Med. Biol
, 2003
"... Abstract. In Xray tomography, the structure of a three dimensional body is reconstructed from a collection of projection images of the body. Medical CT imaging does this using an extensive set of projections from all around the body. However, in many practical imaging situations only a small number ..."
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Cited by 11 (4 self)
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Abstract. In Xray tomography, the structure of a three dimensional body is reconstructed from a collection of projection images of the body. Medical CT imaging does this using an extensive set of projections from all around the body. However, in many practical imaging situations only a small number of truncated projections is available from a limited angle of view. Three dimensional imaging using such data is complicated for two reasons: (i) Typically, sparse projection data does not contain sufficient information to completely describe the 3D body, and (ii) Traditional CT reconstruction algorithms, such as filtered backprojection, do not work well when applied to few irregularly spaced projections. Concerning (i), existing results about the information content of sparse projection data are reviewed and discussed. Concerning (ii), it is shown how Bayesian inversion methods can be used to incorporate a priori information into the reconstruction method, leading to improved image quality over traditional methods. Based on the discussion, a lowdose threedimensional Xray imaging modality is described. Submitted to: Phys. Med. Biol. 1.
FunkHecke Formula For Orthogonal Polynomials On Spheres And On Balls
, 2000
"... Analogues of the FunkHecke formula for spherical harmonics are proved for Dunkl's hharmonics associated to the reection groups and for orthogonal polynomials related to hharmonics on the unit ball. In particular, an analogue and its application is discussed for the weight function (1 jxj ..."
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Cited by 10 (9 self)
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Analogues of the FunkHecke formula for spherical harmonics are proved for Dunkl's hharmonics associated to the reection groups and for orthogonal polynomials related to hharmonics on the unit ball. In particular, an analogue and its application is discussed for the weight function (1 jxj on the unit ball in R 1.
Lecture Notes On Orthogonal Polynomials Of Several Variable
 Adv. Theory Spec. Funct. Orthogonal Polynomials
"... Contents 1. Introduction 1 2. General properties 7 3. hharmonics and orthogonal polynomials on the sphere 11 4. Orthogonal polynomials on the unit ball 20 5. Orthogonal polynomials on the simplex 25 6. Classical type product orthogonal polynomials 29 7. Fourier orthogonal expansions 34 8. N ..."
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Cited by 7 (0 self)
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Contents 1. Introduction 1 2. General properties 7 3. hharmonics and orthogonal polynomials on the sphere 11 4. Orthogonal polynomials on the unit ball 20 5. Orthogonal polynomials on the simplex 25 6. Classical type product orthogonal polynomials 29 7. Fourier orthogonal expansions 34 8. Notes and Literature 42 Reference 43 These lecture notes provide an introduction to orthogonal polynomials of several variables. It will cover the basic theory but deal mostly with examples, paying special attention to those orthogonal polynomials associated with classical type weight functions supported on the standard domains, for which fairly explicit formulae exist. There is little prerequisites for these lecture nodes, a working knowledge of classical orthogonal polynomials of one variable satises. 1. Introduction 1.1. Denition: one var