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141
Interpolation of Scattered Data: Distance Matrices and Conditionally Positive Definite Functions
 CONSTRUCTIVE APPROXIMATION
, 1986
"... Among other things, we prove that multiquadric surface interpolation is always solvable, thereby settling a conjecture of R. Franke. ..."
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Cited by 296 (3 self)
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Among other things, we prove that multiquadric surface interpolation is always solvable, thereby settling a conjecture of R. Franke.
Shape Distributions
 ACM Transactions on Graphics
, 2002
"... this paper, we propose and analyze a method for computing shape signatures for arbitrary (possibly degenerate) 3D polygonal models. The key idea is to represent the signature of an object as a shape distribution sampled from a shape function measuring global geometric properties of an object. The pr ..."
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Cited by 205 (1 self)
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this paper, we propose and analyze a method for computing shape signatures for arbitrary (possibly degenerate) 3D polygonal models. The key idea is to represent the signature of an object as a shape distribution sampled from a shape function measuring global geometric properties of an object. The primary motivation for this approach is to reduce the shape matching problem to the comparison of probability distributions, which is simpler than traditional shape matching methods that require pose registration, feature correspondence, or model fitting
Matching 3D Models with Shape Distributions
"... Measuring the similarity between 3D shapes is a fundamental problem, with applications in computer vision, molecular biology, computer graphics, and a variety of other fields. A challenging aspect of this problem is to find a suitable shape signature that can be constructed and compared quickly, whi ..."
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Cited by 194 (7 self)
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Measuring the similarity between 3D shapes is a fundamental problem, with applications in computer vision, molecular biology, computer graphics, and a variety of other fields. A challenging aspect of this problem is to find a suitable shape signature that can be constructed and compared quickly, while still discriminating between similar and dissimilar shapes. In this paper, we propose and analyze a method for computing shape signatures for arbitrary (possibly degenerate) 3D polygonal models. The key idea is to represent the signature of an object as a shape distribution sampled from a shape function measuring global geometric properties of an object. The primary motivation for this approach is to reduce the shape matching problem to the comparison of probability distributions, which is a simpler problem than the comparison of 3D surfaces by traditional shape matching methods that require pose registration, feature correspondence, or model fitting. We find that the dissimilarities be...
Recovering Edges in IllPosed Inverse Problems: Optimality of Curvelet Frames
, 2000
"... We consider a model problem of recovering a function f(x1,x2) from noisy Radon data. The function f to be recovered is assumed smooth apart from a discontinuity along a C2 curve – i.e. an edge. We use the continuum white noise model, with noise level ɛ. Traditional linear methods for solving such in ..."
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Cited by 58 (14 self)
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We consider a model problem of recovering a function f(x1,x2) from noisy Radon data. The function f to be recovered is assumed smooth apart from a discontinuity along a C2 curve – i.e. an edge. We use the continuum white noise model, with noise level ɛ. Traditional linear methods for solving such inverse problems behave poorly in the presence of edges. Qualitatively, the reconstructions are blurred near the edges; quantitatively, they give in our model Mean Squared Errors (MSEs) that tend to zero with noise level ɛ only as O(ɛ1/2)asɛ → 0. A recent innovation – nonlinear shrinkage in the wavelet domain – visually improves edge sharpness and improves MSE convergence to O(ɛ2/3). However, as we show here, this rate is not optimal. In fact, essentially optimal performance is obtained by deploying the recentlyintroduced tight frames of curvelets in this setting. Curvelets are smooth, highly anisotropic elements ideally suited for detecting and synthesizing curved edges. To deploy them in the Radon setting, we construct a curveletbased biorthogonal decomposition
ANALYTIC SOLUTION TO THE BUSEMANNPETTY PROBLEM ON SECTIONS OF CONVEX BODIES
, 1999
"... We derive a formula connecting the derivatives of parallel section functions of an originsymmetric star body in Rn with the Fourier transform of powers of the radial function of the body. A parallel section function (or (n − 1)dimensional Xray) gives the ((n − 1)dimensional) volumes of all hyp ..."
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Cited by 43 (8 self)
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We derive a formula connecting the derivatives of parallel section functions of an originsymmetric star body in Rn with the Fourier transform of powers of the radial function of the body. A parallel section function (or (n − 1)dimensional Xray) gives the ((n − 1)dimensional) volumes of all hyperplane sections of the body orthogonal to a given direction. This formula provides a new characterization of intersection bodies in Rn and leads to a unified analytic solution to the BusemannPetty problem: Suppose that K and L are two originsymmetric convex bodies in Rn such that the ((n − 1)dimensional) volume of each central hyperplane section of K is smaller than the volume of the corresponding section of L; is the (ndimensional) volume of K smaller than the volume of L? In conjunction with earlier established connections between the BusemannPetty problem, intersection bodies, and positive definite distributions, our formula shows that the answer to the problem depends on the behavior of the (n − 2)nd derivative of the parallel section functions. The affirmative answer to the BusemannPetty problem for n ≤ 4 and negative answer for n ≥ 5 now follow from the fact that convexity controls the second derivatives, but does not control the derivatives of higher orders.
Mathematics of thermoacoustic tomography
 European Journal Applied Mathematics
"... The paper presents a survey of mathematical problems, techniques, and challenges arising in the Thermoacoustic (also called Photoacoustic or Optoacoustic) Tomography. 1 ..."
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Cited by 34 (6 self)
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The paper presents a survey of mathematical problems, techniques, and challenges arising in the Thermoacoustic (also called Photoacoustic or Optoacoustic) Tomography. 1
Reconstructing Planar Domains From Their Moments
 Inverse Problems
, 2000
"... . In many areas of science and engineering it is of interest to find the shape of an object or region from indirect measurements which can actually be distilled into moments of the underlying shapes we seek to reconstruct. In this paper, we describe a theoretical framework for the reconstruction of ..."
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Cited by 26 (10 self)
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. In many areas of science and engineering it is of interest to find the shape of an object or region from indirect measurements which can actually be distilled into moments of the underlying shapes we seek to reconstruct. In this paper, we describe a theoretical framework for the reconstruction of a class of planar semianalytic domains from their moments. A part of this class, known as quadrature domains, can approximate, arbitrarily closely, any bounded domain in the complex plane, and is therefore of great practical importance. We provide an exact reconstruction algorithm of quadrature domains. Some numerical demonstrations of the proposed algorithms will be presented. In addition, relations of the present theory to computerassisted tomography and a geophysical inverse problem will be briefly discussed. 1. Introduction The theoretical subject of this paper is the truncated L problem of moments in two variables and some of its ramifications. The practical aspects of the paper are ...
Range descriptions for the spherical mean Radon transform, preprint 2006, arXiv: math. AP/0606314
"... The transform considered in the paper averages a function supported in a ball in R n over all spheres centered at the boundary of the ball. This Radon type transform arises in several contemporary applications, e.g. in thermoacoustic tomography and sonar and radar imaging. Range descriptions for suc ..."
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Cited by 19 (11 self)
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The transform considered in the paper averages a function supported in a ball in R n over all spheres centered at the boundary of the ball. This Radon type transform arises in several contemporary applications, e.g. in thermoacoustic tomography and sonar and radar imaging. Range descriptions for such transforms are important in all these areas, for instance when dealing with incomplete data, error correction, and other issues. Four different types of complete range descriptions are provided, some of which also suggest inversion procedures. Necessity of three of these (appropriately formulated) conditions holds also in general domains, while the complete discussion of the case of general domains will require another publication.
Generalizations of the BusemannPetty problem for sections of convex bodies
 J. Funct. Anal
"... Abstract. We present generalizations of the BusemannPetty problem for dual volumes of intermediate central sections of symmetric convex bodies. It is proved that the answer is negative when the dimension of the sections is greater than or equal to 4. For 2 and 3dimensional sections, both negative ..."
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Cited by 16 (4 self)
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Abstract. We present generalizations of the BusemannPetty problem for dual volumes of intermediate central sections of symmetric convex bodies. It is proved that the answer is negative when the dimension of the sections is greater than or equal to 4. For 2 and 3dimensional sections, both negative and positive answers are given depending on the orders of dual volumes involved, and certain cases remain open. For bodies of revolution, a complete solution is obtained in all dimensions. Let M be a compact convex set of dimension i in Rn that contains the origin in its relative interior. When i = n, M is called a convex body. For 1 ≤ k ≤ i, when a subspace η ⊂ Rn has dimension n − i + k, the intersection M ∩ η is a kdimensional compact convex set in general. The kth dual volume Ṽk(M) is defined as the average
Multivariate Gini Indices
 Journal of Multivariate Analysis
, 1995
"... The Gini index and the Gini mean difference of a univariate distribution are extended to measure the disparity of a general dvariate distribution. We propose and investigate two approaches, one based on the distance of the distribution from itself, the other on the volume of a convex set in (d + 1) ..."
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Cited by 16 (6 self)
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The Gini index and the Gini mean difference of a univariate distribution are extended to measure the disparity of a general dvariate distribution. We propose and investigate two approaches, one based on the distance of the distribution from itself, the other on the volume of a convex set in (d + 1)space, named the lift zonoid of the distribution. When d = 1, this volume equals the area between the usual Lorenz curve and the line of zero disparity, up to a scale factor. We get two definitions of the multivariate Gini index, which are different (when d#1) but connected through the notion of the lift zonoid. Both notions inherit properties of the univariate Gini index, in particular, they are vector scale invariant, continuous, bounded by 0 and 1, and the bounds are sharp. They vanish if and only if the distribution is concentrated at one point. The indices have aceteris paribus property and are consistent with multivariate extensions of the Lorenz order. Illustrations with data conclude...