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70
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 195 (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
On stable numerical differentiation
 Mathem. of Computation
, 1968
"... Abstract. A new approach to the construction of finitedifference methods is presented. It is shown how the multipoint differentiators can generate regularizing algorithms with a stepsize h being a regularization parameter. The explicitly computable estimation constants are given. Also an iterative ..."
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Cited by 39 (22 self)
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Abstract. A new approach to the construction of finitedifference methods is presented. It is shown how the multipoint differentiators can generate regularizing algorithms with a stepsize h being a regularization parameter. The explicitly computable estimation constants are given. Also an iteratively regularized scheme for solving the numerical differentiation problem in the form of Volterra integral equation is developed. 1.
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.
Fast Xray and beamlet transforms for threedimensional data
 in Modern Signal Processing
, 2002
"... Abstract. Threedimensional volumetric data are becoming increasingly available in a wide range of scientific and technical disciplines. With the right tools, we can expect such data to yield valuable insights about many important phenomena in our threedimensional world. In this paper, we develop t ..."
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Cited by 13 (8 self)
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Abstract. Threedimensional volumetric data are becoming increasingly available in a wide range of scientific and technical disciplines. With the right tools, we can expect such data to yield valuable insights about many important phenomena in our threedimensional world. In this paper, we develop tools for the analysis of 3D data which may contain structures built from lines, line segments, and filaments. These tools come in two main forms: (a) Monoscale: the Xray transform, offering the collection of line integrals along a wide range of lines running through the image — at all different orientations and positions; and (b) Multiscale: the (3D) beamlet transform, offering the collection of line integrals along line segments which, in addition to ranging through a wide collection of locations and positions, also occupy a wide range of scales. We describe different strategies for computing these transforms and several basic applications, for example in finding faint structures buried in noisy data. 1.
Stable numerical differentiation: when is it possible
 Jour. Korean SIAM
"... Abstract. Two principally different statements of the problem of stable numerical differentiation are considered. It is analyzed when it is possible in principle to get a stable approximation to the derivative f ′ given noisy data fδ. Computational aspects of the problem are discussed and illustrate ..."
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Cited by 11 (8 self)
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Abstract. Two principally different statements of the problem of stable numerical differentiation are considered. It is analyzed when it is possible in principle to get a stable approximation to the derivative f ′ given noisy data fδ. Computational aspects of the problem are discussed and illustrated by examples. These examples show the practical value of the new understanding of the problem of stable differentiation. 1.
Static Arbitrage Bounds on Basket Option Prices
, 2002
"... We consider the problem of computing upper and lower bounds on the price of a European basket call option, given prices on other similar baskets. Although this problem is very hard to solve exactly in the general case, we show that in some instances the upper and lower bounds can be computed via sim ..."
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Cited by 11 (1 self)
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We consider the problem of computing upper and lower bounds on the price of a European basket call option, given prices on other similar baskets. Although this problem is very hard to solve exactly in the general case, we show that in some instances the upper and lower bounds can be computed via simple closedform expressions, or linear programs. We also introduce an efficient linear programming relaxation of the general problem based on an integral transform interpretation of the call price function. We show that this...
Formula for the radius of the support of the potential in terms of the scattering data
 Jour. of Phys. A
, 1998
"... Abstract. Let q(r) , r = x  , x ∈ R 3, be a realvalued squareintegrable compactly supported function, and [0, a] be the smallest interval containing the support of q(r). Let A(α ′ , α) = A(α ′ · α) be the corresponding scattering amplitude at a fixed positive energy, k2 � = 1. Let δℓ be the p ..."
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Cited by 7 (6 self)
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Abstract. Let q(r) , r = x  , x ∈ R 3, be a realvalued squareintegrable compactly supported function, and [0, a] be the smallest interval containing the support of q(r). Let A(α ′ , α) = A(α ′ · α) be the corresponding scattering amplitude at a fixed positive energy, k2 � = 1. Let δℓ be the phase 2ℓ + 1 shifts at k = 1. It is proved that lim δℓ ℓ→ ∞ e 1 � 2ℓ = a, provided that q(r) does not change sign in some, arbitrary small, neighborhood of a. 1. Introduction. The aim of this paper is to give a partial justification of the modified conjecture due to A.G. Ramm [R1, p.356, formula (7)]. Let us make the following assumption. Assumption (A): the potential q(r) , r = x  , is spherically symmetric, realvalued, � a
Tomographic reconstruction of piecewise smooth images
 in Proc. IEEE Comput. Soc. Conf. Comput. Vis. Pattern Recognit
"... In computed tomography, direct inversion of the Radon transform requires more projections than are practical due to constraints in scan time and image accessibility. Therefore, it is necessary to consider the estimation of reconstructed images when the problem is underconstrained, i.e., when a uniq ..."
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Cited by 5 (0 self)
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In computed tomography, direct inversion of the Radon transform requires more projections than are practical due to constraints in scan time and image accessibility. Therefore, it is necessary to consider the estimation of reconstructed images when the problem is underconstrained, i.e., when a unique solution does not exist. To resolve ambiguities among solutions, it is necessary to place additional constraints on the reconstructed image. In this paper, we present a surface evolution technique to model the reconstructed image as piecewise smooth. We model the reconstructed image as two regions that are each smoothly varying in intensity and are separated by a smooth surface. We define a cost functional to penalize deviation from piecewise smoothness while ensuring that the projections of the estimated image match the measured projections. From this functional, we derive an evolution for the modeled image intensity and an evolution for the surface, thereby defining a variational tomographic estimation technique. We show example reconstructions to highlight the performance of the proposed method on real medical images. 1.
Some new velocity averaging results
 SIAM J. Math. Anal
"... Abstract. Let (R, µ) be a nonatomic finite measure space and E = Lr (R, µ) a Lebesgue space over R. Then we consider tempered distributions f and g (depending on x ∈ Rn and v ∈ R), for which divx(af) = g in S ′(Rn, E). Here a: R − → Rn is a bounded function of v (a velocity field) satisfying a nond ..."
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Cited by 3 (2 self)
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Abstract. Let (R, µ) be a nonatomic finite measure space and E = Lr (R, µ) a Lebesgue space over R. Then we consider tempered distributions f and g (depending on x ∈ Rn and v ∈ R), for which divx(af) = g in S ′(Rn, E). Here a: R − → Rn is a bounded function of v (a velocity field) satisfying a nondegeneracy condition. We study the regularity of the average ¯ f = ∫ R f(·, v)ψ(v) dµ(v) ∈ S ′ (Rn) (with ψ ∈ Lr ′ (R, µ) a suitable weight function) when f and g are bounded in Banach space valued Besov spaces. We also present some compactness results for sequences of averages.