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

Abstract. A new approach to the construction of finite-difference methods is presented. It is shown how the multi-point 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.

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 single-photon 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 three-dimensional case to the two-dimensional 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.

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.

Abstract. Three-dimensional 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 three-dimensional world. In this paper, we develop tools for the analysis of 3-D data which may contain structures built from lines, line segments, and filaments. These tools come in two main forms: (a) Monoscale: the X-ray 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 (3-D) 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.

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 closed-form 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...

Abstract. Let q(r) , r = |x | , x ∈ R 3, be a real-valued square-integrable 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, real-valued, � a

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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.

We pesent Shannon sampling theory for functions defined on T \Theta IR, where T denotes the circle group, prove a new estimate for the aliasing error, and apply the result to parallel-beam diffraction tomography. The class of admissible sampling lattices is characterized and general sampling conditions are derived which lead to the identification of new efficient sampling schemes. Corresponding results for x-ray tomography are obtained in the high-frequency limit. 1. INTRODUCTION Sampling theorems provide interpolation formulas for functions whose Fourier transform is compactly supported. If the Fourier transform does not have compact support, a so-called aliasing error occurs. In this paper we pesent a new estimate for the aliasing error for functions defined on T \Theta IR, where T denotes the circle group, and work out its application to computed tomography. In computed tomography (CT) an object is exposed to radiation which is measured after passing through the object. From the...