Institute for Systems Research
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Item On the Converse to Pompeiu's Problem(1997) Berenstein, Carlos A.; ISRThis is a reprint of a 1976 paper that appears in an inaccessible Brazilian journal and has become very looked after. It deals with the problem of determining a convex plane domain from the existence of infinitely many over determined Neumann eigenvalues. Recent related work in magneto hydrodynamics of Vogelius and other applications are closely related to this result. The more general result appears in J. Analyse Math 1980 and Crelle l987. See Zalcmain's bibliographic survey of pompeiu problem for other references.Item Radon Transform, Wavelets, and Applications(1996) Berenstein, Carlos A.; ISRNotes of a graduate course given in Venice, Italy, during June 1996 organized by CIME, directed to graduate students to show the interplay of different kinds of Radon transforms and medical and material science problemsItem Do Solid Tori Have the Pompeiu Property?(1996) Berenstein, Carlos A.; Khavinson, Dmitry; ISRWe show that solid tori in Rn satisfy the Pompeiu property: This problem remains open for dimensions n ﺠ4.Item Wavelet-Based Multiresolution Local Tomography(1995) Rashid-Farrokhi, F.; Liu, K.J. Ray; Berenstein, Carlos A.; Walnut, D.; ISRWe develop an algorithm to reconstruct the wavelet coefficients of an image from the Radon transform data. The proposed method uses the properties of wavelets to localize the Radon transform and can be used to reconstruct a local region of the cross section of a body, using almost completely local data which significantly reduces the amount of exposure and computations in X-ray tomography. This property which distinguishes our algorithm from the previous algorithms is based on the observation that for some wavelet basis with sufficiently many vanishing moments, the ramp-filtered version of the scaling function as well as the wavelet function has extremely rapid decay. We show that the variance of the elements of the null- space is negligible in the locally reconstructed image. Also we find an upper bound for the reconstruction error in terms of the amount of data used in the algorithm. To reconstruct a local region 20 pixels in radius in a 256 X 256 image we require 12.5% of full exposure data while the previous methods can reduce the amount of exposure only to 40% for the same case.Item Further Results on MAP Optimality and Strong Consistency of Certain Classes of Morphological Filters(1994) Sidiropoulos, N.D.; Baras, John S.; Berenstein, Carlos A.; ISRIn two recent papers [1], [2], Sidiropoulos et al. have obtained statistical proofs of Maximum A Posteriori} (MAP) optimality and strong consistency of certain popular classes of Morphological filters, namely, Morphological Openings, Closings, unions of Openings, and intersections of Closings, under i.i.d. (both pixel-wise, and sequence-wide) assumptions on the noise model. In this paper we revisit this classic filtering problem, and prove MAP optimality and strong consistency under a different, and, in a sense, more appealing set of assumptions, which allows the explicit incorporation of geometric and Morphological constraints into the noise model, i.e., the noise may now exhibit structure; Surprisingly, it turns out that this affects neither the optimality nor the consistency of these field-proven filters.Item Local Inversion of the Radon Transform in Even Dimensions Using Wavelets(1993) Berenstein, Carlos A.; Walnut, D.; ISRWe use the theory of the continuous wavelet transform to derive inversion formulas for the Radon transform. These formulas are almost local for even dimensions in the sense that for a given mean square error we can decide which lines near a point must be used to approximate the function at the point within the given error.Item Interpolating Varieties for Spaces of Meromorphic Functions(1992) Berenstein, Carlos A.; Li, Bao Q.; ISRVarious interesting results on interpolation theory of entire functions with given growth conditions have been obtained by imposing conditions on multiplicity varieties and weights. All the results discussed in the literature are limited to the space of entire functions. In this paper, we shall extend and generalize the interpolation problem of entire functions to meromorphic functions. The analytic conditions sufficient and necessary for a given multiplicity variety to be interpolating for meromorphic functions with given growth conditions will be obtained. Moreover, purely geometric characterization of interpolating varieties will be given for slowly decreasing radial weights which enable us to determine whether or not a given multiplicity variety is an interpolating variety by direct calculation. when weights grow so rapidly as to allow infinite order functions in the considered space, the geometric conditions would become more delicate. For such weights p(z), we also find purely geometric sufficient as well as necessary conditions provided that log p(exp r) is convex. As corollaries of our results, one obtains the corresponding results for the interpolation of entire functions.Item Computer Assisted Tomography Applied to Plasma Electron Distribution Functions(1992) Li, S.; Lin, Qipeng; Coplan, M.A.; Moore, J.H.; Berenstein, Carlos A.; ISRWe consider several possible instruments based on Computer Tomography to determine space plasma distribution functions.Item Optimal Filtering of Digital Binary Images Corrupted by Union/Intersection(1992) Sidiropoulos, N.D.; Baras, John S.; Berenstein, Carlos A.; ISRWe model digital binary image data as realizations of a bounded discrete random set, a mathematical object which can be directly defined on a finite lattice. We consider the problem of estimating realizations of discrete random sets distorted by a degradation process which can be described by a union/intersection model. First we present an important structural result concerning the probabilistic specification of discrete random sets defined on a finite lattice. Then we formulate the optimal filtering problem for the case of discrete random sets. Two distinct filtering approaches are pursued. For images which feature strong spatial statistical variations we propose a simple family of spatially varying filters, which we call mask filters, and, for each degradation model, derive explicit formulas for the optimal Mask filter. We also consider adaptive mask filters, which can be effective in a more general setting. For images which exhibit a stationary behavior, we consider the class of Morphological filters. First we provide some theoretical justification for the popularity of certain Morphological filtering schemes. In particular, we show that if the signal is smooth, then these schemes are optimal (in the sense of providing the MAP estimate of the signal) under a reasonable worst-case statistical scenario. Then we show that, by using an appropriate (under a given degradation model) expansion of the optimal filter, we can obtain universal characterizations of optimality which do not rely on strong assumptions regarding the spatial interaction of geometrical primitives of the signal and the noise. This approach corresponds to a somewhat counter- intuitive use of fundamental morphological operators; however it is exactly this mode of the use that enables us to arrive at characterizations of optimality in terms of the fundamental functionals of random set theory, namely the generating functionals of the signal and the noise.Item Range of the k-Dimensional Radon Transform in Real Hyperbolic Spaces(1991) Berenstein, Carlos A.; Tarabusi, E.C.; ISRCharacterizations of the range of the totally geodesic k- dimensional Radon transform on the n-dimensional hyperbolic space are given both in terms of moment conditions and as the kernel of a differential operator.