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We carry out deconvolution by transforming the data into a new general discrete Radon domain that can handle any assumed boundary conditions for the associated matrix inversion problem. For each associated component (projection), one can then apply deconvolution routines to smaller (and possibly better) conditioned matrix inversion problems than the matrix inversion problem for the entire image. We demonstrate this new scheme by adaptively deconvolving these components using a combination of regularized inversion and wavelet filtering techniques. This procedure allows us to provide image estimates based on a generalized ridgelet frame. We then devise methods for carrying out this scheme locally to provide estimates based on generalized multiscaled ridgelets which are then filtered and combined to form an estimate from a curvelet-like domain. The techniques presented here suggest a whole new paradigm for developing deconvolution algorithms that incorporate leading deconvolution schemes. Various experimental results show that our methods can perform significantly better than standard deconvolution techniques.

The blind source separation problem is discussed in this article. Focusing on the assumption of independency of the sources in the time-frequency domain, we present a mathematical formulation for the estimation problem of the number of sources. The proposed method uses the quotient of complex valued time-frequency information of only two observed signals to detect the number of sources. No fewer number of observed signals than the detected number of sources is needed to separate sources. The assumption on sources is quite general independence in the time-frequency plane, which is different from that of independent component analysis. We propose algorithms with feedback and give numerical simulations to show the method works well even for noisy case.

In this paper, we discuss the inverse problem of identifying the connectivity and the conductivity of the links between adjacent pair of nodes in a network, in terms of an input-output map. To do t...

This article extends previous results on the Pompeiu problem with moments. On the Heisenberg group the results previously considered for L2(Hn) have been extended to the function spaces Lp(Hn) for 1 ≤ p ≤ ∞. In the case of L∞(Hn), the conclusions recover differential conditions comparable to those observed in the Euclidean space Cn.

Our aim in this paper is to demonstrate how a result of [Agranovsky et al., J. Analyse Math. 57 (1992) 281–296] on Hn can be extended to use more general integral conditions. In particular, we show that the conclusions of [Agranovsky et al., J. Analyse Math. 57 (1992) 281–296] remain valid when a weight zm is included in the integral conditions. This question relates to a result of Zalcman for Euclidean space, which replaces a requirement for two spheres with the possibility of using two moments on a single sphere without any stipulation on the radius. As a precursor to investigating moments on Hn, we consider how this result extends to Cn.

A general uniqueness theorem is proved for meromorphic functions in Cn which share three distinct small functions with their linear partial differential polynomials. As a consequence, a necessary and sufficient condition in terms of shared values for a meromorphic function to be a solution of a linear partial differential equation of constant coefficients is obtained.

Multivariate residue calculus (in the spirit of J. Lipman) is developed from the computational point of view (for example with several variants of the classical Transformation Law), and used in order to make totally explicit the Bezout identity (and therefore the algebraic Nullstellensatz) in K [ X 1 , . . . , X n ], where K is an infinite field of arbitrary characteristic. Such identities provide sharp size estimates for the denominator and the 'divisors" in the Bezout identity when K is the quotient field of a factorial regular ring A equipped with a size (such as Z or F p [τ1 , . . . , τ q ]). The estimates obtained by the authors in a previous work in the case A = Z are sharpened here, while analytic techniques are replaced with an algebraic approach.

We seek a new non-destructive testing technique to probe a linear crack in a two-dimensional circular region. The detection method utilizes steady-state electrical boundary measurements. First we give the criterion to determine the presence of a radial surface breaking crack. Then a probing procedure is developed to locate the exterior crack tip on the boundary. Once the exterior crack tip is located, we proceed to estimate the crack length applying the numerical algorithm we have constructed in [BCW]. Stability and sensitivity of our method, which depend on the positions of current source electrodes, are also discussed following the computational experiments. A function which is constructed from voltage boundary measurements is used to investigate the stability and sensitivity property of the method.

Nonlinear regression and nonlinear regularization are two powerful approaches to segmentation and nonlinear filtering. In this correspondence, we propose a hybrid approach that effectively combines the best of both worlds and can be efficiently implemented via the Viterbi algorithm.

We prove that regular functions of one quaternionic variahle which satisfy a large class of differential equations cannot have compact singulanties. This result is equivalent to the fact that a large family of 8 × 4 matrices has torsion-free cokernel. The result (obvrons in inc complex case) eaisly extends to Clifford algebray.

Our main objective in this note is to show how multidimensional residues can be used to make explicit division formulas in polynomial algebras; we will also discuss division problems with Laurent polynomials and detail, in some particular cases, the methods that can be used to compute the complete sum of residues of a rational function; we will finally indicate the relation between the search of effective bounds and the numerical computation of multidimensional integrals.

The paper is devoted to the following problem. Consider the set of all radial functions with centers at the points of a closed surface inRn. Are such functions complete in the spaceLq(Rn)? It is shown that the answer is positive if and only ifq is not less than 2n/(n + 1). A similar question is also answered for Riemannian symmetric spaces of rank 1. Relations of this problem with the wave and heat equations are also discussed.

We study the relation between convolution operators and the totally geodesic Radon transform on hyperbolic spaces. As an application we show that the linearized inverse conductivity problem in the ...

Let K be a commutative field; an algorithmic approach to residue symbols defined on a Noetherian K-algebra R has been developed. It is used to prove an effective Nullstellensatz for polynomials defined over infinite factorial rings A equipped with a size. This result extends (and slightly improves) the previous work of the authors in the case A = Z. Let p1, . . . , pM be polynomials in n variables with coefficients in an integral domain A, and respective degrees D1 ≥ D2 ≥ · · · ≥ DM , with no common zeros in an integral closure of the quotient field K of A. It follows from the versions of the Hilbert Nullstellensatz in [B], [CGH], [Ko] that one can find an element r0 ∈ A\{0} and polynomials qj ∈ A[x] such that r0 = M ∑ j=1 qjpj (1) with a priori estimates on the degrees max j deg qj ≤ (3/2)D1 · · ·Dμ, where μ = min{n,M} and ι = max{j : 1 ≤ j < μ − 1, Dj = 2}. When A = Z, using analytic methods, and mainly integral representation formulas and multidimensional residues in C, one can show [BGVY, Section 5] that system (1) can be solved with the estimates  max j deg qj ≤ n(2n+ 1)(3/2) μ ∏ j=1 Dj , maxh(qj) ≤ κ(n)D 1 ( μ ∏ j=1 Dj )8 (h+ logM +D1 logD1), (2) for the Mahler size h(qj) in terms of the maximal Mahler size h of the original polynomials pj . Recall that the Mahler size of p ∈ Z[x1, . . . , xn] is given by h(p) = ∫ [0,1]n log |p(e1 , . . . , en)| dθ . For A = Z, one can also recover the estimate for log |r0|, implicit in (2), from the Arithmetic Bezout Theorem (see [Ph2],[BGS, Theorem 5.4.4]), which shows that Received by the editors April 15, 1996. 1991 Mathematics Subject Classification. Primary 14Q20; Secondary 13F20, 14C17, 32C30.

We consider in this paper interpolation problems for weighted spaces of entire and meromorphic functions. Various (analytic and geometric) conditions necessary and sufficient for multiplicity varieties to be interpolating varieties will be given.

A necessary and sufficient geometric characterization and a necessary and sufficient analytic characterization of interpolating varieties for the space of entire functions will be obtained in the paper, which as an application will also give a generalization of the well-known Pólya-Levinson density theorem.

We consider binary digital images as realizations of a uniformly bounded discrete random set, a mathematical object that can be defined directly on a finite lattice. In this setting we show that it is possible to move between two equivalent probabilistic model specifications. We formulate a restricted version of the discrete-case analog of a Boolean random-set model, obtain its probability mass function, and use some methods of morphological image analysis to derive tools for its statistical inference.

In this paper, we describe necessary and sufficient conditions for the linear dependence of N entire (or smooth) functions in in terms of vanishing of several, conveniently generalized, Wronskians, and prove the version of the well-known Borel theorem about the linear dependency of exponential functions.

We prove in this paper that a given discrete variety $V$ in $\bold C^n$ is an interpolating variety for a weight $p$ if and only if $V$ is a subset of the variety $\{\xi \in \bold C^n: f_1(\xi)=f_2(\xi)= \cdots=f_n(\xi)=0\}$ of $m$ functions $f_1,\ldots,f_m$ in the weighted space the sum of whose directional derivatives in absolute value is not less than $\epsilon\exp(-Cp(\zeta)),\quad \zeta\in V$ for some constants $\epsilon$, $C>0$. The necessary and sufficient conditions will be also given in terms of the Jacobian matrix of $f_,\ldots,f_m.$ As a corollary, we solve an open problem posed by Berenstein and Taylor about interpolation for discrete varieties.

We obtain a Morera type characterization for the square integrable CR functions on the Heisenberg group.