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22
CramérRao Bounds for Parametric Shape Estimation in Inverse Problems
 IEEE Trans. on Image Processing
, 2003
"... We address the problem of computing fundamental performance bounds for estimation of object boundaries from noisy measurements in inverse problems, when the boundaries are parameterized by a finite number of unknown variables. Our model applies to multiple unknown objects, each with its own unknown ..."
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Cited by 9 (2 self)
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We address the problem of computing fundamental performance bounds for estimation of object boundaries from noisy measurements in inverse problems, when the boundaries are parameterized by a finite number of unknown variables. Our model applies to multiple unknown objects, each with its own unknown gray level, or color, and boundary parameterization, on an arbitrary known background. While such fundamental bounds on the performance of shape estimation algorithms can in principle be derived from the CramerRao lower bounds, very few results have been reported due to the di#culty of computing the derivatives of a functional with respect to shape deformation. In this paper, we provide a general formula for computing CramerRao lower bounds in inverse problems where the observations are related to the object by a general linear transform, followed by a possibly nonlinear and noisy measurement system.
CramérRao Bounds for 2D Target Shape Estimation in Nonlinear Inverse Scattering Problems with Application to Passive Radar
 IEEE Trans. on Antennas and Propagat
, 2001
"... We present new methods for computing fundamental performance limits for twodimensional (2D) parametric shape estimation in nonlinear inverse scattering problems with an application to passive radar imaging. We evaluate CramrRao lower bounds (CRB) on shape estimation accuracy using the domain der ..."
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Cited by 7 (3 self)
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We present new methods for computing fundamental performance limits for twodimensional (2D) parametric shape estimation in nonlinear inverse scattering problems with an application to passive radar imaging. We evaluate CramrRao lower bounds (CRB) on shape estimation accuracy using the domain derivative technique from nonlinear inverse scattering theory. The CRB provides an unbeatable performance limit for any unbiased estimator, and under fairly mild regularity conditions is asymptotically achieved by the maximum likelihood estimator (MLE). The resultant CRBs are used to define an asymptotic global confidence region, centered around the true boundary, in which the boundary estimate lies with a prescribed probability. These global confidence regions conveniently display the uncertainty in various geometric parameters such as shape, size, orientation, and position of the estimated target and facilitate geometric inferences. Numerical simulations are performed using the layer approach and the Nystrm method for computation of domain derivatives and using Fourier descriptors for target shape parameterization. This analysis demonstrates the accuracy and generality of the proposed methods. Index TermsCramrRao bounds, Fourier descriptors, global confidence regions, nonlinear inverse scattering, passive radar imaging, shape estimation. I.
Electromagnetic Scattering From an Orthotropic Medium
 J. Int. Eq. Appl
, 1997
"... We investigate electromagnetic wave propagation in an inhomogeneous anisotropic medium. For the case of an orthotropic medium we derive the LippmannSchwinger equation, which is equivalent to a system of strongly singular integral equations. Uniqueness and existence of a solution is shown and we exa ..."
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Cited by 7 (2 self)
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We investigate electromagnetic wave propagation in an inhomogeneous anisotropic medium. For the case of an orthotropic medium we derive the LippmannSchwinger equation, which is equivalent to a system of strongly singular integral equations. Uniqueness and existence of a solution is shown and we examine the regularity of the solution by means of integral equations. We prove the infinite Fr'echet differentiability of the scattered field in its dependence on the refractive index of the anisotropic medium and we derive a characterization of the Fr'echet derivatives as a solution of an anisotropic scattering problem. 1 Introduction. Integral equation methods play a central role in the study of electromagnetic scattering problems. This is primarily due to the fact that the mathematical formulation of scattering problems leads to equations defined over unbounded domains, and hence by the reformulation in terms of integral equations one can replace the problem over an unbounded domain by one...
Optimal shape design for elliptic equations via BIEmethods
, 1998
"... this paper we shall study shape optimization problems for 2dimensional simply connected bounded domains\Omega\Gamma where the domains under consideration satisfying a condition of starshapeness with respect to a neighbourhood U ffi (x 0 ) = fy 2 IR ..."
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Cited by 6 (4 self)
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this paper we shall study shape optimization problems for 2dimensional simply connected bounded domains\Omega\Gamma where the domains under consideration satisfying a condition of starshapeness with respect to a neighbourhood U ffi (x 0 ) = fy 2 IR
Inverse Obstacle Scattering Using Reduced Data
, 1999
"... : The classical result of Schiffer in acoustic scattering is that a knowledge of the far field pattern for all observation directions and all incident directions at a fixed wave number k uniquely determines the soundsoft scattering obstacle D. This is widely believed to be far greater information ..."
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Cited by 4 (2 self)
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: The classical result of Schiffer in acoustic scattering is that a knowledge of the far field pattern for all observation directions and all incident directions at a fixed wave number k uniquely determines the soundsoft scattering obstacle D. This is widely believed to be far greater information than is required for uniqueness and more recent results have been obtained that considerably weaken the amount of data needed provided certain restrictions are placed on the scatterer. Most of this work has concentrated on reducing the number of incident waves required for a unique determination. This paper will take another approach and seeks to determine sufficient information to recover the obstacle from measurements of the far field at isolated points. Our approach will be constructive and some numerical reconstructions will be presented. ams (mos) subject classification primary 81U40, 65R30; secondary 35J05. 1. Introduction The standard problem in inverse obstacle scattering for timeh...
CramérRao Bounds for Parametric Estimation of Target Boundaries in Nonlinear Inverse Scattering Problems
 IEEE Trans. on Antennas and Propagat
, 1999
"... We present new methods for computing fundamental performance limits for parametric shape estimation in inverse scattering problems, such as passive radar imaging. We evaluate CramerRao lower bounds (CRB) on shape estimation accuracy using the domain derivative technique from nonlinear inverse scatte ..."
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Cited by 3 (3 self)
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We present new methods for computing fundamental performance limits for parametric shape estimation in inverse scattering problems, such as passive radar imaging. We evaluate CramerRao lower bounds (CRB) on shape estimation accuracy using the domain derivative technique from nonlinear inverse scattering theory. The CRB provides an unbeatable performance limit for any unbiased estimator, and under fairly mild regularity conditions, is asymptotically achieved by the maximum likelihood estimator (MLE), hence serving as a predictor of the high signalto noise ratio performance of the MLE. Furthermore, the resultant CRB's are used to define a global confidence region, centered around the true boundary, in which the boundary estimate lies with a prescribed probability. These global confidence regions conveniently display the uncertainty in various geometric parameters such as shape, size, orientation, and position of the estimated target, and facilitate geometric inferences. Numerical simula...
Domain Sensitivity Analysis for Acoustic Scattering Problems
, 2000
"... We consider scattering problems described by the Dirichlet boundary value problem for the scalar Helmholtz equation in exterior domains\Omega := IR 2 n\Omega i : Such problems include the scattering of acoustic waves from a soundsoft obstacle or the scattering of electromagnetic waves from perfe ..."
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Cited by 3 (3 self)
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We consider scattering problems described by the Dirichlet boundary value problem for the scalar Helmholtz equation in exterior domains\Omega := IR 2 n\Omega i : Such problems include the scattering of acoustic waves from a soundsoft obstacle or the scattering of electromagnetic waves from perfectly conducting cylinders. The paper concerns with the characterization of the Gateaux derivative of the farfield pattern with respect to the shape of the scatterer. Using a modification of the method of adjoint problems [4] we propose a representation of the domain derivative of the farfield pattern, which is very suitable for a numerical application in case of nonsmooth boundaries. This method can be easily applied to exterior problems with other kinds of boundary conditions and boundary singularities like edges and vertices (see [1] where complete proves will be given for the results presented here). 1 Formulation of the problem We consider exterior domains\Omega := IR 2 n\Omega i ...
On The Far Field In Obstacle Scattering
 SIAM J. Appl. Math
, 1999
"... . We will establish far field identities for the scattering of timeharmonic acoustic waves from soundsoft and soundhard obstacles and will illustrate two applications of these identities for the inverse obstacle scattering problem. Firstly, we will provide an alternate proof for the Fr'echet diff ..."
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Cited by 3 (1 self)
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. We will establish far field identities for the scattering of timeharmonic acoustic waves from soundsoft and soundhard obstacles and will illustrate two applications of these identities for the inverse obstacle scattering problem. Firstly, we will provide an alternate proof for the Fr'echet differentiability of the far field pattern with respect to the boundary. Secondly, we will connect the identities with a new method for the approximate solution of the inverse obstacle scattering problem which was recently suggested by Potthast. Key words. Helmholtz equation, acoustic waves, obstacle scattering, far field pattern, inverse scattering, Fr'echet derivative, point sources AMS subject classifications. 35J05, 35P25, 35R25, 35R30, 45A05, 45P05 1. Introduction. Let D ae IR 3 be a bounded domain with a connected boundary @D of class C 2 and outward unit normal . Consider the exterior Dirichlet problem: Given a continuous function f on @D, find a solution v 2 C 2 (IR 3 n ¯ D) ...
Integral Equation Methods in Inverse Obstacle Scattering
 ANZIAM Journal
, 1995
"... In this survey we consider a regularized Newton method for the approximate solution of the inverse problem to determine the shape of an obstacle from a knowledge of the far field pattern for the scattering of timeharmonic acoustic or electromagnetic plane waves. Our analysis is in two dimensions an ..."
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Cited by 2 (1 self)
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In this survey we consider a regularized Newton method for the approximate solution of the inverse problem to determine the shape of an obstacle from a knowledge of the far field pattern for the scattering of timeharmonic acoustic or electromagnetic plane waves. Our analysis is in two dimensions and the numerical scheme is based on the solution of boundary integral equations by a Nystrom method. We include an example of the reconstruction of a planar domain with a corner both to illustrate the feasibility of the use of radial basis functions for the reconstruction of boundary curves with local features and to connect the presentation to some of the research work of Professor David Elliott. 1 The inverse obstacle problem The purpose of this survey is to introduce the reader to inverse obstacle scattering problems for timeharmonic waves by considering the case of two dimensions, i.e., scattering of timeharmonic acoustic or electromagnetic waves by infinitely long cylindrical obstacle...
Parametric level set methods for inverse problems, (anticipated
, 2010
"... In this paper, a parametric level set method for reconstruction of obstacles in general inverse problems is considered. General evolution equations for the reconstruction of unknown obstacles are derived in terms of the underlying level set parameters. We show that using the appropriate form of para ..."
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Cited by 2 (2 self)
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In this paper, a parametric level set method for reconstruction of obstacles in general inverse problems is considered. General evolution equations for the reconstruction of unknown obstacles are derived in terms of the underlying level set parameters. We show that using the appropriate form of parameterizing the level set function results a significantly lower dimensional problem, which bypasses many difficulties with traditional level set methods, such as regularization, reinitialization and use of signed distance function. Moreover, we show that from a computational point of view, low order representation of the problem paves the path for easier use of Newton and quasiNewton methods. Specifically for the purposes of this paper, we parameterize the level set function in terms of adaptive compactly supported radial basis functions, which used in the proposed manner provides flexibility in presenting a larger class of shapes with fewer terms. Also they provide a “narrowbanding ” advantage which can further reduce the number of active unknowns at each step of the evolution. The performance of the proposed approach is examined in three examples of inverse problems, i.e., electrical resistance tomography, Xray computed tomography and diffuse optical tomography.