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'Learning Ace makes it easier for me focus on what I want to learn and not what the average search engine suggests I should learn.' Radio Science, Volume 31, Number 1, Pages 51-65, January-February 1996 Wavelet-based methods for the nonlinear inverse scattering problem using the extended Born approximation Eric L. Miller Center for Electromagnetics Research, Department of Electrical and Computer Engineering Northeastern University, Boston, Massachusetts Alan S. Willsky Laboratory for Information and Decision Systems, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts Abstract. In this paper, we present an approach to the nonlinear inverse scattering problem using the extended Born approximation (EBA) on the basis of methods from the fields of multiscale and statistical signal processing. By posing the problem directly in the wavelet transform domain, regularization is provided through the use of a multiscale prior statistical model. Using the maximum a posteriori (MAP) framework, we introduce the relative Cram r-Rao bound (RCRB) as a tool for analyzing the level of detail in a reconstruction supported by a data set as a function of the physics, the source-receiver geometry, and the nature of our prior information. The MAP estimate is determined using a novel implementation of the Levenberg-Marquardt algorithm in which the RCRB is used to achieve a substantial reduction in the effective dimensionality of the inversion problem with minimal degradation in performance. Additional reduction in complexity is achieved by taking advantage of the sparse structure of the matrices defining the EBA in scale space. An inverse electrical conductivity problem arising in geophysical prospecting applications provides the vehicle for demonstrating the analysis and algorithmic techniques developed in this paper. 1. Introduction The desire to characterize the composition of a medium on the basis of observations of scattered ra- diation is a common problem in a variety of applica- tion areas [Kak and Slaley, 1987, Bates et al., 1991, Torres-Verdi'n and Hahashy, 1994]. Despite the ubiq- uity of such inverse scattering problems, generating a solution can be quite difficult because of the com- putational burden associated with the nonlinearity of the problem and the fact that these problems are highly ill posed. In this paper, we present a col- lection of methods for overcoming these difficulties based upon techniques drawn from the disciplines of multiscale and statistical signal processing. We em- Copyright 1996 by the American Geophysical Union. Paper number 95RS03130. 0048-6604/96/95RS-03130508.00 ploy estimation-theoretic analysis techniques to iden- tify those degrees of freedom in a wavelet representa- tion of the quantity to be reconstructed for which the data provide significant information. Such a formula- tion represents a natural framework for the analysis of issues such as the trade-off between reconstruction accuracy and resolution, as well as the development of bounds on our ability to localize spatial anomalies in the region of interest. Direct incorporation of this information into a nonlinear inversion algorithm cou- pled with a multiscale implementation of the forward scattering model results in substantial computational savings with little loss in reconstruction fidelity. We apply our method to an inverse electrical conductiv- ity problem encountered in geophysical exploration applications. The extended Born approximation (EBA) devel- oped by Hahashy e! al. [1993] is used to lower the computational complexity of the forward modeling portion of our inverse scattering algorithm. The 51 . Their power lies in the fact that they only require a small number of coefficients to represent gen- eral functions and large data sets sparsity The authors are listed alphabetically. Email: {richb, volkan, duarte, chinmay}@rice.edu, Web: dsp.rice.edu/cs. This work was supported under N00014-03-1-0444. Department of Mathematics and Computer Science, Clarkson University, Potsdam, NY 13699-5815 (bolltem@clarkson.edu). The work of this author has been -based methods and relate them to a first stage of point/local detection and a second stage of extended pattern detection. One sparsity The authors are listed alphabetically. Email: {richb, volkan, duarte, chinmay}@rice.edu, Web: dsp.rice.edu/cs. This work was supported Learning Ace is a 100% student-focused web destination that leverages our proprietary adaptive search and recommendation technologies and our instructor/publisher relationships to develop an insanely valuable resource to students at each learning point of their coursework. Discover, Browse and Learn. Source.


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