Focusing and imaging using eigenfunctions of the scattering operator

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Abstract
An inverse scattering method that uses eigenfunctions of the scattering operator is presented. This approach provides a unified framework that encompasses eigenfunction methods of focusing and quantitative image reconstruction in arbitrary media. Scattered acoustic fields are described using a compact, normal operator. The eigenfunctions of this operator are shown to correspond to the far-field patterns of source distributions that are directly proportional to the position-dependent contrast of a scattering object. Conversely, the eigenfunctions of the scattering operator specify incident-wave patterns that focus on these effective source distributions. These focusing properties are employed in a new inverse scattering method that represents unknown scattering media using products of numerically calculated fields of eigenfunctions. A regularized solution to the nonlinear inverse scattering problem is shown to result from combinations of these products, so that the products comprise a natural basis for efficient and accurate reconstructions of unknown inhomogeneities. The corresponding linearized problem is solved analytically, resulting in a simple formula for the low-pass-filtered scattering potential. The linear formula is analytically equivalent to known filtered-backpropagation formulas for Born inversion, and, at least in the case of small scattering objects, has advantages of computational simplicity and efficiency. A similarly efficient and simple formula is derived for the nonlinear problem in which the total acoustic pressure can be determined based on an estimate of the medium. Computational results illustrate focusing of eigenfunctions on discrete and distributed scattering media, quantitative imaging of inhomogeneous media using products of retransmitted eigenfunctions, inverse scattering in an inhomogeneous background medium, and reconstructions for data corrupted by noise.

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