A layer-stripping procedure for solving three-dimensional Schrödinger equation inverse scattering problems is developed. This method operates by recursively reconstructing the potential from the jump in the scattered field at the wave front, and then using the reconstructed potential to propagate the wave front and the scattered field further into the inhomogeneous region. It is thus a generalization of algorithms that have been developed for one-dimensional inverse scattering problems. Although the procedure has not yet been numerically tested, the corresponding one-dimensional algorithms have performed well on synthetic data. The procedure is applied to a two-dimensional inverse seismic problem. Connections between simplifications of this method and Born approximation inverse scattering methods are also noted.
The performance of the eigenimage filter is compared with those of several other filters as applied to magnetic resonance image (MRI) scene sequences for image enhancement and segmentation. Comparisons are made with principal component analysis, matched, modified-matched, maximum contrast, target point, ratio, log-ratio, and angle image filters. Signal-to-noise ratio (SNR), contrast-to-noise ratio (CNR), segmentation of a desired feature (SDF), and correction for partial volume averaging effects (CPV) are used as performance measures. For comparison, analytical expressions for SNRs and CNRs of filtered images are derived, and CPV by a linear filter is studied. Properties of filters are illustrated through their applications to simulated and acquired MRI sequences of a phantom study and a clinical case; advantages and weaknesses are discussed. The conclusion is that the eigenimage filter is the optimal linear filter that achieves SDF and CPV simultaneously.
Abstruct-Layer stripping algorithms for inverse scattering problems are very fast but have the reputation of being numerically unstable, especially when applied to noisy data. The goal of this paper is to provide an explicitly discrete framework for layer stripping algorithms for the two-dimensional (2-D) Schrodinger equation inverse scattering problem. We determine when 2-D layer stripping algorithms are numerically stable, explain why they are stable, and specify exactly the (discrete) problem they solve when they are stable. We reformulate the 2-
D Schrodinger equation as a multichannel two-component wave system by Fourier transforming the Schrodinger equation in the lateral spatial variable. Discretization results in new 2-D layer stripping algorithms which incorporate multichannel transmission effects; this leads to an important new feasibility condition onimpulse reflection response data for stability of these algorithms. A 2-D discrete Schrodinger equation is defined, and analogous results are obtained. Numerical examples illustrate the new results, especially how rendering noisy data feasible stabilizes layer stripping algorithms.
The Schur algorithm and its times-domain counterpart, the fast Cholseky recursions, are some efficient signal processing algorithms which are well adapted to the study of inverse scattering problems.These algorithms use a layer stripping approach to reconstruct a lossless scattering medium described by symmetric two-component wave equations which model the interaction of right and left propagating waves.In this paper, the Schur and fast Cholesky recursions are presented and are used to study several inverse problems such as the reconstruction of nonuniform lossless transmission lines, the inverse problem for a layered acoustic medium, and the linear least-squares estimation of stationary stochastic processes. The inverse scattering problem for asymmetric two-component wave equations corresponding to lossy media is also examined and solved by using two coupled sets of Schur recursions. This procedure is then applied to the inverse problem for lossy transmission lines.-3-
Numerical results of implementing a two-dimensional layer stripping algorithm to solve the two-dimensional Schrodinger equation inverse potential problem are presented and discussed. This is the first exact (all multiple scattering and diffraction effects are included) numerical solution of a multi-dimensional Schrtidinger equation inverse potential problem, excluding optimization-based approaches. The results are as follows: (1) the layer stripping algorithm successfully reconstructed the potential from scattering data measured on a plane (as it would be in many applications); (2) the algorithm avoids multiple scattering errors present in Born approximation reconstructions; and (3) the algorithm is insensitive to small amounts of noise in the scattering data. Simplifications of layer stripping and invariant imbedding algorithms under the Born approximation are also discussed.
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