Inverse scattering and shape reconstruction.

Investigations of new and improved solutions to inverse problems are considered. Three of the solutions are concerned with inverse scattering. The other two solutions deal with reconstructing binary images from few projections and determining the shape and orientation of a three-dimensional object from silhouettes. In addition, a review of solutions to direct and inverse scattering problems is presented. An inverse scattering algorithm for reconstructing variable refractive index distributions is examined. The inversion algorithm is based on an expression for the wave function which explicitly incorporates the inverse scattering data. It is claimed that this considerably increases the efficiency of the algorithm. The algorithm is implemented in two-dimensional space and examples of reconstructions of objects from computer-generated scattering data are presented. The problem of determining the shape of a two-dimensional impenetrable obstacle from a set of measurements of its far-field scattering amplitude is considered. The problem is formulated as a non-linear operator equation which is solved by an iterative method. The use of the null-field method to solve the direct problem leads to efficient evaluation of the Fréchet derivative of the non-linear operator. Computational implementations confirm the numerical accuracy of the algorithm. An extension to the Rayleigh-Gans (Born) approximation is examined. The extension involves incorporating a high frequency approximation to the wave field into the conventional Rayleigh-Gans (Born) approximation. Numerical implementation of an algorithm based on this extension to the Rayleigh-Gans (Born) approximation indicates that its reconstruction accuracy is generally superior to that of the conventional Rayleigh-Gans (Born) approximation. Efficient algorithms for reconstructing a binary cross-section (each of whose pixel amplitudes is either zero or unity) from few one-dimensional projections are introduced and illustrated by example. It is shown that only two projections are needed to reconstruct a convex cross-section. Non-convex cross-sections need more projections but far fewer than are necessary to reconstruct grey-scale images. When presented with noisy one-dimensional projections, the algorithms remain useful, although their performance improves with the number of given projections. Determination of a three-dimensional object's shape and orientation from its silhouettes is studied, on the understanding that the relative orientations of the given silhouettes are unknown a priori. The result of this study is an algorithm which could be suitable for incorporation into a robot's vision system. The algorithm is based on a method for determining the orientation of an object from its two-dimensional projections. To overcome the reduced information content of silhouettes as compared with two-dimensional projections, a self consistency check is introduced. Numerical implementations of the algorithm confirm that it can generate usefully accurate estimates of the orientations and shapes of technologically non-trivial objects