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Development of an algorithm for diffraction tomography based on sensitivity analysis and the optimization of a topology

Gonzalo R. Feijoo
Sandia National Laboratories
California
November 14, 2003
 
In this talk I will present two methods for the reconstruction of the position and shape of scatterers in an acoustic domain. These methods are inspired on ideas from shape optimization. In the first part of the talk, the problem of reconstructing the shape of a single rigid scatterer will be posed as a constrained optimization problem with the shape as the “design variable” (borrowing terminology from mathematical programming) and the acoustic equations as the constraint. The cost functional is the mismatch between the measurements and the results from a simulation. An iterative method is proposed to solve this optimization problem efficiently.
 
In the second part of the talk, the previous approach is extended to consider the more general problem where the number of scatterers in the domain is not known. A new method is proposed which relies on the definition of a function, called the topological derivative, that has support in the image and at every point quantifies the sensitivity (or derivative) of the scattered field to the introduction of an infinitesimal scatterer at that point. This function is an extension of the concept of shape differentiation that is used by the first algorithm. It will be shown that the expression for the topological derivative can be calculated analytically. As a result, the proposed scheme is not iterative. For both methods, no assumptions like the Born or Rytov approximations are made. It will be shown through several numerical experiments that this last method is capable of reconstructing both the position and the shape of scatterers in the domain with very good quality.
 

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