Development of an
algorithm for diffraction tomography based on
sensitivity analysis and the optimization of
Gonzalo R. Feijoo
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.