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A new imaging method in diffraction tomography based on the topological derivative

The development and implementation of a new method in diffraction tomography, which is based on the optimization of a topology, is discussed. The method 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. If the scatterers are rigid, the expression for the topological derivative is calculated analytically. As a result, the proposed scheme is not iterative. Furthermore, no assumptions such as the Born or Rylov approximations are made. It is shown through several numerical experiments that the image produced by this function is capable of reconstructing both the position and the shape of scatterers in the domain with excellent agreement.