by Simone Genovesi and Emanuele Salerno
At the Signal and Images lab, ISTICNR, we are developing a new algorithm to reconstruct the permittivity range profile of a layered medium from microwave backscattering data. The algorithm is based on a particle swarm strategy to optimize a specific edgepreserving objective functional. Our technique is able to efficiently find the global optimum of the objective functional, while preserving the discontinuities in the reconstructed profile.
Inverse scattering is relevant to a very large class of problems, where the unknown structure of a scattering object is estimated by measuring the scattered field produced by known probing waves. As illustrated in Figure 1, several application areas are involved, such as civil and industrial engineering, nondestructive testing, medical imaging and subsurface inspection. However, a truly satisfactory solution to the inverse scattering problem has not so far been found owing to its intrinsic illposedness. Common inversion techniques are either highly sensitive to noise or produce oversmoothed profiles by enforcing global constraints. This is a major drawback, since the discontinuities carry essential information on possible heterogeneous inclusions in the building material, and maintaining them in the reconstructed profile is very important. Moreover, the nonlinear relationship between the scattered field and the object function, and the robustness of the inversion algorithms, are still open issues and most scattering tomography techniques are not sufficiently reliable to solve practical problems.

Our current purpose is to reconstruct onedimensional permittivity range profiles of architectural objects from microwave backscattering data on a certain frequency range. The complete iterative procedure involves a forward solver and an optimizer. The former computes the backscattered field from the currently proposed solution. The latter uses the results from the solver plus some a priori information to drive the optimization process. We have chosen to discretize the wall into a finite number of homogeneous and lossless layers of equal thickness (see Figure 2). It is assumed that the total thickness of the wall is known and the incidence of the probing waves is normal.

In order to estimate the wall permittivity as a function of the depth coordinate z, we build a functional containing two distinct terms. The first is a suitable distance between the measured and the calculated backscattering data. The second is a combination of a quadratic, firstorder, smoothness constraint and an explicit smoothnessbreaking term, which preserves possible abrupt permittivity variations where these are likely to occur.
Our unknowns are the permittivity values of all the layers and the locations of possible discontinuities. To optimize the objective functional, we are implementing a particle swarm algorithm. Particle swarm optimization (PSO) is an evolutionary computation technique inspired by the social behaviour of flocks of birds and swarms of insects. In PSO, each solution is represented by an agent, or particle, which explores the multidimensional solution space.
During the search procedure, the agents change their positions over time by flying around in the solution space. Since we adopt a fully connected swarm topology, each agent knows the best location found by the rest of the swarm, and is able to adjust its velocity according to its own experience and the experience of all the other agents. As a result, any agent is stochastically attracted towards both its own best location and the best location found by the swarm, and is able to evaluate the status of its current location. The initialization of the position and velocity of each agent is random. We are now running some preliminary simulations to test different swarm sizes and other initialization procedures, and to tune the parameters involved in the velocity update (eg particle inertia, and social and cognitive rate).

Our first results have been encouraging. Figure 3 shows a simulated profile with large discontinuities, and its reconstruction from noisy data as obtained by our swarm algorithm. The wall was assumed to be 24cm thick and was subdivided into twelve layers. Note that the permittivity values are large (this structure could model a concrete wall with an internal air inclusion). This means that our test object is strongly scattering, and many other inversion techniques would have failed to reconstruct it.
This research is conducted within the framework of a project financed jointly by the Italian National Research Council (CNR) and the Italian Ministry of Education and Research.
Please contact:
Simone Genovesi, ISTICNR, Italy
Email: simone.genovesiisti.cnr.it