POCS Approach for imaging dielectric objects

In this report, I consider the problem of reconstructing a tomographic image of a dielectric object from backscattering measurements. A data acquisition system coherently measures the electric field scattering at microwave frequencies, and a linear algorithm reconstructs a band-pass version of the object function. The first-order approximations usually adopted to linearize the inverse scattering problem greatly limit the possibility of imaging the object with sufficient fidelity. For objects with sensible contrast, the linear assumptions on the scattering equations are justified if the wavelength is similar to the size of the probed object. In many cases, the spatial resolution thus obtained is not satisfactory. In our case, furthermore, the passband does not contain the low-frequency part of the object function. A solution to these problems is to adopt fully nonlinear data models. This approach, however, normally leads to very costly algorithms. Another approach is to use a linear data model and a nonlinear reconstruction algorithm. An iterative Projection Onto Convex Sets (POCS) algorithm is proposed here, which exploits the measurement data, and additional information on the compact-support, the reality and the positivity of the solutions. This algorithm is studied both theoretically and experimentally, and its relationships with the Gerchberg method and the Projected Landweber method are pointed out