Solving Inverse Scattering Problems Using Truncated Cosine Fourier Series Expansion Method

The aim of inverse scattering problems is to extract the unknown parameters of a medium from measured back scattered fields of an incident wave illuminating the target. The unknowns to be extracted could be any parameter affecting the propagation of waves in the medium. Inverse scattering has found vast applications in different branches of science such as medical tomography, non-destructive testing, object detection, geophysics, and optics (Semnani & Kamyab, 2008; Cakoni & Colton, 2004). From a mathematical point of view, inverse problems are intrinsically ill-posed and nonlinear (Colton & Paivarinta, 1992; Isakov, 1993). Generally speaking, the ill-posedness is due to the limited amount of information that can be collected. In fact, the amount of independent data achievable from the measurements of the scattered fields in some observation points is essentially limited. Hence, only a finite number of parameters can be accurately retrieved. Other reasons such as noisy data, unreachable observation data, and inexact measurement methods increase the ill-posedness of such problems. To stabilize the inverse problems against ill-posedness, usually various kinds of regularizations are used which are based on a priori information about desired parameters. (Tikhonov & Arsenin, 1977; Caorsi, et al., 1995). On the other hand, due to the multiple scattering phenomena, the inverse-scattering problem is nonlinear in nature. Therefore, when multiple scattering effects are not negligible, the use of nonlinear methodologies is mandatory. Recently, inverse scattering problems are usually considered in global optimization-based procedures (Semnani & Kamyab, 2009; Rekanos, 2008). The unknown parameters of each cell of the medium grid would be directly considered as the optimization parameters and several types of regularizations are used to overcome the ill-posedness. All of these regularization terms commonly use a priori information to confine the range of mathematically possible solutions to a physically acceptable one. We will refer to this strategy as the direct method in this chapter. Unfortunately, the conventional optimization-based methods suffer from two main drawbacks. The first is the huge number of the unknowns especially in 2-D and 3-D cases 22