Iterative approaches using forward solver

In conventional cost-function approach, which is the one followed in this thesis, minimizes the differences between measured scattered fields (only outside the OI) and the scattered fields that are calculated from a possible solution. This approach is computationally complex because the system of equations has to be built at each iteration. In order to calculate the scattered field, when the electric contrast or the size of the OI is small, one may use the well-known Born approximation [69]. Another popular approximation is the Rytov approximation [69]. The Rytov approximation gave better results in imaging an object with high contrast. Basically, the result of linearizing the inverse problem is a significant loss of accuracy in predicting dielectric properties. These methods are computationally efficient and can obtain images in a short runtime, but they usually fail when a complex media with high contrast scatterers

exist [70–72]. Forward solver based on Integral Equations (IE) such as the the Born Iterative Method (BIM) [73], Distorted Born Iteration Method (DBIM) [74], Local Shape Function (LSF) [75,76] have also been implemented. Various attempts were made to reduce the reconstruction problem complexity by taking into account different approximations and simplifications, such as the dual-mesh scheme [77], conformal mesh reconstruction [23], adjoint technique [24], frequency-hopping reconstruction algorithms [78], and iterative multi-scaling approach [79–81]. In addition, methods for solving nonlinear Partial Differential Equations (PDE) such as Finite-Difference Time-Domain(FDTD) method appears to be more appropriate model for the EM scattered fields and it has been used in this thesis.

In order to minimize the cost-function and retrieve the unknown objects from the measurements, different deterministic (local optimization) and stochastic (global optimization) approaches have been proposed.

i. Deterministic approaches

These techniques proceed by minimizing the cost-function using the Newton type minimizations. These techniques always require the selection of some kind of regular- ization terms. Examples of such deterministic algorithms are the Modified-Newton method [20,82], Gauss-Newton (GN) inversion [83–87], Inexact-Newton (IN) [88,89], Quasi-Newton method [90], Newton-Kantorovich (NK) [91], and Levenberg-Marquardt (LM) inversion [92]. The Gauss-Newton technique (or NK method) is also equivalent

to the DBIM, as shown in [93]. The ill-posedness is usually treated by employing different regularization techniques. Various regularization techniques such as Tikhonov Regularization [54,74,92,94–98], Krylov subsurface regularization [20,99], Maxwell regularizer (physical regularizer) [57], and MR [84] have been used. These traditional regularization methods, which facilitate the inversion of ill-conditioned matrices are application-independent, which enables these methods to be used for a variety of applications. In addition, these traditional regularizations work well when only a few scatterers with small difference in dielectric properties (contrast) exist. From a computational point of view, deterministic techniques are attractive, however, they can be trapped in local minimum. This means that the local-based optimization imaging techniques are only accurate if the starting trial solution is not far from the real solution or the regularization keeps the search around the global minimum. In many practical cases, it is not possible to guess the proper initial point or regular- ization term, and therefore some inaccuracies in the resulting reconstructed image may appear. In terms of number of frequencies, both multiple frequencies [100–103] or single frequency [76,104–106] approach have been used. The main advantages of deterministic algorithms is their convergence speed. This imaging procedure works well when only a few scatterers with small difference in dielectric properties (contrast) exist. Including a-priori information in theses approaches is quite complex.

ii. Stochastic approaches

In contrast to deterministic approaches, a number of global optimization methods have been utilized in solving non-linear inverse scattering problems. The stochas- tic approaches are potentially able to obtain global minimum which most probably corresponds to a true solution. Without dependency on initial guess they are a better choice when multiple scatterers inside heterogeneous objects are presented. The stochastic approaches include the stochastic search base, such as the Simulated Annealing techniques [107,108], Ant Colony Optimizer (ACO) [109] and population- based evolutionary algorithms such as Neural-Networks (NN) [110], Genetic Algo- rithms (GAs) [13,111–118], Differential Evolution Strategy (DES) [119–122], Particle Swarm Optimization (PSO) [123–126], and more recently the Evolutionary Algorithms (EAs) [127]. These global optimization methods can be evaluated based on different parameters such as the ability to deal with complex cost-functions, the simplicity of use, the number of control parameters, convergence rate, and the possibility of the exploitation of the parallelism by modern PC clusters. One of the advantages of using global optimization methods is that they can escape from local minima through ran- domization, and there is no need for the rigorous regularization (which often results in smoothing effects). Furthermore, including somea-priori information such as physical and geometrical structure is quite easy and very flexible. Despite all the advantages the heavy computational load is a major drawback inherent in stochastic approaches. Therefore, mainly they have been utilized in Two-Dimension (2D) imaging approaches;

however, there has been same efforts initiated for implementing Three-Dimensional (3D) imaging using these approaches [64,96,128–130].

iii. Hybrid approaches

Besides “bare” techniques, a number of hybrid approaches have been implemented to improve the convergence and accuracy. Basically, hybrid methods are integrating the stochastic and deterministic approaches. Moreover, the stochastic approach starts from trial solution to find the right solution and then the deterministic approach starts from this initial data and the solution is quickly reached. Some examples for these approaches include the hybrid GA and LM [131] and hybrid GA and CG [113]. In [132] we used the multi-resolution strategies and zooming procedure with hybridization of qualitative and quantitative techniques in order to enhance spatial resolution only in those regions of interest (this hybrid technique will be explained in Chapter 3). Hybrid methods also include the combination of two stochastic methods such as GA and NN [133] or two deterministic methods such as hybrid extended Born approximation and a gradient procedure [134]. Furthermore, the hybrid methods can be devised by combining the qualitative and quantitative stochastic method such as the hybrid of linear sampling and Ant Colony [135,136]. The integration of the stochastic and deterministic methods can be made stronger optimization. For example, the Memetic Algorithm (MA) [137–139] is the result of combining the stochastic and deterministic methods.