Quantitative Reconstruction of Complex Permittivity Distributions by Means of Microwave Tomography

This paper concerns an iterative method for solving an inverse electromagnetic scattering problem: the quantitative reconstruction of complex permittivity distribution of inhomogeneous objects. During the last decade, intensive research and interest have been focussed on reconstruction algorithms based on the so-called Diffraction Tomography with applications in microwaves or ultrasound [1–16]. The main motivations for using such approaches are to obtain explicit formulas for solving the imaging problem, to take advantage of existing numerical algorithms (Fast Fourier Transform) and to implement the algorithms on PC’s or minicomputers for imaging system purposes (planar microwave camera [5], experiments with ultrasounds and microwaves [11][13][14][16], cylindrical microwave system [15]). However, Diffraction Tomography is subjected to various limitations which include artefacts due to diffraction effects in strongly inhomogeneous media [7][10][16][17][18].The final objective of microwave imagery is to determine the complex permittivity profile (permittivity and conductivity profiles) of the object under investigation. An increasing number of papers [17–25] have been devoted to a such non-linear inverse problem. Different solutions based on Moment Methods [19–25] have been explored, but, convergence depends on contrasting objects. Furthermore, stability is very sensitive to the observation point locations and measurement accuracy (due to the ill-conditioning of the matrix which has to be inverted). Stability and sensivity are why the use of an iterative scheme is important: effects of ill-conditioning can be significantly reduced by enforcing the convergence with “a priori” information (object external shape, upper and lower bounds of complex permittivity, presence of different media,…). Concerning the uniqueness of object reconstruction, it has been proved [24] that if the background medium is dissipative, then the inverse scattering problem has a unique solution.