In this dissertation, a spectral inverse technique, applying the moment method procedure with a series expansion for the induced field in each enlarged cell, is suggested to reconstruct permittivity profiles of inhomogeneous dielectric objects and to reduce the ill-posedness inherent to inverse scattering problems.
In order to make electromagnetic imaging feasible in practical applications, it is indispensable to obtain an adequate reconstruction algorithm accounting for the effects of refraction and diffraction accurately. By virtue of the rapid developments of high speed and large-storage computers, an inversion scheme in the configuration domain has been obtained by applying the moment-method procedures of direct scattering problems in the reverse sequence. While this inversion scheme provides superresolution and simultaneous reconstruction of complex permittivity distribution, its reconstructed profiles are affected by extremely large error even if quite a small noise contaminates the measurement field. This difficulty may be affected by ill-posedness inherent in inverse scattering problems. In order to analyze its physical reasons and then regularize its effect suitably, the above inversion scheme is modified to be applicable in the spectral domain. The key advantage in the spectral scheme is that several variables involved in the inverse scattering problem such as the measurement location, the basis function, the cell geometry, etc. are explicitly separated. Since this ill-posedness in the inversion scheme becomes worse as the discretization of the dielectric object becomes finer, spectral inversion scheme by using the moment-method procedures is expected to reduce the large reconstruction error by enlarging the size of each cell.
A spectral inversion scheme, based on the moment-method procedure with series-expanded field within each cell by dividing the dielectric object into a small number of rectangular cells with large area, is developed to regularize the ill-posedness inherent in inverse scattering problems. One of the interesting features on the presented scheme is that the relative dielectric constant may be obtained by averaging over each cell. This averaging is expected to play an important role in regularizing the high-frequency effect due to noise.
By performing numerical simulations for various type of dielectric objects when no noise is contaminated the scattered field, it is demonstrated that this inverse technique provides close reconstruction of permittivity profiles regardless of measurement location, the size and shape of dielectric object and permittivity profile. Compared to the previous scheme of the pulse basis, the presented scheme is shown to reduce significantly the computation time.
When Gaussian random noise contaminates the scattered field, the reconstruction by the pulse basis scheme render it impossible to recognize the given profile due to a number of large fluctuations in the reconstructed profile. It is well known that if the size of each cell decreases, the more accurate high spatial-frequency components of the scattered field are required to reconstruct the super-resolution profile. But, since the higher spatial-frequency components are very sensitive to the contaminated noise, large errors in the reconstructed profiles are unavoidable if negligible noise in the scattered field. This difficulties are called the ill-posedness inherent in inverse scattering problems. Hence, enlarging the size of each cell may be considered as one of compromise on the regularization of the noise effect with less fine resolution. Therefore, the reconstruction of larger cell approach yields less fluctuation than that of pulse basis. Furthermore, it is interesting to note that the ill-posedness is reduced even more by employing the averaging over the cell. Since finite truncation of the expansion coefficients of field distribution in each cell results in larger reconstruction error near the cell boundary, weighting the value near center of cell more strongly than the cell boundary provides good results.
Reduction of ill-posedness by the enlargement of the cell size and the averaging over the cell with a suitable weighting function is expected to play an important role in regularizing the noise effect for the more complicated inverse scattering problems.
