In the electromagnetic inverse diffraction problem, the first order Born approximation in the integral equation for the scattered field may be used. It gives a well-known Fourier transform relation eliminating the nonlinearity between the scattered fields and the physical properties of objects by replacing the total field inside the integral by the incident field. In the last decade, this linearizing approximation has been intensively studied in the medical imaging and nondistructive evaluation of materials since it offers numerically efficient and stable inversion algorithm. However, it is limited in the weakly scattering object.
In this dissertation, the limitation of the first order Born approximation is explicitly shown by using the reconstruction by projection, and an improved Born inversion for dielectric profile reconstruction of dielectric cylinder is suggested. It improvess the validity of the first order Born approximation up to about ten times in the multiplication of the size of the object and the square root of the relative dielectric constant minus one.
One may introduce a projection function defined as the one dimensional inverse Fourier transform of the scattered far fields measured for multi-frequency incident waves. Then the convolution back-projection scheme, well-known in the X-ray computerized tomography, may be used to obtain the cross-sectional images or the quantitative distribution of the relative dielectric constants in the cross-section. The projection function obtained from the inverse Fourier transform is larger than the real size since the propagation velocity of the electromagnetic wave inside the dielectric object is slow due to higher dielectric constant. The projection may be deformed from the refraction and diffraction of the wave within the object. In the case of weak scattering objects, the slowness and deformation is sufficiently small and may be neglected. However, for the strong scattering objects, this phenominon is pronounced and the projection function deviates from the original one, which limits the validity of the first order Born approximation.
The projection obtained through the inverse Fourier transform of the time-harmonic fields is basically equal to the range profile due to the irradiation of the impulse plane wave on the objects, since impulse is consisted of infinitely many time-harmonic fields. Therefore, one may find the starting points of the object and the external boundary of the object by rotating the object. One may correct the extended projection obtained from the direct inverse Fourier transform of the scattered fields by the measured external boundary. It is shown via this correction that the reconstructed image closely predicts the original up to about 10 times strong scatterer than the ordinary Born inversion. Numerical examples and the X-band microwave measured reconstruction through this improved Born inversion are presented to compared with the ordinary Born inversion.
Furthermore, it is shown that the similar improvent may also be possible for the first order Rytov inversion that is compared with the first order Born inversion. The actual inversion system such as the water-immersed imaging system and the subsurface radar is operated in the dissipated medium and the effects of the dissipativeness of the background medium and scatterer are investigated by using the concept of the point spread function in the band-limited system. Bi-static scheme is also included in its Fourier transformation formulation and its improved Born reconstruction is tested numerically.