Physical optics(PO) and Born approximation yield the Fourier transform relation between the object function of the scatterer and the scattered field for reconstructing images of the conducting and the dielectric objects, respectively.
For the inversion of the conducting objects, Bojarski's identity has been used, which utilizes the back-scattered field of the wide frequency band in the far zone of the object obtained by either rotating the object or the measurement point. PO approximation, however, is not valid for the low frequency signals and the limited high frequency band signals are used for the Bojarski's identity. This formulation of Bojarski is extended to two and three dimensional bistatic configurations for which the Fourier transform reconstruction is equivalent to the back-projection algorithm of the X-ray tomography and interpreted as a frequency and angular diversity.
Iterative methods utilizing the Newton-like numerical algorithms are employed to reconstruct the smaller conducting object comparable to the wavelength since it requires handling of the large-size matrix inversion. This suffers also from the proper choice of the initial shape since the iteration process may be trapped in one of the local minima of the cost function.
Angular spectral domain inversion is formulated for the reconstruction of complex permittivity profiles. This method is shown to be less sensitive than the configuration domain inversion to the ill-posedness in a sense that a small error (or noise) in the scattered field causes a large error in the inverted polarization current.
One may apply this angular spectral domain formulation to the reconstruction of the conducting object. Representing the scattered field in the closed surface surrounding the unknown object by the summation of the angular modes(cylindrical modes for the two dimensional and spherical modes for the three dimensional case), the Fourier transform of the object shape function is shown to be equal to the summation of all the phase-shifted angular modal coefficients under the PO approximation. One may truncate the angular modes by the effective propagating modes, where its number may be obtained from the maximum size of the scatterer. Since the maximum size of the scatterer may be estimated from the number of the effective propagating modes obtainable from the measured scattered field, the proper frequency band to meet the PO approximation may be selected and the minimum measurement points to acquire the effective propagating modes are obtained from the Nyquist sampling theorem.
This allows us to reconstruct the shape of the conducting object from either the far or the near scattered fields with mutiple incident waves of single or multiple frequencies. Numerical examples of the two-dimensional conducting cylinders with the different shapes are shown and compared with those of the Bojarski's method.
The reconstructed images by the angular modes are far more accurate than those of Bojarski's method. The reconstruction via the angular mode decomposition of the scattered field may be extended to the three dimensional object. Spherical angular modal expansion of the scattered field is used for the reconstruction of the conducting sphere numerically and depolarization term is shown to degrade the reconstruction quality. It is shown that how the two orthogonal polarizations of the incident electric fields improve the reconstructed image quality quantitatively.
