Shapes of large conducting objects are reconstructed from the scattered fields by using the physical optics approximation for large objects. For smaller objects where the physical optics approximation may not be applied, iterative methods utilizing Newton-like or Simulated Annealing algorithm may be applied to obtain the shape function by minimizing the cost function defined by the summation over the squared magnitude of the difference between the measured field and the field calculated from the assumed shape of the conductor. With multiple incidences and much more data points than unknowns of the shape function, the inversion is stabilized in this iterative method.
Iterative inverse scattering using Levenberg-Marquardt(LM) algorithm, which is one of the steepest descent Newton-like algorithms, suffers from the proper choices of the initial shape and its center since this process may be trapped in one of the local minima of the cost function. One may shows the profiles of these local minima of the cost function by defining the cost function in terms of the angular modal coefficient of the scattered field for a circular conducting cylinder. One may show numerically that there exists many local minima, that the depth of the local minima increases, and its interval decreases as the number of angular modes of the measured scattered field decreases. This requires the initial shape closer to the original shape, otherwise the iterative inversion traps into one of the local minima. Noise in the scattered field is shown to make the global minimum and the local minima shifted and its interval and shapes distorted.
It is shown that one needs all the effective propagating modes and multiple incident waves to illuminate the object without shadow spots in order to have the stable inversion in presence of noise. Addition of the exponentially small higher order modes, however, does not improve the stability of the inversion.
The number of the effective propagating modes is proportional to the maximum radius of the conducting object from the coordinate origin. Utilizing this principle, one may find the center of the scatterer by shifting the origin of the coordinate system such that it gives the minimum number of effective propagating modes. With two orthogonal incident waves, one may find the approximate elliptic shape of the conductor from the respective total scattering cross section, which is used as the initial shape for the iterative method
With the center and the initial shape estimated from the measured scattered field, one may show numerically that LM algorithm reconstructs a concave conducting cylinder of its cross section up to 1.5$\lambda$ by 1.5$\lambda$ , where $\lambda$ is the free space wavelength. It is shown that closer initial shape is needed to reconstruct its original shape for the larger object by LM algorithm, since it may trap into one of the local minima of the cost function.
Simulated annealing(SA) algorithm is useful in the reconstruction of more complex scatterer where the center and the initial shape are difficult to find closely to the original scatterer, such as the multiple conductors. Its computing time and reconstruction accuracy is improved by employing a hybrid algorithm of SA and LM.
