Object Restoration by Zero Location

Abstract

The relationship between the location of zeros of scattered fields and the corresponding object information in one dimension is investigated using the theory of entire functions. Each zero of a Hadamard factor in the scattered field is shown to contribute a complex harmonic in the object space. Using this formalism, we investigate how noise in the scattered field and data confined to a spectrum of finite extent affect object reconstruction and show that this inverse process is unstable. We indicate how the task of input restoration can be regularized by introducing a global parametrization containing a modulus bound and expressing the total energy of the object wave in terms of the zero configuration in the scattered field. This leads to object reconstruction by solving a set of nonlinear algebraic equations. The method is illustrated with a computer simulation of a practical example.