Object Restoration by Zero Location

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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.

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