Polygon reconstruction from moments using array processing

Abstract

We prove a set of results showing that the vertices of any simply-connected planar polygonal region can be reconstructed from a finite number of its complex moments using array processing. In particular, we derive and illustrate several new algorithms for the reconstruction of the vertices of simply-connected polygons from moments. These results find applications in a variety of apparently disparate areas such as computerized tomography and inverse potential theory, where in the former it is of interest in estimating the shape of an object from a finite number of its projections; while in the latter, the objective is to extract the shape of a gravitating body from measurements of its exterior logarithmic potentials at a finite number of points. The applications of the algorithms, we develop, to tomography hence expose a seemingly deep connection between the fields of tomography and array processing. This connection implies that a host of numerical algorithms such as MUSIC, Min-norm, and Prony are now available for application to tomographic reconstruction problems.<>