Reconstructing polygons from moments with connections to array processing

Abstract

In this paper we establish 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. 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 to estimate 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. We show that the problem of polygonal vertex reconstruction from moments can in fact be posed as an array processing problem, and taking advantage of this relationship, we derive and illustrate several new algorithms for the reconstruction of the vertices of simply-connected polygons from moments.

Fourier, Gauss, Fraunhofer, Porod and the shape from moments problem

“…This result is also related to the so-called "shape from moments" problem which is to find the ordered vertices of a polygon given an appropriate set of the polygon moments [4] [5]. As shown by Milanfar [4], in the complex plane with z = x + iy, the result of Davis can be written…”

“…In the second step we have used (5) and in the last step the definition of h (z) in terms of its Fourier transform. The β's cancel in the coefficient of the exponents in line three and hence we reproduce the result of Davis that the integral over a polygon of ∂ 2 z h (z) is the sum of h (z) evaluated at the vertices of the polygon times coefficients which depend only on the vertices and not on h (z) .…”

“…We use (5) to show that the volume in three dimensions of a parallelpiped formed by three vectors a i is given by det [a ij ] where a ij the 3×3 matrix of the components of the a i , i.e,.the first column is a 1i , the second a 2i and the third a 3i with of course, i = 1, 2, 3. If the a i are infinitesimals given by d R i = ∂ si Rds i with no sum on i, then det [dR ij ] d 3 s forms the fundamental volume element dv for integration.…”

“…From a probabilistic point of view the Fourier transform of a shape properly normalized can be considered as the characteristic function or moment generating function of the shape. Hence the Fourier transform of the shape is intimately related to the moments of the shape and hence to the "shape from moments problem", which has been studied recently for polygons by Golub, Milanfar and others [4] [5] and is related to the overall problem of pattern recognition. The Fourier transform of an area bounded by a smooth curve or a polygon in the plane can be shown to imply Hopf's Umlaufsatz [6], which states that the tangent vector (or the normal vector or any linear combination thereof) to a smooth or piecewise smooth closed orientable simply connected curve in the plane rotates by ±2π as it goes completely around the curve.…”

Shape from moments as an inverse problem

“…An intriguing inverse problem proposed in [9] suggests the reconstruction of a planar polygon from a set of its complex moments. Considering an indicator function being 1 in the interior of the polygon and 0 elsewhere, these moments are global functions created by integrating the power function z k over the plane, and weighted by this indicator function.…”