Shape reconstruction from moments: Theory, algorithms, and applications

In many areas of science and engineering, it is of interest to find the shape of an object or region from indirect measurements. For instance, in geophysical prospecting, gravitational or magnetic field measurements made on the earth's surface are used to detect an oil reservoir deep inside the earth. In a different application, X-ra;li attenuation measurements are used in Computer Assisted Tomography (CAT) to reconstruct the shape and density of biological or inorganic materials for diagnostic and other purposes. It turns out that in these two rather disparate areas of application, among many others, the partial information can actually be distilled into moments of the underlying shapes we seek to reconstruct. Moments of a shape convey geometric information about it. For instance; the area, center of mass, and moments of inertia of an object give a rough idea of how large it is, where it is located, how round it is, and in which direction it is elongated. For simple shapes such as an ellipse, this information is sufficient to uniquely specify the shape. However, it is well-known that, for a general shape, the infinite set of moments of the object is required to uniquely specify it. Remarkable exceptions are simple polygons, and a more general class of shapes called quadrature domains that are described by semi-algebraic curves. These exceptions are of great practical importance in that they can be used to approximate, arbitrarily closely, any bounded domain in the plane. In this paper, we will describe our efforts directed at developing the mathematical basis, including some stable and efficient numerical techniques for the reconstruction of (or approximation by) these classes of shapes given "measured" moments.