A stable numerical method for inverting shape from moments. (English) Zbl 0956.65030

Golub, Gene H.; Milanfar, Peyman; Varah, James

SIAM J. Sci. Comput. 21, No. 4, 1222-1243 (1999).

The reconstruction of a polygon in the complex plane from its moments is studied. The paper presents a stable method. In its outline (a Hankel-matrix generalized eigenvalue problem), the key points are the clarification of the sensitivity (with numerical illustrations) and improved conditioning (via shifts of moments and their diagonal scaling). Motivation is beyond any doubt: Similar problems are currently very ill conditioned though often needed in practice. In order to underline that, an explicit application is offered at the end (in a geophysical reconstruction of an anomalous domain from its gravimetric measurements) and several others are listed (notably: tomography).
In a historical comment a duality of the problem in question to a 2-D numerical quadrature (in a way generalizing the Motzkin and Schoenberg formula for triangular regions) is pointed out, and a remark is added recalling its connection (and a broader grasp in comparison) with the matrix pencil solution of certain signal decomposition problems.

Reviewer: Miloslav Znojil (Řež)

MSC:

65F15	Numerical computation of eigenvalues and eigenvectors of matrices
44A60	Moment problems
65F35	Numerical computation of matrix norms, conditioning, scaling
65E05	General theory of numerical methods in complex analysis (potential theory, etc.)
94A12	Signal theory (characterization, reconstruction, filtering, etc.)
86A20	Potentials, prospecting

Keywords:

inverting shape frome moments; Hankel-matrix generalized eigenvalue problem; conditioning; geophysical reconstruction; scaling; gravimetric measurements; matrix pencil solution; signal decomposition problems

Software:

mctoolbox