Shape reconstruction from moments: theory, algorithms, and applications

Peyman Milanfar; Mihai Putinar; James Varah; Bjoern Gustafsson; Gene H. Golub

doi:10.1117/12.406519

13 November 2000 Shape reconstruction from moments: theory, algorithms, and applications

Peyman Milanfar, Mihai Putinar, James Varah, Bjoern Gustafsson, Gene H. Golub

Proceedings Volume 4116, Advanced Signal Processing Algorithms, Architectures, and Implementations X; (2000) https://doi.org/10.1117/12.406519
Event: International Symposium on Optical Science and Technology, 2000, San Diego, CA, United States

Abstract

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-ray 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.

Citation Download Citation

Peyman Milanfar, Mihai Putinar, James Varah, Bjoern Gustafsson, and Gene H. Golub "Shape reconstruction from moments: theory, algorithms, and applications", Proc. SPIE 4116, Advanced Signal Processing Algorithms, Architectures, and Implementations X, (13 November 2000); https://doi.org/10.1117/12.406519

ACCESS THE FULL ARTICLE

INSTITUTIONAL
Select your institution to access the SPIE Digital Library.

SELECT YOUR INSTITUTION

PERSONAL
Sign in with your SPIE account to access your personal subscriptions or to use specific features such as save to my library, sign up for alerts, save searches, etc.

PERSONAL SIGN IN

No SPIE Account? Create one

PURCHASE THIS CONTENT

SUBSCRIBE TO DIGITAL LIBRARY

50 downloads per 1-year subscription

Members: $195

Non-members: $335 ADD TO CART

25 downloads per 1 - year subscription

Members: $145

Non-members: $250 ADD TO CART

PURCHASE SINGLE ARTICLE

Includes PDF, HTML & Video, when available