Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-15T15:17:22.143Z Has data issue: false hasContentIssue false
  • English
  • Français

Inverting the statistics of the circle transform

Published online by Cambridge University Press:  14 July 2016

Peter Waksman
Peter Waksman*
Affiliation:
University of Rochester
*
∗Present address: 194 School Street, Acton, MA 01720, USA. Research supported by NSF Grant No. DMS-8602025.
Get access Rights & Permissions [Opens in a new window]

Abstract

For a plane domain the circle transform assigns to each circle its length of intersection with the domain. The problem is to determine the geometry of the domain given the mean of these intersection lengths as the circles' centers vary and given how that mean varies with circle radius. We also consider the mean square and higher moments of the intersection lengths — as a function of the circle radius. The discussion includes an identification of the geometric content of the mean when centers are inside of a disk and of the second moment when the centers are arbitrary points in the plane. The latter is equivalent to the distribution of distance between pairs of points of the domain.

Keywords

MOMENTSDISTANCE DISTRIBUTION
Type
Research Papers
Information
Journal of Applied Probability , Volume 25 , Issue 4 , December 1988 , pp. 708 - 716
Copyright
Copyright © Applied Probability Trust 1988 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Blaschke, W. (1949) Vorlesungen Über Integralgeometrie. Chelsea, New York.Google Scholar
Bott, R. and Tu, L. W. (1982) Differential Forms in Algebraic Topology. Springer-Verlag, New York.Google Scholar
Helgason, S. (1980) The Radon Transform. Progress in Mathematics 5, Birkhauser, Boston.CrossRefGoogle Scholar
Mallows, C. L. and Clark, J. M. C. (1970) Linear-intercept distributions do not characterize plane sets. J. Appl. Prob. 7, 240–244.Google Scholar
Salzman, L. (1980) Offbeat integral geometry. Amer. Math. Monthly 87, 161–174.Google Scholar
Santaló, L. A. (1976) Integral Geometry and Geometric Probability. Encyclopedia of Mathematics, Vol. 1. Addison-Wesley, Reading, Mass. Google Scholar
Strichartz, R. S. (1982) Radon inversion — variations on a theme. Amer. Math. Monthly 89(6).Google Scholar
Waksman, P. (1985) Plane polygons and a conjecture of Blaschke's. Adv. Appl. Prob. 17, 774–793.CrossRefGoogle Scholar
Waksman, P. (1986) Hypothesis testing in integral geometry. Trans. Amer. Math. Soc. 296, 507–520.CrossRefGoogle Scholar
Waksman, P. and Milstein, J. (1988) Equilibrium and dynamics of a Hartline–Ratliff type equation. Unpublished.Google Scholar