
Statistics Papers
Document Type
Journal Article
Date of this Version
2-1990
Publication Source
SIAM Journal on Applied Mathematics
Volume
50
Issue
1
Start Page
288
Last Page
306
DOI
10.1137/0150017
Abstract
This paper studies sets S in Rn which are uniquely reconstructible from their hyperplane integral projections Pi(xi ;S) = ∬ . . . ∫ΧS ( {x1, . . . ,xi, . . . ,xn) dx1 . . . dxi - 1 dxi + 1 . . .dxn onto the n coordinate axes of Rn. It is shown that any additive set S = {x = (x1, . . .,xn) : ∑i = 1n fi(xi)≧0}, where each fi(xi) is a bounded measurable function, is uniquely reconstructible. In particular, balls are uniquely reconstructible. It is shown that in R2 all uniquely reconstructible sets are additive. For n≧3, Kemperman has shown that there are uniquely reconstructible sets in Rn of bounded measure that are not additive. It is also noted for n≧3 that neither of the properties of being additive and being a set of uniqueness is closed under monotone pointwise limits.
A necessary condition for S to be a set of uniqueness is that S contain no bad configuration. A bad configuration is two finite sets of points T1 in Int(S) and T2 in Int(Sc), where Sc=Rn - S, such that T1 and T2 have the same number of points in any hyperplane xi = c for 1≦ i ≦n, and all c ∈ R2. We show that this necessary condition is sufficient for uniqueness for open sets S in R2.
The results show that prior information about a density f in R2 to be reconstructed in tomography (namely if f is known to have only values 0 and 1) can sometimes reduce the problem of reconstructing f to knowing only two projections of f. Thus even meager prior information can in principle be of enormous value in tomography.
Copyright/Permission Statement
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
Keywords
projections, uniqueness, additive sets, probability
Recommended Citation
Fishburn, P. C., Lagarias, J. C., Reeds, J. A., & Shepp, L. A. (1990). Sets Uniquely Determined by Projections on Axes I. Continuous Case. SIAM Journal on Applied Mathematics, 50 (1), 288-306. http://dx.doi.org/10.1137/0150017
Date Posted: 27 November 2017
This document has been peer reviewed.