The Radon transform on Z₂^k

Persi W. Diaconis and Ronald Lewis Graham

Vol. 118 (1985), No. 2, 323–345

DOI: 10.2140/pjm.1985.118.323

Abstract

Suppose G is a finite group and f is a function mapping G into the set of real numbers R. For a subset S ⊆ G, define the Radon transform F_S off mapping G into R by:

where S + x denotes the set {s + x : s ∈ S}. Thus, the Radon transform can be thought of as a way of replacing f by a “smeared out” version of f. This form of the transform represents a simplified model of the kind of averaging which occurs in certain applied settings, such as various types of tomography and recent statistical averaging techniques.

A fundamental question which arises in connection with the Radon transform is whether or not it is possible to invert it, i.e., whether one can recover (in principle) the function f from knowledge of F_S.

In this paper we investigate this problem in detail for several special classes of groups, including the group of binary n-tuples under modulo 2 addition.

Mathematical Subject Classification 2000

Primary: 44A15

Secondary: 92A07

Milestones

Received: 24 September 1984

Published: 1 June 1985