A Refinement of the Radon Transform and Its Inverse

TLDR: In this paper, the Radon transform of a function is defined as an integration over planes whose normals vary over the entire unit sphere, and it is shown that only the transform over a hemisphere, which can consist of disconnected parts, is required to reconstruct the original function.

Abstract: The Radon transform of a function is defined as an integration over planes whose normals vary over the entire unit sphere. The space is actually covered twice because the distance of the plane from the origin is allowed to be positive or negative. The usual inverse transform requires knowledge of the transform evaluated over the entire sphere. However, we shall show that only the transform over a hemisphere, which can consist of disconnected parts, is required to reconstruct the original function . Thus the redundancy of the double-covering is removed and only one-half of the transform is needed to recover the original function. In essence we have introduced optical coordinates. We then consider function f(x) obtained by applying the inverse Radon transform to an arbitrary function which has the same arguments as the Radon transform but is not, in general, a Radon transform. On applying the Radon transform to f(x) we find that only part of the arbitrary function, to which the inverse was applied, is reproduced. Thus the Radon transform has a left inverse but not a right inverse. However, by restricting the range of variables in the transform space, a right and left inverse can be obtained which are the same. Finally, we give Parseval’s theorem in terms of the refined Radon transform. Though we modify the older proofs for obtaining the Radon transform and its inverse, for the sake of a self-contained paper we also use new elementary proofs based on relations which we have derived between one­-dimensional and three-dimensional delta functions. We expect that our result will have consequences in tomography and other applications. We ourselves will use the result to obtain the exact fields for the scalar three-dimensional wave equation and Maxwell’s equations from fields in the wave zone, and, conversely, fields in the wave zone from the exact causal fields. In fact, the principal reason for our writing the present paper is to cast the Radon transform and its inverse in a form suitable for these applications. Though we shall prove our result for the three-dimensional case only, the proof for the general case can be inferred from our proof.