Admissible Complexes for the Projective X-Ray Transform over a Finite Field

TL;DR

This paper investigates the injectivity of the projective X-ray transform over finite fields and classifies minimal line sets ensuring injectivity, revealing the role of doubly ruled quadric surfaces.

Contribution

It formulates and solves an admissibility problem for the Radon transform over finite fields, identifying minimal sets of lines for injectivity.

Findings

Injectivity of the X-ray transform over finite fields is established.

Minimal sets of lines for injectivity are classified.

Doubly ruled quadric surfaces are key to the classification.

Abstract

We consider the X-ray transform in a projective space over a finite field. It is well known (after E. Bolker) that this transform is injective. We formulate an analog of I.M. Gelfand's admissibility problem for the Radon transform, which asks for a classification of all minimal sets of lines for which the restricted Radon transform is injective. The solution involves doubly ruled quadric surfaces.