A Scrapbook of Inadmissible Line Complexes For the X-ray Transform

TL;DR

This paper investigates the structure of inadmissible line complexes in a finite field model of the X-ray transform in three dimensions, providing a detailed enumeration and illustration of these complexes.

Contribution

It offers a manual count and detailed structural analysis of inadmissible complexes, advancing understanding of invertibility conditions in finite field X-ray transforms.

Findings

Counted inadmissible complexes by hand

Illustrated possible structures of inadmissible complexes

Potential applications in AI for enumerating complexes

Abstract

We consider a finite field model of the X-ray transform that integrates functions along lines in dimension 3, within the context of finite fields. The admissibility problem asks for minimal sets of lines for which the restricted transform is invertible. Graph theoretic conditions are known which characterize admissible collections of lines, and these have been counted using a brute force computer program. Here we perform the count by hand and, at the same time, produce a detailed illustration of the possible structures of inadmissible complexes. The resulting scrapbook may be of interest in an artificial intelligence approach to enumerating and illustrating admissible complexes in arbitrary dimensions (arbitrarily large ambient spaces, with transforms integrating over subspaces of arbitrary dimensions.)