Solving Admissibility for the Spatial X-Ray Transform On the Two Element Field

The admissibility problem in integral geometry asks for which collections of affine subspaces the Radon transform remains injective. In the discrete setting, this becomes a purely combinatorial question about recovering a function on a finite vector space from its sums over a prescribed family of affine subspaces. In this paper, we study the spatial X-ray transform (line transform) over the finite vector spaces $\mathbb{Z}{2}^{n}$ and give a complete structural and enumerative description of admissible line complexes in $\mathbb{Z}{2}^{4}$. We prove that any admissible line complex in $\mathbb{Z}{2}^{4}$ can be obtained by taking a disjoint union of one or more odd cycles and attaching trees to the cycle vertices. Using this structural description, we carry out a systematic case-by-case enumeration of all admissible complexes in $\mathbb{Z}{2}^{4}$ and derive an exact total count. We then generalize our approach to an algorithm that applies to $\mathbb{Z}{2}^{n}$ for arbitrary $n$, and we then implement it to obtain the total number of admissible complexes in $\mathbb{Z}{2}^{5}$. Our results extend previous small-dimensional classifications and provide an algorithmic framework for studying admissibility in higher dimensions. Beyond their intrinsic combinatorial interest, these structures model discrete sampling schemes for tomographic imaging, and they suggest further connections between admissibility, incidence matrices, and spectral properties of the associated graphs.

💡 Research Summary

This paper tackles the admissibility problem for the discrete spatial X-ray transform over the finite vector space Z₂ⁿ. In this context, the transform sums a function over affine lines (pairs of points). A collection of lines, called a line complex, is “admissible” if the transform restricted to it is injective, meaning the original function can be uniquely recovered from its sums over these lines. The problem is inherently combinatorial, equivalent to the injectivity of the point-line incidence matrix.

The authors’ primary achievement is a complete structural characterization and exact enumeration of admissible line complexes in Z₂⁴. They prove that admissibility is governed by the graph G(C) whose vertices are points of Z₂ⁿ and edges are the lines in C. The main structural theorems establish that admissible complexes must avoid three fundamental obstructions: 1) any vertex not incident to a line (an omitted point), 2) any connected component that is a tree, and 3) the presence of any even-length cycle within the graph. Conversely, a complex covering all points, containing no even cycles, and having no tree components is guaranteed to be admissible.

A crucial finer result shows that within a single connected component of an admissible complex, there can be at most one odd cycle. Consequently, the structure of any admissible complex in Z₂⁴ is precisely a disjoint union of components, each consisting of a single odd cycle (of length 3, 5, …, or 15) with (possibly empty) trees attached to its vertices. Leveraging this clear picture, the authors perform a systematic case-by-case enumeration using combinatorial tools like Cayley’s formula for labeled trees. They derive the exact total number of admissible line complexes in Z₂⁴, which is 984,014,621,487,058,560 (approximately 9.84 × 10¹⁷).

Moving beyond dimension 4, the paper generalizes this enumerative strategy into an algorithm valid for Z₂ⁿ for any n. The algorithm operates by generating all possible sets of disjoint odd cycles that will form the “core skeleton” of the complex, and then counting the number of ways to attach the remaining vertices to these cycle vertices as labeled rooted forests. The authors implement this algorithm to compute the total number of admissible complexes in Z₂⁵.

The results are validated through random sampling and rank computations of incidence matrices. The work extends previous classifications in lower dimensions (like Z₂³) and provides a powerful algorithmic framework for studying admissibility in higher dimensions. Beyond its combinatorial interest, this research has implications for designing discrete sampling schemes in tomographic imaging and suggests connections to the spectral theory of the associated graphs.