A Discrete Radon Transform Based on the Area of Cube-Plane Intersection

TL;DR

This paper introduces an exact and efficient discrete Radon transform for voxelized data based on a closed-form formula for cubes, with applications in shape analysis and optimal transport.

Contribution

It derives a closed-form polynomial expression for the Radon transform of cubes and proposes a regularized, computationally efficient discrete Radon transform suitable for high-dimensional data.

Findings

Exact Radon transform formula for cubes in arbitrary dimensions

Improved numerical stability with regularized transform using slabs

Demonstrated effectiveness in shape matching, classification, and Wasserstein barycenters

Abstract

The Radon transform is a fundamental tool for analyzing data in tomographic imaging, optimal transport, crystallography, and geometric analysis. Numerical computations require an accurate discretization. To deal with voxelized images and objects, we derive a closed-form, piecewise polynomial expression for the Radon transform of an axis-aligned cube in arbitrary dimension $d$. Building on this formula, we propose a discrete Radon transform in $\mathbb{R}^d$ that is both analytically exact for voxelized data and computationally efficient. For improved numerical stability, we introduce a regularized variant replacing the Radon transform of a cube, i.e.\ the $(d-1)$-dimensional area of the intersection between that cube and a hyperplane, by the $d$-dimensional volume of the intersection between the cube and a thin slab around the hyperplane. Numerical experiments demonstrate the effectiveness of the proposed approach in several applications including 3D shape matching, classification, and sliced Wasserstein barycenters. The computational efficiency in higher dimensions is verified by a comparison with Monte Carlo integration.