A Discrete Radon Transform Based on the Area of Cube-Plane Intersection
This work addresses the need for precise numerical computations in fields such as tomographic imaging and optimal transport, though it is incremental as it builds on existing discretization methods.
The authors tackled the problem of accurately discretizing the Radon transform for voxelized data by deriving a closed-form, piecewise polynomial expression for the transform of an axis-aligned cube in arbitrary dimensions, resulting in a method that is analytically exact and computationally efficient, with numerical experiments showing effectiveness in applications like 3D shape matching and classification.
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.