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

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.

Figures (12)

• Figure 1: (a): Projections $t_i$ of the corners $k_i$ of the cube to the red line in direction $\theta$. (b): Planes $H_{\boldsymbol{\theta}}(t)$ for $t=t_2$ in blue, $t=t_3$ in green, and $t\in(t_2,t_3)$ in yellow, with three or five corners.
• Figure 2: The function $A_{\boldsymbol{\theta}}^{{\boldsymbol{a}}}(t)$ for ${\boldsymbol{a}}=(a,a,a)^\top$, $a>0$, is the 3D Radon transform of the cube. Functions depending on $t$ for $\boldsymbol{\theta} = 1/\sqrt{3} (1,1,1)^\top \in \mathbb S^2$ (purple) as reference, and varying $\boldsymbol{\theta}\in\mathbb S^{2}$ in the degenerate (dashed gray) and general case (orange).
• Figure 3: The function $\mathbb S^2 \ni \boldsymbol{\theta} \mapsto A_{\boldsymbol{\theta}}^{{\boldsymbol{a}}}(t)$ for ${\boldsymbol{a}}=(1/2,1/2,1/2)^\top$, which is the 3D Radon transform of the cube. (a): $t = 0$. (b): $t = 1/7$.
• Figure 4: Selection of the first to fives samples of the ModelNet10 dataset for (a) bathtubs, (b) beds, (c) chairs, (d) desks, (e) dresser, and (f) monitors.
• Figure 5: Selection of the first to fives samples of the ModelNet10 dataset for (a) nightstands, (b) sofas, (c) tables, and (d) toilets.

Theorems & Definitions (11)

• Theorem 1
• Lemma 1
• proof
• proof : Proof of Theorem \ref{['thm:area']}
• Corollary 1: 2D Radon transform
• Corollary 2: 3D Radon transform
• Remark 1: Optimal directions
• Corollary 3
• proof
• Remark 2: Related results