On an Analytical Inversion Formula for the Modulo Radon Transform

TL;DR

The paper develops an analytical inversion for the modulo Radon transform (MRT), enabling one-shot high dynamic range tomography by unfolding the modulo-folded Radon data. It derives a Poisson boundary-value problem that links the Laplacian of the Radon transform to its MRT, and proves existence and uniqueness of a weak solution, yielding an explicit MRT inversion formula that combines unfolded Radon data with the classical filtered back projection. By discretizing with Fourier methods, the authors propose the LMU-FBP algorithm, a two-stage procedure that first performs Laplacian-based modulo unfolding (LMU) and then applies a discrete FBP to recover the image, with an optional LMU_+ step for exact recovery when errors are sufficiently small. Numerical experiments on smooth and non-smooth phantoms and realistic walnut data demonstrate that LMU-FBP can handle non-bandlimited MRT data and often outperforms US-FBP, indicating strong practical viability for HDR tomography and motivating further theoretical recovery guarantees for discrete data.

Abstract

This paper proves a novel analytical inversion formula for the so-called modulo Radon transform (MRT), which models a recently proposed approach to one-shot high dynamic range tomography. It is based on the solution of a Poisson problem linking the Laplacian of the Radon transform (RT) of a function to its MRT in combination with the classical filtered back projection formula for inverting the RT. Discretizing the inversion formula using Fourier techniques leads to our novel Laplacian Modulo Unfolding - Filtered Back Projection algorithm, in short LMU-FBP, to recover a function from fully discrete MRT data. Our theoretical findings are finally supported by numerical experiments.

Figures (4)

• Figure 1: Utilized test data for numerical experiments. (a) Smooth phantom from Rieder2003. (b) Shepp-Logan phantom from Shepp1974. (c) Walnut Radon data from Siltanen2015. (d) Radon data of (a). (e) Radon data of (b). FBP reconstruction of (c) serving as ground truth.
• Figure 2: Numerical experiments with smooth phantom. (a) Noisy modulo Radon data with $\lambda = 0.015$ and $\delta = 0.05\cdot\lambda$. (b) US-FBP on (a). (c) LMU-FBP on (a).
• Figure 3: Numerical experiments with Shepp-Logan phantom. (a) Noisy modulo Radon data with $\lambda = 0.06$ and $\nu = 0.05\cdot\lambda$. (b) US-FBP on (a). (c) LMU$_+$-FBP on (a).
• Figure 4: Numerical experiments with walnut dataset. (a) Noisy modulo Radon data with $\lambda = 0.05$ and $\nu = 0.05\cdot\lambda$. (b) US-FBP on (a). (c) LMU-FBP on (a). (d) 2-times downsampled noisy modulo Radon data. (e) US-FBP on (d). (f) LMU-FBP on (d).

Theorems & Definitions (6)

• definition thmcounterdefinition: Modulo Radon transform
• proposition thmcounterproposition: Poincaré inequality
• theorem thmcountertheorem: Inversion formula for $\mathcal{R}^\lambda$
• lemma thmcounterlemma: Weak solution
• proof
• proof : Theorem \ref{['theo:inversion_mrt']}