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Fourier-Domain Inversion for the Modulo Radon Transform

Matthias Beckmann, Ayush Bhandari, Meira Iske

TL;DR

The paper tackles single-shot HDR X-ray tomography by exploiting the modulo Radon transform (MRT) to overcome detector dynamic-range limits. It introduces a Fourier-domain MRT inversion framework with two complementary pipelines: a sequential OMP-FBP method that recovers Radon projections and applies standard filtered back projection, and a direct Fourier reconstruction (OMP-DFR) method that operates entirely in the frequency domain using a discrete differentiation property. The authors establish rigorous recovery guarantees under compact $\lambda$-exceedance and sampling conditions, and demonstrate substantial reductions in sampling requirements and computational complexity, while remaining agnostic to the folding threshold $\lambda$. Numerical simulations and hardware experiments validate HDR reconstruction with enhanced quantization efficiency and robustness to noise, outperforming or matching traditional RT approaches under limited dynamic-range data. Overall, the work broadens HDR tomography capabilities by connecting MRT with efficient FFT-based inversion, enabling practical single-shot HDR imaging with principled guarantees.

Abstract

Inspired by the multiple-exposure fusion approach in computational photography, recently, several practitioners have explored the idea of high dynamic range (HDR) X-ray imaging and tomography. While establishing promising results, these approaches inherit the limitations of multiple-exposure fusion strategy. To overcome these disadvantages, the modulo Radon transform (MRT) has been proposed. The MRT is based on a co-design of hardware and algorithms. In the hardware step, Radon transform projections are folded using modulo non-linearities. Thereon, recovery is performed by algorithmically inverting the folding, thus enabling a single-shot, HDR approach to tomography. The first steps in this topic established rigorous mathematical treatment to the problem of reconstruction from folded projections. This paper takes a step forward by proposing a new, Fourier domain recovery algorithm that is backed by mathematical guarantees. The advantages include recovery at lower sampling rates while being agnostic to modulo threshold, lower computational complexity and empirical robustness to system noise. Beyond numerical simulations, we use prototype modulo ADC based hardware experiments to validate our claims. In particular, we report image recovery based on hardware measurements up to 10 times larger than the sensor's dynamic range while benefiting with lower quantization noise ($\sim$12 dB).

Fourier-Domain Inversion for the Modulo Radon Transform

TL;DR

The paper tackles single-shot HDR X-ray tomography by exploiting the modulo Radon transform (MRT) to overcome detector dynamic-range limits. It introduces a Fourier-domain MRT inversion framework with two complementary pipelines: a sequential OMP-FBP method that recovers Radon projections and applies standard filtered back projection, and a direct Fourier reconstruction (OMP-DFR) method that operates entirely in the frequency domain using a discrete differentiation property. The authors establish rigorous recovery guarantees under compact -exceedance and sampling conditions, and demonstrate substantial reductions in sampling requirements and computational complexity, while remaining agnostic to the folding threshold . Numerical simulations and hardware experiments validate HDR reconstruction with enhanced quantization efficiency and robustness to noise, outperforming or matching traditional RT approaches under limited dynamic-range data. Overall, the work broadens HDR tomography capabilities by connecting MRT with efficient FFT-based inversion, enabling practical single-shot HDR imaging with principled guarantees.

Abstract

Inspired by the multiple-exposure fusion approach in computational photography, recently, several practitioners have explored the idea of high dynamic range (HDR) X-ray imaging and tomography. While establishing promising results, these approaches inherit the limitations of multiple-exposure fusion strategy. To overcome these disadvantages, the modulo Radon transform (MRT) has been proposed. The MRT is based on a co-design of hardware and algorithms. In the hardware step, Radon transform projections are folded using modulo non-linearities. Thereon, recovery is performed by algorithmically inverting the folding, thus enabling a single-shot, HDR approach to tomography. The first steps in this topic established rigorous mathematical treatment to the problem of reconstruction from folded projections. This paper takes a step forward by proposing a new, Fourier domain recovery algorithm that is backed by mathematical guarantees. The advantages include recovery at lower sampling rates while being agnostic to modulo threshold, lower computational complexity and empirical robustness to system noise. Beyond numerical simulations, we use prototype modulo ADC based hardware experiments to validate our claims. In particular, we report image recovery based on hardware measurements up to 10 times larger than the sensor's dynamic range while benefiting with lower quantization noise (12 dB).
Paper Structure (12 sections, 4 theorems, 55 equations, 10 figures, 2 tables, 3 algorithms)

This paper contains 12 sections, 4 theorems, 55 equations, 10 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Revisiting Modulo Radon Transform
  3. Fourier Based Modulo Radon Inversion
  4. Sequential Reconstruction Approach
  5. Theoretical Recovery Guarantees
  6. Direct Fourier Reconstruction Approach
  7. Overview of Direct Fourier Reconstruction Methods
  8. Runtime Complexity
  9. Numerical and Hardware Experiments
  10. Numerical Experiments
  11. Hardware Validation
  12. Conclusion

Key Result

Lemma 1

Let $\mathbf{z} = (z_1,\dots,z_J) \in \mathbb C^J$ with $z_i \neq z_j$ for all $i \neq j$. Moreover, let $\mathcal{S} \subseteq \{0,\ldots,L-1\}$. Then, the reduced Vandermonde matrix $\mathbf{V}_\mathcal{S}(\mathbf{z}) \in \mathbb C^{J \times |\mathcal{S}|}$ of $\mathbf{V}(\mathbf{z}) \in \mathbb C

Figures (10)

  • Figure 1: Overview of the computational imaging pipeline for single-shot HDR tomography via the modulo Radon transform (MRT) Bhandari2020Beckmann2020Beckmann2022.
  • Figure 2: Juxtaposition of OMP-FBP method (top) and OMP-DFR method (bottom).
  • Figure 3: Numerical experiments with the Shepp-Logan phantom and various experiment parameters. (a) Noisy modulo Radon data with $\lambda = 0.175$ and $\nu = 0.01 \cdot \lambda$. (b) US-FBP on (a). (c) OMP-FBP on (a). (d) OMP-NFFT on (a). (e) Noisy modulo Radon data with $\lambda = 0.175$ and $\nu = 0.01 \cdot \lambda$. (f) OMP-FBP on (e). (g) OMP-NFFT on (e). (h) Noisy modulo Radon data with $\lambda = 0.175$, $\sigma = 0.025 \cdot \overline{p}_\boldsymbol{\theta}$ and $\nu = 0.025 \cdot \lambda$. (i) OMP-FBP on (h). (j) OMP-NFFT on (h). (k) Noisy modulo Radon data with $\lambda = 0.175$, $\sigma = 0.08 \cdot \overline{p}_\boldsymbol{\theta}$ and $\nu = 0.1 \cdot \lambda$. (l) OMP-FBP on (k). (m) OMP-NFFT on (k). (n) Noisy modulo Radon data with $\lambda = 0.025$ and shot noise in $[-0.2,0.2]$. (o) OMP-FBP on (n). (p) OMP-NFFT on (n).
  • Figure 4: Investigation of reconstruction quality for decreasing $\lambda$ and fixed $\mathsf{OF}$.
  • Figure 5: Investigation of reconstruction quality for noise with decreasing SNR.
  • ...and 5 more figures

Theorems & Definitions (9)

  • Definition 1: Compact $\lambda$-exceedance
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Proposition 1
  • proof