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Noise-Robust One-Bit Diffraction Tomography and Optimal Dose Fractionation

Pengwen Chen, Albert Fannjiang

TL;DR

This work develops a noise-robust framework for 1-bit diffraction tomography using coded apertures, blending random-matrix theory with forward-scattering modeling and iterative reconstruction methods. By formulating the problem as a 3D phase retrieval task and employing power and shifted inverse power iterations, the authors analyze how Poisson noise (NSR) and dose distribution affect reconstruction quality. A key finding is that optimal dose fractionation occurs at NSR ≈ 1, independent of total dose, implying many low-dose views maximize information gain in 1-bit measurements. The study also demonstrates that 2-phase and 4-phase random masks can match continuous-phase performance, and discusses practical considerations for dose economy and future extensions to adaptive thresholds and hybrid measurement models.

Abstract

This study presents a noise-robust framework for 1-bit diffraction tomography, a novel imaging approach that relies on intensity-only binary measurements obtained through coded apertures. The proposed reconstruction scheme leverages random matrix theory and iterative algorithms to effectively recover 3D object structures under high-noise conditions. A key contribution is the numerical investigation of dose fractionation, revealing optimal performance at a signal-to-noise ratio near 1, {\em independent of the total dose}. This finding addresses the question: How to distribute a given level of total radiation energy among different tomographic views in order to optimize the quality of reconstruction?

Noise-Robust One-Bit Diffraction Tomography and Optimal Dose Fractionation

TL;DR

This work develops a noise-robust framework for 1-bit diffraction tomography using coded apertures, blending random-matrix theory with forward-scattering modeling and iterative reconstruction methods. By formulating the problem as a 3D phase retrieval task and employing power and shifted inverse power iterations, the authors analyze how Poisson noise (NSR) and dose distribution affect reconstruction quality. A key finding is that optimal dose fractionation occurs at NSR ≈ 1, independent of total dose, implying many low-dose views maximize information gain in 1-bit measurements. The study also demonstrates that 2-phase and 4-phase random masks can match continuous-phase performance, and discusses practical considerations for dose economy and future extensions to adaptive thresholds and hybrid measurement models.

Abstract

This study presents a noise-robust framework for 1-bit diffraction tomography, a novel imaging approach that relies on intensity-only binary measurements obtained through coded apertures. The proposed reconstruction scheme leverages random matrix theory and iterative algorithms to effectively recover 3D object structures under high-noise conditions. A key contribution is the numerical investigation of dose fractionation, revealing optimal performance at a signal-to-noise ratio near 1, {\em independent of the total dose}. This finding addresses the question: How to distribute a given level of total radiation energy among different tomographic views in order to optimize the quality of reconstruction?
Paper Structure (20 sections, 2 theorems, 85 equations, 9 figures, 2 algorithms)

This paper contains 20 sections, 2 theorems, 85 equations, 9 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Plan and contribution
  3. Random-matrix theorem
  4. Noise robustness
  5. Forward scattering model
  6. Power iteration and inverse power iteration
  7. Shifted inverse power iteration
  8. Noise-to-signal ratio (NSR)
  9. Testing the algorithms
  10. Optimal fractionation of radiation dose
  11. Dose as the mean photon number
  12. Alternative definition of dose
  13. Dose fractionation simulation
  14. Conclusion and discussion
  15. Discrete tomography
  16. ...and 5 more sections

Key Result

Theorem 2.1

null Let ${\mathcal{A}}$ be an $M\times N$ i.i.d. complex Gaussian matrix and $f_{\min}$ a minimizer of nul3. Suppose Then with an overwhelming probability, the relative error bound holds for some constant $c_0$, where $\|\cdot\|_{\rm F}$ denotes the Frobenius norm. Here and below, the over-line notation denotes the complex conjugation.

Figures (9)

  • Figure 1: Coded aperture diffraction tomography: Diffraction patterns of an object in various orientations are measured with the same random mask (see Section \ref{['sec:forward']})
  • Figure 2: $216\times 216$ image $\Longrightarrow$$36\times 36\times 36$ object.
  • Figure 3: Correlation versus computation time (in second) with 5 independent trials with the power method (dashed line) and the inverse power method (solid line). The number of diffraction patterns is $m=3\rho n$.
  • Figure 4: (a) The two leading eigenvalues and (b) the correlations as function of $\rho$ at NSR =1 where $R_1$ and $R_2$ are respective correlations of the two leading eigenvectors with the original object. The number of diffraction patterns is $m=3\rho n$.
  • Figure 5: The correlation of the RPP reconstruction with $3n$ diffraction patterns vs NSR $\in (0,1)$ and the flattened magnitude $|f|$ with (bottom) and without (top) a beam stop. The beam stop covers the $1\%$ center area of each diffraction pattern. Different direction sets ${\mathcal{T}}$ are independently selected for different NSRs.
  • ...and 4 more figures

Theorems & Definitions (4)

  • Theorem 2.1
  • Remark 2.1
  • Proposition E.1
  • proof