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Total Variation-Based Reconstruction and Phase Retrieval for Diffraction Tomography with an Arbitrarily Moving Object

Robert Beinert, Michael Quellmalz

TL;DR

This work tackles 3D optical diffraction tomography under the Born approximation for a rigid object undergoing time-dependent rotation $R_t$. It develops and compares reconstruction strategies for both complex-field data and magnitude-only data, including a moving-axis filtered backpropagation, a nonuniform Fourier sampling with TV-regularized CG inversion, and an all-at-once Hybrid Input-Output phase retrieval scheme with TV regularization. The results show that TV-regularized and primal-dual reconstructions provide superior robustness to noise compared with traditional backpropagation, while phase retrieval via HIO+TV enables accurate recovery from intensity measurements. The methods are motivated by practical imaging scenarios with moving objects and phase-less measurements, and the work provides practical algorithms and numerical evidence for reliable 3D diffraction tomography.

Abstract

We consider the imaging problem of the reconstruction of a three-dimensional object via optical diffraction tomography under the assumptions of the Born approximation. Our focus lies in the situation that a rigid object performs an irregular, time-dependent rotation under acoustical or optical forces. In this study, we compare reconstruction algorithms in case i) that two-dimensional images of the complex-valued wave are known, or ii) that only the intensity (absolute value) of these images can be measured, which is the case in many practical setups. The latter phase-retrieval problem can be solved by an all-at-once approach based utilizing a hybrid input-output scheme with TV regularization.

Total Variation-Based Reconstruction and Phase Retrieval for Diffraction Tomography with an Arbitrarily Moving Object

TL;DR

This work tackles 3D optical diffraction tomography under the Born approximation for a rigid object undergoing time-dependent rotation . It develops and compares reconstruction strategies for both complex-field data and magnitude-only data, including a moving-axis filtered backpropagation, a nonuniform Fourier sampling with TV-regularized CG inversion, and an all-at-once Hybrid Input-Output phase retrieval scheme with TV regularization. The results show that TV-regularized and primal-dual reconstructions provide superior robustness to noise compared with traditional backpropagation, while phase retrieval via HIO+TV enables accurate recovery from intensity measurements. The methods are motivated by practical imaging scenarios with moving objects and phase-less measurements, and the work provides practical algorithms and numerical evidence for reliable 3D diffraction tomography.

Abstract

We consider the imaging problem of the reconstruction of a three-dimensional object via optical diffraction tomography under the assumptions of the Born approximation. Our focus lies in the situation that a rigid object performs an irregular, time-dependent rotation under acoustical or optical forces. In this study, we compare reconstruction algorithms in case i) that two-dimensional images of the complex-valued wave are known, or ii) that only the intensity (absolute value) of these images can be measured, which is the case in many practical setups. The latter phase-retrieval problem can be solved by an all-at-once approach based utilizing a hybrid input-output scheme with TV regularization.
Paper Structure (5 sections, 2 theorems, 22 equations, 5 figures, 1 algorithm)

This paper contains 5 sections, 2 theorems, 22 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Reconstruction methods
  3. Phase retrieval
  4. Conclusion
  5. Acknowledgement

Key Result

Theorem 1

Let $f\in L^1(\mathbb{R}^3)$ have a compact support in $\{{\bm x} \in \mathbb{R}^3 : \|{\bm x}\|_2 \le r_{\mathrm{M}} \}$, and let the Lebesgue measure of be positive. Then $f$ is uniquely determined by $u^{\mathrm{tot}}_t(\cdot,\cdot,r_{\mathrm{M}})$, $t \in [0,T]$.

Figures (5)

  • Figure 1: Experimental setup with measurement plane located at $x_3=r_{\mathrm M}$, see BeiQue22.
  • Figure 2: Reconstructions with moving axis (exact data). Top left: Ground truth of 3D phantom $f$. Top right: Filtered backpropagation (PSNR 28.55, SSIM 0.781). Bottom left: CG Reconstruction (PSNR 33.75, SSIM 0.962). Bottom right: Primal-dual with TV and regularization parameter $\lambda=0.01$ (PSNR 34.54, SSIM 0.991).
  • Figure 3: Reconstructions with moving axis (5 % Gaussian noise). Left: Filtered backpropagation (PSNR 21.38, SSIM 0.078). Center: CG Reconstruction (PSNR 23.75, SSIM 0.186). Right: Primal-dual with TV and $\lambda=0.05$ (PSNR 34.14, SSIM 0.900).
  • Figure 4: Phase retrieval (exact data). Left: HIO and CG method (PSNR 28.41, SSIM 0.697). Right: HIO primal-dual method (PSNR 34.58, SSIM 0.973).
  • Figure 5: Phase retrieval (5 % Gaussian noise). Left: HIO and CG method (PSNR 23.47, SSIM 0.579). Right: HIO primal-dual method with $\lambda=0.05$ (PSNR 34.00, SSIM 0.991).

Theorems & Definitions (2)

  • Theorem 1: Unique inversion, BeiQue22
  • Theorem 2: Filtered Backpropagation, KirQueRitSchSet21