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Application and Evaluation of the Common Circles Method

Michael Quellmalz, Mia Kvåle Løvmo, Simon Moser, Franziska Strasser, Monika Ritsch-Marte

Abstract

We investigate the application of the common circle method for estimating sample motion in optical diffraction tomography (ODT) of sub-millimeter sized biological tissue. When samples are confined via contact-free acoustical force fields, their motion must be estimated from the captured images. The common circle method identifies intersections of Ewald spheres in Fourier space to determine rotational motion. This paper presents a practical implementation, incorporating temporal consistency constraints to achieve stable reconstructions. Our results on both simulated and real-world data demonstrate that the common circle method provides a computationally efficient alternative to full optimization methods for motion detection.

Application and Evaluation of the Common Circles Method

Abstract

We investigate the application of the common circle method for estimating sample motion in optical diffraction tomography (ODT) of sub-millimeter sized biological tissue. When samples are confined via contact-free acoustical force fields, their motion must be estimated from the captured images. The common circle method identifies intersections of Ewald spheres in Fourier space to determine rotational motion. This paper presents a practical implementation, incorporating temporal consistency constraints to achieve stable reconstructions. Our results on both simulated and real-world data demonstrate that the common circle method provides a computationally efficient alternative to full optimization methods for motion detection.
Paper Structure (22 sections, 4 theorems, 55 equations, 12 figures, 1 algorithm)

This paper contains 22 sections, 4 theorems, 55 equations, 12 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Diffraction Tomography
  3. Fourier Diffraction Theorem
  4. Motion of the object
  5. Common Circle Method
  6. Infinitesimal Common Circle Method
  7. Reconstruction of the Rotation
  8. Infinitesimal Common Circle Method
  9. Estimate of the Angular Velocity
  10. Temporal Regularization of the Angular Velocity
  11. Rotation Matrix
  12. Regularized Direct Common Circle Method
  13. Numerics
  14. Data Preprocessing
  15. Reconstruction Steps
  16. ...and 7 more sections

Key Result

Theorem 1

Let $s,t\in[0,T]$. Assume that there exist unique angles $\varphi,\psi \in [0,2\pi)$ and $\theta\in [0,\pi]$ such that where Then the incremental rotation is

Figures (12)

  • Figure 1: Illustration of the common circles as intersections of the Ewald spheres. Left: Two hemispheres $\mathcal{H}_0$ (green) and $\mathcal{H}_t$ (red) intersect in a common circle. The north pole of both hemispheres is at $\boldsymbol{0}$. Right: For real-valued (lossless) interactions, a dual common circle at the intersection of $\mathcal{H}_0$ (green) and $-\mathcal{H}_t$ (red) exists. Data must agree on the two intersections.
  • Figure 2: (a) Ground truth refractive index $n({\boldsymbol x})$ of the phantom. (b) Absolute value of the preprocessed data $|m_t({\boldsymbol x})|$ for $t=1$ simulated with the BPM with added noise. (c) Reconstruction of the refractive index $n({\boldsymbol x})$ for noisy simulated data using the Rytov approximation and the rotations reconstructed via the common circle method.
  • Figure 3: Error of the reconstructed rotation (in degree) at each frame $t$ for the neuroblastoma phantom of \ref{['fig:bpm']} with exact (a) or noisy data (b), (c).
  • Figure 4: Absolute value of the transformed and preprocessed measurements $|m_t({\boldsymbol x})|$ for the three-beads object dataset for time steps $t\in\{0,5,10\}$ (left to right).
  • Figure 5: Reconstructed rotations for three-beads object. The four plots are the components of the quaternions. Blue: Common circle method. Red: Optimization approach Mos25.
  • ...and 7 more figures

Theorems & Definitions (5)

  • Theorem 1: ElbQueSchSte23
  • Theorem 2: ElbQueSchSte23
  • Lemma 3.1
  • proof
  • Lemma 3.2