Application and Evaluation of the Common Circles Method

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

math.NAFeb 19, 2026

Application and Evaluation of the Common Circles Method

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

Organizations: TU Berlin · Medical University of Innsbruck

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.

Figures & tables

Panel of Figure 1 — 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.

Panel of Figure 2 — 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.

Panel of Figure 3 — Figure 3: Error of the reconstructed rotation (in degree) at each frame $t$ for the neuroblastoma phantom of Figure 2 with exact (a) or noisy data (b), (c).

Panel of Figure 4 — 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).

Panel of Figure 5 — Figure 5: Reconstructed rotations for three-beads object. The four plots are the components of the quaternions. Blue: Common circle method. Red: Optimization approach [ 27 ] .

Panel of Figure 6 — Figure 6: Three-beads object. Left: Rotation angle $\angle(R_{t}R_{0}^{\top})$ in degrees. Right: Angle $d(R_{t}^{\mathrm{CC}},R_{t}^{\mathrm{Opt}})$ in degrees between the rotation $R_{t}^{\mathrm{CC}}$ reconstructed with the common circle method and the Optimization approach $R_{t}^{\mathrm{Opt}}$ [ 27 ] . The average over all time steps is 10.3 °.

Panel of Figure 7 — Figure 7: Reconstructed refractive index $n({\boldsymbol{x}})$ for the Three-Beads object. Left: Rytov reconstruction with rotations from common circle method (PSNR 21.3, SSIM 0.669). Center: Rytov reconstruction with rotations from [ 27 ] (PSNR 21.3, SSIM 0.682). Right: Reconstruction with the optimization approach.

Panel of Figure 8 — Figure 8: Reconstructed refractive index $n({\boldsymbol{x}})$ for the Three-Beads object incorporating the reconstructed translations from [ 27 ] . Left: Rytov reconstruction with rotations from common circle method (PSNR 24.3, SSIM 0.704). Right: Rytov reconstruction with rotations from [ 27 ] (PSNR 27.0, SSIM 0.744).

Panel of Figure 9 — Figure 9: Visualization of Neuroblastoma dataset and reconstructed translations.

Panel of Figure 10 — Figure 10: Reconstructed rotations for the Neuroblastoma. The four plots are the components of the quaternions. Blue: Common circle method. Red: Optimization approach [ 27 ] .

Panel of Figure 11 — Figure 11: Reconstructed rotations for the Neuroblastoma. Left: Rotation angle $d(R_{t},R_{0})$ in degrees. Right: Angle $d(R_{t}^{\mathrm{CC}},R_{t}^{\mathrm{Opt}})$ in degrees between the rotation $R_{t}^{\mathrm{CC}}$ reconstructed with the common circle method and $R_{t}^{\mathrm{Opt}}$ with the Optimization approach [ 27 ] , the average over all time steps is 7.7 °.

Panel of Figure 12 — Figure 12: Reconstructed refractive index $n({\boldsymbol{x}})$ for the Neuroblastoma with different methods. Columns are sections through the coordinate planes (left to right: xy-plane, xz-plane, yz-plane).

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Event-based motion estimation is central to tasks that demand high temporal resolution and robustness to fast motion. Existing methods typically rely on iterative optimization or repeated hypothesis comparison, offsetting the sensor's low-latency advantage. We propose Oriented Distance Field Motion Estimation (ODF Motion Estimation), which replaces this optimization with a single averaging step over a precomputed field of event distance vectors, combined with an adaptive event-count selection strategy and a parameter-free trail filter. On public and self-collected datasets, ODF motion estimation reaches sub-pixel accuracy at the lowest latency among compared methods. We validate its generality on two downstream applications rather than treating them as separate contributions. First, the estimated trajectory is converted into a blur kernel and paired with a compact iterative-unfolding network, trained on simulated motion-estimation noise, for real-time non-blind image deblurring, attaining competitive or superior PSNR/SSIM with under 1M parameters. Second, the same precomputed field is repurposed for directional event filtering in a low-power asynchronous pupil and glint tracker, sustaining stable tracking for tens of seconds while lowering a near-eye module's power draw.

Abstract

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