Motion Detection in Diffraction Tomography by Common Circle Methods

TL;DR

The paper tackles the challenge of reconstructing unknown rigid motion (rotations $R_t$ and translations ${\boldsymbol d}_t$) of a scattering object in optical diffraction tomography under the Born approximation. It develops two motion-detection approaches inspired by cryo-EM: a direct common circle method that uses intersections of hemispheres to recover relative rotations, and an infinitesimal variant that estimates angular velocity $\boldsymbol{\omega}_t$ and integrates to obtain $R_t$, leveraging the Fourier diffraction theorem. Translations are recovered from the complex measurements by exploiting phase information along common circles (and their duals), with a least-squares and phase-unwrapping framework. Numerical experiments on realistic phantoms demonstrate that joint motion estimation and subsequent 3D reconstruction are feasible, even with moving rotation axes and translations, highlighting the method’s potential for diffraction tomography with optically trapped samples. The work establishes a rigorous motion-reconstruction framework, connecting common-circle geometry, stereographic projection, and higher-order constraints to enable phase-aware motion estimation in diffraction-based imaging.

Abstract

The method of common lines is a well-established reconstruction technique in cryogenic electron microscopy (cryo-EM), which can be used to extract the relative orientations of an object given tomographic projection images from different directions. In this paper, we deal with an analogous problem in optical diffraction tomography. Based on the Fourier diffraction theorem, we show that rigid motions of the object, i.e., rotations and translations, can be determined by detecting common circles in the Fourier-transformed data. We introduce two methods to identify common circles. The first one is motivated by the common line approach for projection images and detects the relative orientation by parameterizing the common circles in the two images. The second one assumes a smooth motion over time and calculates the angular velocity of the rotational motion via an infinitesimal version of the common circle method. Interestingly, using the stereographic projection, both methods can be reformulated as common line methods, but these lines are, in contrast to those used in cryo-EM, not confined to pass through the origin and allow for a full reconstruction of the relative orientations. Numerical proof-of-the-concept examples demonstrate the performance of our reconstruction methods.

Figures (13)

• Figure 1: Left: A common line pair between two planes $P_{R_s}$ and $P_{R_t}$ does not determine the relative angle between them uniquely. Right: Knowing pairwise common lines between three planes, we can determine their orientations uniquely (except for degenerate cases). Courtesy of Denise Schmutz from Schm17.
• Figure 2: Experimental setup of transmission imaging in optical diffraction tomography.
• Figure 3: Illustration of the common circles. Two hemispheres $\mathcal{H}_0$ and $\mathcal{H}_t$ intersect in a common circle arc. The north pole of the hemispheres is at $\boldsymbol{0}$.
• Figure 4: Intersection of two hemispheres $\mathcal{H}_s$ and $\mathcal{H}_t$ in the plane through the centers of the hemispheres and the origin. The circular arc $\mathcal{H}_s\cap\mathcal{H}_t$ lies in the plane $\mathcal{V}_{s,t}=\{{\boldsymbol x}\in\mathbb{R}^3: \left<{\boldsymbol x},(R_s-R_t){\boldsymbol e}^3\right>=0\}$ perpendicular to the line between the centers. It is spanned by ${\boldsymbol v}_{s,t}^j$, $j=1,2$ in \ref{['th:YsYt_param']}. The basis ${\boldsymbol w}_{s,t}^j$, $j=1,2$ of $\mathbb{R}^2$ in \ref{['th:gamma']} is illustrated by the orthogonal projection $\tilde{{\boldsymbol w}}_{s,t}^j$ of ${\boldsymbol v}_{s,t}^j$ to the tangent plane $\mathcal{T}_s$ of $\mathcal{H}_s$ at $\boldsymbol{0}$. They are explicitly related by ${\boldsymbol w}_{s,t}^j=P(R_s^\top\tilde{{\boldsymbol w}}_{s,t}^j)$.
• Figure 5: Scaled squared energies $\nu_s$ (left) and $\nu_t$ (right), see \ref{['eq:nu']}, for a characteristic function $f$ of an ellipsoid in $\mathbb{R}^3$. For the relative rotation $R_s^\top R_t = Q^{(3)}(\frac{\pi}{6})Q^{(2)}(\frac{\pi}{4})Q^{(3)}(\frac{2\pi}{3})$, we show the paths of the corresponding two elliptic arcs ${\boldsymbol \gamma}_{s,t}$ and ${\boldsymbol \gamma}_{t,s}$ (solid lines), where ${\boldsymbol \gamma}_{t,s}^-$ denotes the reversed elliptic arc ${\boldsymbol \gamma}_{t,s}^-(\beta)\coloneqq{\boldsymbol \gamma}_{t,s}(-\beta)$, and their dual arcs $\boldsymbol{{\boldsymbol \gamma}}^*_{s,t}$ and $\boldsymbol{{\boldsymbol \gamma}}^*_{t,s}$ (dashed), cf. \ref{['th:dualCircle']}. The relations \ref{['eq:commonCircleIdentity']} and \ref{['eq:commonCircleIdentityDual']} are verified in the center: The top plot shows that the graphs of the scaled squared energy $\nu_s\circ{\boldsymbol \gamma}_{s,t}(\beta)$ along the elliptic arcs in blue and $\nu_t\circ {\boldsymbol \gamma}_{t,s}^-(\beta)$ in red coincide for all $\beta$. The same can be seen in the bottom for the dual arcs.

Theorems & Definitions (41)

• Lemma 3.1: Parameterization of the common circular arcs
• Lemma 3.2: Parameterization by elliptic arcs
• Proposition 1: Representation of ${\boldsymbol \gamma}_{s,t}$ via the Euler angles of $R_s^\top R_t$
• Proposition 2: Parameterization in dual case
• Proposition 3: Special cases $R_s{\boldsymbol e}^3=\pm R_t{\boldsymbol e}^3$
• Theorem 1: Reconstruction of Euler angles
• Remark 1: Methods of common circles and common lines
• Remark 2: Stereographic projection
• Lemma 4.1: Infinitesimal common circle relation
• Theorem 2: Reconstruction of the angular velocity $\boldsymbol{\omega}_t$