Inverse Problems of Trapped Objects

TL;DR

The paper tackles reconstructing the time-dependent rigid motion of a trapped particle from attenuation projections to enable $3$D tomography. They formulate a forward model $\mathcal{J}[T,R]$ for attenuation projections and derive a Fourier-domain identity, enabling translation-free rotation analysis via a reduced map $\tilde{\mathcal{J}}$ that depends on the rotation $R(t)$. The core contributions include a translation-centering approach that recovers $P(T)$, a differentiation-based method to extract cylindrical rotation components $v(t)$ and $\omega_3(t)$ and a third-order equation to recover the cylindrical radius $\alpha(t)$, with explicit handling of nonuniqueness. Numerical experiments on simulated data demonstrate the feasibility of recovering $P(T)$, $v(t)$, $\omega_3(t)$, and $\alpha(t)$ from attenuation projections, while highlighting limitations in symmetric or special-motion cases.

Abstract

Optical and acoustical trapping has been established as a tool for holding and moving microscopic particles suspended in a liquid in a contact-free and non-invasive manner. Opposed to standard microscopic imaging where the probe is fixated, this technique allows imaging in a more natural environment. This paper provides a method for estimating the movement of a transparent particle which is maneuvered by tweezers, assuming that the inner structure of the probe is not subject to local movements. The mathematical formulation of the motion estimation shows some similarities to Cryo-EM single particle imaging, where the recording orientations of the probe need to be estimated.

Figures (6)

• Figure 1: Illumination in the $e_3$-direction of the contour plot of the attenuation coefficient defined in \ref{['eq:defnumericsattcoeff']} for $u(x)=42$ and the resulting attenuation projection image.
• Figure 2: Workflow. AM=Attenuation Mappings, RAM=Reduced Attenuation Mappings.
• Figure 3: (A) and (C): Object rotated with respect to $R$ at two different time steps. (B) and (D): Reflected object rotated with respect to $\check{R}$ at the same time steps. The attenuation projection images are the same.
• Figure 4: Contour plot of the attenuation coefficient defined in \ref{['eq:defnumericsattcoeff']} for $u(x)=42$ and the resulting attenuation projection images at time steps $0,0.25$ and $0.425$.
• Figure 9: Absolute errors in the reconstructions of $\varphi$ (the crosses), $\omega_3$ (the triangles), and $\alpha$ (the squares).

Theorems & Definitions (24)

• Definition 1
• Definition 2
• Definition 3
• Lemma 2.1
• Proof 1
• Definition 4
• Lemma 3.1
• Proof 2
• Proposition 1
• Proof 3