Abstract:
This thesis investigates the mathematical conditions under which the unknown, timedependent rigid motion of a microscopic particle can be uniquely reconstructed from continuous transmission measurements in optical tomography. The motivation stems from optical and acoustic trapping, an established technique for holding and rotating biological samples suspended in a liquid in a contact-free and non-invasive manner. It allows imaging in a more natural environment than standard microscopy, where the sample must be fixed. In these experimental setups, the rotation of the trapped particle is induced by acoustic radiation forces or optical tweezers, providing measurement data from multiple directions and thus enabling a three-dimensional tomographic reconstruction of the sample’s internal structure. However, since the induced motion depends on the sample’s unknown internal properties, it cannot be perfectly controlled and must instead be recovered directly from the measurement data. Once the motion is recovered, the reconstruction of the sample’s internal structure reduces to a classical inverse problem, which lies outside the primary focus of this work.
Under the reasonable assumption that the induced motion is rigid, i.e., consisting of a rotation and translation, we model the optical imaging of a trapped particle starting from Maxwell’s equations. Depending on the physical scale and the scattering properties of the sample, we derive two approximating models: diffraction tomography and parallel-beam tomography.
The optical diffraction tomography (ODT or simply DT) model is based on the Born approximation and describes the wave nature of light for weakly scattering samples. This setting is further simplified by assuming purely rotational motion without translations. One way to recover the rotation is the infinitesimal common circle method, which is based on the Fourier diffraction theorem. We develop the concept of DT-asymmetry and prove that a DT-asymmetric object uniquely determines the angular velocity, which fully describes the rotation. Moreover, we show that the set of DT-symmetric objects is nowhere dense,
ensuring that unique reconstruction via the infinitesimal common circle method is generically feasible.
For larger samples where scattering can be neglected, optical projection tomography (OPT) utilizing the parallel-beam (PB) model provides an appropriate description of the imaging process. For this model, we demonstrate how the sample translation can be recovered directly from the measurement data by tracking the two-dimensional center of the projection images over time. A central contribution of this part of the thesis is the development of the infinitesimal common line method for reconstructing rotational motion. The method is based on the Fourier slice theorem and can be viewed as a continuous-time analog of the common line technique used in single-particle cryogenic electron microscopy (cryo-EM) to estimate particle orientations. A numerical proof of principle validates this approach.
We show that not all rotational motions can be reconstructed, as certain degenerate motions fail to provide sufficient information within the infinitesimal equations. Restricting our focus to non-degenerate motions, we introduce the structural condition of PB-asymmetry. We establish that for PB-asymmetric objects undergoing non-degenerate rotations, all rotation parameters can be recovered up to unavoidable model-induced ambiguities. Moreover, similar to the diffraction tomography case, we prove that PB-asymmetry is a generic property of admissible samples.

Zoom-Link:
univienna.zoom.us/j/65622265250
Meeting ID: 656 2226 5250
Passcode: 862625