Uniqueness of Angular Velocity Reconstruction in Parallel-Beam and Diffraction Tomography
Peter Elbau, Denise Schmutz
TL;DR
The paper tackles the problem of uniquely reconstructing an unknown rotational motion from time-dependent tomographic measurements under two models: diffraction tomography with the Born approximation and parallel-beam tomography. It introduces DT- and PB-asymmetry, proving that these conditions guarantee unique recovery of the angular velocity via infinitesimal common circle and infinitesimal common line methods, respectively. It is shown that the sets of DT-symmetric and PB-symmetric objects are nowhere dense in the admissible object space, implying that uniqueness holds for generic objects. Practically, this establishes a theoretical basis for using motion reconstruction as a preprocessing step in 3D refractive-index tomography of optically or acoustically trapped particles, and provides a framework for constructing asymmetric test objects for robust simulations.
Abstract
This work addresses the problem of uniquely determining a rotational motion from continuous time-dependent measurements within the frameworks of parallel-beam and diffraction tomography. The motivation stems from the challenge of imaging trapped biological samples manipulated and rotated using optical or acoustic tweezers. We analyze the conditions under which the rotation of the unknown sample can be uniquely recovered using the infinitesimal common line and circle method, respectively. We provide explicit criteria for the sample's structure and the induced motion that guarantee unique reconstruction of all rotation parameters. Moreover, we demonstrate that the set of objects for which uniqueness fails is nowhere dense.