Motion-Enabled Tomography of Gaussian-Mixture Particles under Unknown Rigid-Body Motion with Parabolic Trajectories

Abstract

Reconstructing objects with unknown position and/or orientation arises in applications spanning geology to cryo-ET. When the relative motion between the object and the sensing apparatus provides sufficient angular coverage, computerized tomography offers a powerful means of reconstruction. For such scenarios, we propose a parametric spatiotemporal model applied to Gaussian Mixture Models (GMM), in which each constituent Gaussian is parameterized by its own angular velocity, parabolic trajectory, and morphology. GMM are a suitable means of reconstruction because they (i) admit accurate approximations in object space and (ii) have a closed form expression under the ray transform; enabling efficient forward predictions and analysis in data space. By decoupling the projectile motion from the rotation and morphology, the full optimization problem can be reduced to two lower-dimensional sub-problems that are easier to optimize. The corresponding algorithm --- which is applicable to objects in Euclidean space of arbitrary dimension --- is validated on an extensive set of noisy 2D simulations. Its performance provides promise for further experimentation in more complex scenarios, such as higher dimensions, objects with deformable body mechanics, and on physical datasets.

Submitted to SIAM SIIMS. 38 pages, Noisy 2D reconstruction included