Motion-Enabled Tomography via Gaussian Mixture Models

TL;DR

This paper introduces a novel Gaussian Mixture Model-based approach for reconstructing moving objects in tomography, enabling efficient and accurate spatiotemporal analysis.

Contribution

It proposes a parametric GMM framework with closed-form ray transform expressions, facilitating precise and efficient motion-enabled tomography reconstruction.

Findings

Successfully reconstructed a 2D 5-Gaussian GMM with intersecting trajectories.

The method is applicable to objects in arbitrary Euclidean dimensions.

Provides a foundation for future work with noisy data, 3D objects, and non-rigid dynamics.

Abstract

Recovering physical properties of objects in motion is a core task across scientific and industrial applications. 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, projectile motion, and geometry. 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 exact gradient computations in data space. By decoupling the reconstruction problem into two sub-inverse problems, we characterize solutions as minimizers of task-specific objective functions that are derived and solved by utilizing the properties of (ii). The resulting algorithm we provide is applicable to objects in Euclidean space of arbitrary dimension. We validate the method on a simulated 2D problem, achieving accurate reconstruction of a 5-Gaussian GMM with intersecting trajectories. This also provides a foundation for further experimentation in settings with noisy data, 3D objects, and non-rigid body dynamics.