Three-Dimensional Extended Target Tracking and Shape Learning Based on Double Fourier Series and Expectation Maximization

This paper investigates the problem of tracking targets with unknown but fixed 3D star-convex shapes using point cloud measurements. While existing methods typically model shape parameters as random variables evolving according to predefined prior models, this evolution process is often unknown in practice. We propose a particular approach within the Expectation Conditional Maximization (ECM) framework that circumvents this limitation by treating shape-defining quantities as parameters estimated directly via optimization. The objective is the joint estimation of target kinematics, extent, and orientation in 3D space. Specifically, the 3D shape is modeled using a radial function estimated via double Fourier series (DFS) expansion, and orientation is represented using the compact, singularity-free axis-angle method. The ECM algorithm facilitates this joint estimation: an Unscented Kalman Smoother infers kinematics in the E-step, while the M-step estimates DFS shape parameters and rotation angles by minimizing regularized cost functions, promoting robustness and smoothness. The effectiveness of the proposed algorithm is substantiated through two experimental evaluations.