Ellipsoidal extended target estimation fusion via Wasserstein barycentric coordinates

The ellipsoidal extended target estimation fusion problem is challenging due to the need to consider the target extension. Current methods consider the Wasserstein barycenter, which incorporates the underlying geometry of the Gaussian distributions measure space. However, these methods do not optimize the weights for each sensor. In this paper, we propose the Wasserstein Barycentric Coordinates Fusion (WBCF) method which can adaptively select fusion weights by utilizing prior information about the target extension. While the theory of Wasserstein barycentric coordinates is well-established for discrete distributions with identical supports, the general case involving arbitrary probability distributions presents a significant challenge due to the computational complexity arising from solving a non-convex and non-concave optimization problem. In the context of Gaussian distributions, this paper derives closed-form expressions for the derivatives of the objective function. Moreover, by reformulating the problem as a bi-level optimization problem, we propose practical algorithms for WBCF with minimal computational overhead. Finally, we demonstrate the performance of WBCF in simulated extended target tracking scenarios. By utilizing the optimization method, numerical results showcase the efficiency of the proposed method in terms of estimation accuracy and computational time.