This paper considers the fusion of multiple estimates of a spatially extended object, where the object extent is modeled as an ellipse that is parameterized by the orientation and semi-axes lengths. For this purpose, we propose a novel systematic approach that employs a distance measure for ellipses, i.e., the Gaussian Wasserstein distance, as a cost function. We derive an explicit expression for the Minimium Mean Gaussian Wasserstein distance (MMGW) estimate. Based on the concept of a MMGW estimator, we develop efficient methods for the fusion of extended target estimates. The proposed fusion methods are evaluated in a simulated experiment and the benefits of the novel methods are discussed.
This work is also inspired by the Minimum Mean OSPA estimators @cite_7 @cite_32 @cite_17 @cite_4 for multiple target tracking @cite_31 , where optimal estimates with respect to the OSPA distance @cite_22 are calculated. In this context, the concept of a Wasserstein barycenter @cite_28 @cite_30 @cite_13 @cite_12 @cite_9 is also related to this work.
@cite_30 @cite_4 @cite_22 @cite_7 @cite_28 @cite_9 @cite_32 @cite_31 @cite_13 @cite_12 @cite_17
2006228023 1881351112 2089834727 2101295974
The two title concepts have been evolving rather rapidly, but independent of each other. The Wasserstein barycenter, on one hand, has mostly made its appearance in image processing as it can describe a measure of similarity between images. Its minimization might, for example, suggest the best match in image alignment. On the other hand, MMOSPA estimation has been applied largely to multi-target tracking. The Optimal Sub-Pattern Assignment (OSPA) measures the distance between two sets and the Mean OSPA (MOSPA) can be minimized to give the Minimum MOPSA (MMOSPA), which improves MMSE estimation of the target locations when the labeling of the targets in the set is not important. Approximate and exact algorithms have evolved for both Wasserstein barycenters and MMOSPA estimation. Here, we draw connections between the two perspectives and elaborate how they can benefit from each other. This paper proposes a new metric based on the Optimal Sub-pattern Assignment (OSPA) metric for evaluating the performance of multiple target tracking algorithms. It is shown in this paper that by considering the properties of false tracks, missing tracks and miss-detections, the minimization of all distances between tracks enables performance evaluation of multi-target algorithms in a more comprehensive manner than is possible with the OSPA metric for track (OSPAT) introduced by Both maximum likelihood estimation as well as minimum mean optimal subpattern assignment (MMOSPA) estimation have been shown to provide meaningful estimates in instances of target identity uncertainty when the number of targets present is known. Maximum likelihood measurement to track association (2D assignment) has been widely studied and is reviewed in this paper. However, it is widely believed that approximate MMOSPA estimation can not be performed in real time except when considering a very small number of targets. This paper demonstrates the MMOSPA estimator arises as a special case of a minimum mean Wasserstein metric estimator when the number of targets is unknown. Additionally, it is shown that approximate MMOSPA estimates can be calculated in microseconds to miliseconds without extensive optimization, making MMOSPA estimation a practicable alternative to more traditional estimators. Performance evaluation of multi-target tracking algorithms is of great practical importance in the design, parameter optimization and comparison of tracking systems. The goal of performance evaluation is to measure the distance between two sets of tracks: the ground truth tracks and the set of estimated tracks. This paper proposes a mathematically rigorous metric for this purpose. The basis of the proposed distance measure is the recently formulated consistent metric for performance evaluation of multi-target filters, referred to as the OSPA metric. Multi-target filters sequentially estimate the number of targets and their position in the state space. The OSPA metric is therefore defined on the space of finite sets of vectors. The distinction between filtering and tracking is that tracking algorithms output tracks and a track represents a labeled temporal sequence of state estimates, associated with the same target. The metric proposed in this paper is therefore defined on the space of finite sets of tracks and incorporates the labeling error. Numerical examples demonstrate that the proposed metric behaves in a manner consistent with our expectations.