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
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We define a notion of barycenter for random probability measures in the Wasserstein space. We study the population barycenter in terms of existence and uniqueness. Using a duality argument, we give a precise characterization of the population barycenter for compactly supported measures, and we make a connection between averaging in the Wasserstein space and taking the expectation of optimal transport maps. Then, the problem of estimating this barycenter from @math independent and identically distributed random probability measures is considered. To this end, we study the convergence of the empirical barycenter proposed in Agueh and Carlier (2011) to its population counterpart as the number of measures @math tends to infinity. To illustrate the benefits of this approach for data analysis and statistics, we show the usefulness of averaging in the Wasserstein space for curve and image warping. In this setting, we also study the rate of convergence of the empirical barycenter to its population counterpart for some semi-parametric models of random densities. 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. The concept of a miss-distance, or error, between a reference quantity and its estimated controlled value, plays a fundamental role in any filtering control problem. Yet there is no satisfactory notion of a miss-distance in the well-established field of multi-object filtering. In this paper, we outline the inconsistencies of existing metrics in the context of multi-object miss-distances for performance evaluation. We then propose a new mathematically and intuitively consistent metric that addresses the drawbacks of current multi-object performance evaluation metrics. Most area-defense formulations follow from the assumption that threats must first be identified and then neutralized. This is reasonable, but inherent to it is a process of labeling: threat A must be identified and then threat B, and then action must be taken. This manuscript begins from the assumption that such labeling (A & B) is irrelevant. The problem naturally devolves to one of Random Finite Set (RFS) estimation: we show that by eschewing any concern of target label we relax the estimation procedure, and it is perhaps not surprising that by such a removal of constraint (of labeling) performance (in terms of localization) is enhanced. A suitable measure for the estimation of unla-beled objects is the Mean OSPA (MOSPA). We derive a general algorithm which provided the optimal estimator which minimize the MOSPA. We call such an estimator a Minimum MOSPA (MMOSPA) estimator. In this paper, we introduce a notion of barycenter in the Wasserstein space which generalizes McCann's interpolation to the case of more than two measures. We provide existence, uniqueness, characterizations, and regularity of the barycenter and relate it to the multimarginal optimal transport problem considered by Gangbo and Świech in [Comm. Pure Appl. Math., 51 (1998), pp. 23–45]. We also consider some examples and, in particular, rigorously solve the Gaussian case. We finally discuss convexity of functionals in the Wasserstein space. We present new algorithms to compute the mean of a set of empirical probability measures under the optimal transport metric. This mean, known as the Wasserstein barycenter, is the measure that minimizes the sum of its Wasserstein distances to each element in that set. We propose two original algorithms to compute Wasserstein barycenters that build upon the subgradient method. A direct implementation of these algorithms is, however, too costly because it would require the repeated resolution of large primal and dual optimal transport problems to compute subgradients. Extending the work of Cuturi (2013), we propose to smooth the Wasserstein distance used in the definition of Wasserstein barycenters with an entropic regularizer and recover in doing so a strictly convex objective whose gradients can be computed for a considerably cheaper computational cost using matrix scaling algorithms. We use these algorithms to visualize a large family of images and to solve a constrained clustering problem. We expand upon existing literature regarding using Minimum Mean Optimal Sub-Pattern Assignment (MMOSPA) estimates in multitarget tracking, noting its advantages in comparison to Maximum Likelihood (ML) and Minimum Mean Squared Error (MMSE) estimation, and look at the practical computation of MMOSPA estimates. We demonstrate the use of MMOSPA estimation in a two-target tracking scenario as well as outside of tracking in a radar angular superresolution scenario. This paper proposes a new definition of the averaging of discrete probability distributions as a barycenter over the Monge-Kantorovich optimal transport space. To overcome the time complexity involved by the numerical solving of such problem, the original Wasserstein metric is replaced by a sliced approximation over 1D distributions. This enables us to introduce a new fast gradient descent algorithm to compute Wasserstein barycenters of point clouds. This new notion of barycenter of probabilities is likely to find applications in computer vision where one wants to average features defined as distributions. We show an application to texture synthesis and mixing, where a texture is characterized by the distribution of the response to a multi-scale oriented filter bank. This leads to a simple way to navigate over a convex domain of color textures. Equilibrium multi-population matching (matching for teams) is a problem from mathematical economics which is related to multi-marginal optimal transport. A special but important case is the Wasserstein barycenter problem, which has applications in image processing and statistics. Two algorithms are presented: a linear programming algorithm and an efficient nonsmooth optimization algorithm, which applies in the case of the Wasserstein barycenters. The measures are approximated by discrete measures: convergence of the approximation is proved. Numerical results are presented which illustrate the efficiency of the algorithms. 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.