This article considers the Wasserstein barycentric coordinates problem for Gaussian distributions which is the inverse problem of the Wasserstein barycenter problem. These coordinates take into account the underlying geometry of the measure space of Gaussian distributions and are thus meaningful for applications such as diffusion analysis and distributed information fusion. When the probability supports are discrete and identical, the theory of Wasserstein barycentric coordinates is well developed. However, for general probability distributions, the computation of Wasserstein barycentric coordinates is intractable since the technical hurdles involve solving a non-convex and non-concave optimization problem. For Gaussian distributions, we derive the closed-form expression of the derivatives for the objective function and propose a projected gradient descent method to solve the problem. Finally, we illustrate its application in diffusion tensor image (DTI) denoising including simulated DTI with different noise levels and DTI of the human brain.

This paper addresses the problem of drop-point constrained state modeling in the set-valued framework, where only drop-point information of state trajectories is available. Due to the lack of complete trajectory information, existing estimation methods with linear equality constraints struggle to achieve effective state prediction. This paper primarily investigates the integration of drop-point constraint information into the entire system state evolution process via convex optimization projection in the set-valued framework, aiming to reconstruct a linear dynamic system model. Subsequently, by employing multiple affine transformations of ellipsoids and designing a weight matrix, the smoothness of state trajectories is enhanced to better align with real motion patterns. Finally, through simulation experiments, we validate the significant advantages of the reconstructed system model over traditional unconstrained system models under the set-valued framework. Additionally, we demonstrate the impact of drop-point constraints on state trajectory evolution under different initial point conditions with the same drop point.

This paper addresses the localization challenge in cooperative multi-agent wireless sensor networks, specifically focusing on range-based localization. To enhance robustness against outliers in range measurements, we employ the Huber function, leading to the formulation of a robust yet nonconvex optimization problem with coupled agent variables. Confronted with this nonconvex optimization challenge, particularly in largescale networks, we reformulate the problem using Lagrange duality and conjugate theory. This restructuring yields subproblems characterized by smooth strong convexity for dual variables and a simplified form for primal variables, thereby facilitating an efficient solution. Building upon this reformulation, we introduce a novel distributed primal-dual algorithm that employs coordinate descent and proximal minimization techniques within an iterative framework. This approach furnishes closed-form solutions for both primal and dual variables. Theoretically, our method ensures not only the convergence of the sequence of objective function values but also, by leveraging the KurdykaŁojasiewicz property, we establish the guaranteed global convergence of the location estimates sequence to a critical point of the original objective function. Notably, our proposed approach exhibits lower computational complexity, communication cost, and storage space compared to existing methods. Numerical experiments underscore the superiority of the proposed method in terms of robustness and localization accuracy when compared to the other methods in the literature.

The distributed detection fusion is investigated for conditionally dependent sensor networks with channel errors. When the joint probability density functions of the sensor observations are dependent and high dimensional, it is known to be a challenging problem. This paper deals with this problem under Monte Carlo framework. The Bayesian cost function is approximated by Monte Carlo importance sampling. Necessary conditions for optimal sensor rules and optimal fusion rule are derived in the sense of minimizing the approximated Bayesian cost function, respectively. A Gauss-Seidel/person-by-person optimization algorithm is developed to search the optimal sensor rules. It is proved that the discretized algorithm is finitely convergent. Since the error rate of Monte Carlo integration is regardless of dimensionality, the complexity of the new algorithm is much less than that of the previous algorithm based on Riemann sum approximation. The proposed method allows us to design the sensor networks with a higher dimensional joint probability density function of the sensor observations. The typical examples with dependent observations and channel errors are examined. The results of numerical examples demonstrate the effectiveness of the new algorithm.

Multiple-source localization problem based on acoustic energy measurements is investigated by set-membership estimation theory. When the probability density function of measurement noise is unknown-but-bounded, multiple-source localization is a difficult problem since not only the acoustic energy measurement is a complicated nonlinear function of multiple sources, but also the multiple sources bring about a high-dimensional state estimation problem. The main contribution of this paper is as follows. Firstly, to deal with the nonlinear function, it is linearized by the first-order Taylor expansion with a remainder error. The point is that the bounding box of the remainder is derived in each iteration based on the convex bounding set of the state. Especially, when the state can be bounded in a cylinder, the remainder bound can be achieved analytically. Secondly, based on the separate property of the nonlinear observation function, an efficient estimation procedure is developed to deal with the high-dimensional state estimation problem by using an alternately optimization iterative algorithm. In the process of iteration, the remainder bound requires to be known on-line. Finally, a typical numerical example in multiple-source localization demonstrates the effectiveness of the set-membership localization algorithms. In particular, it shows that when the noise is non-Gaussian, the set-membership localization algorithm performs better than the maximum likelihood localization algorithm.

The set-membership information fusion problem is investigated for general multisensor nonlinear dynamic systems. Compared with linear dynamic systems and point estimation fusion in mean squared error sense, it is a more challenging nonconvex optimization problem. Usually, to solve this problem, people try to find an efficient or heuristic fusion algorithm. It is no doubt that an analytical fusion formula should be much significant for raising accuracy and reducing computational burden. However, since it is a more complicated than the convex quadratic optimization problem for linear point estimation fusion, it is not easy to get the analytical fusion formula. In order to overcome the difficulty of this problem, two popular fusion architectures are considered: centralized and distributed set-membership information fusion. Firstly, both of them can be converted into a semidefinite programming problem which can be efficiently computed, respectively. Secondly, their analytical solutions can be derived surprisingly by using decoupling technique. It is very interesting that they are quite similar in form to the classic information filter. In the two analytical fusion formulae, the information of each sensor can be clearly characterized, and the knowledge of the correlation among measurement noises across sensors are not required. Finally, multi-algorithm fusion is used to minimize the size of the state bounding ellipsoid by complementary advantages of multiple parallel algorithms. A typical numerical example in target tracking demonstrates the effectiveness of the centralized, distributed, and multi-algorithm set-membership fusion algorithms. In particular, it shows that multi-algorithm fusion performs better than the centralized and distributed fusion.

The estimation fusion problem and posterior Cramer-Rao bound (PCRB) are presented for multi-sensor nonlinear systems with uncertain observations. In order to effectively deal with the difficulties caused by uncertainty, a novel method is proposed by introducing 0-1 latent variables. It has two nice properties. Firstly, the derived estimation fusion method can take full advantage of the character of the nonlinear function and uncertain observations. Secondly, the uncertain system with a discrete variable can be approximated by a continuous system, where the discrete distribution of the latent variable is approximated by a continuous one, then the PCRB can be achieved by a limiting process of PCRB for the continuous system. Since the derived PCRB has an analytical expression, it can reduce the computational burden much more. A typical numerical example in target tracking demonstrates the effectiveness of the estimation fusion method and the proposed PCRB for the uncertain observation systems.

A novel random data association algorithm is proposed in the framework of multiple hypothesis tracking which can be equivalent to an NP-hard multidimensional assignment problem. The key idea of this new algorithm is to relax the multidimensional assignment problem to a linear programming problem. The solution of the linear programming problem may be treated as the probability of potential tracks, which avoids the computation difficulties of rounding the probability solution to 0/1 satisfying the constraints. Then all tracks and measurements can be integrated to a new whole dynamic system with random coefficient matrices. Moreover, the random coefficient matrix Kalman filtering is applied to the integrated dynamic system to derive the state estimates of the tracks. The significant advantage of the random Kalman filter-based multiple hypothesis tracking data association is that the computation complexity is much less than that of Lagrangian relaxation of the multidimensional assignment problem. Simulation demonstrates the random data association algorithm works well and the running time can be shortened greatly compared with the Lagrangian relaxation algorithm of the multidimensional assignment problem.