Fusion of Gaussian mixture models for possible mismatches of sensor model

Abstract

This paper addresses estimation fusion in the presence of possible mismatches of sensor model. The main concerns of the paper lie in two aspects. One is to improve the filter performance of the single sensor when there are possible mismatches about the sensor model. The other one is to adopt a good fusion scheme to combine local estimates. For these purposes, the measurement process of the local sensor is modeled by multiple models firstly, and the IMM (interacting multiple model) estimator is implemented to produce estimates for individual models. Next, we describe the local estimate by a Gaussian mixture rather than a single Gaussian density in the baseline IMM filter. Such a GMM (Gaussian mixture model) representation of the system state allows us to keep the detailed information about the local tracker, which contributes to the further fusion if treated properly. Finally, the fusion of two Gaussian mixtures is done in the probabilistic framework, and a close-form solution is derived without complex numerical operations. Simulation results demonstrate the effectiveness of the proposed approach.

Introduction

Distributed estimation fusion is an important research topic in the area of data fusion. The potential advantage of fusing the redundant and diverse sensor information contributes to increasing the reliability and flexibility of the whole multi-sensor system. Lots of valuable research results have been achieved [1], [2], [3], [4], [5], [6]. Among them, Ref. [1] summarizes practical algorithms for multi-sensor multi-target tracking in surveillance systems. In [2], some innovative multi-sensor multi-target applications and their solutions are discussed. Ref. [3] covers the basic theory and the most advanced developments in multi-sensor data fusion research area. In [4], the fusion architectures and rules are given concerning about optimal linear estimation fusion, and performance comparisons of different fusion rules are made via simulation experiments in [5]. Considering the cross-correlation of local estimation errors, Ref. [6] develops easily computable formulas for cases with linear and nonlinear observations.

However, in some practical tracking applications, sensor reports may be disturbed by some unknown and unobservable factors. If the filter is designed based on the ideal measurement model (regardless of possible model mismatch), it is hard to obtain a good filter result. Consequently, there may be a great difference among estimates from different local trackers. At this time, the covariance information is convincing no longer, which cannot reflect the accuracy of local estimates well. In this case, the direct fusion of local estimates will not produce a satisfactory result.

In [7], [8], a method combining the CI (Covariance Intersection) and CU (Covariance Union) (CI/CU) was developed, in which the Mahalanobis distance was used to identify the dissimilarity among local estimates. In [9], a probability framework based on the GMM was proposed to fuse possibly biased local estimates. The exact probability that local sensor is biased is needed to construct the Gaussian component of the mixture model. When the prior information is unknown exactly, the performance of the method may experience an evident decrease. In [10], a fast and robust approach (IT-FFGCC) was proposed according to the information-theoretic metric, which defines a parameter indicating how the local estimates differ from each other.

In this paper, we propose a novel approach (called IMM–GMM) to deal with the possible mismatch of the sensor model. The proposed approach develops a GMM representation for the system state based on the IMM estimator [11], [12], [13], and implements the fusion of two Gaussian mixtures in a probabilistic framework to achieve a better fusion performance. The IMM estimator is implemented to obtain an adaptive estimate. Note that in the IMM framework, a set of models is selected to represent the possible system patterns, and the final fused estimate is the weighted combination of estimates based on individual models. Here, the meaning of the word ‘adaptive’ lies in two aspects. One is that the weights of individual models can be adapted to reflect how the individual models match the measurement data. The second one is that the model set can be assumed variable, and be made adaptively based on measurements to capture the characteristics of online models. In our problem, we use the word ‘adaptive’ just meaning the former, and have no issue involved concerning about the MSA (model set adaptation) [13].

The contributions of the paper mainly lie in two aspects. One is to improve the filter performance of the single sensor. The other one is to adopt a suitable fusion scheme to combine local estimates. To cope with the situation where the sensor model may be possibly inaccurate, we model the measurement process for sensors in the framework of multiple models firstly; next, we describe the local estimate by a Gaussian mixture rather than a single Gaussian density, where the parameters for each Gaussian component is obtained based on the IMM recursion. As we known, the Gaussian mixture model [14], [15] severs as an important representation of the system state in many applications such as data fusion [16], [17], pattern recognition [18], [19], and supervised learning of multimedia data [20], [21]. Here, such a GMM representation of system state allows us to keep the detailed information about the local tracker, which contributes to the further fusion if treated properly. Finally, two Gaussian mixtures are fused according to the normalized product in a probabilistic framework [22], [23]. Moreover, the fusion of two Gaussian mixtures is of a closed form without complex numerical operations.

The rest of the paper is organized in the following. In Section 2, we formulate the estimation fusion in the presence of possible mismatches of sensor model, and review some main related work. Section 3 shows how we obtain the GMM representation of the local estimate, and implement the fusion of Gaussian mixture models in the probabilistic framework. Simulation results are given in Section 4 to show the performance of the proposed approach. We conclude in Section 5.