Mixture Random Hypersurface Models for Tracking Multiple Extended Objects Marcus Baum, Benjamin Noack, and Uwe D. Hanebeck Abstract— This paper presents a novel method for tracking multiple extended objects. The shape of a single extended object is modeled with a recently developed approach called Random Hypersurface Model (RHM) that assumes a varying number of measurement sources to lie on scaled versions of the shape boundaries. This approach is extended by introducing a so- called Mixture Random Hypersurface Model (Mixture RHM), which allows for modeling multiple extended targets. Based on this model, a Gaussian-assumed Bayesian tracking method that provides the means to track and estimate shapes of multiple extended targets is derived. Simulations demonstrate the performance of the new approach. Keywords: Multiple Extended Object Tracking, Shape Tracking, Random Hypersurface Model I. I NTRODUCTION In contrast to a point target, an extended object may be the origin of several measurements from different measurement sources on its surface (see Fig. 1). Essentially, there are two major scenarios that illustrate this issue. First, continuously evolving sensor technologies provide advanced resolution capabilities that can result into several measurements of one target during a single scan. The measurement sources vary from scan to scan and their locations depend on the shape of the target but also on more complex target-dependent prop- erties. Second, a group of point targets can also be treated as a single entity when there are strong interdependencies between the individual group members. In this work, we consider the problem of tracking multiple extended objects (see Fig. 1), where the goal is to estimate a shape approximation of each extended target in addition to its kinematic parameters [1]–[3]. Note that this is a non- trivial task, as it is required to deal with measurements that may stem from different objects whose extents are unknown and are part of the estimation problem. In order to model the shape of a single extended target, we employ a recent approach called Random Hypersurface Model (RHM) [3], [4]. An RHM assumes that measurement sources lie on scaled versions of the shape boundary, which allows for estimating the form and the extent of the shape. In doing so, the target can be modeled as a basic shape, such as an ellipse [3], or even as an arbitrary star-convex shape [4]. So far, RHMs are restricted to a single extended target. The main contribution of this paper is a method for modeling and tracking multiple extended targets based on Marcus Baum, Benjamin Noack, and Uwe D. Hanebeck are with the Intelligent Sensor-Actuator-Systems Laboratory (ISAS), Institute for Anthropomatics, Karlsruhe Institute of Technology (KIT), Germany. marcus.baum@kit.edu, noack@kit.edu, uwe.hanebeck@ieee.org = Measurement source = Measurement Fig. 1: Two closely-spaced extended objects. RHMs. For this purpose, we first introduce the novel concept of Mixture Random Hypersurface Models (Mixture RHMs), which are an extension of RHMs to multiple extended targets. Based on this model, we derive a Bayesian method for track- ing multiple extended targets and we suggest a particular implementation as a Gaussian-assumed density filter. The remainder of this paper is structured as follows: First, we give a brief overview of related approaches for extended object tracking in Section II. Afterwards, in Section III we discuss how a single extended object can be modeled with RHMs. Subsequently, we introduce the basic idea of Mixture Random Hypersurface Models in Section IV-A and derive a Bayesian estimator for Mixture RHMs in Section V. Simulations in Section VI demonstrate the feasibility of the approach. Finally, this paper is concluded in Section VII. II. RELATED WORK Spatial distribution models [1], [5] assume that each mea- surement source is randomly drawn from a known object- dependent probability distribution. In [1], [5], [6], spatial distribution models have been utilized to track multiple extended targets, e.g., stick targets, in a cluttered envi- ronment. Recently, spatial distribution models have been integrated into Probability Hypothesis Density (PHD) filters for tracking multiple extended objects [7], [8]. In [9], a Sequential Monte-Carlo (SMC) approach for tracking ex- tended objects based on border parameterization is presented. A recent approach for modeling elliptic target extents is based on random symmetric positive definite matrices [10], [11], which has also been integrated into the Probabilistic Multiple-Hypothesis Tracker (PMHT) framework [12] for tracking multiple extended targets. A thorough comparison of the random matrix approach with Random Hypersurface Models can be found in [13].