Hamiltonian-based data clustering and classification

对于任何扩展实体，如车队、人群、尘埃云，甚至近距离看到的刚性车辆，数据集群是数据处理中的一个基本步骤，是数据识别、分类和跟踪程序的核心。这种漫射实体的观测通常由适当定义的空间中的点来表示。这些点可以表示与数据相关联的位置、活动或其他属性的观察。通常，理解这样的数据的初始任务是找到点之间的关系，以将观测划分成具有定义要跟踪的实体的相似特征的组(即，簇)。这种聚类算法通常基于观察空间中的点的相对位移和/或基于已知参考对象的库来执行分类。如果预先知道集群的数量，如果要分类的对象属于一组已知对象，并且如果集群在时间上是稳定的，则传统的集群算法更容易实现，并且更可靠。如果不是这样的话，算法还必须解决所谓的簇验证问题，并且必须自适应地生成用于执行分类的对象库。这种常规方法对噪声、欠采样、回波的存在和临时数据丢失非常敏感，这将是典型的不合作、扩散观测的情况。提出了一种新的分割和跟踪扩展对象的方法，该方法以稀疏观测为特征，提出了一种新的动态聚类算法：将聚类识别为与一个聚类函数的参考值对应的水平集，其核心思想是从观测数据构造聚类函数，并将该聚类函数作为哈密顿动力系统的生成器，其轨迹描述了这些簇。虽然聚类函数的概念是标准的，但哈密顿动力学的使用提供了一个独特的视角和几个优点。其中包括计算星团的在线几何特征、用降阶模型表示其动力学以及识别其动力学行为的可能性。

For any extended entities such as convoys of vehicles, crowds of people, dust clouds or even rigid vehicles seen at close range, data clustering is an essential step in data processing and it is at the core of data recognition, classification and tracking procedures. Observations of such diffuse entities will generally be represented by points in a properly defined space. These points may represent observations of position, activity or other attributes, associated with the data. Over time, further points will be observed that need not necessarily correspond to any of the same features seen previously.Conventionally, the initial task in understanding such data is to find relations between the points to partition the observations into groups (i.e. the clusters) with similar features that define the entity to be tracked. Such clustering algorithms generally perform the classification on the basis of the relative displacements of the points in the observation space and/or on the basis of a library of known reference objects. Conventional clustering algorithms are easier to implement, and more reliable, if the number of clusters is known in advance, if the objects to be classified belong to a set of known objects, and if the cluster is stable in time. If this is not the case, the algorithm has to solve also the so-called cluster validation problem and has to adaptively generate a library of objects against which to perform the classification.Such convential approaches are sensitive to noise, under-sampling, presence of echoes and temporary data drop-outs, which would be typical of situations of uncooperative, diffuse observations. This proposal concerns a new approach to segmentation and tracking such extended objects characterised by sparse observations over extended times.We propose to develop a novel dynamic clustering algorithm: the clusters are identified as the level sets corresponding to a reference value of a clustering function.The core idea is to construct the clustering function from observations and to regard the clustering function as the generator of a Hamiltonian dynamical system, the trajectories of which describe the clusters. While the notion of clustering function is standard, the use of Hamiltonian dynamics provides an original perspective and several advantages. These include the possibility to compute on-line geometric features of the clusters, to represent their dynamics with reduced order models, and to identify their dynamical behaviour.