Hamiltonian-based data clustering and classification

对于任何扩展的实体，如车队、人群、尘埃云甚至近距离看到的刚性车辆，数据聚类是数据处理的重要步骤，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用一个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的位置、活动或其他属性的观测值。随着时间的推移，将观察到更多的点，这些点不一定与前面看到的任何相同特征相对应。通常，理解此类数据的初始任务是找到点之间的关系，将观测值划分为具有相似特征的组（即集群），这些特征定义了要跟踪的实体。这种聚类算法通常基于观测空间中点的相对位移和/或基于已知参考对象库进行分类。如果预先知道聚类的数量，待分类的对象属于已知对象的集合，并且聚类在时间上是稳定的，那么传统的聚类算法更容易实现，也更可靠。如果不是这种情况，算法还必须解决所谓的集群验证问题，并且必须自适应地生成一个对象库来执行分类。这种传统的方法对噪声、采样不足、回波的存在和临时数据丢失很敏感，这将是不合作、分散观测的典型情况。本文提出了一种新的方法来分割和跟踪这些扩展对象，这些扩展对象的特征是在较长的时间内观察稀疏。我们建议开发一种新的动态聚类算法：将聚类识别为与聚类函数的参考值相对应的水平集。其核心思想是根据观测构造聚类函数，并将聚类函数视为哈密顿动力系统的生成器，其轨迹描述了聚类。虽然聚类函数的概念是标准的，但使用哈密顿动力学提供了一个原始的视角和几个优点。这些包括在线计算集群几何特征的可能性，用降阶模型表示它们的动态，以及识别它们的动态行为。

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.