MMOSPA-based track extraction in the PHD filter - a justification for k-means clustering

Displaying tracks is an essential part of a multi-target tracking system. Recently, it was proposed to extract tracks with respect to the Optimal Sub-Pattern Assignment (OSPA) metric, i.e., the traditionally used squared error loss is replaced with an OSPA loss, which leads to the so-called Minimum Mean OSPA (MMOSPA) estimate. So far, work concentrated on traditional trackers that maintain probability densities for the targets. In this paper, we aim at extracting the MMOSPA estimate from a Probability Hypothesis Density (PHD) as used within the PHD filter. We elaborate that the PHD in general does not contain enough information to determine the exact MMOSPA estimate. However, we then show that if the loss function has a specific form, it is indeed possible to extract point estimates from a PHD that are optimal w.r.t. the underlying unknown random finite set. We discuss two specific loss functions that fulfill this condition and are potentially close to the OSPA loss, a nearest neighbor loss and a kernel distance loss. It turns out that track extraction based on the nearest neighbor loss can be performed with the well-known k-means algorithm. Simulations show when the estimates based on the nearest neighbor and the kernel loss are close to the MMOSPA estimate.