Advances in the Application of Stochastic Geometry in Robotics

Stochastic geometry is an established branch of mathematics that studies uncertainty in geometric structures [1], [2] and is, thus, a befitting framework for autonomous robotic mapping and the well known Simultaneous Localization and Mapping (SLAM) problem, where the fundamental concern is succinctly captured in the title of the 1988 seminal paper by Durrant-Whyte, ”Uncertain geometry in robotics” [3]. The theory of random sets has long been used by statisticians in many diverse applications including agriculture, geology, and epidemiology [1], [2], [4]–[6]. In addition, there has been substantial work by probabilists and statisticians, in point process filtering, such as that by Ikoma et al [7]. Applications of this are random point pattern methods for multiple object recognition in image analysis [8], [9], and recent work based on random set analysis by Vo et al [10], which laid the foundations for set based multi-object visual tracking by Hoseinnezhad et al [11], [12]. The application of random sets in multi-target tracking has led to the development of Finite Set Statistics (FISST) which provides the basis for novel filters such as the Probability Hypothesis Density (PHD) filter [13]– [15] and the Cardinalized (C)-PHD filter [16] which recently attracted considerable research interest as well as deployment in commercial applications. As noted in the field of multi-target filtering by Mahler ([17], page 571):