Signal Detection on Euclidean Groups: Applications to DNA Bends, Robot Localization, and Optical Communication

Three problems from disparate application areas are presented and solved here using a unified framework: 1) estimating the bend angle induced in DNA by a bound ligand such as a transcription factor or anti-cancer drug; 2) determining the intent of a mobile robot by observing its trajectories corrupted by environmental noise; 3) estimating the bit-error probability function associated with phase noise in optical communication systems, and the associated problem of filter design. In all three problems, probability densities on the group of proper rigid-body motions of the plane contain a hidden signal that needs to be detected in order to advance the particular application area. Stochastic differential equations and corresponding Fokker-Planck equations describing random processes that evolve on this group (the Euclidean group) are used to model each of these problems, and methods from harmonic analysis and Lie theory are used to write approximate solutions. From these `forward´ models the desired `signal´ (i.e., an element, or a path, in the Euclidean group is extracted) to infer desired physical parameters from data.