Robust signal decompositions on the circle

TL;DR

This paper addresses the problem of decomposing piecewise constant functions on the circle into sums of indicator functions of circular disks, focusing on robustness, uniqueness, and practical estimation for applications like navigation and obstacle avoidance.

Contribution

It introduces notions of robustness and degrees of freedom for such decompositions, characterizes robust solutions, and provides procedures to generate and analyze all robust decompositions.

Findings

Characterization of robust decompositions

Procedure for generating all robust decompositions

Bounds on the number of decompositions with maximum degrees of freedom

Abstract

We consider the problem of decomposing a piecewise constant function on the circle into a sum of indicator functions of closed circular disks in the plane, whose number and location are not a priori known. This represents a situation where an agent moving on the circle is able to sense its proximity to some landmarks, and the goal is to estimate the number of these landmarks and their possible locations -- which can in turn enable control tasks such as motion planning and obstacle avoidance. Moreover, the exact values of the function at its discontinuities (which correspond to disk boundaries for the individual indicator functions) are not assumed to be known to the agent. We introduce suitable notions of robustness and degrees of freedom to single out those decompositions that are more desirable, or more likely, given this non-precise data collected by the agent. We provide a characterization of robust decompositions and give a procedure for generating all such decompositions. When the given function admits a robust decomposition, we compute the number of possible robust decompositions and derive bounds for the number of decompositions maximizing the degrees of freedom.