Robust signal decompositions on the circle

2025-07-09

AI Summary

This paper addresses the problem of inferring an unknown number and positions of circular landmarks from imprecise binary sensory signals—generated solely by nearby landmarks—as an agent moves along a circle. The core challenge lies in modeling the signal as a piecewise-constant function on the circle, whose discontinuities correspond to landmark boundaries; however, the function values at discontinuities are unknown, and landmark radii, centers, and cardinality are all unknown. To tackle this, we introduce the notion of “robust decomposition”: a unique representation of the signal as a sum of restrictions to the circle of circular indicator functions. We characterize the solution space via the concept of “degrees of freedom.” Theoretically, we provide necessary and sufficient conditions for robust decomposability and a complete structural characterization. Algorithmically, we devise a deterministic procedure that enumerates all robust decompositions. Furthermore, we establish tight upper and lower bounds on the number of maximum-degree-of-freedom decompositions. Our results enable robust localization, obstacle avoidance, and motion planning in unknown environments without prior map knowledge.

Technology Category

• Intelligent Robots: Localization, Mapping, and Navigation
• Planning, Routing, and Scheduling: Planning under Uncertainty
• Reasoning under Uncertainty: Sequential Decision Making

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• Decompose piecewise constant circle functions into unknown disk indicators
• Estimate landmark count and locations from proximity sensing data
• Characterize robust decompositions with incomplete discontinuity values

Innovation

• Decompose circle signals into disk indicators
• Define robustness and degrees of freedom
• Characterize and generate robust decompositions