Shapes from Echoes: Uniqueness from Point-to-Plane Distance Matrices
This addresses ambiguities in localization for acoustic sensing and structure-from-sound applications, but it is incremental as it builds on prior computational work by focusing solely on uniqueness.
The paper tackles the problem of localizing points and planes from point-to-plane distances, which models applications like acoustic SLAM and microphone localization, by providing a complete characterization of uniqueness, enumerating equivalence classes and algebraically characterizing transformations in 2D and 3D.
We study the problem of localizing a configuration of points and planes from the collection of point-to-plane distances. This problem models simultaneous localization and mapping from acoustic echoes as well as the notable "structure from sound" approach to microphone localization with unknown sources. In our earlier work we proposed computational methods for localization from point-to-plane distances and noted that such localization suffers from various ambiguities beyond the usual rigid body motions; in this paper we provide a complete characterization of uniqueness. We enumerate equivalence classes of configurations which lead to the same distance measurements as a function of the number of planes and points, and algebraically characterize the related transformations in both 2D and 3D. Here we only discuss uniqueness; computational tools and heuristics for practical localization from point-to-plane distances using sound will be addressed in a companion paper.