# Shapes from Echoes: Uniqueness from Point-to-Plane Distance Matrices

## TL;DR

This paper characterizes the uniqueness of localizing points and planes from point-to-plane distances, providing a complete mathematical understanding of when configurations are distinguishable based on acoustic echoes.

## Contribution

It offers a complete characterization of the conditions under which point and plane configurations are uniquely identifiable from distance measurements, extending previous work on localization ambiguities.

## Key findings

- Enumerates equivalence classes of configurations with identical distance measurements.
- Provides algebraic characterization of transformations leading to indistinguishable configurations.
- Focuses on theoretical conditions for uniqueness in 2D and 3D.

## Abstract

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.