Can a Ground-Based Vehicle Hear the Shape of a Room?

TL;DR

This paper investigates whether a ground-based vehicle can reconstruct room wall positions from echoes of a single sound event, using microphone arrays and algebraic methods, under specific geometric conditions.

Contribution

It provides theoretical proofs that room reconstruction is almost always possible with certain microphone configurations and assumptions, solving the echo-matching problem with algebraic techniques.

Findings

Reconstruction possible with three non-collinear microphones if no floor/ceiling echoes.

Reconstruction possible with four non-coplanar microphones if walls can have any inclination.

Uses Cayley-Menger determinant and algebraic methods to solve echo-matching problem.

Abstract

Assume that a ground-based vehicle moves in a room with walls or other planar surfaces. Can the vehicle reconstruct the positions of the walls from the echoes of a single sound event? We assume that the vehicle carries some microphones and that a loudspeaker is either also mounted on the vehicle or placed at a fixed location in the room. We prove that the reconstruction is almost always possible if (1) no echoes are received from floors, ceilings or sloping walls and the vehicle carries at least three non-collinear microphones, or if (2) walls of any inclination may occur, the loudspeaker is fixed in the room and there are four non-coplanar microphones. The difficulty lies in the echo-matching problem: how to determine which echoes come from the same wall. We solve this by using a Cayley-Menger determinant. Our proofs use methods from computational commutative algebra.