A Drone Can Hear the Shape of a Room

We show that one can reconstruct the shape of a room with planar walls from the first-order echoes received by four nonplanar microphones placed on a drone with generic position and orientation. Both the cases where the source is located in the room and on the drone are considered. If the microphone positions are picked at random, then with probability one, the location of any wall is correctly reconstructed as long as it is heard by four microphones. Our algorithm uses a simple echo sorting criterion to recover the wall assignments for the echoes. We prove that if the position and orientation of the drone on which the microphones are mounted do not lie on a certain set of dimension at most 5 in the six-dimensional space of all drone positions and orientations, then the wall assignment obtained through our echo sorting criterion must be the right one and thus the reconstruction obtained through our algorithm is correct. Our proof uses methods from computational commutative algebra.