Can a drone hear the shape of a room?

Suppose that some microphones are placed on a drone inside a room with planar walls/floors/ceilings. A loudspeaker emits a sound impulse and the microphones receive several delayed responses corresponding to the sound bouncing back from each planar surface. These are the first-order echoes. In this talks, we will discuss the problem of reconstructing the shape of the room from the first-order echoes. The time delay for each echo determines the distance from the microphone to a mirror image of the source reflected across a wall. Since we do not know which echo corresponds to which wall, the distances are unlabeled. The problem is to figure out under which circumstances, and how, one can find out the correct distance-wall assignments and reconstruct the wall positions. 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 is generic, 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. This is joint work with Gregor Kemper.