Reconstructing a room from noisy echoes
When a loudspeaker produces a short pulse of sound in a room with planar walls, a microphone listening to the echoes can be used to obtain the multiset of distances from its position to the walls. In the ideal case, the exact values of the distances are recovered from the echoes. In that case, four microphones placed in generic positions are sufficient for the wall positions to be reconstructable from the echoes. In particular, we have shown that, when the microphones are placed on a drone in a non-planar configuration, then the wall reconstruction problem is well-posed for all but a set of measure zero of drone positions. Extending this result to the reconstruction of walls in the non-ideal case is not straightforward, as the existence of exceptional drone positions for which the problem is not well-posed make the non-ideal case ill-conditioned. We will discuss this issue and suggest a potential solution. This is joint work with Gregor Kemper from TU Munich.