Shapes from Echoes (22rit002)


(Technical University of Munich)

(Purdue University)


The Banff International Research Station will host the "Shapes from Echoes" workshop in Banff from March 20, 2022 to March 27, 2022.

Can a drone hear the shape of a room? Can it work out its own location in the room from the echoes of a sound event? Can a car or another ground-based vehicle do the same? These types of questions are being studied by Mireille Boutin and Gregor Kemper at BIRS. Specifically, they assume that four microphones are placed on a drone. Then a loudspeaker, either at a fixed location or also mounted on the drone, emits a single sound impulse. With some delay, depending on the distance traveled by the sound, the microphones receive the echoes bouncing off the walls. Is the information contained in the delays enough to reconstruct the shape of the room? In a 2020 paper, Boutin and Kemper proved that the answer is "yes" with probability one. Their argument used methods from computational commutative algebra.

Now they are focusing on the case of ground-based vehicles, which, counterintuitively, is harder, and on the reverse question of whether and how a drone can determine its own location from echoes bouncing off the walls of a room it already knows. They are also working on practical questions of how to deal with measurement errors in the delay times. These issues need to be dealt with for real-world applications.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).