We show that one can reconstruct the shape of a room with planar walls from the first-order echoes received by four non-planar 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 6-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.We show that one can reconstruct the shape of a room with planar walls from the first-order echoes received by four non-planar 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 6-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.

Video illustration

References:

1. •M. Boutin, G. Kemper, “A Drone Can Hear the Shape of a Room,”,” SIAM J. on Applied Algebra and Geometry, 4:1(2020), 123–140.  (arXiv version, PDF)
   

2. •K.S. Cohen, “Drones Can Determine the Shape of a Room by Listening,” SIAM Nugget, Feb 6 (2020).
   

Code:

1. •Shape from Echoes MAGMA Code, Purdue University Research Repository, published on 01 Feb 2019, doi:10.4231/Q3AN-YR91.
   

Press Coverage:

1. • Press release from TU Munich, with accompanying video,
   

2. • Press release from Purdue,
   

3. • SIAM nugget article
   

4. • Inverse article
   

5. • Drone Life article
   

6. • News Atlas Article
   

7. • Drone DJ article
   

8. • Robotics tomorrow article
   

9. • Eurek Alert article
   

10. • Washington Technology article
    

11. • Mirage News article (in German)
    

12. • Electronics Weekly article
    

13. • Tech Explorist article
    

14. • Venture Beat article
    

15. • Siecle Digital article (in French)
    

16. • Phys dot org article
    

17. • ...

Photo Credit: Brian Powell