Imagine a loudspeaker is placed in a room with a few microphones. When the loudspeaker emits a sound impulse, the microphones receive several delayed responses as the sound reverberates from each wall in the room. These first-order echoes--heard after sound impulses have bounced only once on a wall--then bounce back from each wall to create second-order echoes and so on.
"The microphones listen to a short sound impulse bouncing on finite planar surfaces -- or the 'walls,'" Boutin, a professor of mathematics and electrical and computer engineering at Purdue University, explains . "When a microphone hears a sound that has bounced on a wall, the time difference between the emission and reception of the sound is recorded. This time difference corresponds to the distance traveled by the sound during that time."
The authors use a known modeling technique to focus on first-order echoes. This method interprets bounced sound as coming from a virtual source behind the wall instead of from the source, thus allowing a virtual source point to represent each wall.
However, the microphones cannot determine the distance that corresponds to each virtual source point, i.e., each wall. In response, Boutin and her colleagues designed a method to label the distances that correlate with each wall, a process they call "echo sorting."
This study demonstrates that reconstructing a room from first-order echoes acquired by four microphones is a theoretical problem that is well-posed under generic conditions. "This is a first step towards solving the corresponding real-world problem," Boutin observes. "If the problem was not well-posed, then a practical solution would require more information. But since we know that it is well-posed, we can move on to the next step: finding a way to reconstruct the room when the echo measurements are noisy."
While the mathematical framework simply requires a rigid configuration of non-coplanar microphones, the research has a range of other potential applications. "These microphones can be placed inside a room or on any vehicle, such as a car, an underwater vehicle, or a person's helmet," Gregor Kemper, a professor in the Department of Mathematics at Technische Universität München, explains. The authors' journal paper poses examples with stationary, indoor sound sources as well as sources placed on vehicles that may get rotated and translated due to movement; these latter sources present significantly more complicated situations.
Achieving computational economy for such problems is an important goal for Boutin and Kemper. Their method requires a computer algebra system to perform symbolic computations, which can become more computationally complex for other variations of the problem, thus limiting its expansion to similar problems. "Finding a less computationally expensive technique to prove the same results would be desirable, especially if this method turned out to be applicable to other cases," Kemper says. "Our mathematical framework is suitable for surface-based vehicles, but the actual computations necessary for the proof present challenges. We hope other teams will explore this issue."
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SIAM Journal on Applied Algebra and Geometry