If you stand in the middle of Lausanne Cathedral in Switzerland and snap your fingers, the sound waves will project outward until they bounce off the stone walls, arches, altars, and statues of the Gothic church, and return, in a series of echoes, to your ears. From the noise, your brain might be able to infer the large size and openness of the sanctuary. But with a few microphones and a newly developed mathematical algorithm, scientists can now get more much information than that: They can determine the precise shape of the room.

The algorithm—based on the sort of echolocation that bats and dolphins use to navigate—could be incorporated into cell phone apps to determine room dimensions for architectural or design purposes, says the study's lead author, electrical engineer Ivan Dokmanic of the Swiss Federal Institute of Technology in Lausanne. It could also be used to develop more realistic echoes in video games and virtual reality simulations and to eliminate the echo from phone calls.

The study of how to model the relationship between echoes and physical surrounding is relatively new, with scientists turning to the problem within the past decade. But sorting out individual echoes from a recording is technically challenging. Dokmanic and his colleagues started with a much simpler structure than Lausanne Cathedral: a small, empty lecture room at his university. They placed four microphones at random spots in the room; the point was to make sure that their calculations worked with any configuration of mics. "People working on this problem in the past have typically used multiple sound sources and a whole lot of microphones in very constrained, precisely measured arrangements," Dokmanic says. "But we wanted to be able to use as little machinery as possible to do this."

The researchers had someone stand in the center of the room and snap their fingers or pop a balloon. Then they developed a mathematical algorithm to analyze the recordings from each microphone. Their method first eliminated echoes that had bounced off more than one wall or off small objects within the room in order to simplify the problem. The researchers then made an assumption: Every echo could be considered mathematically equivalent to a sound that emanated from a mirror image of the source. This mathematical trick gave them a new way of looking at the problem and manipulating the data that turned out to be the key to sifting through the sounds. It allowed a mathematical program to sort out which echoes came from the same walls, and then the placement and angles of the walls, giving the researchers the distances and angles between walls.

But the model still needed a real-world trial. So Dokmanic and his colleagues took the setup to Lausanne Cathedral. "The cathedral is not one of those rooms that satisfies our modeling assumptions of an empty box," he says. "It's kind of a nightmare scenario for the algorithm, with lots of small objects giving off echoes."

Even in the Gothic structure, echoes from the popping of a balloon were correctly interpreted by the algorithm, which sorted out reverberations from the main walls and determined their distances and angles from the balloon, while ignoring echoes reflected from the smaller surfaces, the team reports online today in the *Proceedings of the National Academy of Sciences*.

For now, the algorithm is a series of equations and mathematical matrices on the researcher's computers, but integrating the calculations in other programs would open up real-world applications, Dokmanic says. Cell phone apps—requiring a few cell phones in one room—could be programmed to spit out room dimensions faster than it takes to pull out a tape measure and do it manually. The information assists designers and help sound engineers determine where to place speakers to minimize echo. And backtracking from a room shape could work, too. In a criminal investigation, sound recordings of a gunshot could be combined with known room data to find the exact position of a gunman.

"This is a very elegant method, and I don't see much room for improvement in terms of the mathematics," says electrical engineer Jason Filos of Imperial College London, who was not involved in the new work. But Tapio Lokki, a virtual acoustic researcher at Aalto University in Finland, says there's still room for new models to get more detailed information out of echoes. "They have found a few [walls] here, but what do the corners look like? What is the roughness of the surfaces? These kinds of questions are much more challenging," he says, and such information would advance the understanding and minimization of echoes. "But the field is progressing and I think in 5 years we will know much more."