Sensor networks provide fertile ground for information theorists to go looking for good problems to think about. Some of those problems, like relay channels, or distributed compression, are old and well known; the sensor networking application has led to renewed interest in them. Others are not so old and less well known. In this talk, I will discuss some work we are doing on problems of the latter kind.
My talk is organized as follows. I will first argue that for an important class of distributed sensing/control applications, the signals observed/controlled by the network are best modeled as solutions of a wave PDE, instead of the classical bandlimited model pervasive in DSP. Then, I will formulate a new source coding problem, distributed compression of spatial waves. I will present solutions to two distributed DSP problems that come up in the study of coding strategies for our problem: reconstruction of the waves themselves out of measurements collected at a finite number of locations in space, as well as of the corresponding polyhedral spatial domains containing those waves. I will conclude by revisiting what these results say about the question of "can you hear the shape of a drum?", a problem in differential geometry that had remained open for a long time until it was settled in 1992.
Parts of this talk are based on joint work with Mingbo Zhao (Cornell/ECE), Georgios N. Lilis (Cornell/ECE), and Joseph M. Rosenblatt (UIUC/Math).
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