Topological Signal Processing

What is topological signal processing? The very title indicates a melding of different viewpoints — that of the topologist and that of the engineer. This text provides a nice exposition of the topological ideas used to extract information from signals and the practical details of signal processing. Thrown in is a welcome dose of perspective on how these topics are natural allies. Finding signals in mounds of data requires organizing and filtering the data and then recognizing relevant information. Of course, one of the major issues in signal processing is handling noise, or deformation, and this is exactly what topology is good at doing. Hence, using topology to do signal processing should be a natural consideration.

It is worth mentioning that one of Robinson’s stated goals is to make sheaf theory a player in the game of signal processing. He is clear that not only should using topology be a natural consideration for signal processing, but in particular the structure of sheaves should be the natural encoding structure. This drives his expository choices. For those concerned about this, he informs us that by focusing on sheaves over cellular spaces the proofs of the necessary results are much easier than in full-blown sheaf theory. I don’t know enough to judge those statements, but I can say that the use of sheaves here didn’t seem too complicated. In fact, after reading the text, it does seem natural!

Robinson’s intended audience is first year graduate students in both engineering and mathematics, and advanced undergraduates. He reaches this mark. There’s nice background on simplicial complexes, functors and cohomology, sheaves, the Euler characteristic, and persistence; plenty of theory. But the book has a very strong practical bent too. In each chapter, after developing some theory, the topics are used in case studies which introduce the applied nature of the subject. There are ten case studies overall, and each one provides a detailed look at some practical situation. For example, one study considers detecting information about rotating targets using the example of a ceiling fan. Another tracks pollution through a water system. Another finds the location of a radio receiver using ambient radio signals. In other words, the case studies are interesting situations.

Throughout the text there are numerous examples and diagrams. Each chapter also ends with some open questions. These features make the book quite readable. There are also some exercises scattered throughout as well.

In closing, this slim volume packs a lot into two hundred pages. It invites one to pick it up, crack the spine, and start reading. Once begun, the manageable size of the sections, the many examples, the many case studies keep one reading.

Michele Intermont is an associate professor of mathematics at Kalamazoo College in Kalamazoo, MI.