Michael Robinson’s book, Topological Signal Processing, is an ambitious view into the development and application of algebraic topology to the field of signal processing. Following an introduction and motivational chapter, the book contains five chapters devoted to key concepts related to signal processing: parametrization, signals, detection, transforms, and noise. Each of these later chapters begins with a well-designed introduction to some topics from algebraic topology followed by case studies of specific problems from signal processing and open questions. The design of the book is powerful, forcing the author and reader to repeatedly traverse what should feel like a large divide between abstract, pure algebraic topology and explicit, practical problems in signal processing. The author’s clear and intuitive writing style makes this journey enjoyable, with a wealth of information included in a relatively short book at approximately 200 pages including appendices.
The book addresses some classical topics in algebraic topology, focusing mainly on sheaves over cellular spaces and sheaf cohomology, and also including CW and simplicial complexes, the Euler characteristic, and the formalism of categories and functors. It also delves into the realm of persistence sheaves, a relatively new subject enabling the extraction of topological structure from noisy data. Throughout, the author defines topics relative to the more general algebraic topological setting but rapidly focuses on the structures most directly related to signal processing. While each of the final five chapters starts with a fairly broad topic in algebraic topology, they require only a few sections of development and focus refinement to apply the theory to case studies of problems from signal processing.
Despite the interesting format and clear writing, it would be difficult to use this book for its intended audience of advanced undergraduates and first-year graduate students in mathematics and engineering. On the positive side, the students should benefit from the straightforward discussion of some of the basics of algebraic topology. The author nimbly covers a wealth of information, often using figures and carefully constructed examples to efficiently represent key information. Unfortunately, exercises, which are embedded in the text, are sparse in some sections and nonexistent in the remaining sections. As an introduction to concepts from algebraic topology, this approach seems to be too minimal for students to form a solid foundation and an instructor would need to supplement the text with many additional exercises. The case studies present a similar problem. While the presented applications are interesting and serve to both illustrate the use of algebraic topology in signal processing and motivate a discussion of remaining open problems, there are very few student exercises. More importantly, data and algorithm implementations are not given. A hands-on exploration of the examples would require further research into the original sources. As written, the case studies appear to be extended abstracts for papers written by the author and others, although admittedly more truthful than the typical abstract in discussing practical difficulties and limitations.
Where this book excels is in addressing the three goals outlined by the author on the first page. These include showing that topological invariants provide useful information for system analysis and design, showing that signal processing concepts correspond to algebraic topological concepts, and advocating for using sheaves in signal processing. This book offers an enjoyable exposition for a reader with a background in algebraic topology. Rooting the case studies in sheaves and related structures presents a nice introduction to signal processing for those more familiar with algebraic topology. Similarly, a researcher in signal processing, especially one granted access to existing code in this area, should get a lot out of reading this book and applying the algorithms to data sets similar to those presented in the case studies. Perhaps future editions of the book or online supplements could provide the additional data, code, and exercises to make a more in-depth exploration possible for others.