Topological signal processing.

*(English)*Zbl 1294.94001
Mathematical Engineering. Berlin: Springer (ISBN 978-3-642-36103-6/hbk; 978-3-642-36104-3/ebook). xvi, 208 p. (2014).

The author gives three major goals for this book: firstly to show that topological invariants provide qualitative information about signals that is both relevant and practical, second to show that the signal processing concepts of filtering, detection, and noise correspond respectively to the concepts of sheaves, functoriality and sequences, and third to advocate for the use of sheaf theory in signal processing. The author proceeds through a mixture of theoretical exposition and practical examples drawn from case studies.

The case studies range from tracking, to the extraction of topological information from intersections in coverage. They are introduced in the first chapter with references to the location in the book where each case is described in more detail. In chapter two, “Parametrization”, the author introduces some basic notions from topology of which the notions of cell complex and manifold essential for the sequel. This chapter describes the case study of signal manifolds for localization, tracking and navigation.

In chapter three, “Signals”, sheaves make their appearance. The local sections of a sheaf being interpreted as the signal detected. An edge in a cell complex represents a region in which two detectors are able to detect a given signal. It is important to note that much of the subsequent discussion deals with sheaves over cell complexes so that definitions such as that of a pull back are slightly non standard. The notion of a topological filter is first introduced here, although the precise definition is delayed until chapter four. Examples of topological filters include thresholding and angle valued filters.

Chapter four, “Detection”, introduces sheaf cohomology. This allows the author to define a topological filter as a diagram of sheaf morphisms \[ \mathcal S_1 \overset{m_1}{\leftarrow} \mathcal S_2 \overset{m_2}{\rightarrow} \mathcal S_3 \] when the morphism \(m_1\) induces an isomorphism on sheaf cohomology. \(\mathcal S_1\) is the input, \(\mathcal S_3\) the output and \(\mathcal S_2\) is the internal state of the filter. The case study here is that of a sampling filter. Chapter 5, “Transforms”, introduces the notion of the Euler integral, which as its name suggests relates to the Euler characteristic and the author shows, in the case study, how the Euler integral detects shapes. Chapter 6, “Noise”, introduces persistent cohomology as a detector. The major example here is the recovery of topological information from a point cloud.

Each chapter concludes with a section on open questions.

The target audience is practitioners so that the theoretical notions are covered with the practitioner in mind with motivations emphasized.

The case studies range from tracking, to the extraction of topological information from intersections in coverage. They are introduced in the first chapter with references to the location in the book where each case is described in more detail. In chapter two, “Parametrization”, the author introduces some basic notions from topology of which the notions of cell complex and manifold essential for the sequel. This chapter describes the case study of signal manifolds for localization, tracking and navigation.

In chapter three, “Signals”, sheaves make their appearance. The local sections of a sheaf being interpreted as the signal detected. An edge in a cell complex represents a region in which two detectors are able to detect a given signal. It is important to note that much of the subsequent discussion deals with sheaves over cell complexes so that definitions such as that of a pull back are slightly non standard. The notion of a topological filter is first introduced here, although the precise definition is delayed until chapter four. Examples of topological filters include thresholding and angle valued filters.

Chapter four, “Detection”, introduces sheaf cohomology. This allows the author to define a topological filter as a diagram of sheaf morphisms \[ \mathcal S_1 \overset{m_1}{\leftarrow} \mathcal S_2 \overset{m_2}{\rightarrow} \mathcal S_3 \] when the morphism \(m_1\) induces an isomorphism on sheaf cohomology. \(\mathcal S_1\) is the input, \(\mathcal S_3\) the output and \(\mathcal S_2\) is the internal state of the filter. The case study here is that of a sampling filter. Chapter 5, “Transforms”, introduces the notion of the Euler integral, which as its name suggests relates to the Euler characteristic and the author shows, in the case study, how the Euler integral detects shapes. Chapter 6, “Noise”, introduces persistent cohomology as a detector. The major example here is the recovery of topological information from a point cloud.

Each chapter concludes with a section on open questions.

The target audience is practitioners so that the theoretical notions are covered with the practitioner in mind with motivations emphasized.

Reviewer: Jonathan Hodgson (Swarthmore)

##### MSC:

94-02 | Research exposition (monographs, survey articles) pertaining to information and communication theory |

94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |

55N30 | Sheaf cohomology in algebraic topology |