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Computational Homology by Kaczynski, Mischaikow, Mrozek

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Monograph not a textbook

This is one of the early books in the field of computational topology. There is a lot of good in this book but I had a few complaints.

A thorough presentation of all the mathematics is given including proofs of all theorems. That's not easy to handle for a novice. In fact, a graduate level course in modern algebra, as well as some point-set topology, seems to be required for the student to follow the proofs. I liked the introduction for a first-time student in the beginning of the book. After that, the book becomes very "dense". The homology theory is developed for cubical complexes instead of the traditional simplicial complexes as necessary for studying digital images. I liked this approach (but not the notation) so much that I used it later in my own book. Numerous exercises are provided.

The algorithms are fully written, in pseudocode. They are easy to follow as long as you understand the mathematics but some prior experience with algorithms might be necessary. A software package, called CHomP, is available online.


What's missing

Even though the issues of topological features of images, i.e., the homology theory of n-dimensional images, are well covered, some important, applied content is missing.

First is persistent homology. This is the main tool for dealing with noise.

Second is geometry, i.e., measuring objects. You need that too for evaluating noise.

Third is Gray scale images. You can't just get away with binary images, in real life.

Website contains examples and downloads, but the projects provided are exclusively geared toward academic research. There are also just so few of them! Try Examples of image analysis.


Contents

Part I Homology

  • 1 Preview
    • 1.1 Analyzing Images
    • 1.2 Nonlinear Dynamics
    • 1.3 Graphs
    • 1.4 Topological and Algebraic Boundaries
    • 1.5 Keeping Track of Directions
    • 1.6 Mod 2 Homology of Graphs
  • 4 Chain Maps and Reduction Algorithms
    • 4.1 Chain Maps
    • 4.2 Chain Homotopy
    • 4.3 Internal Elementary Reductions
    • 4.4 KMS Reduction Algorithm
  • 5 Preview of Maps
    • 5.1 Rational Functions and Interval Arithmetic
    • 5.2 Maps on an Interval
    • 5.3 Constructing Chain Selectors
    • 5.4 Maps of $\Gamma ^1$
  • 7 Computing Homology of Maps
    • 7.1 Producing Multivalued Representation
    • 7.2 Chain Selector Algorithm
    • 7.3 Computing Homology of Maps
    • 7.4 Geometric Preboundary Algorithm

Part II Extensions

  • 8 Prospects in Digital Image Processing
    • 8.1 Images and Cubical Sets
    • 8.2 Patterns from Cahn–Hilliard
    • 8.3 Complicated Time-Dependent Patterns
    • 8.4 Size Function

Part III Tools from Topology and Algebra

  • 14 Syntax of Algorithms
    • 14.1 Overview
    • 14.2 Data Structures
    • 14.3 Compound Statements
    • 14.4 Function and Operator Overloading
    • 14.5 Analysis of Algorithms