This site contains: mathematics courses and book; covers: image analysis, data analysis, and discrete modelling; provides: image analysis software. Created and run by Peter Saveliev.

From Intelligent Perception

Computational Homology by Kaczynski, Mischaikow, Mrozek is one of the early books in the field of computational topology. There is a lot of good in this book and there some inadequacies.

1 Math

Thorough presentation of all the mathematics is given, including the proofs of all theorems. That's not easy to handle for a novice. In fact, a graduate level course in modern algebra is required for the student to follow the proofs. Moreover, this is not a good place for someone to get started with homology even though all modern algebra and point-set topology background is presented in the appendix. The introduction is good as a motivation for the first time student.

The homology theory presented is for cubical complexes. That's is necessary for studying digital images. The notation chosen is painfully awkward.

Numerous exercises are provided.

2 Algorithms

Algorithms are fully written, in pseudocode. They are easy to follow if you understand the math. Some prior experience with algorithms may be necessary.

The algorithms are analyzed.

Software, called CHomP, is available. However you can't just start using it. Prior experience with C++ is required. The latest releases of CHomP are a challenge to get running. Too many moving parts!

3 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.

Bottom line, this is monograph not a textbook.

4 Contents

4.1 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

• 2 Cubical Homology
  • 2.1 Cubical Sets
  • 2.2 The Algebra of Cubical Sets
  • 2.3 Connected Components and $H_0(X)$
  • 2.4 Elementary Collapses
  • 2.5 Acyclic Cubical Spaces
  • 2.6 Homology of Abstract Chain Complexes
  • 2.7 Reduced Homology

• 3 Computing Homology Groups
  • 3.1 Matrix Algebra over $Z$
  • 3.2 Row Echelon Form
  • 3.3 Smith Normal Form
  • 3.4 Structure of Abelian Groups
  • 3.5 Computing Homology Groups
  • 3.6 Computing Homology of Cubical Sets
  • 3.7 Preboundary of a Cycle—Algebraic Approach

• 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 $\Gamma ^1$

• 6 Homology of Maps
  • 6.1 Representable Sets
  • 6.2 Cubical Multivalued Maps
  • 6.3 Chain Selectors
  • 6.4 Homology of Continuous Maps
  • 6.5 Homotopy Invariance

• 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

4.2 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

• 9 Homological Algebra
  • 9.1 Relative Homology
  • 9.2 Exact Sequences
  • 9.3 The Connecting Homomorphism
  • 9.4 Mayer–Vietoris Sequence
  • 9.5 Weak Boundaries

• 10 Nonlinear Dynamics
  • 10.1 Maps and Symbolic Dynamics
  • 10.2 Differential Equations and Flows
  • 10.3 Wa˙zewski Principle
  • 10.4 Fixed-Point Theorems
  • 10.5 Degree Theory
  • 10.6 Complicated Dynamics
  • 10.7 Computing Chaotic Dynamics

• 11 Homology of Topological Polyhedra
  • 11.1 Simplicial Homology
  • 11.2 Comparison of Cubical and Simplicial Complexes
  • 11.3 Homology Functor

4.3 Part III Tools from Topology and Algebra

• 12 Topology
  • 12.1 Norms and Metrics in $R^d$
  • 12.2 Topology
  • 12.3 Continuous Maps
  • 12.4 Connectedness
  • 12.5 Limits and Compactness

• 13 Algebra
  • 13.1 Abelian Groups
  • 13.2 Fields and Vector Spaces
  • 13.3 Homomorphisms
  • 13.4 Free Abelian Groups

• 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