Mathematics of Digital Images: Creation, Compression, Restoration, Recognition

Description

This book is quite academic in tone, but practical in content. It is more of a math book that uses imaging in its examples than a book about imaging that uses math as a tool. It does a good job of starting from the beginning in any mathematical topic it explains, going through an explanation of the theory including proofs, and almost always showing at least one imaging example to explain each mathematical topic. Exercises are included, but these are not generally proofs in the classical sense. Instead, you may be asked to draw a diagram or image proving a theorem, or be asked to explain how a particular image proves a theorem. Answers to selected exercises are in the back of the book. Because this book has such good explanations on subjects such as the SVD and information theory, it might be useful to students that are not that interested in imaging simply because the analogies made to imaging make the mathematical theory quite clear. However, the last two parts of this six part book are very much aimed at those who are interested in image processing. I notice that the table of contents is not shown here, so I do that next:

Part I - THE PLANE
1. Isometries
Introduction; Isometries and their sense; The classification of isometries
2. How Isometries combine
Reflections are the key; Some useful compositions; The image of a line of symmetry; The dihedral group; Appendix on groups;
3. The seven braid patterns
Constructing braid patterns
4. Plane patterns and symmetries
Translations and nets; Cells; The five net types;
5. The 17 plane patterns
Preliminaries; The general parallelogram net; The centered rectangular net; The square net; The hexagonal net; Examples of the 17 plane pattern types; Scheme for identifying pattern types;
6. More plane truth
Equivalent symmetry groups; Plane patterns classified; Tilings and Coxeter Graphs; Creating plane patterns;

Part II - MATRIX STRUCTURES
7. Vectors and matrices
Vectors and handedness; Matrices and determinants; Further products of vectors in 3-space; The matrix of a transformation; Permutations and proof of determinant rules;
8. Matrix algebra
Introduction to eigenvalues; Rank and some ramifications; Similarity to a diagonal matrix; The Singular Value Decomposition;

Part III - Here’s to Probability
9. Probability
Sample spaces; Baye’s Theorem; Random variables; A census of distributions; Mean inequalities;
10. Random Vectors
Random Vectors; Functions of a random vector; The ubiquity of normal/Gaussian vectors; Correlation and its elimination;
11. Sampling and inference
Statistical inference; The Bayesian approach; Simulation; Markov Chain Monte Carlo

Part IV- Information, Error, and belief
12. Entropy and coding
The idea of entropy; COdes and binary trees; Huffman text compression; Huffman code redundancy; Arithmetic codes; Prediction by partial matching; LZW Compression; Entropy and minimum description length;
13. Information and error correction
Channel capacity; Error-correcting codes; Probabilistic decoding; Bayesian nets in computer vision;

Part V- Transforming the Image
14. The Fourier Transform
The DFT; The CFT; DFT connections;
15. Transforming Images
The Fourier Transform in two dimensions; Filters; Deconvolution and image restoration; Compression
16. Scaling
Nature, fractals, and compression; Wavelets; The Discrete Wavelet Transform; Wavelet relatives

Part VI - See, Edit, and Reconstruct
17. B-Spline Wavelets
Splines from boxes; The step to subdivision; The wavelet subdivision; The wavelet formulation; Band matrices for finding Q,A, and B; Surface wavelets;
18. Further methods
Neural networks; Self-organizing nets; Information Theory revisited; Tomography

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