图像分析数学：创作、压缩、复原、识别
Book Description
This major revision of the author's popular book still focuses on foundations and proofs, but now exhibits a shift away from Topology to Probability and Information Theory (with Shannon's source and channel encoding theorems) which are used throughout. Three vital areas for the digital revolution are tackled (compression, restoration and recognition), establishing not only what is true, but why, to facilitate education and research. It will remain a valuable book for computer scientists, engineers and applied mathematicians.

Mathematics of Digital Images: Creation, Compression, Restoration, Recognition
Compression, restoration and recognition are three of the key components of digital imaging. The mathematics needed to understand and carry out all these components are explained here in a style that is at once rigorous and practical with many worked examples, exercises with solutions, pseudocode, and sample calculations on images. The introduction lists fast tracks to special topics such as Principal Component Analysis, and ways into and through the book, which abounds with illustrations. The first part describes plane geometry and pattern-generating symmetries, along with some on 3D rotation and reflection matrices. Subsequent chapters cover vectors, matrices and probability. These are applied to simulation, Bayesian methods, Shannon's information theory, compression, filtering and tomography. The book will be suited for advanced courses or for self-study. It will appeal to all those working in biomedical imaging and diagnosis, computer graphics, machine vision, remote sensing, image processing and information theory and its applications
Table of contents

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

 
Review
"Recognition explains the mathematics needed to carry out various aspects of digital imaging through examples, exercises with solutions, pseudocode and sample calculations on images. Suitable for a course or tutorial."
Spectra
"This book explains the mathematics needed to understand and carry out these components in a style at once rigorous and practical, with many worked examples, exercises and solutions, pseudocode, and sample calculations on images."
Bookshelf
"This book covers a lot of ground!"
Jason Dowling, IAPR Newsletter
"...offers both theory and practical applications and exercises. College-level courses will want to consider this as a classroom text on the subject, but specialty libraries will also find it a popular pick for advanced self-study." - California Bookwatch Diane C. Donovan, Midwest Book Review
Customer Reviews

Most Helpful Customer Reviews
      1 of 1 people found the following review helpful:

A good book on mathematics applied to image processing, April 13, 2007

By  calvinnme "Texan refugee" (Fredericksburg, Va) - See all my reviews
     

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:

A popular pick for advanced self-study., March 12, 2007

By  Midwest Book Review (Oregon, WI USA) - See all my reviews

College-level collections strong in either science or computer science - into the intermediate studies levels - will want to add MATHEMATICS OF DIGITAL IMAGES: CREATION, COMPRESSION, RESTORATION, RECOGNITION to their collections. These three elements are key to digital imaging - and the math needed to carry out all these components are explored in a textbook which offers both theory and practical applications and exercises. College-level courses will want to consider this as a classroom text on the subject, but specialty libraries will also find it a popular pick for advanced self-study.

Diane C. Donovan
California Bookwatch