Friday, 18 March 2011

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

Mathematics of Digital Images: Creation, Compression, Restoration, Recognition
| 2006-09-18 00:00:00 | | 0 | Computer Graphics

This is a major revision of the author's successful book Mathematics of Computer Graphics. It still focuses on foundations and proofs, but now exhibits a shift towards digital image compression, restoration, and recognition. Topology is replaced by Probability and Information Theory (with Shannon's source and channel encoding Theorems) which are used throughout. Several fractal methods are given in the service of Compression, along with linear transforms (hence FFT, DCT, JPEG, wavelets etc), recent neural methods, and the ubiquitous vector quantisation. Optimising for the Human Visual System is a subtheme. The superiority of Pyramid methods with respect to entropy is proved. Restoration offers convolution/deconvolution against noise and blurr. Recognition explores not only the Hough and Radon transforms, Statistical feature extraction, and Neural classification, but also Tomography, the recovery of 3-D images from 2-D data. It extends finally to multiple fractal dimensions in medical and other nature-related images.

User review
A must have
A through review of the basic concepts of image processing and their (not basic at all) mathematical foundations. Also includes THE BEST introduction to linear algebra and DSP I have encountered so far.

User review
A good book on mathematics applied to image processing
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:


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;


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

User review
A popular pick for advanced self-study.
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

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