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.

Part I. The Plane. Isometries. How isometries combine. The braid patterns. Plane patterns and their symmetries. The 17 plane patterns. More plane truth. Part II. Further Foundations . Matrix and vector algebra. Probability. Noiseless entropy encoding. Noise and simulation. Information theory. Part III. Image Compression. Linear transforms. Pyramid encoding. Fractal image compression. Neural methods. Quantisation. Part IV. Restoration and Recognition. Transforms, filters, convolution. Statistical pattern recognition. Neural classification. Tomography. Fractal dimensions.