Mathematical Imaging is currently a rapidly growing field in applied mathematics, with an increasing need for theoretical mathematics.
This book, the second of two volumes, emphasizes the role of mathematics as a rigorous basis for imaging sciences. It provides a comprehensive and convenient overview of the key mathematical concepts, notions, tools and frameworks involved in the various fields of gray-tone and binary image processing and analysis, by proposing a large, but coherent, set of symbols and notations, a complete list of subjects and a detailed bibliography. It establishes a bridge between the pure and applied mathematical disciplines, and the processing and analysis of gray-tone and binary images. It is accessible to readers who have neither extensive mathematical training, nor peer knowledge in Image Processing and Analysis.
It is a self-contained book focusing on the mathematical notions, concepts, operations, structures, and frameworks that are beyond or involved in Image Processing and Analysis. The notations are simplified as far as possible in order to be more explicative and consistent throughout the book and the mathematical aspects are systematically discussed in the image processing and analysis context, through practical examples or concrete illustrations. Conversely, the discussed applicative issues allow the role of mathematics to be highlighted.
Written for a broad audience – students, mathematicians, image processing and analysis specialists, as well as other scientists and practitioners – the author hopes that readers will find their own way of using the book, thus providing a mathematical companion that can help mathematicians become more familiar with image processing and analysis, and likewise, image processing and image analysis scientists, researchers and engineers gain a deeper understanding of mathematical notions and concepts.
Part 5 Twelve Main Geometrical Frameworks for Binary Images 1
Chapter 21 The Set-Theoretic Framework 3
Chapter 22 The Topological Framework 9
Chapter 23 The Euclidean Geometric Framework 23
Chapter 24 The Convex Geometric Framework 37
Chapter 25 the Morphological Geometric Framework 47
Chapter 26 The Geometric and Topological Framework 57
Chapter 27 The Measure-Theoretic Geometric Framework 71
Chapter 28 The Integral Geometric Framework 89
Chapter 29 The Differential Geometric Framework 111
Chapter 30 The Variational Geometric Framework 129
Chapter 31 The Stochastic Geometric Framework 135
Chapter 32 The Stereological Framework 159
Part 6 Four Specific Geometrical Framework for Binary Images 177
Chapter 33 The Granulometric Geometric Framework 179
Chapter 34 The Morphometric Geometric Framework 189
Chapter 35 The Fractal Geometric Framework 211
Chapter 36 The Textural Geometric Framework 229
Part 7 Four 'Hybrid' Framework for Gray-Tone and Binary Images 241
Chapter 37 The Interpolative Framework 243
Chapter 38 The Bounded-Variation Framework 253
Chapter 39 The Level Set Framework 269
Chapter 40 The Distance-Map Framework 281
Concluding Discussion and Perspectives 295
Tables of Notations and Symbols 303
Table of Acronyms 341
Table of Latin Phrases 347
Index of Authors 435
Index of Subjects 445