Mathematical Foundations of Image Processing and Analysis, Volume 2

✍ Scribed by Pinoli, Jean-Charles

• Publisher: Wiley-ISTE
• Year: 2014
• Tongue: English
• Leaves: 491
• Series: Digital signal and image processing series
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Content: Cover
Title Page
Copyright
Contents
Preface
Introduction
PART 5: Twelve Main Geometrical Frameworks for Binary Images
Chapter 21: The Set-Theoretic Framework
21.1. Paradigms
21.2. Mathematical concepts and structures
21.2.1. Mathematical disciplines
21.3. Main notions and approaches for IPA
21.3.1. Pixels and objects
21.3.2. Pixel and object separation
21.3.3. Local finiteness
21.3.4. Set transformations
21.4. Main applications for IPA
21.4.1. Object partition and object components
21.4.2. Set-theoretic separation of objects and object removal. 21.4.3. Counting of separate objects21.4.4. Spatial supports border effects
21.5. Additional comments
Historical comments and references
Bibliographic notes and additional readings
Further topics and readings
Some references on applications to IPA
Chapter 22: The Topological Framework
22.1. Paradigms
22.2. Mathematical concepts and structures
22.2.1. Mathematical disciplines
22.2.2. Special classes of subsets of Rn
22.2.3. Fell topology for closed subsets
22.2.4. Hausdorff topology for compact subsets
22.2.5. Continuity and semi-continuity of set transformations. 22.2.6. Continuity of basic set-theoretic and topological operations22.3. Main notions and approaches for IPA
22.3.1. Topologies in the spatial domain Rn
22.3.2. The Lebesgue-(Čech) dimension
22.3.3. Interior and exterior boundaries
22.3.3.1. Topologically regular objects
22.3.4. Path-connectedness
22.3.5. Homeomorphic objects
22.4. Main applications to IPA
22.4.1. Topological separation of objects and object removal
22.4.1.1. (Path)-connected components
22.4.2. Counting of separate objects
22.4.3. Contours of objects
22.4.4. Metric diameter
22.4.5. Skeletons of proper objects. 22.4.6. Dirichlet-Voronoi's diagrams22.4.7. Distance maps
22.4.8. Distance between objects
22.4.9. Spatial support's border effects
22.5. Additional comments
Historical comments and references
Bibliographic notes and additional readings
Further topics and readings
Some references on applications to IPA
Chapter 23: The Euclidean Geometric Framework
23.1. Paradigms
23.2. Mathematical concepts and structures
23.2.1. Mathematical disciplines
23.2.2. Euclidean dimension
23.2.3. Matrices
23.2.4. Determinants
23.2.5. Eigenvalues, eigenvectors and trace of a matrix. 23.2.6. Matrix norms23.3. Main notions and approaches for IPA
23.3.1. Affine transformations
23.3.2. Special groups of affine transformations
23.3.3. Linear and affine sub-spaces and Grassmannians
23.3.4. Linear and affine spans
23.4. Main applications to IPA
23.4.1. Basic spatial transformations
23.4.1.1. Reflected objects
23.4.2. Hyperplanes
23.4.3. Polytopes
23.4.4. Minkowski addition and subtraction
23.4.5. Continuity and semi-continuities of Euclidean transformations
23.5. Additional comments
Historical comments and references
Commented bibliography and additional readings.