Cover -- Title Page -- Copyright -- Contents -- Preface -- Introduction -- Elements of Mathematical Terminology -- Part 1. An Overview of Image Processing and Analysis (IPA) -- Chapter 1. Gray-Tone Images -- 1.1. Intensity images, pixels and gray tones -- 1.2. Scene, objects, context, foreground and background -- 1.3. Simple intensity image formation process models -- 1.3.1. The multiplicative image formation process model -- 1.3.2. The main human brightness perception laws -- 1.4. The five main requirements for a relevant imaging approach -- 1.5. Additional comments -- Chapter 2. Gray-Tone Image Processing and Analysis -- 2.1. Image processing -- 2.1.1. Image enhancement -- 2.1.2. Image restoration -- 2.1.3. Image inpainting -- 2.1.4. Image warping, registration and morphing -- 2.2. Image analysis -- 2.2.1. Image features -- 2.2.2. Image feature detection and extraction -- 2.2.3. Image segmentation -- 2.3. Image comparison -- 2.3.1. Image pattern analysis, recognition and formation -- 2.3.2. Image quality measure -- 2.4. Importance of Human Vision -- 2.5. Additional comments -- Chapter 3. Binary Images -- 3.1. Scene, objects and context -- 3.1.1. Types of collection of objects -- 3.1.2. Types of perturbations -- 3.2. Binary and multinary images -- 3.2.1. Binary images -- 3.2.2. Multinary images -- 3.3. Additional comments -- Chapter 4. Binary Image Processing and Analysis -- 4.1. Binary image processing -- 4.1.1. Binary image processing methods -- 4.2. Binary image analysis -- 4.2.1. Object feature detection and extraction -- 4.3. Binary image and object description -- 4.3.1. Binary image and object descriptors -- 4.3.2. Properties of the binary image and object descriptor -- 4.4. Object comparison -- 4.5. Object analysis, recognition and formation -- 4.5.1. Object recognition -- 4.5.2. Object formation.

4.6. Additional comments -- Chapter 5. Key Concepts and Notions for IPA -- 5.1. Dimensionality -- 5.1.1. Dimension in Physics -- 5.1.2. Dimension in Mathematics -- 5.1.3. Dimension in imaging sciences and technologies -- 5.2. Continuity and discreteness -- 5.3. Scale, resolution and definition -- 5.3.1. Scale -- 5.3.2. Resolution -- 5.3.3. Image definition -- 5.4. Domains -- 5.5. Ranges -- 5.5.1. Pointwise ranges -- 5.5.2. Local ranges -- 5.5.3. Global ranges -- 5.5.4. Constrained ranges -- 5.6. Additional comments -- Chapter 6. Mathematical Imaging Frameworks -- 6.1. Mathematical imaging frameworks -- 6.1.1. Mathematical imaging paradigms -- 6.1.2. Mathematical imaging frameworks -- 6.1.3. Mathematical imaging approaches -- 6.2. Image representation and image modeling -- 6.2.1. Imaging representation -- 6.2.2. Imaging modeling -- 6.3. A mathematical imaging methodology -- 6.4. Additional comments -- Part 2. Basic Mathematical Reminders for Gray-Tone and Binary Image Processing and Analysis -- Chapter 7. Basic Reminders in Set Theory -- 7.1. Mathematical disciplines -- 7.2. Sets and elements -- 7.2.1. Membership -- 7.2.2. Relations and operations between sets -- 7.2.3. Power sets -- 7.3. Order and equivalence relations on sets -- 7.3.1. Order relations on sets -- 7.3.2. Lattices and complete lattices -- 7.3.3. Equivalence relations on sets -- 7.4. Mappings between sets -- 7.5. Mapping composition, involutions, and idempotent mappings -- 7.5.1. Fixed elements of a mapping -- 7.5.2. Injections, surjections and bijections -- 7.5.3. Single-valued mappings, multivalued mappings and correspondences -- 7.5.4. Monotonic mappings between ordered sets -- 7.6. Cardinality -- 7.7. Cover -- 7.8. Additional comments -- Chapter 8. Basic Reminders in Topology and Functional Analysis -- 8.1. Mathematical disciplines.

8.2. Topological spaces -- 8.2.1. Neighborhood systems -- 8.2.2. Open and closed sets, interiors, closures and boundaries -- 8.2.3. Kuratowski's closure axioms -- 8.2.4. Topologies and topological bases -- 8.2.5. Continuous mappings, homeomorphisms and embeddings -- 8.2.6. Topological separations -- 8.3. Metric spaces -- 8.3.1. Metrics -- 8.3.2. Balls -- 8.3.3. Convergent sequences -- 8.3.4. Complete metric spaces -- 8.3.5. Uniform continuity -- 8.3.6. Lipschitz mappings -- 8.3.7. Hölder mappings -- 8.3.8. Equivalence of metrics -- 8.3.9. Distance-preserving mapping and isometries -- 8.3.10. Locally bounded mapping -- 8.4. Some particular kinds of points in topological and metric spaces -- 8.5. Some particular kinds of subsets in topological and metric spaces -- 8.5.1. Dense and meagre sets -- 8.5.2. Connected and path-connected sets -- 8.5.3. Bounded sets -- 8.5.4. Chebyshev sets -- 8.6. Some particular topological spaces -- 8.6.1. Compact spaces -- 8.6.2. Separable spaces -- 8.6.3. Baire spaces -- 8.6.4. Polish spaces -- 8.6.5. Alexandrov spaces -- 8.7. Lipschitz and Gromov-Hausdorff distances -- 8.7.1. Distortion of a mapping and a correspondence -- 8.7.2. Lipschitz distances -- 8.7.3. Gromov-Hausdorff distances -- 8.7.4. Lipschitz and Gromov-Hausdorff convergences -- 8.8. Topological vector spaces -- 8.8.1. Vector spaces -- 8.8.2. Vector algebras -- 8.8.3. General topological vector spaces -- 8.8.4. Normed vector spaces and Banach spaces -- 8.8.5. Inner vector spaces, Euclidean spaces and Hilbert spaces -- 8.8.6. Operators -- 8.8.7. Reflexive topological vector spaces -- 8.8.8. Riesz-Fréchet's representation theorem -- 8.8.9. Lax-Milgram's theorem -- 8.8.10. Weak formulation of problems -- 8.8.11. Fréchet spaces -- 8.9. Additional comments.

Part 3. The Main Mathematical Notions for the Spatial and Tonal Domains -- Chapter 9. The Spatial Domain -- 9.1. Paradigms -- 9.2. Mathematical structures -- 9.3. Main approaches for IPA -- 9.3.1. Pixels -- 9.3.2. Pixels in the continuous setting -- 9.3.3. Pixels in the discrete setting -- 9.3.4. Point and cell discrete representations for pixels -- 9.4. Main applications to IPA -- 9.4.1. The continuous case -- 9.4.2. The discrete case and the adjacency relationships -- 9.5. Additional comments -- Chapter 10. The Tonal Domain -- 10.1. Paradigms -- 10.2. Mathematical concepts and structures -- 10.2.1. Mathematical disciplines -- 10.2.2. Gray tones -- 10.2.3. The tonal domains -- 10.2.4. Gray-tone vector space and algebra -- 10.2.5. Gray-tone norms and gray-tone inner products -- 10.2.6. Gray-tone order modulus -- 10.2.7. Gray-tone Riesz space -- 10.2.8. The gray-tone positive cone -- 10.3. Main approaches for IPA -- 10.3.1. Classical linear operations -- 10.3.2. General linear operations -- 10.3.3. The multiplicative homomorphic operations -- 10.3.4. The logarithmic-ratio operations -- 10.3.5. The logarithmic operations -- 10.3.6. The homomorphic logarithmic operations -- 10.3.7. The isomorphic definition of the product operation -- 10.4. Main applications for IPA -- 10.4.1. Tonal affinities -- 10.4.2. Monotonic tonal transformations -- 10.4.3. Positive tonal transformations -- 10.5. Additional comments -- Part 4. Ten Main Functional Frameworks for Gray Tone Images -- Chapter 11. The Algebraic and Order Functional Framework -- 11.1. Paradigms -- 11.2. Mathematical structures and notions for IPA -- 11.2.1. Gray-tone functions -- 11.2.2. The gray-tone function vector space and vector algebra -- 11.2.3. The gray-tone function vector lattice -- 11.2.4. The gray-tone function normed vector lattice.

11.2.5. Extended gray-tone functions -- 11.3. Main approaches for IPA -- 11.3.1. Classical linear image processing -- 11.3.2. General linear image processing -- 11.4. Main applications for IPA -- 11.4.1. Gray-tone image darkening and whitening -- 11.4.2. Gray-tone image dynamic range maximization -- 11.4.3. Gray-tone image denoising -- 11.4.4. Gray-tone function addition and physical superposition -- 11.4.5. Gray-tone function subtraction and physical dissociation -- 11.5. Additional comments -- Chapter 12. The Morphological Functional Framework -- 12.1. Paradigms -- 12.2. Mathematical concepts and structures -- 12.2.1. Mathematical disciplines -- 12.2.2. Local maxima and minima of a gray-tone function -- 12.2.3. Semi-continuity of extended gray-tone functions -- 12.2.4. Examples of semi-continuous gray-tone functions -- 12.3. Main approaches for IPA -- 12.3.1. Mathematical morphology -- 12.3.2. Morphological dilation and erosion -- 12.3.3. Morphological opening and closing -- 12.3.4. Functional structuring functions -- 12.3.5. Image rank filtering -- 12.4. Main applications for IPA -- 12.4.1. Edge detection -- 12.4.2. Image softening -- 12.4.3. Image segmentation -- 12.4.4. Rank filtering -- 12.5. Additional comments -- Chapter 13. The Integral Functional Framework -- 13.1. Paradigms -- 13.2. Mathematical structures -- 13.2.1. Mathematical disciplines -- 13.2.2. Lebesgue-Bochner gray-tone function spaces -- 13.2.3. Locally p-integrable gray-tone functions -- 13.3. Main approaches for IPA -- 13.3.1. Integral operators -- 13.3.2. Integral transformations -- 13.3.3. Lebesgue pixels -- 13.3.4. Orthogonality and correlation -- 13.3.5. Integral equations -- 13.4. Main applications for IPA -- 13.4.1. Image softening -- 13.4.2. Image hardening -- 13.5. Additional comments.

Chapter 14. The Convolutional Functional Framework.