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 objects -- 21.4.4. 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 operations -- 22.3. 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 diagrams -- 22.4.7. 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 norms -- 23.3. 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 -- Further topics and readings -- Some references on applications to IPA -- Chapter 24: The Convex Geometric Framework -- 24.1. Paradigms -- 24.2. Mathematical concepts and structures -- 24.2.1. Mathematical disciplines -- 24.3. Main notions and approaches for IPA -- 24.3.1. Convex objects -- 24.3.1.1. Convex objects and extremal pixels -- 24.3.1.2. Convex objects and hyperplanes -- 24.3.1.3. Convex hulls -- 24.3.2. Hausdorff topology for compact convex objects -- 24.3.3. Compact poly-convex objects -- 24.3.4. Star-shaped objects -- 24.3.5. Simplices -- 24.4. Main applications to IPA.

24.4.1. Convex deficiency set and concavities -- 24.4.2. Functions related to convex and star-shaped objects -- 24.4.3. Delaunay triangulation -- 24.5. Additional comments -- Historical comments -- Bibliographic notes and additional readings -- Further topics and readings -- Some references on applications to IPA -- Chapter 25: The Morphological Geometric Framework -- 25.1. Paradigms -- 25.2. Mathematical concepts and structures -- 25.2.1. Mathematical disciplines -- 25.3. Mathematical notions and approaches for IPA -- 25.3.1. Morphological dilation and erosion -- 25.3.2. Morphological closing and opening -- 25.3.3. Set properties of morphological dilation, erosion, closing and opening -- 25.3.4. Morphological regular objects -- 25.3.5. Continuity of the morphological operations -- 25.4. Main notions and approaches for IPA -- 25.4.1. Morphological transformations -- 25.4.2. Parallel objects -- 25.4.3. Federer sets -- 25.5. Main applications to IPA -- 25.5.1. Object contours and morphological boundaries -- 25.5.2. Object filtering and morphological smoothing -- 25.5.3. Morphological skeleton -- 25.5.4. Ultimate erosion -- 25.5.5. Morphing -- 25.6. Additional comments -- Historical comments and references -- Bibliographic notes and additional readings -- Further topics and readings -- Applications to image processing and analysis -- Chapter 26: The Geometric and Topological Framework -- 26.1. Paradigms -- 26.2. Mathematical concepts and structures -- 26.2.1. Mathematical disciplines -- 26.2.2. Manifolds or locally Euclidean spaces -- 26.2.3. Manifolds with border -- 26.2.4. Submanifolds -- 26.2.5. Compact and closed manifolds -- 26.2.6. Lipschitz manifolds and Lipschitz sets -- 26.3. Mathematical approaches for IPA -- 26.3.1. Unit ball and unit cube, torii and annulii -- 26.3.2. Points, curves and surfaces -- 26.3.3. Hypersurfaces.

26.3.4. Homeomorphic and homotopic objects -- 26.4. Main applications to IPA -- 26.4.1. Contour -- 26.4.1.1. Contour in dimension 2 -- 26.4.1.2. Contour in dimension 3 -- 26.4.2. Topological content -- 26.4.3. The Lebesgue-(Čech) dimension of homeomorphic or homotopic objects -- 26.4.4. The Descartes-Euler-Poincaré's number and the Betti numbers -- 26.4.5. Some particular basic manifolds -- 26.5. Additional comments -- Historical comments and references -- Bibliographic notes and suggested readings -- Further topics and readings -- Some references on applications to Image Analysis -- Chapter 27: The Measure-Theoretic Geometric Framework -- 27.1. Paradigms -- 27.2. Mathematical concepts and structures -- 27.2.1. Mathematical disciplines -- 27.2.2. The Gauss measure -- 27.2.3. The Peano-Jordan measures -- 27.2.4. Measures and contents -- 27.2.5. Outer measures and Borel sets -- 27.2.6. Finite and σ-finite measures -- 27.2.7. Null sets, negligible sets and complete measures -- 27.2.8. Atoms and atomic measures -- 27.2.9. The n-dimensional Lebesgue measure -- 27.2.9.1. The volume of the n-dimensional unit ball -- 27.2.10. The m-dimensional Hausdorff measure -- 27.2.11. Jordan sets -- 27.3. Main approaches for IPA -- 27.3.1. Rectifiable objects -- 27.3.1.1. Besicovitch-Federer's decomposition theorem -- 27.3.2. Parallel dilated objects -- 27.3.3. The Minkowski contents -- 27.3.3.1. The m-dimensional Minkowski content -- 27.3.3.2. The outer and two-sided n-dimensional Minkowski content -- 27.3.4. The Fréchet-Nikodym-Aronszajn distance -- 27.3.5. Caccioppoli sets -- 27.4. Applications to IPA -- 27.4.1. Perimeter measures -- 27.4.1.1. The Minkowski-Steiner's formula -- 27.4.1.2. Caccioppoli-De Giorgi's perimeter -- 27.4.2. Invariant measures -- 27.4.2.1. Haar measures -- 27.4.3. The m-dimensional Favard measure -- 27.4.3.1. Purely unrectifiable sets.

27.4.4. Comparison of objects -- 27.5. Additional comments -- Historical comments and references -- Bibliographic notes and additional readings -- Further topics and readings -- Some references on applications to IPA -- Chapter 28: The Integral Geometric Framework -- 28.1. Paradigms -- 28.2. Mathematical concepts and structures -- 28.2.1. Mathematical disciplines -- 28.2.2. Geometric functionals -- 28.2.3. Intrinsic volumes and Minkowski functionals on compact convex objects -- 28.2.3.1. The Steiner polynomials -- 28.2.3.2. Properties of intrinsic volumes and Minkowski functionals -- 28.2.4. Content functionals on finite unions of compact convex objects -- 28.2.4.1. Properties of Minkowski functionals -- 28.2.5. Hadwiger's characterization theorem -- 28.2.6. Particular m-dimensional content functionals -- 28.2.6.1. Integral of mean curvature and mean breadth for compact convex objects -- 28.2.7. Continuity of geometric functionals -- 28.3. Main approaches for IPA -- 28.3.1. The Favard measure and Cauchy-Crofton's formulas -- 28.3.2. Cauchy-Crofton's formulas for compact, poly-convex objects -- 28.3.3. Cauchy-Crofton's formulas for a k-dimensional countably rectifiable manifold -- 28.3.4. Intersections with lower dimensional affine subspaces -- 28.3.5. The covariogram of a measurable object -- 28.3.5.1. Lipschitzian covariograms -- 28.4. Applications to IPA -- 28.4.1. p-dimensional affine sections -- 28.4.2. m-dimensional content functionals for n=1, 2 and 3 -- 28.4.3. Steiner's formulas for n=1, 2 and 3 -- 28.4.4. Cauchy-Crofton's formulas in dimension 2 and 3 -- 28.4.4.1. Cauchy-Crofton's formulas in dimension 3 -- 28.4.4.2. Cauchy-Crofton's formulas in dimension 2 -- 28.4.5. Feret diameters and areas -- 28.4.6. Other diameters -- 28.4.7. Cauchy's projection formulas -- 28.4.7.1. Cauchy's projection formula for convex bodies.

28.4.7.2. Cauchy's total projection formulae for curve length.