Mutational and Morphological Analysis: Tools for Shape Evolution and Morphogenesis

ISBN-13: 9781461272007 / Angielski / Miękka / 2012 / 425 str.

Jean-Pierre Aubin

Opis

The analysis, processing, evolution, optimization and/or regulation, and control of shapes and images appear naturally in engineering (shape optimization, image processing, visual control), numerical analysis (interval analysis), physics (front propagation), biological morphogenesis, population dynamics (migrations), and dynamic economic theory. These problems are currently studied with tools forged out of differential geometry and functional analysis, thus requiring shapes and images to be smooth. However, shapes and images are basically sets, most often not smooth. J.-P. Aubin thus constructs another vision, where shapes and images are just any compact set. Hence their evolution -- which requires a kind of differential calculus -- must be studied in the metric space of compact subsets. Despite the loss of linearity, one can transfer most of the basic results of differential calculus and differential equations in vector spaces to mutational calculus and mutational equations in any mutational space, including naturally the space of nonempty compact subsets. "Mutational and Morphological Analysis" offers a structure that embraces and integrates the various approaches, including shape optimization and mathematical morphology. Scientists and graduate students will find here other powerful mathematical tools for studying problems dealing with shapes and images arising in so many fields.

Recenzje

"This monograph collects various tools needed for the analysis of shape evolution, viability problems, image processing, visual control, biological morphogenesis, wave-front propagation, etc. It consists of four parts:1 -- mutational analysis (a sort of calculus in metric spaces); 2 -- morphological and set-valued analysis; 3 -- geometrical and algebraic morphology; 4 -- differential inclusions (an outline only). The large list of references contains 472 items, moreover, exhaustive bibliographical comments are provided. The concept of mutational space explored in the first two chapters is a framework for developing a sort of differential calculus for maps between general metric spaces. The third chapter contains a description of mutational structures in hyperspaces, especially in the space of nonvoid compact subsets of Euclidean spaces... The present monograph offers a structure that embraces and integrates various approaches to shape evolution and may be useful for...scientists working with images arising in engineering, interval analysis, physics, biological morphogenesis, population dynamics and dynamic economic theory." -Zentralblatt Math

Spis treści

I Mutational Analysis in Metric Spaces.- 1 Mutational Equations.- 1.1 Transitions on Metric spaces.- 1.2 Mutations of Single-Valued Maps.- 1.3 Primitives of Mutations.- 1.4 Mutational Cauchy-Lipschitz’s Theorem.- 1.5 Contingent Transitions.- 1.6 Mutational Nagumo’s Theorem.- 1.6.1 Characterization of Viable Subsets.- 1.6.2 Upper Semicontinuity of Solution Maps.- 1.6.3 Closure of a Viability Domain.- 1.6.4 ?-Limit sets.- 1.7 Viability Kernels and Capture Basins.- 1.7.1 Viability Kernels and Capture Basin.- 1.7.2 Particular Solutions to Mutational Equations.- 1.7.3 Exit and Hitting Functions.- 1.8 Epimutations of Extended Functions.- 1.8.1 Extended Functions.- 1.8.2 Contingent Epiderivatives.- 1.8.3 Contingent Epimutations.- 1.8.4 The Fermat Rule.- 1.8.5 Epimutation of the Distance to a Set.- 1.9 Lyapunov Functions.- 1.9.1 Lower-Semicontinuous Lyapunov Functions.- 1.9.2 The Characterization Theorem.- 1.9.3 Construction of Lyapunov Functions.- 1.10 Approximation of Mutational Equations.- 1.10.1 Euler Schemes.- 1.10.2 Viable Subsets under a Discrete System.- 1.10.3 The Viability Kernel Algorithm.- 2 Mutational Analysis.- 2.1 Mutations of Set-Valued Maps.- 2.2 The Mutational Invariant Manifold Theorem.- 2.2.1 The Decomposable Case.- 2.2.2 The General Case.- 2.3 Control of Mutational Systems.- 2.3.1 Feedback Maps.- 2.3.2 Stabilization.- 2.3.3 Dynamical Feedbacks.- 2.3.4 Optimal Control.- 2.4 Inverse Function Theorems on Metric Spaces.- 2.4.1 Zeros of Functions.- 2.4.2 The Constrained Inverse Function Theorem.- 2.4.3 The Inverse Set-Valued Map Theorem.- 2.5 Newton’s Method.- 2.6 Calculus of Contingent Transition Sets.- 2.6.1 Contingent Transitions to Subsets defined by Equality and Inequality Constraints.- 2.6.2 Contingent Transitions to Intersections and Inverse Images.- 2.7 Doss Integrals on Metric Spaces.- II Morphological and Set-Valued Analysis.- 3 Morphological Spaces.- 3.1 Power Maps.- 3.1.1 Set-Valued Maps.- 3.1.2 Embedding Power Spaces into Vector Spaces.- 3.1.3 Inverse Images and Cores.- 3.1.4 Composition of Maps.- 3.2 The Space of Nonempty Compact Subsets.- 3.2.1 Pompeiu-Hausdorff Topology on the Set of Compact Subsets.- 3.2.2 Support Functions.- 3.2.3 Pompeiu-Hausdorff Distance on the Set of Compact Convex Subsets.- 3.3 Minkowski Operations on Subsets of a Vector Space.- 3.3.1 Dilations and Erosions.- 3.3.2 Minkowski Contents and the Isoperimetric Inequality.- 3.4 Structuring Transitions.- 3.4.1 Structuring Transitions of Power Spaces.- 3.4.2 Basic Concepts of Mathematical Morphology.- 3.4.3 Structuring Mutations of Power Maps.- 3.5 Shape Transitions.- 3.5.1 Shape Transitions on a Vector Space.- 3.5.2 Shape Transitions on a Subset of a Vector Space.- 3.5.3 Shape Transitions on Power Spaces.- 3.5.4 Shape Mutations of Power Maps.- 3.5.5 Shape Derivatives.- 3.5.6 Shape Transitions on ?-Algebra.- 3.6 Mutation of Level Sets of Smooth Functions.- 3.7 Morphological Transitions.- 3.7.1 Morphological Transitions on Compact Sets.- 3.7.2 Morphological Transitions on a Closed Subset.- 3.7.3 Morphological Tubes.- 3.7.4 Morphological Mutations of Power Maps.- 3.7.5 Graphical Mutations of Set-Valued Maps.- 3.8 Equivalent Morphological Transitions.- 3.9 Semi-Permeable Sets.- 3.10 The Aumann and Doss Integrals of a Set-Valued Map.- 4 Morphological Dynamics.- 4.1 Morphological Equations.- 4.1.1 Morphological Primitives.- 4.1.2 Morphological Cauchy-Lipschitz’s Theorem.- 4.1.3 Morphological Equation for Interval Analysis.- 4.1.4 Steiner Morphological Equation.- 4.1.5 Morphological Nagumo’s Theorem.- 4.1.6 Morphological Equilibrium.- 4.1.7 Travelling Waves of Graphical Equations.- 4.1.8 The Morphological Invariant Manifold Theorem.- 4.2 Contingent Sets to Families of Compact Subsets.- 4.2.1 Paratingent Cones.- 4.2.2 Intersectability.- 4.2.3 Confinement.- 4.3 Intersectable and Confined Tubes.- 4.3.1 Viability of Tubes Governed by Morphological Equations.- 4.3.2 Intersectable Tubes.- 4.3.3 Confined Tubes.- 4.4 Epimutation of a Marginal Function.- 4.5 Asymptotic Stability of a Target.- 4.5.1 Asymptotic Targeting.- 4.5.2 Dissipative Systems.- 4.6 Morphological Control and Application to Visual Control.- 4.6.1 Morphological Controlled Problems.- 4.6.2 Example: Visual Control.- 5 Set-Valued Analysis.- 5.1 Graphical and Epigraphical Sums and Differences.- 5.1.1 Graphical sums and differences of Maps.- 5.1.2 Episums and Epidifferences of Functions.- 5.1.3 Toll Sets.- 5.2 Limits of Sets.- 5.2.1 Definitions.- 5.2.2 Calculus of Limits.- 5.2.3 Painlevé-Kuratowski and Pompeiu-Hausdorff Limits.- 5.2.4 Graphical Convergence of Maps.- 5.2.5 Epilimits.- 5.2.6 Semicontinuous Maps.- 5.2.7 The Marginal Selection.- 5.3 Graphical Derivatives of Set-Valued Maps.- 5.3.1 Contingent Derivatives.- 5.3.2 Contingent Epiderivatives.- 5.3.3 Derivatives of Distance Functions to a Map.- 5.4 Morphological Mutations and Contingent Derivatives.- 5.5 Examples of Contingent Derivatives.- 5.5.1 Derivatives of Level-Set Tubes.- 5.5.2 Derivatives of Morphological Tubes.- 5.5.3 Contingent Derivative of the Transport of a Set-Valued Map.- 5.6 Morphological Primitives.- 5.7 Graphical Primitives.- 5.8 Contingent Infinitesimal Generator of a Koopman Process.- 5.9 Jump Maps of Distributions.- 5.9.1 Weak Derivatives: Distribution and Contingent Derivatives.- 5.9.2 Vector Distributions.- 5.9.3 Upper Jump Map of a Distribution.- III Geometrical and Algebraic Morphology.- 6 Morphological Geometry.- 6.1 Projectors and Proximal Normals.- 6.1.1 Projections and Proximal Normals.- 6.1.2 Skeleta.- 6.1.3 Monotonicity Properties of the Projector.- 6.1.4 Normals.- 6.1.5 The Convex Core of a Closed Subset.- 6.2 Derivatives of Distance Functions.- 6.3 Derivatives of Projectors.- 6.4 Discriminating Domains of Hamiltonians.- 6.4.1 Dual Characterization of Semi-Permeability.- 6.4.2 Cardaliaguet’s Discriminating Domains and Kernels.- 6.5 Dual Characterizations.- 6.5.1 Convex Processes and their Transposes.- 6.5.2 Codifferentials.- 6.5.3 Subdifferentials and Generalized Gradients.- 6.5.4 Codifferential of Level-Set Tubes.- 6.5.5 Codifferential of Morphological Primitives.- 6.5.6 Cardaliaguet’s Solutions to Front Propagation Problems.- 6.5.7 Dual Formulation of Graphical Derivatives.- 6.5.8 Dual Formulation of Frankowska’s Solutions to Hamilton-Jacobi Equations.- 6.6 Chronector and Brachynormals.- 6.6.1 Hitting time.- 6.6.2 Chronector and Brachynormals.- 6.6.3 Derivative of the Chronector.- 6.7 Morphological Analysis on Grids: Digitalization.- 6.7.1 Gauge of Structuring Elements.- 6.7.2 Digital Distances.- 6.7.3 Projections and Normal Proximals.- 7 Morphological Algebra.- 7.1 Dioids, Lattices and their Morphisms.- 7.1.1 Dioids.- 7.1.2 Lattices.- 7.1.3 Morphisms of Dioids and Lattices.- 7.1.4 Quasi-Inverses.- 7.1.5 Noetherian Idealoids.- 7.2 Examples of Morphological Morphisms.- 7.2.1 Morphisms Associated with a Set-Valued Map.- 7.2.2 Viability Kernels and Absorption Basins.- 7.2.3 Topological Properties.- 7.2.4 Limit Sets.- 7.2.5 Basins of Attraction.- 7.3 Galois Transform.- 7.4 Vicarious Temporal Logic.- 7.4.1 Nonconsistent Logic Associated with a Closing.- 7.4.2 The Algebra of Closed Subsets.- 7.4.3 Vicarious Temporal Frames.- IV Appendix.- 8 Differential Inclusions: A Tool-Box.- 8.1 Set Topologies.- 8.1.1 Hausdorff Topology on the Set of Closed Subsets.- 8.1.2 Hausdorff-Lebesgue Topology.- 8.1.3 The Oriented Topology.- 8.2 Variational Equations and the Coarea Formula.- 8.2.1 Linear Systems.- 8.2.2 The Variational Equation.- 8.2.3 The Coarea Theorem.- 8.3 The Gronwall and Filippov Estimates.- 8.3.1 The Gronwall Lemma.- 8.3.2 The Filippov Theorem.- 8.4 Viability Theory at a Glimpse.- 8.5 Differential Inclusions for Maximal Monotone Maps.- 8.5.1 Monotone and Maximal Monotone Maps.- 8.5.2 Yosida Approximations.- 8.5.3 The Crandall-Pazy Theorem.- 8.5.4 Nonhomogeneous Differential Inclusions.- Biblographical Comments.