Mutational and Morphological Analysis: Tools for Shape Evolution and Morphogenesis

by Aubin, Jean-Pierre

• ISBN13: 9780817639358
• ISBN10: 0817639357
• Format: Hardcover
• Copyright: 1999-02-01
• Publisher: Springer Verlag

Summary

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.

Table of Contents

Preface	xiii	(4)
Acknowledgments	xvii	(2)
Introduction	xix	(10)
Outline of the Book	xxix
I Mutational Analysis in Metric Spaces	1	(98)
1 Mutational Equations	3	(60)
Introduction	3	(4)
1.1 Transitions on Metric spaces	7	(6)
1.2 Mutations of Single-Valued Maps	13	(2)
1.3 Primitives of Mutations	15	(6)
1.4 Mutational Cauchy-Lipschitz's Theorem	21	(3)
1.5 Contingent Transitions	24	(4)
1.6 Mutational Nagumo's Theorem	28	(10)
1.6.1 Characterization of Viable Subsets	29	(6)
1.6.2 Upper Semicontinuity of Solution Maps	35	(1)
1.6.3 Closure of a Viability Domain	36	(1)
1.6.4 (Omega)-Limit sets	37	(1)
1.7 Viability Kernels and Capture Basins	38	(6)
1.7.1 Viability Kernels and Capture Basin	39	(2)
1.7.2 Particular Soluation to Mutational Equations	41	(1)
1.7.3 Exit and Hitting Functions	42	(2)
1.8 Epimutations of Extended Functions	44	(9)
1.8.1 Extended Functions	45	(2)
1.8.2 Contingent Epiderivatives	47	(1)
1.8.3 Contingent Epimutations	48	(2)
1.8.4 The Fermat Rule	50	(2)
1.8.5 Epimutations of the Distance to a Set	52	(1)
1.9 Lyapunov Functions	53	(4)
1.9.1 Lower-Semicontinuous Lyapunov Functions	53	(1)
1.9.2 The Characterization Theorem	54	(2)
1.9.3 Construction of Lyapunov Functions	56	(1)
1.10 Approximation of Mutational Equations	57	(6)
1.10.1 Euler Schemes	57	(3)
1.10.2 Viable Subsets under a Discrete System	60	(1)
1.10.3 The Viability Kernel Algorithm	61	(2)
2 Mutational Analysis	63	(36)
Introduction	63	(3)
2.1 Mutations of Set-Valued Maps	66	(3)
2.2 The Mutational Invariant Manifold Theorem	69	(6)
2.2.1 The Decomposable Case	70	(4)
2.2.2 The General Case	74	(1)
2.3 Control of Mutational Systems	75	(7)
2.3.1 Feedback Maps	75	(3)
2.3.2 Stabilization	78	(1)
2.3.3 Dynamical Feedbacks	79	(2)
2.3.4 Optimal Control	81	(1)
2.4 Inverse Function Theorems on Metric Spaces	82	(5)
2.4.1 Zeros of Functions	82	(1)
2.4.2 The Constrained Inverse Function Theorem	83	(3)
2.4.3 The Inverse Set-Valued Map Theorem	86	(1)
2.5 Newton's Method	87	(1)
2.6 Calculus of Contingent Transition Sets	88	(4)
2.6.1 Contingent Transitions to Subsets defined by Equality and Inequality Constraints	89	(1)
2.6.2 Contingent Transitions to Intersections and Inverse Images	90	(2)
2.7 Doss Integrals on Metric Spaces	92	(7)
II Morphological and Set-Valued Analysis	99	(166)
3 Morphological Spaces	101	(65)
Introduction	101	(4)
3.1 Power Maps	105	(14)
3.1.1 Set-Valued Maps	108	(4)
3.1.2 Embedding Power Spaces into Vector Spaces	112	(5)
3.1.3 Inverse Images and Cores	117	(2)
3.1.4 Compositions of Maps	117	(2)
3.2 The Space of Nonempty Compact Subsets	119	(8)
3.2.1 Pompeiu-Hausdorff Topology on the Set of Compact Subsets	119	(4)
3.2.2 Support Functions	123	(3)
3.2.3 Pompeiu-Hausdorff Distance on the Set of Compact Convex Subsets	126	(1)
3.3 Minkowski Operations on Subsets of a Vector Space	127	(4)
3.3.1 Dilations and Erosions	128	(2)
3.3.2 Minkowski Contents and the Isoperimetric Inequality	130	(1)
3.4 Structuring Transitions	131	(4)
3.4.1 Structuring Transitions of Power Spaces	132	(1)
3.4.2 Basic Concepts of Mathematical Morphology	133	(1)
3.4.3 Structuring Mutations of Power Maps	134	(1)
3.5 Shape Transitions	135	(9)
3.5.1 Shape Transitions on a Vector Space	135	(2)
3.5.2 Shape Transitions on a Subset of a Vector Space	137	(1)
3.5.3 Shape Transitions on Power Spaces	138	(1)
3.5.4 Shape Mutations of Power Maps	139	(1)
3.5.5 Shape Derivatives	140	(1)
3.5.6 Shape Transitions on Sigma-Algebra	141	(3)
3.6 Mutation of Level Sets of Smooth Functions	144	(3)
3.7 Morphological Transitions	147	(9)
3.7.1 Morphological Transitions on Compact Sets	147	(5)
3.7.2 Morphological Transitions on a Closed Subsets	152	(1)
3.7.3 Morphological Tubes	153	(1)
3.7.4 Morphological Mutations of Power Maps	153	(2)
3.7.5 Graphical Mutations of Set-Valued Maps	155	(1)
3.8 Equivalent Morphological Transitions	156	(2)
3.9 Semi-Permeable Sets	158	(4)
3.10 The Aumann and Doss Integrals of a Set-Valued Map	162	(4)
4 Morphological Dynamics	166	(39)
Introduction	166	(4)
4.1 Morphological Equations	170	(8)
4.1.1 Morphological Primitives	170	(1)
4.1.2 Morphological Cauchy-Lipschitz's Theorem	171	(1)
4.1.3 Morphological Equation for Interval Analysis	172	(1)
4.1.4 Steiner Morphological Equation	173	(1)
4.1.5 Morphological Nagumo's Theorem	174	(1)
4.1.6 Morphological Equilibrium	175	(1)
4.1.7 Travelling Waves of Graphical Equations	176	(1)
4.1.8 The Morphological Invariant Manifold Theorem	177	(1)
4.2 Contingent Sets to Families of Compact Subsets	178	(8)
4.2.1 Paratingent Cones	178	(1)
4.2.2 Intersectability	179	(3)
4.2.3 Confinement	182	(4)
4.3 Intersectable and Confined Tubes	186	(5)
4.3.1 Viability of Tubes Governed by Morphological Equations	186	(1)
4.3.2 Intersectable Tubes	187	(2)
4.3.3 Confined Tubes	189	(2)
4.4 Epimutation of a Marginal Function	191	(7)
4.5 Asymptotic Stability of a Target	198	(4)
4.5.1 Asymptotic Targeting	198	(3)
4.5.2 Dissipative Systems	201	(1)
4.6 Morphological Control and Application to Visual Control	202	(3)
4.6.1 Morphological Controlled Problems	202	(1)
4.6.2 Example: Visual Control	203	(2)
5 Set-Valued Analysis	205	(60)
Introduction	205	(5)
5.1 Graphical and Epigraphical Sums and Differences	210	(9)
5.1.1 Graphical sums and differences of Maps	210	(3)
5.1.2 Episums and Epidifferences of Functions	213	(2)
5.1.3 Toll Sets	215	(4)
5.2 Limits of Sets	219	(12)
5.2.1 Definitions	219	(1)
5.2.2 Calculus of Limits	220	(2)
5.2.3 Painleve-Kuratowski and Pompeiu-Hausdorff Limits	222	(2)
5.2.4 Graphical Convergence of Maps	224	(1)
5.2.5 Epilimits	225	(1)
5.2.6 Semicontinuous Maps	226	(2)
5.2.7 The Marginal Selection	228	(3)
5.3 Graphical Derivatives of Set-Valued Maps	231	(7)
5.3.1 Contingent Derivatives	231	(4)
5.3.2 Contingent Epiderivatives	235	(2)
5.3.3 Derivatives of Distance Functions to a Map	237	(1)
5.4 Morphological Mutations and Contingent Derivatives	238	(3)
5.5 Examples of Contingent Derivatives	241	(5)
5.5.1 Derivatives of Level-Set Tubes	241	(2)
5.5.2 Derivatives of Morphological Tubes	243	(2)
5.5.3 Contingent Derivative of the Transport of a Set-Valued Map	245	(1)
5.6 Morphological Primitives	246	(7)
5.7 Graphical Primitives	253	(2)
5.8 Contingent Infinitesimal Generator of a Koopman Process	255	(6)
5.9 Jump Maps of Distributions	261	(4)
5.9.1 Weak Derivatives: Distribution and Contingent Derivatives	261	(1)
5.9.2 Vector Distributions	262	(1)
5.9.3 Upper Jump Map of a Distribution	263	(2)
III Geometrical and Algebraic Morphology	265	(90)
6 Morphological Geometry	267	(52)
Introduction	267	(4)
6.1 Projectors and Proximal Normals	271	(10)
6.1.1 Projections and Proximal Normals	271	(3)
6.1.2 Skeleta	274	(1)
6.1.3 Monotonicity Properties of the Projector	275	(3)
6.1.4 Normals	278	(2)
6.1.5 The Convex Core of a Closed Subset	280	(1)
6.2 Derivatives of Distance Functions	281	(3)
6.3 Derivatives of Projectors	284	(5)
6.4 Discriminating Domains of Hamiltonians	289	(5)
6.4.1 Dual Characterization of Semi-Permeability	289	(3)
6.4.2 Cardaliaguet's Discriminating Domains and Kernels	292	(2)
6.5 Dual Characterizations	294	(10)
6.5.1 Convex Processes and their Transposes	294	(1)
6.5.2 Codifferentials	295	(1)
6.5.3 Subdifferentials and Generalized Gradients	296	(2)
6.5.4 Codifferential of Level-Set Tubes	298	(2)
6.5.5 Codifferential of Morphological Primitives	300	(1)
6.5.6 Cardaliaguet's Solutions to Front Propagation Problems	301	(1)
6.5.7 Dual Formulation of Graphical Derivatives	302	(1)
6.5.8 Dual Formulation of Frankowska's Solutions to Hamilton-Jacobi Equations	303	(1)
6.6 Chronector and Brachynormals	304	(6)
6.6.1 Hitting time	305	(1)
6.6.2 Chronector and Brachynormals	306	(3)
6.6.3 Derivative of the Chronector	309	(1)
6.7 Morphological Analysis on Grids: Digitalization	310	(9)
6.7.1 Gauge of Structuring Elements	310	(2)
6.7.2 Digital Distances	312	(4)
6.7.3 Projections and Normal Proximals	316	(3)
7 Morphological Algebra	319	(36)
Introduction	319	(2)
7.1 Dioids, Lattices and their Morphisms	321	(10)
7.1.1 Dioids	321	(3)
7.1.2 Lattices	324	(1)
7.1.3 Morphisms of Dioids and Lattices	325	(3)
7.1.4 Quasi-Inverses	328	(1)
7.1.5 Noetherian Idealoids	329	(2)
7.2 Examples of Morphological Morphisms	331	(12)
7.2.1 Morphisms Associated with a Set-Valued Map	332	(1)
7.2.2 Viability Kernels and Absorption Basins	333	(4)
7.2.3 Topological Properties	337	(3)
7.2.4 Limit Sets	340	(1)
7.2.5 Basins of Attraction	341	(2)
7.3 Galois Transform	343	(7)
7.4 Vicarious Temporal Logic	350	(5)
7.4.1 Nonconsistent Logic Associated with a Closing	350	(1)
7.4.2 The Algebra of Closed Subsets	351	(1)
7.4.3 Vicarious Temporal Frames	352	(3)
IV Appendix	355	(29)
8 Differential Inclusions: A Tool-Box	357	(27)
Introduction	357	(1)
8.1 Set Topologies	358	(8)
8.1.1 Hausdorff Topology on the Set of Closed Subsets	358	(4)
8.1.2 Hausdorff-Lebesgue Topology	362	(1)
8.1.3 The Oriented Topology	363	(3)
8.2 Variational Equations and the Coarea Formula	366	(4)
8.2.1 Linear Systems	366	(2)
8.2.2 The Variational Equation	368	(1)
8.2.3 The Coarea Theorem	369	(1)
8.3 The Gronwall and Filippov Estimates	370	(2)
8.3.1 The Gronwall Lemma	370	(1)
8.3.2 The Filippov Theorem	371	(1)
8.4 Viability Theory at a Glimpse	372	(3)
8.5 Differential Inclusions for Maximal Monotone Maps	375	(9)
8.5.1 Monotone and Maximal Monotone Maps	375	(2)
8.5.2 Yosida Approximations	377	(1)
8.5.3 The Crandall-Pazy Theorem	378	(1)
8.5.4 Nonhomogeneous Differential Inclusions	379	(5)
Bibliographical Comments	384	(11)
Bibliography	395	(26)
Index	421