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This work explores the optimization of geometry and topology of shapes through discussions on shape representations, relaxation theory, and optimal transport, highlighting the mathematical tools involved. It delves into applications across various fields, including science, engineering, economics, social sciences, biology, physics, and image processing. The contents are divided into two parts. The first part addresses geometric issues in PDE problems related to the infinity Laplace operator, solutions to free boundary problems amid geometric uncertainties, and distributed and boundary control problems for the semidiscrete Cahn–Hilliard/Navier–Stokes system with nonsmooth Ginzburg–Landau energies. It also covers high-order topological expansions for 2D Helmholtz problems, a new phase field model for approximating interfacial energies in multiphase systems, and the optimization of eigenvalues and eigenmodes using the adjoint method, along with discrete varifolds and surface approximation. The second part focuses on weak Monge–Ampere solutions of the semi-discrete optimal transportation problem, optimal transportation theory with repulsive costs, and Wardrop equilibria. It examines the Lagrangian branched transport model and its equivalence with the Eulerian formulation, nonlinear evolution systems as perturbations of Wasserstein gradient flows, and pressureless Euler equations with a maximal density constraint. Finally, it di

The book explores the intersection of mathematics and imaging applications, divided into two main parts. The first part delves into optimization, inverse problems, shape spaces, computer vision, and computational anatomy. Key topics include a second-order decomposition model for image processing, numerical experimentation, and optimizing spatial and tonal data for PDE-based inpainting. It also addresses image registration through phase-amplitude separation, rotation invariance in exemplar-based image inpainting, and convective regularization for optical flow. Additionally, it presents a variational method for quantitative photoacoustic tomography and bilevel approaches for learning variational imaging models. The second part shifts focus to geometric control and related areas, such as Riemannian geometry, celestial mechanics, and quantum control. It discusses non-degenerate forms of the generalized Euler-Lagrange condition for state-constrained optimal control problems and examines the Purcell three-link swimmer, highlighting geometric and numerical aspects of periodic optimal controls. Other topics include controllability of Keplerian motion with low-thrust systems, higher variational equation techniques for integrability of homogeneous potentials, and an introduction to KAM theory in celestial mechanics. The section concludes with discussions on invariants of contact sub-pseudo-Riemannian structures, time-optimal control fo

Ce cours propose une introduction au traitement d'image mathématique déterministe, abordant les problématiques clés telles que le débruitage, le filtrage, la restauration, la segmentation, le rehaussement et le défloutage, ainsi qu'un aperçu des techniques d'acquisition. Les méthodes mathématiques présentées sont principalement déterministes, incluant la transformation de Fourier, les ondelettes, les équations aux dérivées partielles, la morphologie mathématique et les méthodes variationnelles. Des applications comme la stéganographie, la compression et l'inpainting sont brièvement discutées pour illustrer les concepts. Le livre propose également un rappel des notions mathématiques essentielles, se voulant auto-suffisant, et une bibliographie fournie pour permettre aux lecteurs d'approfondir les techniques qui les intéressent. Destiné aux étudiants de MASTER, aux élèves-ingénieurs et aux chercheurs, cet ouvrage vise à leur fournir une compréhension approfondie des techniques mathématiques fondamentales en traitement et analyse d'image.