Variational methods

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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