This book presents a selection of the most recent algorithmic advances in Riemannian geometry in the context of machine learning, statistics, optimization, computer vision, and related fields. The unifying theme of the different chapters in the book is the exploitation of the geometry of data using the mathematical machinery of Riemannian geometry. As demonstrated by all the chapters in the book, when the data is intrinsically non-Euclidean, the utilization of this geometrical information can lead to better algorithms that can capture more accurately the structures inherent in the data, leading ultimately to better empirical performance. This book is not intended to be an encyclopedic compilation of the applications of Riemannian geometry. Instead, it focuses on several important research directions that are currently actively pursued by researchers in the field. These include statistical modeling and analysis on manifolds,optimization on manifolds, Riemannian manifolds and kernel methods, and dictionary learning and sparse coding on manifolds. Examples of applications include novel algorithms for Monte Carlo sampling and Gaussian Mixture Model fitting, 3D brain image analysis,image classification, action recognition, and motion tracking.

This book presents a comprehensive treatise on Riemannian geometric computations and related statistical inferences in several computer vision problems. This edited volume includes chapter contributions from leading figures in the field of computer vision who are applying Riemannian geometric approaches in problems such as face recognition, activity recognition, object detection, biomedical image analysis, and structure-from-motion. Some of the mathematical entities that necessitate a geometric analysis include rotation matrices (e.g. in modeling camera motion), stick figures (e.g. for activity recognition), subspace comparisons (e.g. in face recognition), symmetric positive-definite matrices (e.g. in diffusion tensor imaging), and function-spaces (e.g. in studying shapes of closed contours).

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This first volume focuses on real-time control theory, data assimilation, real-time visualization, high-dimensional state spaces and interaction of different reduction techniques.

This book offers a timely and comprehensive snapshot of research and developments in the field of control engineering. Covering a wide range of theoretical and practical issues, the contributions describes a number of different control approaches, such adaptive control, fuzzy and neuro-fuzzy control, remote and robust control systems, real time an fault tolerant control, among others. Sensors and actuators, measurement systems, renewable energy systems, aerospace systems as well as industrial control and automation, are also comprehensively covered. Based on the proceedings of the 14th APCA International Conference on Automatic Control and Soft Computing, held on July 1-3, 2020, in Bragança, Portugal, the book offers a timely and thoroughly survey of the latest research in the field of control, and a source of inspiration for researchers and professionals worldwide.

This book offers a timely and comprehensive snapshot of research and developments in the fields of dynamic systems and control engineering. Covering a wide range of theoretical and practical issues, the contributions describes a number of different control approaches, such as PID control, adaptive control, nonlinear systems and control, intelligent monitoring and control based on fuzzy and neural systems, robust control systems, and real time control, among others. Sensors and actuators, measurement systems, renewable energy systems, aeronautic and aerospace systems as well as industrial control and automation, are also comprehensively covered. Based on the proceedings of the 15th APCA International Conference on Automatic Control and Soft Computing, held on July 6-8, 2022, in Caparica, Portugal, the book offers a timely and thoroughly survey of the latest research in the fields of dynamic systems and automatic control engineering, and a source of inspiration for researchers and professionals worldwide.

This book constitutes the refereed proceedings of the 40th German Conference on Pattern Recognition, GCPR 2018, held in Stuttgart, Germany, in October 2018. The 48 revised full papers presented were carefully reviewed and selected from 118 submissions. The German Conference on Pattern Recognition is the annual symposium of the German Association for Pattern Recognition (DAGM). It is the national venue for recent advances in image processing, pattern recognition, and computer vision and it follows the long tradition of the DAGM conference series, which has been renamed to GCPR in 2013 to reflect its increasing internationalization. In 2018 in Stuttgart, the conference series celebrated its 40th anniversary.

Processing, Analyzing and Learning of Images, Shapes, and Forms: Part 2, Volume 20, surveys the contemporary developments relating to the analysis and learning of images, shapes and forms, covering mathematical models and quick computational techniques. Chapter cover Alternating Diffusion: A Geometric Approach for Sensor Fusion, Generating Structured TV-based Priors and Associated Primal-dual Methods, Graph-based Optimization Approaches for Machine Learning, Uncertainty Quantification and Networks, Extrinsic Shape Analysis from Boundary Representations, Efficient Numerical Methods for Gradient Flows and Phase-field Models, Recent Advances in Denoising of Manifold-Valued Images, Optimal Registration of Images, Surfaces and Shapes, and much more. - Covers contemporary developments relating to the analysis and learning of images, shapes and forms - Presents mathematical models and quick computational techniques relating to the topic - Provides broad coverage, with sample chapters presenting content on Alternating Diffusion and Generating Structured TV-based Priors and Associated Primal-dual Methods

Processing, Analyzing and Learning of Images, Shapes, and Forms: Part 2, Volume 20, surveys the contemporary developments relating to the analysis and learning of images, shapes and forms, covering mathematical models and quick computational techniques. Chapter cover Alternating Diffusion: A Geometric Approach for Sensor Fusion, Generating Structured TV-based Priors and Associated Primal-dual Methods, Graph-based Optimization Approaches for Machine Learning, Uncertainty Quantification and Networks, Extrinsic Shape Analysis from Boundary Representations, Efficient Numerical Methods for Gradient Flows and Phase-field Models, Recent Advances in Denoising of Manifold-Valued Images, Optimal Registration of Images, Surfaces and Shapes, and much more. Covers contemporary developments relating to the analysis and learning of images, shapes and forms Presents mathematical models and quick computational techniques relating to the topic Provides broad coverage, with sample chapters presenting content on Alternating Diffusion and Generating Structured TV-based Priors and Associated Primal-dual Methods

This book provides an introduction to and survey of recent developments in pseudo-Riemannian geometry, including applications in mathematical physics, by leading experts in the field. Topics covered are: Classification of pseudo-Riemannian symmetric spaces Holonomy groups of Lorentzian and pseudo-Riemannian manifolds Hypersymplectic manifolds Anti-self-dual conformal structures in neutral signature and integrable systems Neutral Kahler surfaces and geometric optics Geometry and dynamics of the Einstein universe Essential conformal structures and conformal transformations in pseudo-Riemannian geometry The causal hierarchy of spacetimes Geodesics in pseudo-Riemannian manifolds Lorentzian symmetric spaces in supergravity Generalized geometries in supergravity Einstein metrics with Killing leaves The book is addressed to advanced students as well as to researchers in differential geometry, global analysis, general relativity and string theory. It shows essential differences between the geometry on manifolds with positive definite metrics and on those with indefinite metrics, and highlights the interesting new geometric phenomena, which naturally arise in the indefinite metric case. The reader finds a description of the present state of the art in the field as well as open problems, which can stimulate further research.

Covariance matrices play important roles in many areas of mathematics, statistics, and machine learning, as well as their applications. In computer vision and image processing, they give rise to a powerful data representation, namely the covariance descriptor, with numerous practical applications. In this book, we begin by presenting an overview of the {\it finite-dimensional covariance matrix} representation approach of images, along with its statistical interpretation. In particular, we discuss the various distances and divergences that arise from the intrinsic geometrical structures of the set of Symmetric Positive Definite (SPD) matrices, namely Riemannian manifold and convex cone structures. Computationally, we focus on kernel methods on covariance matrices, especially using the Log-Euclidean distance. We then show some of the latest developments in the generalization of the finite-dimensional covariance matrix representation to the {\it infinite-dimensional covariance operator} representation via positive definite kernels. We present the generalization of the affine-invariant Riemannian metric and the Log-Hilbert-Schmidt metric, which generalizes the Log-Euclidean distance. Computationally, we focus on kernel methods on covariance operators, especially using the Log-Hilbert-Schmidt distance. Specifically, we present a two-layer kernel machine, using the Log-Hilbert-Schmidt distance and its finite-dimensional approximation, which reduces the computational complexity of the exact formulation while largely preserving its capability. Theoretical analysis shows that, mathematically, the approximate Log-Hilbert-Schmidt distance should be preferred over the approximate Log-Hilbert-Schmidt inner product and, computationally, it should be preferred over the approximate affine-invariant Riemannian distance. Numerical experiments on image classification demonstrate significant improvements of the infinite-dimensional formulation over the finite-dimensional counterpart. Given the numerous applications of covariance matrices in many areas of mathematics, statistics, and machine learning, just to name a few, we expect that the infinite-dimensional covariance operator formulation presented here will have many more applications beyond those in computer vision.