To achieve the complex task of interpreting what we see, our brains rely on statistical regularities and patterns in visual data. Knowledge of these regularities can also be considerably useful in visual computing disciplines, such as computer vision, computer graphics, and image processing. The field of natural image statistics studies the regular
This textbook offers a statistical view on the geometry of multiple view analysis, required for camera calibration and orientation and for geometric scene reconstruction based on geometric image features. The authors have backgrounds in geodesy and also long experience with development and research in computer vision, and this is the first book to present a joint approach from the converging fields of photogrammetry and computer vision. Part I of the book provides an introduction to estimation theory, covering aspects such as Bayesian estimation, variance components, and sequential estimation, with a focus on the statistically sound diagnostics of estimation results essential in vision metrology. Part II provides tools for 2D and 3D geometric reasoning using projective geometry. This includes oriented projective geometry and tools for statistically optimal estimation and test of geometric entities and transformations and their relations, tools that are useful also in the context of uncertain reasoning in point clouds. Part III is devoted to modelling the geometry of single and multiple cameras, addressing calibration and orientation, including statistical evaluation and reconstruction of corresponding scene features and surfaces based on geometric image features. The authors provide algorithms for various geometric computation problems in vision metrology, together with mathematical justifications and statistical analysis, thus enabling thorough evaluations. The chapters are self-contained with numerous figures and exercises, and they are supported by an appendix that explains the basic mathematical notation and a detailed index. The book can serve as the basis for undergraduate and graduate courses in photogrammetry, computer vision, and computer graphics. It is also appropriate for researchers, engineers, and software developers in the photogrammetry and GIS industries, particularly those engaged with statistically based geometric computer vision methods.
Aims and Scope This book is both an introductory textbook and a research monograph on modeling the statistical structure of natural images. In very simple terms, “natural images” are photographs of the typical environment where we live. In this book, their statistical structure is described using a number of statistical models whose parameters are estimated from image samples. Our main motivation for exploring natural image statistics is computational m- eling of biological visual systems. A theoretical framework which is gaining more and more support considers the properties of the visual system to be re?ections of the statistical structure of natural images because of evolutionary adaptation processes. Another motivation for natural image statistics research is in computer science and engineering, where it helps in development of better image processing and computer vision methods. While research on natural image statistics has been growing rapidly since the mid-1990s, no attempt has been made to cover the ?eld in a single book, providing a uni?ed view of the different models and approaches. This book attempts to do just that. Furthermore, our aim is to provide an accessible introduction to the ?eld for students in related disciplines.
During the last few centuries, natural philosophers, and more recently vision scientists, have recognized that a fundamental problem in biological vision is that the sources underlying visual stimuli are unknowable in any direct sense, because of the inherent ambiguity of the stimuli that impinge on sensory receptors. The light that reaches the eye from any scene conflates the contributions of reflectance, illumination, transmittance, and subsidiary factors that affect these primary physical parameters. Spatial properties such as the size, distance and orientation of physical objects are also conflated in light stimuli. As a result, the provenance of light reaching the eye at any moment is uncertain. This quandary is referred to as the inverse optics problem. This book considers the evidence that the human visual system solves this problem by incorporating past human experience of what retinal images have typically corresponded to in the real world.
During the last few centuries, natural philosophers, and more recently vision scientists, have recognized that a fundamental problem in biological vision is that the sources underlying visual stimuli are unknowable in any direct sense, because of the inherent ambiguity of the stimuli that impinge on sensory receptors. The light that reaches the eye from any scene conflates the contributions of reflectance, illumination, transmittance, and subsidiary factors that affect these primary physical parameters. Spatial properties such as the size, distance and orientation of physical objects are also conflated in light stimuli. As a result, the provenance of light reaching the eye at any moment is uncertain. This quandary is referred to as the inverse optics problem. This book considers the evidence that the human visual system solves this problem by incorporating past human experience of what retinal images have typically corresponded to in the real world.
This book focuses on the application and development of information geometric methods in the analysis, classification and retrieval of images and signals. It provides introductory chapters to help those new to information geometry and applies the theory to several applications. This area has developed rapidly over recent years, propelled by the major theoretical developments in information geometry, efficient data and image acquisition and the desire to process and interpret large databases of digital information. The book addresses both the transfer of methodology to practitioners involved in database analysis and in its efficient computational implementation.
This volume contains the proceedings of the AMS Special Session on Algebraic and Geometric Methods in Applied Discrete Mathematics, held on January 11, 2015, in San Antonio, Texas. The papers present connections between techniques from “pure” mathematics and various applications amenable to the analysis of discrete models, encompassing applications of combinatorics, topology, algebra, geometry, optimization, and representation theory. Papers not only present novel results, but also survey the current state of knowledge of important topics in applied discrete mathematics. Particular highlights include: a new computational framework, based on geometric combinatorics, for structure prediction from RNA sequences; a new method for approximating the optimal solution of a sum of squares problem; a survey of recent Helly-type geometric theorems; applications of representation theory to voting theory and game theory; a study of fixed points of tensors; and exponential random graph models from the perspective of algebraic statistics with applications to networks. This volume was written for those trained in areas such as algebra, topology, geometry, and combinatorics who are interested in tackling problems in fields such as biology, the social sciences, data analysis, and optimization. It may be useful not only for experts, but also for students who wish to gain an applied or interdisciplinary perspective.
Over the past 15 years, there has been a growing need in the medical image computing community for principled methods to process nonlinear geometric data. Riemannian geometry has emerged as one of the most powerful mathematical and computational frameworks for analyzing such data. Riemannian Geometric Statistics in Medical Image Analysis is a complete reference on statistics on Riemannian manifolds and more general nonlinear spaces with applications in medical image analysis. It provides an introduction to the core methodology followed by a presentation of state-of-the-art methods. Content includes: - The foundations of Riemannian geometric methods for statistics on manifolds with emphasis on concepts rather than on proofs - Applications of statistics on manifolds and shape spaces in medical image computing - Diffeomorphic deformations and their applications As the methods described apply to domains such as signal processing (radar signal processing and brain computer interaction), computer vision (object and face recognition), and other domains where statistics of geometric features appear, this book is suitable for researchers and graduate students in medical imaging, engineering and computer science. - A complete reference covering both the foundations and state-of-the-art methods - Edited and authored by leading researchers in the field - Contains theory, examples, applications, and algorithms - Gives an overview of current research challenges and future applications
Why are We Writing This Book? Visual data (graphical, image, video, and visualized data) affect every aspect of modern society. The cheap collection, storage, and transmission of vast amounts of visual data have revolutionized the practice of science, technology, and business. Innovations from various disciplines have been developed and applied to the task of designing intelligent machines that can automatically detect and exploit useful regularities (patterns) in visual data. One such approach to machine intelligence is statistical learning and pattern analysis for visual data. Over the past two decades, rapid advances have been made throughout the ?eld of visual pattern analysis. Some fundamental problems, including perceptual gro- ing,imagesegmentation, stereomatching, objectdetectionandrecognition,and- tion analysis and visual tracking, have become hot research topics and test beds in multiple areas of specialization, including mathematics, neuron-biometry, and c- nition. A great diversity of models and algorithms stemming from these disciplines has been proposed. To address the issues of ill-posed problems and uncertainties in visual pattern modeling and computing, researchers have developed rich toolkits based on pattern analysis theory, harmonic analysis and partial differential eq- tions, geometry and group theory, graph matching, and graph grammars. Among these technologies involved in intelligent visual information processing, statistical learning and pattern analysis is undoubtedly the most popular and imp- tant approach, and it is also one of the most rapidly developing ?elds, with many achievements in recent years. Above all, it provides a unifying theoretical fra- work for intelligent visual information processing applications.
