3D rotation analysis is widely encountered in everyday problems thanks to the development of computers. Sensing 3D using cameras and sensors, analyzing and modeling 3D for computer vision and computer graphics, and controlling and simulating robot motion all require 3D rotation computation. This book focuses on the computational analysis of 3D rotation, rather than classical motion analysis. It regards noise as random variables and models their probability distributions. It also pursues statistically optimal computation for maximizing the expected accuracy, as is typical of nonlinear optimization. All concepts are illustrated using computer vision applications as examples. Mathematically, the set of all 3D rotations forms a group denoted by SO(3). Exploiting this group property, we obtain an optimal solution analytical or numerically, depending on the problem. Our numerical scheme, which we call the "Lie algebra method," is based on the Lie group structure of SO(3). This book also proposes computing projects for readers who want to code the theories presented in this book, describing necessary 3D simulation setting as well as providing real GPS 3D measurement data. To help readers not very familiar with abstract mathematics, a brief overview of quaternion algebra, matrix analysis, Lie groups, and Lie algebras is provided as Appendix at the end of the volume.

Understanding Geometric Algebra: Hamilton, Grassmann, and Clifford for Computer Vision and Graphics introduces geometric algebra with an emphasis on the background mathematics of Hamilton, Grassmann, and Clifford. It shows how to describe and compute geometry for 3D modeling applications in computer graphics and computer vision. Unlike similar texts, this book first gives separate descriptions of the various algebras and then explains how they are combined to define the field of geometric algebra. It starts with 3D Euclidean geometry along with discussions as to how the descriptions of geometry could be altered if using a non-orthogonal (oblique) coordinate system. The text focuses on Hamilton's quaternion algebra, Grassmann's outer product algebra, and Clifford algebra that underlies the mathematical structure of geometric algebra. It also presents points and lines in 3D as objects in 4D in the projective geometry framework; explores conformal geometry in 5D, which is the main ingredient of geometric algebra; and delves into the mathematical analysis of camera imaging geometry involving circles and spheres. With useful historical notes and exercises, this book gives readers insight into the mathematical theories behind complicated geometric computations. It helps readers understand the foundation of today's geometric algebra.

3D rotation analysis is widely encountered in everyday problems thanks to the development of computers. Sensing 3D using cameras and sensors, analyzing and modeling 3D for computer vision and computer graphics, and controlling and simulating robot motion all require 3D rotation computation. This book focuses on the computational analysis of 3D rotation, rather than classical motion analysis. It regards noise as random variables and models their probability distributions. It also pursues statistically optimal computation for maximizing the expected accuracy, as is typical of nonlinear optimization. All concepts are illustrated using computer vision applications as examples. Mathematically, the set of all 3D rotations forms a group denoted by SO(3). Exploiting this group property, we obtain an optimal solution analytical or numerically, depending on the problem. Our numerical scheme, which we call the "Lie algebra method," is based on the Lie group structure of SO(3). This book also proposes computing projects for readers who want to code the theories presented in this book, describing necessary 3D simulation setting as well as providing real GPS 3D measurement data. To help readers not very familiar with abstract mathematics, a brief overview of quaternion algebra, matrix analysis, Lie groups, and Lie algebras is provided as Appendix at the end of the volume.

Because circular objects are projected to ellipses in images, ellipse fitting is a first step for 3-D analysis of circular objects in computer vision applications. For this reason, the study of ellipse fitting began as soon as computers came into use for image analysis in the 1970s, but it is only recently that optimal computation techniques based on the statistical properties of noise were established. These include renormalization (1993), which was then improved as FNS (2000) and HEIV (2000). Later, further improvements, called hyperaccurate correction (2006), HyperLS (2009), and hyper-renormalization (2012), were presented. Today, these are regarded as the most accurate fitting methods among all known techniques. This book describes these algorithms as well implementation details and applications to 3-D scene analysis. We also present general mathematical theories of statistical optimization underlying all ellipse fitting algorithms, including rigorous covariance and bias analyses and the theoretical accuracy limit. The results can be directly applied to other computer vision tasks including computing fundamental matrices and homographies between images. This book can serve not simply as a reference of ellipse fitting algorithms for researchers, but also as learning material for beginners who want to start computer vision research. The sample program codes are downloadable from the website: https://sites.google.com/a/morganclaypool.com/ellipse-fitting-for-computer-vision-implementation-and-applications.

Modeling data from visual and linguistic modalities together creates opportunities for better understanding of both, and supports many useful applications. Examples of dual visual-linguistic data includes images with keywords, video with narrative, and figures in documents. We consider two key task-driven themes: translating from one modality to another (e.g., inferring annotations for images) and understanding the data using all modalities, where one modality can help disambiguate information in another. The multiple modalities can either be essentially semantically redundant (e.g., keywords provided by a person looking at the image), or largely complementary (e.g., meta data such as the camera used). Redundancy and complementarity are two endpoints of a scale, and we observe that good performance on translation requires some redundancy, and that joint inference is most useful where some information is complementary. Computational methods discussed are broadly organized into ones for simple keywords, ones going beyond keywords toward natural language, and ones considering sequential aspects of natural language. Methods for keywords are further organized based on localization of semantics, going from words about the scene taken as whole, to words that apply to specific parts of the scene, to relationships between parts. Methods going beyond keywords are organized by the linguistic roles that are learned, exploited, or generated. These include proper nouns, adjectives, spatial and comparative prepositions, and verbs. More recent developments in dealing with sequential structure include automated captioning of scenes and video, alignment of video and text, and automated answering of questions about scenes depicted in images.

Understanding Geometric Algebra: Hamilton, Grassmann, and Clifford for Computer Vision and Graphics introduces geometric algebra with an emphasis on the background mathematics of Hamilton, Grassmann, and Clifford. It shows how to describe and compute geometry for 3D modeling applications in computer graphics and computer vision. Unlike similar texts, this book first gives separate descriptions of the various algebras and then explains how they are combined to define the field of geometric algebra. It starts with 3D Euclidean geometry along with discussions as to how the descriptions of geometry could be altered if using a non-orthogonal (oblique) coordinate system. The text focuses on Hamilton's quaternion algebra, Grassmann's outer product algebra, and Clifford algebra that underlies the mathematical structure of geometric algebra. It also presents points and lines in 3D as objects in 4D in the projective geometry framework; explores conformal geometry in 5D, which is the main ingredient of geometric algebra; and delves into the mathematical analysis of camera imaging geometry involving circles and spheres. With useful historical notes and exercises, this book gives readers insight into the mathematical theories behind complicated geometric computations. It helps readers understand the foundation of today's geometric algebra.

The 2nd International Workshop on Statistical Methods in Video Processing, SMVP 2004, was held in Prague, Czech Republic, as an associated workshop of ECCV 2004, the 8th European Conference on Computer Vision. A total of 30 papers were submitted to the workshop. Of these, 17 papers were accepted for presentation and included in these proceedings, following a double-blind review process. The workshop had 42 registered participants. The focus of the meeting was on recent progress in the application of - vanced statistical methods to solve computer vision tasks. The one-day scienti?c program covered areas of high interest in vision research, such as dense rec- struction of 3D scenes, multibody motion segmentation, 3D shape inference, errors-in-variables estimation, probabilistic tracking, information fusion, optical ?owcomputation, learningfornonstationaryvideodata, noveltydetectionin- namic backgrounds, background modeling, grouping using feature uncertainty, and crowd segmentation from video. We wish to thank the authors of all submitted papers for their interest in the workshop.Wealsowishtothankthemembersofourprogramcommitteeandthe external reviewers for their commitment of time and e?ort in providing valuable recommendations for each submission. We are thankful to Vaclav Hlavac, the General Chair of ECCV 2004, and to Radim Sara, for the local organization of the workshop and registration management. We hope you will ?nd these proceedings both inspiring and of high scienti?c qualit

Linear algebra is one of the most basic foundations of a wide range of scientific domains, and most textbooks of linear algebra are written by mathematicians. However, this book is specifically intended to students and researchers of pattern information processing, analyzing signals such as images and exploring computer vision and computer graphics applications. The author himself is a researcher of this domain. Such pattern information processing deals with a large amount of data, which are represented by high-dimensional vectors and matrices. There, the role of linear algebra is not merely numerical computation of large-scale vectors and matrices. In fact, data processing is usually accompanied with "geometric interpretation." For example, we can think of one data set being "orthogonal" to another and define a "distance" between them or invoke geometric relationships such as "projecting" some data onto some space. Such geometric concepts not only help us mentally visualize abstract high-dimensional spaces in intuitive terms but also lead us to find what kind of processing is appropriate for what kind of goals. First, we take up the concept of "projection" of linear spaces and describe "spectral decomposition," "singular value decomposition," and "pseudoinverse" in terms of projection. As their applications, we discuss least-squares solutions of simultaneous linear equations and covariance matrices of probability distributions of vector random variables that are not necessarily positive definite. We also discuss fitting subspaces to point data and factorizing matrices in high dimensions in relation to motion image analysis. Finally, we introduce a computer vision application of reconstructing the 3D location of a point from three camera views to illustrate the role of linear algebra in dealing with data with noise. This book is expected to help students and researchers of pattern information processing deepen the geometric understanding of linear algebra.

Machine vision is the study of how to build intelligent machines which can understand the environment by vision. Among many existing books on this subject, this book is unique in that the entire volume is devoted to computational problems, which most books so not deal with. One of the main subjects of this book is the mathematics underlying all vision problems - projective geometry, in particular. Since projective geometry has been developed by mathematicians without any regard to machine vision applications, our first attempt is to `tune' it into the form applicable to machine vision problems. The resulting formulation is termed computational projective geometry and applied to 3-D shape analysis, camera calibration, road scene analysis, 3-D motion analysis, optical flow analysis, and conic image analysis.

A salient characteristic of machine vision problems is that data are not necessarily accurate. Hence, computational procedures defined by using exact relationships may break down if blindly applied to inaccurate data. In this book, special emphasis is put on robustness, which means that the computed result is not only exact when the data are accurate but also is expected to give a good approximation in the prescence of noise. The analysis of how the computation is affected by the inaccuracy of the data is also crucial. Statistical analysis of computations based on image data is also one of the main subjects of this book.

This text discusses the mathematical foundations of statistical inference for building 3-dimensional models from image and sensor data that contain noise -- a task involving autonomous robots guided by video cameras and sensors. The text employs a theoretical accuracy for the optimization procedure, which maximizes the reliability of estimations based on noise data. 1996 edition.