Kenichi Kanatani

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

Hamilton, Grassmann, and Clifford for Computer Vision and Graphics

Understanding Geometric Algebra

Rudolf Mester

Dorin Comaniciu

David Suter

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 quality.

ECCV 2004 Workshop SMVP 2004, Prague, Czech Republic, May 16, 2004, Revised Selected Papers

Statistical Methods in Video Processing

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.

Kenichi Kanatani

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