An Invitation to 3-D Vision From Images to Geometric Models

This book is intended to give students at the advanced undergraduate or introduc­ tory graduate level, and researchers in computer vision, robotics and computer graphics, a self-contained introduction to the geometry of three-dimensional (3- D) vision. This is the study of the reconstruction of 3-D models of objects from a collection of 2-D images. An essential prerequisite for this book is a course in linear algebra at the advanced undergraduate level. Background knowledge in rigid-body motion, estimation and optimization will certainly improve the reader's appreciation of the material but is not critical since the first few chapters and the appendices provide a review and summary of basic notions and results on these topics. Our motivation Research monographs and books on geometric approaches to computer vision have been published recently in two batches: The first was in the mid 1990s with books on the geometry of two views, see e. g. [Faugeras, 1993, Kanatani, 1993b, Maybank, 1993, Weng et aI. , 1993b]. The second was more recent with books fo­ cusing on the geometry of multiple views, see e. g. [Hartley and Zisserman, 2000] and [Faugeras and Luong, 2001] as well as a more comprehensive book on computer vision [Forsyth and Ponce, 2002]. We felt that the time was ripe for synthesizing the material in a unified framework so as to provide a self-contained exposition of this subject, which can be used both for pedagogical purposes and by practitioners interested in this field.

1 Introduction.- 1.1 Visual perception from 2-D images to 3-D models.- 1.2 A mathematical approach.- 1.3 A historical perspective.- I Introductory Material.- 2 Representation of a Three-Dimensional Moving Scene.- 3 Image Formation.- 4 Image Primitives and Correspondence.- II Geometry of Two Views.- 5 Reconstruction from Two Calibrated Views.- 6 Reconstruction from Two Uncalibrated Views.- 7 Estimation of Multiple Motions from Two Views.- III Geometry of Multiple Views.- 8 Multiple-View Geometry of Points and Lines.- 9 Extension to General Incidence Relations.- 10 Geometry and Reconstruction from Symmetry.- IV Applications.- 11 Step-by-Step Building of a 3-D Model from Images.- 12 Visual Feedback.- V Appendices.- A Basic Facts from Linear Algebra.- A.1 Basic notions associated with a linear space.- A.1.1 Linear independence and change of basis.- A.1.2 Inner product and orthogonality.- A.1.3 Kronecker product and stack of matrices.- A.2 Linear transformations and matrix groups.- A.3 Gram-Schmidt and the QR decomposition.- A.4 Range, null space (kernel), rank and eigenvectors of a matrix.- A.5 Symmetric matrices and skew-symmetric matrices.- A.6 Lyapunov map and Lyapunov equation.- A.7 The singular value decomposition (SVD).- A.7.1 Algebraic derivation.- A.7.2 Geometric interpretation.- A.7.3 Some properties of the SVD.- B Least-Variance Estimation and Filtering.- B.1 Least-variance estimators of random vectors.- B.1.1 Projections onto the range of a random vector.- B.1.2 Solution for the linear (scalar) estimator.- B.1.3 Affine least-variance estimator.- B.1.4 Properties and interpretations of the least-variance estimator.- B.2 The Kalman-Bucy filter.- B.2.1 Linear Gaussian dynamical models.- B.2.2 A little intuition.- B.2.3 Observability.- B.2.4 Derivation of the Kalmanfilter.- B.3 The extended Kalman filter.- C Basic Facts from Nonlinear Optimization.- C.1 Unconstrained optimization: gradient-based methods.- C.1.1 Optimality conditions.- C.1.2 Algorithms.- C.2 Constrained optimization: Lagrange multiplier method..- C.2.1 Optimality conditions.- C.2.2 Algorithms.- References.- Glossary of Notation.