An Invitation to 3-D Vision

Preface 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 2.1 Three-dimensional Euclidean space 2.2 Rigid body motion 2.3 Rotational motion and its representations 2.4 Rigid body motion and its representations 2.5 Coordinate and velocity transformations 2.6 Summary 2.7 Exercises 2.A Quaternions and Euler angles for rotations 3 Image formation 3.1 Representation of images 3.2 Lenses, light, and basic photometry 3.3 A geometric model of image formation 3.4 Summary 3.5 Exercises 3.A Basic photometry with light sources and surfaces 3.B Image formation in the language of projective geometry 4 Image primitives and correspondence 4.1 Correspondence of geometric features 4.2 Local deformation models 4.3 Matching point features 4.4 Tracking line features 4.5 Summary 4.6 Exercises 4.A Computing image gradients II Geometry of two views 5 Reconstruction from two calibrated views 5.1 Epipolar geometry 5.2 Basic reconstruction algorithms 5.3 Planar scenes and homography 5.4 Continuous motion case 5.5 Summary 5.6 Exercises 5.A Optimization subject to epipolar constraint 6 Reconstruction from two uncalibrated views 6.1 Uncalibrated camera or distorted space? 6.2 Uncalibrated epipolar geometry 6.3 Ambiguities and constraints in image formation 6.4 Stratified reconstruction 6.5 Calibration with scene knowledge 6.6 Dinner with Kruppa 6.7 Summary 6.8 Exercises 6.A From images to Fundamental matrices 6.B Properties of Kruppa's equations 7 Segmentation of multiple moving objects from two views 7.1 Multibody epipolar constraint and Fundamental matrix 7.2 A rank condition for the number of motions 7.3 Geometric properties of the multibody Fundamental matrix 7.4 Multibody motion estimation and segmentation 7.5 Multibody structure from motion 7.6 Summary 7.7 Exercises 7.A Homogeneous polynomial factorization III Geometry of multiple views 8 Multiple-view geometry of points and lines 8.1 Basic notation for (pre-)image and co-image of points and lines 8.2 Preliminary rank conditions of multiple images 8.3 Geometry of point features 8.4 Geometry of line features 8.5 Uncalibrated factorization and stratification 8.6 Summary 8.7 Exercises 8.A Proof for the properties of bilinear and trilinear constraints 9 Extension to general incidence relations 9.1 Incidence relations among points, lines, and planes 9.2 Rank conditions for incidence relations 9.3 Universal rank conditions on the multiple-view matrix 9.4 Summary 9.5 Exercises 9.A Incidence relations and rank conditions 9.B Beyond constraints among four views 9.C Examples for geometric interpretation of the rank conditions 10 Geometry and reconstruction from symmetry 10.1 Symmetry and multiple-view geometry 10.2 Symmetry-based 3-D reconstruction 10.3 Camera calibration from symmetry 10.4 Summary 10.5 Exercises IV Applications 11 Step-by-step building of a 3-D model from images 11.1 Feature selection 11.2 Feature correspondence 11.3 Projective reconstruction 11.4 Upgrade from projective to Euclidean reconstruction 11.5 Reconstruction with partial scene knowledge 11.6 Calibrated reconstruction (reality check) 11.7 Visualization 11.8 Additional techniques for image-based modeling (discussion) 12 Visual feedback 12.1 Motion and shape estimation as a filtering problem 12.2 Virtual insertion in live video 12.3 Visual feedback for autonomous car driving 12.4 Visual feedback for autonomous helicopter landing V Appendices Appendix A Basic facts from linear algebra A.1 Basic notions associated to a linear space A.2 Linear transformations and matrix groups A.3 Gram-Schmidt procedure and $QR$ decomposition A.4 Range, null, 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) Appendix B Least-variance estimation and filtering B.1 Least-variance estimators of random vectors B.2 The Kalman-Bucy filter B.3 The extended Kalman filter Appendix C Basic facts from nonlinear optimization C.1 Unconstrained optimization: gradient based methods C.2 Constrained optimization: Lagrange multiplier method References Glossary of notation Index