Cover -- Title -- Copyright -- Dedication -- Contents -- Foreword -- Preface -- 1 Introduction - a Tour of Multiple View Geometry -- 1.1 Introduction - the ubiquitous projective geometry -- 1.1.1 Affine and Euclidean Geometry -- 1.2 Camera projections -- 1.3 Reconstruction from more than one view -- 1.4 Three-view geometry -- 1.5 Four view geometry and n-view reconstruction -- 1.6 Transfer -- 1.7 Euclidean reconstruction -- 1.8 Auto-calibration -- 1.9 The reward I : 3D graphical models -- 1.10 The reward II: video augmentation -- Part 0 The Background: Projective Geometry, Transformations and Estimation -- Outline -- 2 Projective Geometry and Transformations of 2D -- 2.1 Planar geometry -- 2.2 The 2D projective plane -- 2.2.1 Points and lines -- 2.2.2 Ideal points and the line at infinity -- 2.2.3 Conics and dual conics -- 2.3 Projective transformations -- 2.3.1 Transformations of lines and conics -- 2.4 A hierarchy of transformations -- 2.4.1 Class I: Isometries -- 2.4.2 Class II: Similarity transformations -- 2.4.3 Class III: Affine transformations -- 2.4.4 Class IV: Projective transformations -- 2.4.5 Summary and comparison -- 2.4.6 Decomposition of a projective transformation -- 2.4.7 The number of invariants -- 2.5 The projective geometry of 1D -- 2.6 Topology of the projective plane -- 2.7 Recovery of affine and metric properties from images -- 2.7.1 The line at infinity -- 2.7.2 Recovery of affine properties from images -- 2.7.3 The circular points and their dual -- 2.7.4 Angles on the projective plane -- 2.7.5 Recovery of metric properties from images -- 2.8 More properties of conics -- 2.8.1 The pole-polar relationship -- 2.8.2 Classification of conics -- 2.9 Fixed points and lines -- 2.10 Closure -- 2.10.1 The literature -- 2.10.2 Notes and exercises -- 3 Projective Geometry and Transformations of 3D.

3.1 Points and projective transformations -- 3.2 Representing and transforming planes, lines and quadrics -- 3.2.1 Planes -- 3.2.2 Lines -- 3.2.3 Quadrics and dual quadrics -- 3.2.4 Classification of quadrics -- 3.3 Twisted cubics -- 3.4 The hierarchy of transformations -- 3.4.1 The screw decomposition -- 3.5 The plane at infinity -- 3.6 The absolute conic -- 3.7 The absolute dual quadric -- 3.8 Closure -- 3.8.1 The literature -- 3.8.2 Notes and exercises -- 4 Estimation - 2D Projective Transformations -- 4.1 The Direct Linear Transformation (DLT) algorithm -- 4.1.1 Over-determined solution -- 4.1.2 Inhomogeneous solution -- 4.1.3 Degenerate configurations -- 4.1.4 Solutions from lines and other entities -- 4.2 Different cost functions -- 4.2.1 Algebraic distance -- 4.2.2 Geometric distance -- 4.2.3 Reprojection error - both images -- 4.2.4 Comparison of geometric and algebraic distance -- 4.2.5 Geometric interpretation of reprojection error -- 4.2.6 Sampson error -- Linear cost function -- 4.2.7 Another geometric interpretation -- 4.3 Statistical cost functions and Maximum Likelihood estimation -- 4.4 Transformation invariance and normalization -- 4.4.1 Invariance to image coordinate transformations -- 4.4.2 Non-invariance of the DLT algorithm -- 4.4.3 Invariance of geometric error -- 4.4.4 Normalizing transformations -- 4.5 Iterative minimization methods -- A word about parametrization -- Function specification -- Initialization -- Iteration methods -- 4.6 Experimental comparison of the algorithms -- 4.7 Robust estimation -- 4.7.1 RANSAC -- 4.7.2 Robust Maximum Likelihood estimation -- 4.7.3 Other robust algorithms -- 4.8 Automatic computation of a homography -- 4.8.1 Application domain -- 4.8.2 Implementation and run details -- 4.9 Closure -- 4.9.1 The literature -- 4.9.2 Notes and exercises -- 5 Algorithm Evaluation and Error Analysis.

5.1 Bounds on performance -- 5.1.1 Error in one image -- 5.1.2 Error in both images -- 5.1.3 Optimal estimators (MLE) -- 5.1.4 Determining the correct convergence of an algorithm -- 5.2 Covariance of the estimated transformation -- 5.2.1 Forward propagation of covariance -- 5.2.2 Backward propagation of covariance -- 5.2.3 Over-parametrization -- 5.2.4 Application and examples -- 5.2.5 Error in both images -- 5.2.6 Using the covariance matrix in point transfer -- 5.3 Monte Carlo estimation of covariance -- 5.4 Closure -- 5.4.1 The literature -- 5.4.2 Notes and exercises -- Part I Camera Geometry and Single View Geometry -- Outline -- 6 Camera Models -- 6.1 Finite cameras -- 6.2 The projective camera -- 6.2.1 Camera anatomy -- 6.2.2 Action of a projective camera on points -- 6.2.3 Depth of points -- 6.2.4 Decomposition of the camera matrix -- 6.2.5 Euclidean vs projective spaces -- 6.3 Cameras at infinity -- 6.3.1 Affine cameras -- 6.3.2 Error in employing an affine camera -- 6.3.3 Decomposition of P… -- 6.3.4 A hierarchy of affine cameras -- 6.3.5 More properties of the affine camera -- 6.3.6 General cameras at infinity -- 6.4 Other camera models -- 6.4.1 Pushbroom cameras -- 6.4.2 Line cameras -- 6.5 Closure -- 6.5.1 The literature -- 6.5.2 Notes and exercises -- 7 Computation of the Camera Matrix P -- 7.1 Basic equations -- 7.2 Geometric error -- 7.2.1 Geometric interpretation of algebraic error -- 7.2.2 Estimation of an affine camera -- 7.3 Restricted camera estimation -- Experimental evaluation -- 7.4 Radial distortion -- 7.5 Closure -- 7.5.1 The literature -- 7.5.2 Notes and exercises -- 8 More Single View Geometry -- 8.1 Action of a projective camera on planes, lines, and conics -- 8.1.1 On planes -- 8.1.2 On lines -- 8.1.3 On conics -- 8.2 Images of smooth surfaces -- 8.3 Action of a projective camera on quadrics.

8.4 The importance of the camera centre -- 8.4.1 Moving the image plane -- 8.4.2 Camera rotation -- 8.4.3 Applications and examples -- 8.4.4 Projective (reduced) notation -- 8.4.5 Moving the camera centre -- 8.5 Camera calibration and the image of the absolute conic -- 8.5.1 The image of the absolute conic -- 8.5.2 Orthogonality and Omega -- 8.6 Vanishing points and vanishing lines -- 8.6.1 Vanishing points -- A note on computing vanishing points -- 8.6.2 Vanishing lines -- Computing vanishing lines -- 8.6.3 Orthogonality relationships amongst vanishing points and lines -- 8.7 Affine 3D measurements and reconstruction -- 8.8 Determining camera calibration K from a single view -- 8.8.1 The geometry of the constraints -- 8.9 Single view reconstruction -- 8.10 The calibrating conic -- Orthogonality and the calibrating conic -- 8.11 Closure -- 8.11.1 The literature -- 8.11.2 Notes and exercises -- Part II Two-View Geometry -- Outline -- 9 Epipolar Geometry and the Fundamental Matrix -- 9.1 Epipolar geometry -- 9.2 The fundamental matrix F -- 9.2.1 Geometric derivation -- 9.2.2 Algebraic derivation -- 9.2.3 Correspondence condition -- 9.2.4 Properties of the fundamental matrix -- 9.2.5 The epipolar line homography -- 9.3 Fundamental matrices arising from special motions -- 9.3.1 Pure translation -- 9.3.2 Pure planar motion -- 9.4 Geometric representation of the fundamental matrix -- 9.4.1 Pure planar motion -- 9.5 Retrieving the camera matrices -- 9.5.1 Projective invariance and canonical cameras -- 9.5.2 Projective ambiguity of cameras given F -- 9.5.3 Canonical cameras given F -- 9.6 The essential matrix -- 9.6.1 Properties of the essential matrix -- 9.6.2 Extraction of cameras from the essential matrix -- 9.6.3 Geometrical interpretation of the four solutions -- 9.7 Closure -- 9.7.1 The literature -- 9.7.2 Notes and exercises.

10 3D Reconstruction of Cameras and Structure -- 10.1 Outline of reconstruction method -- 10.2 Reconstruction ambiguity -- 10.3 The projective reconstruction theorem -- 10.4 Stratified reconstruction -- 10.4.1 The step to affine reconstruction -- Translational motion -- Scene constraints -- The infinite homography -- One of the cameras is affine -- 10.4.2 The step to metric reconstruction -- 10.4.3 Direct metric reconstruction using Omega -- 10.5 Direct reconstruction - using ground truth -- 10.6 Closure -- 10.6.1 The literature -- 10.6.2 Notes and exercises -- 11 Computation of the Fundamental Matrix F -- 11.1 Basic equations -- 11.1.1 The singularity constraint -- 11.1.2 The minimum case - seven point correspondences -- 11.2 The normalized 8-point algorithm -- 11.3 The algebraic minimization algorithm -- 11.3.1 Iterative estimation -- 11.4 Geometric distance -- 11.4.1 The Gold Standard method -- 11.4.2 Parametrization of rank-2 matrices -- 11.4.3 First-order geometric error (Sampson distance) -- 11.5 Experimental evaluation of the algorithms -- 11.5.1 Recommendations -- 11.6 Automatic computation of F -- 11.7 Special cases of F-computation -- 11.7.1 Pure translational motion -- 11.7.2 Planar motion -- 11.7.3 The calibrated case -- 11.8 Correspondence of other entities -- 11.9 Degeneracies -- 11.9.1 Points on a ruled quadric -- 11.9.2 Points on a plane -- 11.9.3 No translation -- 11.10 A geometric interpretation of F-computation -- 11.11 The envelope of epipolar lines -- 11.11.1 Verification of epipolar line covariance -- 11.12 Image rectification -- 11.12.1 Mapping the epipole to infinity -- 11.12.2 Matching transformations -- 11.12.3 Algorithm outline -- 11.12.4 Affine rectification -- 11.13 Closure -- 11.13.1 The literature -- 11.13.2 Notes and exercises -- 12 Structure Computation -- 12.1 Problem statement -- 12.2 Linear triangulation methods.

12.3 Geometric error cost function.