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in computervision Richard Hartley and Andrew Zisserman CAMBRIDGE

ENGINEERING LIBRARY Multiple View Geometry in Computer Vision Second Edition Richard Hartley Australian National University, Canberra, Australia Andrew Zisserman University of Oxford, UK C A M B R I D GE UNIVERSITY PRESS

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge, CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 477 Wiiiiamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarcon 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org © Cambridge University Press 2000, 2003 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First Published 2000 Reprinted 2001, 2002 Second Edition 2003 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication data ISBN 0521 54051 8 hardback This book led us intc

Dedication This book is dedicated to Joe Mundy whose vision and constant search for new ideas led us into this field.

Contents Foreword Preface 1 Introduction - the ubiquitous projective geometry Camera projections Reconstruction from more than one view Three-view geometry Four view geometry and n-view reconstruction Transfer Euclidean reconstruction Introduction - a Tour of Multiple View Geometry 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Auto-calibration 1.9 The reward 1: 3D graphical models 1.10 The reward II: video augmentation -•&' page xi xiii 1 1 6 10 12 13 14 16 17 18 19 PART 0: The Background: Projective Geometry, Transformations and Esti mation Outline 2 Projective Geometry and Transformations of 2D Planar geometry The 2D projective plane Projective transformations 2.1 2.2 2.3 2.4 A hierarchy of transformations 2.5 The projective geometry of ID Topology of the projective plane 2.6 2.7 Recovery of affine and metric properties from images 2.8 More properties of conies 2.9 2.10 Closure Fixed points and lines 3 Projective Geometry and Transformations of 3D 3.1 3.2 Points and projective transformations Representing and transforming planes, lines and quadrics v 23 24 25 25 26 32 37 44 46 47 58 61 62 65 65 66

Contents 3.3 3.4 3.5 3.6 3.7 3.8 Twisted cubics The hierarchy of transformations The plane at infinity The absolute conic The absolute dual quadric Closure 4 Estimation - 2D Projective Transformations The Direct Linear Transformation (DLT) algorithm 4.1 4.2 Different cost functions 4.3 4.4 4.5 4.6 4.7 4.8 Automatic computation of a homography 4.9 Statistical cost functions and Maximum Likelihood estimation Transformation invariance and normalization Iterative minimization methods Experimental comparison of the algorithms Robust estimation Closure 5 Algorithm Evaluation and Error Analysis Bounds on performance Covariance of the estimated transformation 5.1 5.2 5.3 Monte Carlo estimation of covariance 5.4 Closure PART I: Camera Geometry and Single View Geometry Outline 6 Camera Models Finite cameras The projective camera Cameras at infinity 6.1 6.2 6.3 6.4 Other camera models 6.5 Closure 7 Computation of the Camera Matrix P 7.1 Basic equations 7.2 Geometric error 7.3 7.4 7.5 Restricted camera estimation Radial distortion Closure 8 More Single View Geometry Images of smooth surfaces 8.1 Action of a projective camera on planes, lines, and conies 8.2 8.3 Action of a projective camera on quadrics 8.4 8.5 The importance of the camera centre Camera calibration and the image of the absolute conic 75 77 79 81 83 85 87 88 93 102 104 110 115 116 123 127 132 132 138 149 150 151 152 153 153 158 166 174 176 178 178 180 184 189 193 195 195 200 201 202 208

Contents 8.6 Vanishing points and vanishing lines 8.7 Affine 3D measurements and reconstruction 8.8 Determining camera calibration K from a single view 8.9 8.10 The calibrating conic 8.11 Closure Single view reconstruction PART II: Two-View Geometry Outline 9 Epipolar Geometry and the Fundamental Matrix 9.1 Epipolar geometry 9.2 The fundamental matrix F 9.3 Fundamental matrices arising from special motions 9.4 Geometric representation of the fundamental matrix 9.5 9.6 9.7 Retrieving the camera matrices The essential matrix Closure 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 10.5 Direct reconstruction - using ground truth 10.6 Closure 11 Computation of the Fundamental Matrix F 11.1 Basic equations 11.2 The normalized 8-point algorithm 11.3 The algebraic minimization algorithm 11.4 Geometric distance 11.5 Experimental evaluation of the algorithms 11.6 Automatic computation of F 11.7 Special cases of F-computation 11.8 Correspondence of other entities 11.9 Degeneracies 11.10 A geometric interpretation of F-computation 11.11 The envelope of epipolar lines 11.12 Image rectification 11.13 Closure 12 Structure Computation 12.1 Problem statement 12.2 Linear triangulation methods 12.3 Geometric error cost function 12.4 Sampson approximation (first-order geometric correction) VI] 213 220 223 229 231 233 237 238 239 239 241 247 250 253 257 259 262 262 264 266 267 275 276 279 279 281 282 284 288 290 293 294 295 297 298 302 308 310 310 312 313 314

Vlll Contents 12.5 An optimal solution 12.6 Probability distribution of the estimated 3D point 12.7 Line reconstruction 12.8 Closure 13 Scene planes and homographies 13.1 Homographies given the plane and vice versa 13.2 Plane induced homographies given F and image correspondences 13.3 Computing F given the homography induced by a plane 13.4 The infinite homography HQO 13.5 Closure 14 Affine Epipolar Geometry 14.1 Affine epipolar geometry 14.2 The affine fundamental matrix 14.3 Estimating FA from image point correspondences 14.4 Triangulation 14.5 Affine reconstruction 14.6 Necker reversal and the bas-relief ambiguity 14.7 Computing the motion 14.8 Closure PART III: Three-View Geometry Outline 15 The Trifocal Tensor 15.1 The geometric basis for the trifocal tensor 15.2 The trifocal tensor and tensor notation 15.3 Transfer 15.4 The fundamental matrices for three views 15.5 Closure 16 Computation of the Trifocal Tensor T 16.1 Basic equations 16.2 The normalized linear algorithm 16.3 The algebraic minimization algorithm 16.4 Geometric distance 16.5 Experimental evaluation of the algorithms 16.6 Automatic computation of T 16.7 Special cases of T-computation 16.8 Closure PART IV: N-View Geometry Outline 17 ^-Linearities and Multiple View Tensors 17.1 Bilinear relations 17.2 Trilinear relations 315 321 321 323 325 326 329 334 338 340 344 344 345 347 353 353 355 357 360 363 364 365 365 376 379 383 387 391 391 393 395 396 399 400 404 406 409 410 411 411 414

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