Multiple View Geometry in Computer VisionA basic problem in computer vision is to understand the structure of a real world scene given several images of it. Techniques for solving this problem are taken from projective geometry and photogrammetry. Here, the authors cover the geometric principles and their algebraic representation in terms of camera projection matrices, the fundamental matrix and the trifocal tensor. The theory and methods of computation of these entities are discussed with real examples, as is their use in the reconstruction of scenes from multiple images. The new edition features an extended introduction covering the key ideas in the book (which itself has been updated with additional examples and appendices) and significant new results which have appeared since the first edition. Comprehensive background material is provided, so readers familiar with linear algebra and basic numerical methods can understand the projective geometry and estimation algorithms presented, and implement the algorithms directly from the book. 
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User Review  ndwork  Overstock.comThe book is phenomenol. It explains how to accomplish things that look like magic. One can create a 3d geometry of a scene with only three images of that scene. It's amazing. The delivery of the ... Read full review
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Contents
Projective Geometry and Transformations of 2D  25 
22 The 2D projective plane  26 
23 Projective transformations  32 
24 A hierarchy of transformations  37 
25 The projective geometry of ID  44 
26 Topology of the projective plane  46 
27 Recovery of affine and metric properties from images  47 
28 More properties of conics  58 
126 Probability distribution of the estimated 3D point  321 
128 Closure  323 
Scene planes and homographies  325 
131 Homographies given the plane and vice versa  326 
132 Plane induced homographies given F and image correspondences  329 
133 Computing F given the homography induced by a plane  334 
134 The infinite homography H  338 
135 Closure  340 
29 Fixed points and lines  61 
210 Closure  62 
Projective Geometry and Transformations of 3D  65 
32 Representing and transforming planes lines and quadrics  66 
33 Twisted cubics  75 
34 The hierarchy of transformations  77 
35 The plane at infinity  79 
36 The absolute conic  81 
37 The absolute dual quadric  83 
38 Closure  85 
Estimation 2D Projective Transformations  87 
41 The Direct Linear Transformation DLT algorithm  88 
42 Different cost functions  93 
43 Statistical cost functions and Maximum Likelihood estimation  102 
44 Transformation invariance and normalization  104 
45 Iterative minimization methods  110 
46 Experimental comparison of the algorithms  115 
47 Robust estimation  116 
48 Automatic computation of a homography  123 
49 Closure  127 
Algorithm Evaluation and Error Analysis  132 
52 Covariance of the estimated transformation  138 
53 Monte Carlo estimation of covariance  149 
54 Closure  150 
Camera Geometry and Single View Geometry  151 
Outline  152 
Camera Models  153 
62 The projective camera  158 
63 Cameras at infinity  166 
64 Other camera models  174 
65 Closure  176 
Computation of the Camera Matrix P  178 
72 Geometric error  180 
73 Restricted camera estimation  184 
74 Radial distortion  189 
75 Closure  193 
More Single View Geometry  195 
82 Images of smooth surfaces  200 
83 Action of a projective camera on quadrics  201 
84 The importance of the camera centre  202 
85 Camera calibration and the image of the absolute conic  208 
86 Vanishing points and vanishing lines  213 
87 Affine 3D measurements and reconstruction  220 
88 Determining camera calibration K from a single view  223 
89 Single view reconstruction  229 
810 The calibrating conic  231 
811 Closure  233 
TwoView Geometry  237 
Outline  238 
Epipolar Geometry and the Fundamental Matrix  239 
92 The fundamental matrix F  241 
93 Fundamental matrices arising from special motions  247 
94 Geometric representation of the fundamental matrix  250 
95 Retrieving the camera matrices  253 
96 The essential matrix  257 
97 Closure  259 
3D Reconstruction of Cameras and Structure  262 
102 Reconstruction ambiguity  264 
103 The projective reconstruction theorem  266 
104 Stratified reconstruction  267 
105 Direct reconstruction using ground truth  275 
106 Closure  276 
Computation of the Fundamental Matrix F  279 
112 The normalized 8point algorithm  281 
113 The algebraic minimization algorithm  282 
114 Geometric distance  284 
115 Experimental evaluation of the algorithms  288 
116 Automatic computation of F  290 
117 Special cases of Fcomputation  293 
118 Correspondence of other entities  294 
119 Degeneracies  295 
1110 A geometric interpretation of Fcomputation  297 
1111 The envelope of epipolar lines  298 
1112 Image rectification  302 
1113 Closure  308 
Structure Computation  310 
122 Linear triangulation methods  312 
123 Geometric error cost function  313 
124 Sampson approximation firstorder geometric correction  314 
125 An optimal solution  315 
Affine Epipolar Geometry  344 
142 The affine fundamental matrix  345 
143 Estimating F from image point correspondences  347 
144 Triangulation  353 
146 Necker reversal and the basrelief ambiguity  355 
147 Computing the motion  357 
148 Closure  360 
ThreeView Geometry  363 
Outline  364 
The Trifocal Tensor  365 
152 The trifocal tensor and tensor notation  376 
153 Transfer  379 
154 The fundamental matrices for three views  383 
155 Closure  387 
Computation of the Trifocal Tensor T  391 
162 The normalized linear algorithm  393 
163 The algebraic minimization algorithm  395 
164 Geometric distance  396 
165 Experimental evaluation of the algorithms  399 
166 Automatic computation of T  400 
167 Special cases of Tcomputation  404 
168 Closure  406 
NView Geometry  409 
Outline  410 
NLinearities and Multiple View Tensors  411 
172 Trilinear relations  414 
173 Quadrilinear relations  418 
17 A Intersections of four planes  421 
175 Counting arguments  422 
176 Number of independent equations  428 
177 Choosing equations  431 
178 Closure  432 
NView Computational Methods  434 
182 Affine reconstruction the factorization algorithm  436 
183 Nonrigid factorization  440 
184 Projective factorization  444 
185 Projective reconstruction using planes  447 
186 Reconstruction from sequences  452 
187 Closure  456 
AutoCalibration  458 
192 Algebraic framework and problem statement  459 
193 Calibration using the absolute dual quadric  462 
194 The Kruppa equations  469 
195 A stratified solution  473 
196 Calibration from rotating cameras  481 
197 Autocalibration from planes  485 
198 Planar motion  486 
199 Single axis rotation turntable motion  490 
1910 Autocalibration of a stereo rig  493 
1911 Closure  497 
Duality  502 
202 Reduced reconstruction  508 
203 Closure  513 
Cheirality  515 
212 Front and back of a camera  518 
213 Threedimensional point sets  519 
214 Obtaining a quasiaffine reconstruction  520 
215 Effect of transformations on cheirality  521 
216 Orientation  523 
217 The cheiral inequalities  525 
218 Which points are visible in a third view  528 
219 Which points are in front of which  530 
2110 Closure  531 
Degenerate Configurations  533 
222 Degeneracies in two views  539 
223 CarlssonWeinshall duality  546 
224 Threeview critical configurations  553 
225 Closure  558 
Appendices  561 
Tensor Notation  562 
Gaussian Normal and 𝓧˛ Distributions  565 
Parameter Estimation  568 
Matrix Properties and Decompositions  578 
Leastsquares Minimization  588 
Iterative Estimation Methods  597 
Some Special Plane Projective Transformations  628 
634  
646  
Common terms and phrases
3space 3D points absolute conic affine camera affine reconstruction affine transformation algebraic algorithm ambiguity autocalibration axis backprojected calibration camera centre camera matrices chapter circular points columns computed configuration consider constraints coordinate frame corresponding points cost function covariance matrix defined degrees of freedom determined distance entries epipolar geometry epipolar line epipole equivalent Euclidean Euclidean transformation example fundamental matrix Gaussian geometric error given homogeneous homography image plane image points inliers internal parameters intersection iterative linear mapping matches matrix F measured method metric reconstruction minimization motion normal nullspace obtained orthogonal outliers parallel pixels planar plane at infinity plane TT point correspondences problem projective camera projective geometry projective reconstruction projective transformation quadric rank RANSAC ratio rays represented result rotation rows scale scene screw axis set of points shown in figure solution space three views transformation H trifocal tensor twisted cubic vanishing line vanishing point vector zero
Popular passages
Page 641  CJ Poelman and T. Kanade. A paraperspective factorization method for shape and motion recovery.
Page 637  Selfcalibration of a ID projective camera and its application to the selfcalibration of a 2D projective camera.
Page 637  RI Hartley. Lines and points in three views and the trifocal tensor.
Page 637  Lee, K. Ottenberg, and M. Nolle, "Analysis and solutions of the three point perspective pose estimation problem," in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp.
Page 634  Chengke, editors, EuropeChina workshop on Geometrical Modelling and Invariants for Computer Vision, pages 214221. Xidan University Press, Xi'an, China, 1995. [4] F.
Page 638  Hartley. In defense of the eightpoint algorithm. IEEE Transactions on Pattern Analysis and Machine Intelligence, 19(6):580593, June 1997.
Page 638  A. Heyden. Projective structure and motion from image sequences using subspace methods. In Scandinavian Conference on Image Analysis.
Page 636  Vision, pages 269275, 1995. 3. R. Cipolla and A. Blake. Surface shape from the deformation of apparent contours. Int. Journal of Computer Vision, 9(2):83112, 1992.