Epipolar Geometry in Stereo, Motion and Object Recognition

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1 Epipolar Geometry in Stereo, Motion and Object Recognition A Unified Approach by GangXu Department of Computer Science, Ritsumeikan University, Kusatsu, Japan and Zhengyou Zhang INRIA Sophia-Antipolis, Sophia-Antipolis, France KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON

2 CONTENTS Forward by Olivier Faugeras Forward by Saburo Tsuji PREFACE xiii xv xvii 1 INTRODUCTION l 1.1 Vision Research Multiple View Problems in Vision Organization of this Book Notation 5 2 CAMERA MODELS AND EPIPOLAR GEOMETRY Modeling Cameras Pinhole Camera and Perspective Projection Perspective Projection Matrix and Extrinsic Parameters Intrinsic Parameters and Normalized Camera The General Form of Perspective Projection Matrix Perspective Approximations Orthographic and Weak Perspective Projections Paraperspective Projection Affine Cameras Epipolar Geometry Under Full Perspective Projection Concepts in Epipolar Geometry Working with Normalized Image Coordinates 29 vn

3 viii EPIPOLAR GEOMETRY Working with Pixel Image Coordinates Working with Camera Perspective Projection Matrices Fundamental Matrix and Epipolar Transformation A General Form of Epipolar Equation for Any Projection Model Intersecting Two Optical Rays The Full Perspective Projection Case Epipolar Geometry Under Orthographic, Weak Perspective, Paraperspective and General AfHne Projections Orthographic and Weak Perspective Projections Paraperspective Projection The General Affine Camera Epipolar Geometry Between Two Images with Lens Distortion Camera Distortion Modelling Computating Distorted Coordinates from Ideal Ones Epipolar Constraint Between Two Images with Distortion Summary 70 2.A Appendix 71 2.A.I Thin and Thick Lens Camera Models 71 2.A.2 Inverse and Pseudoinverse Matrices 75 RECOVERY OF EPIPOLAR GEOMETRY FROM POINTS Determining Fundamental Matrix Under Full Perspective Projection Exact Solution with 7 Point Matches Analytic Method with 8 or More Point Matches Analytic Method with Rank-2 Constraint Nonlinear Method Minimizing Distances of Points to Epipolar Lines Nonlinear Method Minimizing Distances Between Observation and Reprojection Robust Methods 93

4 Contents ix Characterizing the Uncertainty of Fundamental Matrix An Example of Fundamental Matrix Estimation Denning Epipolar Bands by Using the Estimated Uncertainty Determining Fundamental Matrix for Affine Cameras Exact Solution with 4 Point Matches Analytic Method with More than 4 Point Matches Minimizing Distances of Points to Epipolar Lines Minimizing Distances Between Observation and Reprojection An Example of Affine Fundamental Matrix Estimation Charactering the Uncertainty of Affine Fundamental Matrix Determining Motion Equation in the 2D Affine Motion Case Recovery of Multiple Epipolar Equations by Clustering The Problem in Stereo, Motion and Object Recognition Definitions and Assumptions Error Analysis of Motion Parameters Estimating Covariance Matrix The Maximal Likelihood Approach Robust Estimation Using Exponential of Gaussian Distribution A Clustering Algorithm An Example of Clustering Projective Reconstruction Projective Structure from Two Uncalibrated Images Computing Camera Projection Matrices Reconstruction Techniques Use of Projective Structure Affine Reconstruction Affine Structure from Two Uncalibrated Affine Views Relation to Previous Work Experimental Results 159

5 EPIPOLAR GEOMETRY 3.6 Summary A Appendix A.1 Approximate Estimation of Fundamental Matrix from General Matrix A.2 Estimation of Affine Transformation 165 RECOVERY OF EPIPOLAR GEOMETRY FROM LINE SEGMENTS OR LINES Line Segments or Straight Lines Solving Motion Using Line Segments Between Two Views Overlap of Two Corresponding Line Segments Estimating Motion by Maximizing Overlap Implementation Details Reconstructing 3D Line Segments Experimental Results Discussions Determining Epipolar Geometry of Three Views Trifocal Constraints for Point Matches Trifocal Constraints for Line Correspondences Linear Estimation of K, L, and M Using Points and Lines Determining Camera Projection Matrices Image Transfer Summary 204 REDEFINING STEREO, MOTION AND OBJECT RECOGNITION VIA EPIPOLAR GEOMETRY Conventional Approaches to Stereo, Motion and Object Recognition Stereo Motion Object Recognition Correspondence in Stereo, Motion and Object Recognition as ID Search Stereo Matching 209

6 Contents xi Motion Correspondence and Segmentation D Object Recognition and Localization Disparity and Spatial Disparity Space Disparity under Full Perspective Projection in the Parallel Camera Case Disparity under Full Perspective Projection in the General Case Disparity under Weak Perspective and Paraperspective Projections Spatial Disparity Space and Smoothness Correspondence as Search for Surfaces and Contours in SDS Summary IMAGE MATCHING AND UNCALIBRATED STEREO Finding Match Candidates by Correlation Extracting Points of Interest Matching Through Correlation Rotating Correlation Windows Unique Correspondence by Relaxation and Robust Estimation of Epipolar Geometry for Perspective Images Measure of the Support for a Match Candidate Relaxation Process Detection of False Matches Unique Correspondence by Robust Estimation of Epipolar Geometry for Affine Images Discarding Unlikely Match Candidates Generating Local Groups of Point Matches for Clustering Image Matching with the Recovered Epipolar Geometry A Simple Implementation for Matching Corners A Simple Implementation for Matching Edges An Example of Matching Uncalibrated Perspective Images An Example of Matching Uncalibrated Affine Images Summary 243

7 xii EPIPOLAR GEOMETRY 7 MULTIPLE RIGID MOTIONS: CORRESPONDENCE AND SEGMENTATION Problems of Multiple Rigid Motions Determining Epipolar Equations for Multiple Rigid Motions The Algorithm Experimental Results Matching and Segmenting Edge Images with Known Epipolar Equations Representing the Problem in ESDS A Support Measure for Selection from Multiple Candidates Experimental Results Transparent Multiple Rigid Motions Summary and Discussions Appendix: SVD Algorithm for Structure and Motion Recovery with Known Epipolar Equations D OBJECT RECOGNITION AND LOCALIZATION Introduction: 3D Model vs 2D Model, and Single Model View vs. Multiple Model Views Recognition and Localization with Single Model View with Model Views Is a Single View Sufficient for 3D Object Recognition? Matching Model View with Input View as Uncalibrated Stereo Images An Example Recognition and Localization with Multiple Model Views Intersection of Epipolar Lines Basis Vectors Determining Coefficients An Example Summary CONCLUDING REMARKS 291 REFERENCES 293 INDEX 309

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