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Reconstruction from Two Calibrated Views Two-View Geometry

Reconstruction from Two Calibrated Views Two-View Geometry. General Formulation. Given two views of the scene recover the unknown camera displacement and 3D scene structure. Pinhole Camera Model. 3D points Image points Perspective Projection Rigid Body Motion

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Reconstruction from Two Calibrated Views Two-View Geometry

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  1. Reconstruction from Two Calibrated ViewsTwo-View Geometry

  2. General Formulation Given two views of the scene recover the unknown camera displacement and 3D scene structure Invitation to 3D vision

  3. Pinhole Camera Model • 3D points • Image points • Perspective Projection • Rigid Body Motion • Rigid Body Motion + Projective projection Invitation to 3D vision

  4. 3D Structure and Motion Recovery Euclidean transformation measurements unknowns Find such Rotation and Translation and Depth that the reprojection error is minimized Two views ~ 200 points 6 unknowns – Motion 3 Rotation, 3 Translation - Structure 200x3 coordinates - (-) universal scale Difficult optimization problem Invitation to 3D vision

  5. Invitation to 3D vision

  6. Epipolar Geometry • Epipolar constraint • Essential matrix Invitation to 3D vision

  7. l1, l2 l1, l2 Epipolar Geometry • Epipolar lines l1, l2 • Coimages of • epipolarlines • Epipoles Image correspondences l1 l2 Invitation to 3D vision

  8. Characterization of the Essential Matrix • Essential matrix Special 3x3 matrix Theorem 1a(Essential Matrix Characterization) A non-zero matrix is an essential matrix iff its SVD: satisfies: with and and Invitation to 3D vision

  9. Estimating the Essential Matrix • Estimate Essential matrix • Decompose Essential matrix into • Given n pairs of image correspondences: • Find such Rotationand Translationthat the epipolar error is minimized • Space of all Essential Matrices is 5 dimensional • 3 Degrees of Freedom – Rotation • 2 Degrees of Freedom – Translation (up to scale !) Invitation to 3D vision

  10. Estimating Essential Matrix • Denote • Rewrite • Collect constraints from all points Invitation to 3D vision

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