Reconstruction from Two Calibrated Views Two-View Geometry

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Presentation transcript:

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 MASKS © 2004 Invitation to 3D vision

Pinhole Camera Model 3D points Image points Perspective Projection Rigid Body Motion Rigid Body Motion + Projective projection MASKS © 2004 Invitation to 3D vision

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 MASKS © 2004 Invitation to 3D vision

MASKS © 2004 Invitation to 3D vision

Epipolar Geometry Epipolar constraint Essential matrix MASKS © 2004 Invitation to 3D vision

Epipolar Geometry Epipolar lines l1, l2 Coimages of epipolar lines Epipoles Image correspondences l1 l2 MASKS © 2004 Invitation to 3D vision

Characterization of the 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 MASKS © 2004 Invitation to 3D vision

Estimating the Essential Matrix Estimate Essential matrix Decompose Essential matrix into Given n pairs of image correspondences: Find such Rotation and Translation that 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 !) MASKS © 2004 Invitation to 3D vision

Estimating Essential Matrix Denote Rewrite Collect constraints from all points MASKS © 2004 Invitation to 3D vision