# Berkeley ELENG 290T - Epipolar Geometry (32 pages)

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Epipolar Geometry class 11 Multiple View Geometry Comp 290 089 Marc Pollefeys Multiple View Geometry course schedule subject to change Jan 7 9 Intro motivation Projective 2D Geometry Jan 14 16 no class Projective 2D Geometry Jan 21 23 Projective 3D Geometry no class Jan 28 30 Parameter Estimation Parameter Estimation Feb 4 6 Algorithm Evaluation Camera Models Feb 11 13 Camera Calibration Single View Geometry Feb 18 20 Epipolar Geometry 3D reconstruction Feb 25 27 Fund Matrix Comp Structure Comp Planes Homographies Trifocal Tensor Three View Reconstruction Multiple View Geometry Mar 4 6 Mar 18 20 Mar 25 27 MultipleView Reconstruction Bundle adjustment Apr 1 3 Auto Calibration Papers Apr 8 10 Dynamic SfM Papers Apr 15 17 Cheirality Papers Apr 22 24 Duality Project Demos More Single View Geometry Projective cameras and planes lines conics and quadrics PTl P T CP Q cone PQ P T C Camera calibration and vanishing points calibrating conic and the IAC Two view geometry Epipolar geometry 3D reconstruction F matrix comp Structure comp Three questions i Correspondence geometry Given an image point x in the first view how does this constrain the position of the corresponding point x in the second image ii Camera geometry motion Given a set of corresponding image points xi x i i 1 n what are the cameras P and P for the two views iii Scene geometry structure Given corresponding image points xi x i and cameras P P what is the position of their pre image X in space The epipolar geometry a C C x x and X are coplanar The epipolar geometry b What if only C C x are known The epipolar geometry a All points on project on l and l The epipolar geometry b Family of planes and lines l and l Intersection in e and e The epipolar geometry epipoles e e intersection of baseline with image plane projection of projection center in other image vanishing point of camera motion direction an epipolar plane plane containing baseline 1 D family an epipolar line intersection of epipolar plane with image always come in corresponding pairs Example converging cameras Example motion parallel with image plane The fundamental matrix F algebraic representation of epipolar geometry x a l we will see that mapping is singular correlation i e projective mapping from points to lines represented by the fundamental matrix F The fundamental matrix F geometric derivation x H x l e x e H x Fx mapping from 2 D to 1 D family rank 2 The fundamental matrix F algebraic derivation X P x C l P C P P x F e P P note doesn t work for C C F 0 P P I P x X The fundamental matrix F correspondence condition The fundamental matrix satisfies the condition that for any pair of corresponding points x x in the two images T T x Fx 0 x l 0 The fundamental matrix F F is the unique 3x3 rank 2 matrix that satisfies x TFx 0 for all x x i ii iii iv v Transpose if F is fundamental matrix for P P then FT is fundamental matrix for P P Epipolar lines l Fx l FTx Epipoles on all epipolar lines thus e TFx 0 x e TF 0 similarly Fe 0 F has 7 d o f i e 3x3 1 homogeneous 1 rank2 F is a correlation projective mapping from a point x to a line l Fx not a proper correlation i e not invertible Epipolar Line Homography The epipolar line geometry l l epipolar lines k line not through e l F k xl and symmetrically l FT k xl k l k l e Fk l e pick k e since eTe 0 l F e l l FT e l Pure Translation camera motion Fundamental matrix for pure translation Forward motion Fundamental matrix for pure translation F e H e example e 1 0 0 T H K 1RK 0 0 0 F 0 0 1 0 1 0 x T Fx 0 y y x PX K I 0 X 1 K x P X K I t x Z X Y Z T K 1x Z x x Kt Z motion starts at x and moves towards e faster depending on Z pure translation F only 2 d o f xT e xx 0 auto epipolar General motion x T e Hx 0 x T e x 0 x K RK 1x K t Z Geometric representation of F FS F FT 2 x x FA F FT 2 x Fx 0 x T FS x 0 T Fs Steiner conic 5 d o f Fa xa x pole of line ee w r t Fs 2 d o f F FS FA x T FA x 0 Pure planar motion Steiner conic Fs is degenerate two lines Projective transformation and invariance Derivation based purely on projective concepts x Hx x H x F H FH T 1 F invariant to transformations of projective 3 space P P a F F a P P unique not unique canonical form P I 0 P M m F m M Projective ambiguity of cameras given F previous slide at least projective ambiguity this slide not more Show that if F is same for P P and P P there exists a projective transformation H so that P HP and P HP P I 0 P A a P I 0 P A a F a A a A lemma a ka A k 1 A avT Canonical cameras given F F matrix corresponds to P P iff P TFP is skew symmetric X P T Possible choice P I 0 P e F e T FPX 0 X The essential matrix fundamental matrix for calibrated cameras remove K E t R R R T t x T Ex 0 x K E K T FK 5 d o f 3 for R 2 for t up to scale E is essential matrix if and only if two singularvalues are equal and third 0 E Udiag 1 1 0 V T 1 x x K 1x Four possible reconstructions from E only one solution where points is in front of both cameras Next class 3D reconstruction

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