Epipolar Geometry Prof. D. Stricker

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1 Outline 1. Short introduction: points and lines Epipolar Geometry Prof. D. Stricker 2. Two views geometry: Epipolar geometry Relation point/line in two views The geometry of two cameras Definition of the fundamental matrix F With slides from A. Zisserman, S. Lazebnik, Seitz 1

2 The projective plane Why do we need homogeneous coordinates? represent points at infinity, homographies, perspective projection, multi-view relationships What is the geometric intuition? a point in the image is a ray in projective space Projective Transformation: linear transformation that keeps lines. Projective Space: an extension of the Euclidian space where two lines always meet. x = x/1 R 2 y = y/1 (x,y) = (x,y,1) = (kx,ky,k) k 0 i.e. Position Euclidian Coordinates 2D projective Geomety Homogeneous coordinates in P 2 -y (0,0,0) -z (sx,sy,s) (x,y,1) x image plane Each point (x,y) on the plane is represented by a ray (sx,sy,s) all points on the ray are equivalent: (x, y, 1) (sx, sy, s) (x,y,0) = (x/0,y/0,0) = (,,0) Point at Infinity 4 i.e. Direction 2

3 Projective lines What does a line in the image correspond to in projective space? Point and line duality A line l is a homogeneous 3-vector It is to every point (ray) p on the line: l T p=0 l p 1 p 2 l 2 l 1 p A line is a plane of rays through origin all rays (x,y,z) satisfying: ax + by + cz = 0 l T p A line is also represented as a homogeneous 3-vector l What is the line l spanned by rays p 1 and p 2? l is to p 1 and p 2 l = p 1 p 2 l is the plane normal What is the intersection of two lines l 1 and l 2? p is to l 1 and l 2 p = l 1 l 2 Points and lines are dual in projective space given any formula, can switch the meanings of points and lines to get another formula 3

4 Example: Computing vanishing points (from lines) Ideal points and the line at infinity v Intersections of parallel lines q 2 q 1 p 2 p 1 Intersect p 1 q 1 with p 2 q 2 In practice: least squares version Better to use more than two lines and compute the closest point of intersection See notes by Bob Collins for one good way of doing this: Skew matrix of l tangent vector normal direction Example 4

5 Outline Stereo head 1. Short introduction: points and lines 2. Two views geometry: Epipolar geometry Relation point/line in two views The geometry of two cameras Definition of the fundamental matrix F Camera on a mobile vehicle 5

6 Pentagon example left image right image Scenarios The two images can arise from A stereo rig consisting of two cameras the two images are acquired simultaneously or A single moving camera (static scene) the two images are acquired sequentially range map The two scenarios are geometrically equivalent 6

7 The objective Corresponding points are images of the same scene point Given two images of a scene acquired by known cameras compute the 3D position of the scene (structure recovery) Triangulation Basic principle: triangulate from corresponding image points Determine 3D point at intersection of two back-projected rays C C / The back-projected points generate rays which intersect at the 3D scene point 7

8 An algorithm for stereo reconstruction The correspondence problem 1. For each point in the first image determine the corresponding point in the second image (this is a search problem) Given a point x in one image find the corresponding point in the other image 2. For each pair of matched points determine the 3D point by triangulation (this is an estimation problem) This appears to be a 2D search problem, but it is reduced to a 1D search by the epipolar constraint 8

9 General outline of 3D reconstruction Notation 1. Epipolar geometry TODAY The two cameras are P and P /, and a 3D point X is imaged as the geometry of two cameras reduces the correspondence problem to a line search 2. Stereo correspondence algorithms X P P / x x / 3. Triangulation C C / Warning for equations involving homogeneous quantities = means equal up to scale 9

10 Epipolar geometry Epipolar line Given an image point in one view, where is the corresponding point in the other view?? epipolar line C epipole C / baseline A point in one view generates an epipolar line in the other view The corresponding point lies on this line Epipolar constraint Reduces correspondence problem to 1D search along an epipolar line 10

11 Epipolar geometry Nomenclature Epipolar geometry is a consequence of the coplanarity of the camera centres and scene point X x x / C C / The camera centres, corresponding points and scene point lie in a single plane, known as the epipolar plane left epipolar line The epipolar line l / x is the image of the ray through x The epipole e is the point of intersection of the line joining the camera centres with the image plane this line is the baseline for a stereo rig, and the translation vector for a moving camera e X C C / e / x / l / right epipolar line The epipole is the image of the centre of the other camera: e = PC /, e / = P / C 11

12 The epipolar pencil The epipolar pencil X X e e / baseline As the position of the 3D point X varies, the epipolar planes rotate about the baseline. This family of planes is known as an epipolar pencil. All epipolar lines intersect at the epipole. (a pencil is a one parameter family) e e / baseline As the position of the 3D point X varies, the epipolar planes rotate about the baseline. This family of planes is known as an epipolar pencil. All epipolar lines intersect at the epipole. (a pencil is a one parameter family) Epipolar geometry depends only on the relative pose (position and orientation) and internal parameters of the two cameras, i.e. the position of the camera centres and image planes. It does not depend on the scene structure (3D points external to the camera). 12

13 Epipolar geometry example I: parallel cameras Epipolar geometry example II: converging cameras e e / Note, epipolar lines are in general not parallel 13

14 Algebraic representation of epipolar geometry We know that the epipolar geometry defines a mapping x l / Derivation of the algebraic expression Outline Step 1: for a point x in the first image back project a ray with camera P P point in first image epipolar line in second image Step 2: choose two points on the ray and project into the second image with camera P / P / Step 3: compute the line through the two image points using the relation l / = p x q 14

15 choose camera matrices Step 1: for a point x in the first image back project a ray with camera P internal calibration rotation translation from world to camera coordinate frame A point x back projects to a ray first camera world coordinate frame aligned with first camera where Z is the point s depth, since second camera satisfies 15

16 Step 2: choose two points on the ray and project into the second image with camera P / P / Step 3: compute the line through the two image points using the relation l / = p x q Consider two points on the ray Compute the line through the points Z = 0 is the camera centre Z = is the point at infinity Using the identity F F is the fundamental matrix Project these two points into the second view 16

17 The fundamental matrix F Example I: compute the fundamental matrix for a parallel camera stereo rig F is the unique 3x3 rank 2 matrix that satisfies x T Fx=0 for all x x X Y f Z (i) Epipolar lines: l =Fx & l=f T x (ii) Epipoles: on all epipolar lines, thus e T Fx=0, x e T F=0, similarly Fe=0 f (iii) F has 7 d.o.f., i.e. 3x3-1(homogeneous)-1(rank2) (iv) F is a correlation, projective mapping from a point x to a line l =Fx (not a proper correlation, i.e. not invertible) = t x /f (but we are in homogeneous space) reduces to y = y /, i.e. raster correspondence (horizontal scan-lines) 17

18 Example II: compute F for a forward translating camera X Y f Z X Y f Z f f Geometric interpretation? = t z /f (but we are in homogeneous space) 18

19 Summary: Properties of the Fundamental matrix X Y Z f f first image second image 19

20 THANK YOU! 20

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