Epipolar Geometry. Lecture 19: Essential and Fundamental Matrices. This Lecture. Epipolar Geometry. Essential Matrix. Essential Matrix.
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1 Epipolar Geometry Lecture 19: Essential and Fundamental Matrices Epipole : location of cam2 as seen by cam1. Epipole : location of cam1 as seen by cam2. Epipolar Geometry This Lecture Corresponding points lie on conjugate epipolar lines Given a point in one image, how do we determine the corresponding epipolar line to search along in the second image? Essential Matrix Essential Matrix The essential and fundamental matrices are 3x3 matrices that encode the epipolar geometry of two views. P l P P r Motivation: Given a point in one image, multiplying by the essential/fundamental matrix will tell us which epipolar line to search along in the second view. O l p l e l T e r p r Or R,T = rotation, and translation S= E=RS is essential matrix 1
2 Essential Matrix Properties has rank 2 has both a and nullspace (important!!!!) depends only on the EXTRINSIC Parameters (R & T) Longuet-Higgins Makes Sense This relates viewing rays Importance of Longuet-Higgens... This relates 2D film points Note, there is nothing magic about Longuet- Higgins equation. A film point can also be thought of as a viewing ray. They are equivalent. (u,v) 2D film point (u,v,f) 3D point on film plane k(u,v,f) viewing ray into the scene k(x, Y, Z) ray through point P in the scene [hint: k=f/z, and we have u=fx/z, v=fy/z]. Let l be a line in the image: Remember: Using homogeneous coordinates: p r belongs to epipolar line in the image defined by 2
3 Epipoles Remember: Remember: epipoles belong to the epipolar lines And they belong to all the epipolar lines p l belongs to epipolar line in the image defined by We can use this to compute the location of the epipoles. There will be an example, shortly... Essential Matrix Summary Fundamental Matrix Epipolar lines: The essential matrix uses CAMERA coordinates Pixel coord (row,col) To use image coordinates we must consider the INTRINSIC camera parameters: Affine transform matrix Camera (film) coord Epipoles: Fundamental Matrix Fundamental Matrix Properties has rank 2 depends on the INTRINSIC and EXTRINSIC Parameters (f, etc ; R & T) Analogous to essential matrix. The fundamental matrix also tells how pixels (points) in each image are related to epipolar lines in the other image. short version: The same equation works in pixel coordinates too! 3
4 normalize so sum of squares of first two terms is 1 (optional) x = y = x = y = ( ) L = ( ) ( ) x = y = 8.5 4
5 ( ) L= ( ) where is the epipole? F * el = vector in the nullspace of matrix F However, due to noise, F may not be singular. So instead, next best thing is eigenvector associated with smallest eigenvalue of F x = y = 8.5 >> [u,d] = eigs(f * F) u= where is the epipole? d = 1.e8* eigenvector associated with smallest eigenvalue >> uu = u(:,3) uu = ( ) >> uu / uu(3) : to get pixel coords ( ) e r * F = F * er = vector in the nullspace of matrix F However, due to noise, F may not be singular. So instead, next best thing is eigenvector associated with smallest eigenvalue of F >> [u,d] = eigs(f * F ) u= d = 1.e8* eigenvector associated with smallest eigenvalue >> uu = u(:,3) uu = ( ) >> uu / uu(3) : to get pixel coords ( ) 5
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