1 CSE486, Penn State Robert Collins Lecture 19: Essential and Fundamental Matrices CSE486, Penn State Robert Collins Epipolar Geometry image1 image 2 Epipole : location of cam2 as seen by cam1. Epipole : location of cam1 as seen by cam2. CSE486, Penn State Robert Collins Epipolar Geometry image1 image 2 Corresponding points lie on conjugate epipolar lines CSE486, Penn State Robert Collins This Lecture… image1 image 2 Given a point in one image, how do we determine the corresponding epipolar line to search along in the second image? CSE486, Penn State Robert Collins Essential Matrix The essential and fundamental matrices are 3x3 matrices that “encode” the epipolar geometry of two views. 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. CSE486, Penn State Robert Collins Essential Matrix e r e l O r p r P p l O l T P l P r R,T = rotation, and translation S= E=RS is “essential matrix”
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CSE486, Penn StateRobert Collins
Lecture 19:
Essential and Fundamental Matrices
CSE486, Penn StateRobert Collins
Epipolar Geometry
image1 image 2
Epipole : location of cam2as seen by cam1.
Epipole : location of cam1as seen by cam2.
CSE486, Penn StateRobert Collins
Epipolar Geometry
image1 image 2
Corresponding pointslie on conjugate epipolar lines
CSE486, Penn StateRobert Collins
This Lecture…
image1 image 2
Given a point in one image, how do we determinethe corresponding epipolar line to search along inthe second image?
CSE486, Penn StateRobert Collins
Essential Matrix
The essential and fundamental matrices are 3x3 matricesthat “encode” the epipolar geometry of two views.
Motivation: Given a point in one image, multiplyingby the essential/fundamental matrix will tell us which epipolar line to search along in the second view.
CSE486, Penn StateRobert Collins
Essential Matrix
eerreell OOrr
pprr
PP
ppll
OOllTT
PPll PPrr
R,T = rotation, and translation
S=
E=RS is “essential matrix”
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CSE486, Penn StateRobert Collins
Essential Matrix Properties
• has rank 2
has both a left and right nullspace (important!!!!)
• depends only on the EXTRINSIC Parameters (R & T)
CSE486, Penn StateRobert Collins
Longuet-Higgins equation
CSE486, Penn StateRobert Collins
Longuet-Higgins equation
Importance of Longuet-Higgens ...
This relates viewing rays
This relates 2D film points
CSE486, Penn StateRobert Collins
Longuet-Higgins Makes Sense
• Note, there is nothing magic about Longuet-Higgins equation.
• A film point can also be thought of as aviewing 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].
CSE486, Penn StateRobert Collins
Epipolar Lines
• Let l be a line in the image:
• Using homogeneous coordinates:
CSE486, Penn StateRobert Collins
Epipolar Lines
• Remember:
pprr belongs to belongs to epipolarepipolar line in the right image defined by line in the right image defined by
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CSE486, Penn StateRobert Collins
Epipolar Lines
• Remember:
ppll belongs to belongs to epipolarepipolar line in the left image defined by line in the left image defined by
CSE486, Penn StateRobert Collins
Epipoles
• Remember: epipoles belong to the epipolar lines
• And they belong to all the epipolar lines
We can use this to compute the location of the epipoles.There will be an example, shortly...
CSE486, Penn StateRobert Collins
Essential Matrix Summary
Longuet-Higgins equation
Epipolar lines:
Epipoles:
CSE486, Penn StateRobert Collins
Fundamental Matrix
The essential matrix uses CAMERA coordinates
To use image coordinates we must consider theTo use image coordinates we must consider theINTRINSIC camera parameters:INTRINSIC camera parameters:
Pixel coord(row,col)
Affine transform matrixCamera (film) coord
CSE486, Penn StateRobert Collins
Fundamental Matrix
short version: The same equation works inpixel coordinates too!
CSE486, Penn StateRobert Collins
Fundamental Matrix Properties
• has rank 2
• depends on the INTRINSIC and EXTRINSIC Parameters (f, etc ;R & T)
Analogous to essential matrix. The fundamentalmatrix also tells how pixels (points) in each image are related to epipolar lines in the other image.