User:BenFrantzDale/Epipolar geometry

Terminology

 * F (3×3) is the fundamental matrix.
 * E (3×3) is the essential matrix
 * P (3×4) is the Camera matrix (the projection matrix from world to image)
 * C (3×1 or 4×1, homogeneous) $$C=-R^\top T$$ is the center of projection in 3D space.
 * T (3×1 or 4×1, homogeneous) $$T=-RC$$ is the T column of the RT matrix.
 * K is the intrinsic parameters -- the camera matrix

From http://www.ecse.rpi.edu/~rjradke/papers/radkemcn08.pdf:
 * $$P = K R [I | -C]$$

where $$R [I | -C]$$ is the extrinsic parameters.

If you know P, K, and P' then you can compute F many ways (epipolar-4.pdf):
 * $$F = [P'C]_\times P' P^+=[K't]_\times K' R K^{-1} = K'^{-\top} [t]_\times RK^{-1}=\cdots$$

where $$P^+ = [K^{-1}; 0\; 0\; 0]$$.

The essential matrix is given by
 * $$E=[t]_\times R = R[R^\top t]_\times$$

and so
 * $$F = K'^{-\top} E K^{-1}$$

and
 * $$ E = K'^\top F K$$.

Epipolar lines are represented as 3-vectors -- normals to the 3D planes from a camera center such that where that plane intersects the corresponding image plane is the epipolar line on that image.