User:BenFrantzDale/Epipolar geometry
Terminology[edit]
- 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) is the center of projection in 3D space.
- T (3×1 or 4×1, homogeneous) 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:
![{\displaystyle P=KR[I|-C]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/897451eb861e6b8fd9c1a1c60702151cb3781206)
where is the extrinsic parameters.
![{\displaystyle R[I|-C]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6f12f26fbc6074601c06665ff198e8e0828340b)
If you know P, K, and P' then you can compute F many ways (epipolar-4.pdf):
![{\displaystyle F=[P'C]_{\times }P'P^{+}=[K't]_{\times }K'RK^{-1}=K'^{-\top }[t]_{\times }RK^{-1}=\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ae02bc1bd6963bcf115104aa1c8934bdbc5adce)
where .
![{\displaystyle P^{+}=[K^{-1};0\;0\;0]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8fea065ec064cd20431adbedd3ef9ae620c55a18)
The essential matrix is given by
![{\displaystyle E=[t]_{\times }R=R[R^{\top }t]_{\times }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57055412716fee5910bef509bef28a7a248cf0c9)
and so
and
- .
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.