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Fundamental Matrix



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Fundamental Matrix

The fundamental matrix encodes all given geometrical constraints between a set of two stereo-images. Given a point $ x_1$ in the first view and its corresponding point $ x_2$ in the second view in homogenous coordinates, $ F$ fulfills the following equation


$ F$ is a 3×3 matrix with rank 2. Details about the derivation can be found in [HZ00]. Some of the most important properties of $ F$ are:

Transpose
If $ F$ is the fundamental matrix of a pair of cameras (C,C‘), then $ F^T$ is the fundamental matrix of the pair in the opposite order (C‘,C).
Epipolar lines
For any point x in the first image, we can find the corresponding epipolar line $ l'$ with $ l' = Fx_1$ and also
$ l = F^Tx_2$.
Epipole
for any point $ x$ other than the epipole $ e$, $ l' = Fx_1$ contains the epipole $ e'$. Thus $ e'$ satisfies
$ e'^T(Fx)=(e'^TF)x = 0$ for all $ x$. It follows that $ e'^TF=0$, i.e. $ e'$ is the left null-space of F. Similarly $ Fe = 0$, i.e. is the right null-space of F.

To compute $ F$ you can use corresponding points to solve the homogeneous linear Equation 2.15. The standard technique is to use pre-conditioning followed by symmetric matrix eigendecomposition to solve for the nine elements of $ F$ up to an undetermined scale factor.

In case of known camera calibration parameters the essential matrix $ E$ is the equivalent to the fundamental matrix $ F$. In this case the rotation and translation between the cameras can be computed, up to an unknown scale factor. The images are now related by the following equation


where $ E$ is the essential matrix, $ x_1'$ and $ x_2'$ are ideal image coordinates. The ideal image coordinates $ x_1',x_2'$have to be calculated from the original image coordinates $ x_1,x_2$, so that $ x_1'$ and $ x_2'$ are the projected coordinates for an ideal camera. An ideal camera has focal distances
$ f_x = f_y = 1$, image center
$ x_0 = y_0 = 0$ and homogenous z-coordinate = 1. Thus the camera calibration matrix is the identical matrix $ I_{3x3}$. The essential matrix can be written as


where $ R$ is the rotation matrix, $ T$ is the translation vector between the two cameras and $ [T]_x$ is defined as


For more information about the essential matrix have a look at [Fau93,HZ00].

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Up: Stereo Geometry