{"id":1992,"date":"2014-02-05T17:32:40","date_gmt":"2014-02-05T17:32:40","guid":{"rendered":"https:\/\/www.anagram.at\/en\/diplomarbeit\/fundamental-matrix\/"},"modified":"2014-02-05T17:32:40","modified_gmt":"2014-02-05T17:32:40","slug":"fundamental-matrix","status":"publish","type":"page","link":"https:\/\/www.anagram.at\/en\/diplomarbeit\/fundamental-matrix\/","title":{"rendered":"Fundamental Matrix"},"content":{"rendered":"


\n
\n Next:<\/b> Rectification<\/a>
\n Up:<\/b> Stereo Geometry<\/a>
\n Previous:<\/b> Epipolar Geometry<\/a>
\n<\/p>\n

<\/p>\n

Fundamental Matrix
\n<\/h2>\n

\nThe fundamental matrix encodes all given geometrical constraints between a set of two stereo-images. Given a point \"$ in the first view and its corresponding point \"$ in the second view in homogenous coordinates, \"$ fulfills the following equation<\/p>\n<\/p>\n\n

<\/p>\n\n\n
\"$displaystyle<\/td>\n\n(2.15)<\/td>\n<\/tr>\n<\/table>\n<\/div>\n


<\/p>\n\n

\n\"$ is a 3×3 matrix with rank 2. Details about the derivation can be found in [HZ00<\/a>]. Some of the most important properties of \"$ are:<\/p>\n<\/p>\n

\n
Transpose<\/strong><\/dt>\n
If \"$ is the fundamental matrix of a pair of cameras (C,C’), then \"$ is the fundamental matrix of the pair in the opposite order (C’,C).\n<\/dd>\n
Epipolar lines<\/strong><\/dt>\n
For any point x in the first image, we can find the corresponding epipolar line \"$ with \"$ and also
\n\"$.\n<\/dd>\n
Epipole<\/strong><\/dt>\n
for any point \"$ other than the epipole \"$, \"$ contains the epipole \"$. Thus \"$ satisfies
\n\"$ for all \"$. It follows that \"$, i.e. \"$ is the left null-space of F. Similarly \"$, i.e. is the right null-space of F.\n<\/dd>\n<\/dl>\n

\nTo compute \"$ you can use corresponding points to solve the homogeneous linear Equation 2.15<\/a>. The standard technique is to use pre-conditioning followed by symmetric matrix eigendecomposition to solve for the nine elements of \"$ up to an undetermined scale factor. <\/p>\n

\nIn case of known camera calibration parameters \tthe essential matrix \"$ is the equivalent to the fundamental matrix \"$. 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<\/p>\n<\/p>\n\n

<\/p>\n\n\n
\"$displaystyle<\/td>\n\n(2.16)<\/td>\n<\/tr>\n<\/table>\n<\/div>\n


<\/p>\n\nwhere \"$ is the essential matrix, \"$ and \"$ are ideal image coordinates. The ideal image coordinates \"$have to be calculated from the original image coordinates \"$, so that \"$ and \"$ are the projected coordinates for an ideal camera. An ideal camera has focal distances
\n\"$, image center
\n\"$ and homogenous z-coordinate = 1. Thus the camera calibration matrix is the identical matrix \"$. The essential matrix can be written as<\/p>\n\n

<\/p>\n\n\n
\"$displaystyle<\/td>\n\n(2.17)<\/td>\n<\/tr>\n<\/table>\n<\/div>\n


<\/p>\n\nwhere \"$ is the rotation matrix, \"$ is the translation vector between the two cameras and \"$ is defined as <\/p>\n\n

<\/p>\n\n\n
\"$displaystyle<\/td>\n\n(2.18)<\/td>\n<\/tr>\n<\/table>\n<\/div>\n


<\/p>\n\n

\nFor more information about the essential matrix have a look at [Fau93<\/a>,HZ00<\/a>].<\/p>\n<\/p>\n\n

Next:<\/b> Rectification<\/a>
\n Up:<\/b> Stereo Geometry<\/a>
\n<\/p>\n

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