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Concept
Essential matrix
Résumé
In computer vision, the essential matrix is a matrix, that relates corresponding points in stereo images assuming that the cameras satisfy the pinhole camera model.
More specifically, if and are homogeneous in image 1 and 2, respectively, then
if and correspond to the same 3D point in the scene.
The above relation which defines the essential matrix was published in 1981 by H. Christopher Longuet-Higgins, introducing the concept to the computer vision community. Richard Hartley and Andrew Zisserman's book reports that an analogous matrix appeared in photogrammetry long before that. Longuet-Higgins' paper includes an algorithm for estimating from a set of corresponding normalized image coordinates as well as an algorithm for determining the relative position and orientation of the two cameras given that is known. Finally, it shows how the 3D coordinates of the image points can be determined with the aid of the essential matrix.
The essential matrix can be seen as a precursor to the fundamental matrix, . Both matrices can be used for establishing constraints between matching image points, but the fundamental matrix can only be used in relation to calibrated cameras since the inner camera parameters (matrices and ) must be known in order to achieve the normalization. If, however, the cameras are calibrated the essential matrix can be useful for determining both the relative position and orientation between the cameras and the 3D position of corresponding image points. The essential matrix is related to the fundamental matrix with
This derivation follows the paper by Longuet-Higgins.
Two normalized cameras project the 3D world onto their respective image planes. Let the 3D coordinates of a point P be and relative to each camera's coordinate system. Since the cameras are normalized, the corresponding image coordinates are
and
A homogeneous representation of the two image coordinates is then given by
and
which also can be written more compactly as
and
where and are homogeneous representations of the 2D image coordinates and and are proper 3D coordinates but in two different coordinate systems.
Source officielle
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