Essential matrix on arvau

Wikipedia / Essential matrix

In computer vision, the essential matrix matrix that relates corresponding points in stereo images assuming that the cameras satisfy the pinhole camera model.

Function

are homogeneous normalized image coordinates in image 1 and 2, respectively, then

correspond to the same 3D point in the scene (not an "if and only if" due to the fact that points that lie on the same epipolar line in the first image will get mapped to the same epipolar line in the second image).

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 is known. Finally, it shows how the 3D coordinates of the image points can be determined with the aid of the essential matrix.

Use

The essential matrix can be seen as a precursor to the fundamental matrix) 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

Derivation and definition

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 relative to each camera's coordinate system. Since the cameras are normalized, the corresponding image coordinates are

A homogeneous representation of the two image coordinates is then given by

which also can be written more compactly as

are proper 3D coordinates but in two different coordinate systems.

Another consequence of the normalized cameras is that their respective coordinate systems are related by means of a translation and rotation. This implies that the two sets of 3D coordinates are related as

is a 3-dimensional translation vector.

The essential matrix is then defined as:

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is the matrix representation of the cross product will transform points in the 2nd view to the 1st view.

we are only interested in the orientations of the normalized image coordinates (See also: Triple product from left and right with the 3D coordinates of point P in the two different coordinate systems:

.
is a rotation matrix.
Properties of the matrix representation of the cross product.

are > 0, otherwise they are not visible in both cameras. This gives

which is the constraint that the essential matrix defines between corresponding image points.

Properties

matrix can be an essential matrix for some stereo cameras. To see this notice that it is defined as the matrix product of one rotation matrix and one skew-symmetric matrix. The skew-symmetric matrix must have two singular values which are equal and another which is zero. The multiplication of the rotation matrix does not change the singular values which means that also the essential matrix has two singular values which are equal and one which is zero. The properties described here are sometimes referred to as internal constraints of the essential matrix.

can be seen as an element of a projective space, that is, two such matrices are considered equivalent if one is a non-zero scalar multiplication is defined as

has a well-defined "scaling". It depends on the application which position is the more relevant.

The constraints can also be expressed as

and

Here, the last equation is a matrix constraint, which can be seen as 9 constraints, one for each matrix element. These constraints are often used for determining the essential matrix from five corresponding point pairs.

have three degrees of freedom each, in total six. If the essential matrix is considered as a projective element, however, one degree of freedom related to scalar multiplication must be subtracted leaving five degrees of freedom in total.

Estimation

Given a set of corresponding image points it is possible to estimate an essential matrix which satisfies the defining epipolar constraint for all the points in the set. However, if the image points are subject to noise, which is the common case in any practical situation, it is not possible to find an essential matrix which satisfies all constraints exactly.

Depending on how the error related to each constraint is measured, it is possible to determine or estimate an essential matrix which optimally satisfies the constraints for a given set of corresponding image points. The most straightforward approach is to set up a total least squares problem, commonly known as the eight-point algorithm.

Extracting rotation and translation

is seen as a projective element rather than having a well-determined scaling.

Finding one solution

is based on performing a SVD, see Hartley & Zisserman's book. without an SVD, for example, following Longuet-Higgins' paper.

gives

diagonal matrix with

which, according to the internal constraints of the essential matrix, must consist of two identical and one zero value. Define

with

and make the following ansatz

may not completely fulfill the constraints when dealing with real world data (f.e. camera images), the alternative

with

may help.

Proof

do satisfy the defining equation for the essential matrix

. Since

, since

According to the general properties of the matrix representation of the cross product.

if necessary.

Finding all solutions

is also undefined. It must lie in the null space since

above. For the translation vector this only causes a change of sign, which has already been described as a possibility. For the rotation, on the other hand, this will produce a different transformation, at least in the general case.

for the chosen translation direction.

It turns out, however, that only one of the four classes of solutions can be realized in practice. Given a pair of corresponding image coordinates, three of the solutions will always produce a 3D point which lies behind at least one of the two cameras and therefore cannot be seen. Only one of the four classes will consistently produce 3D points which are in front of both cameras. This must then be the correct solution. Still, however, it has an undetermined positive scaling related to the translation component.

satisfy the internal constraints of the essential matrix corresponding to the smallest singular value.

3D points from corresponding image points

, if the essential matrix is known and the corresponding rotation and translation transformations have been determined.