In
computer vision
Computer vision is an interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate tasks that the human ...
, the essential matrix is a
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
,
that relates
corresponding points in
stereo images assuming that the cameras satisfy the
pinhole camera model.
Function
More specifically, if
and
are homogeneous
''normalized'' image coordinates 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
Andrew Zisserman (born 1957) is a British computer scientist and a professor at the University of Oxford, and a researcher in computer vision. As of 2014 he is affiliated with DeepMind.
Education
Zisserman received the Part III of the Mathem ...
's book reports that an analogous matrix appeared in
photogrammetry
Photogrammetry is the science and technology of obtaining reliable information about physical objects and the environment through the process of recording, measuring and interpreting photographic images and patterns of electromagnetic radiant ima ...
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.
Use
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 essential 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
:
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 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.
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
:
where
is a
rotation matrix and
is a 3-dimensional translation vector.
The essential matrix is then defined as:
:
where
is the
matrix representation of the cross product with
.
Note: Here, the transformation
will transform points in the 2nd view to the 1st view.
For the definition of
we are only interested in the orientations of the normalized image coordinates (See also:
Triple product).
As such we don't need the translational component when substituting image coordinates into the essential equation.
To see that this definition of
describes a constraint on corresponding image coordinates multiply
from left and right with the 3D coordinates of point P in the two different coordinate systems:
:
# Insert the above relations between
and
and the definition of
in terms of
and
.
#
since
is a rotation matrix.
# Properties of the
matrix representation of the cross product.
Finally, it can be assumed that both
and
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
Not every arbitrary
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, both
. The skew-symmetric matrix must have two
singular values In mathematics, in particular functional analysis, the singular values, or ''s''-numbers of a compact operator T: X \rightarrow Y acting between Hilbert spaces X and Y, are the square roots of the (necessarily non-negative) eigenvalues of the self ...
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.
If the essential matrix
is multiplied by a non-zero scalar, the result is again an essential matrix which defines exactly the same constraint as
does. This means that
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 of the other. This is a relevant position, for example, if
is estimated from image data. However, it is also possible to take the position that
is defined as
:
where
, and then
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.
The essential matrix has five or six degrees of freedom, depending on whether or not it is seen as a projective element. The rotation matrix
and the translation vector
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 The eight-point algorithm is an algorithm used in computer vision to estimate the essential matrix or the fundamental matrix related to a stereo camera pair from a set of corresponding image points. It was introduced by Christopher Longuet-Higgin ...
.
Extracting rotation and translation
Given that the essential matrix has been determined for a stereo camera pair -- for example, using the estimation method above -- this information can be used for determining also the rotation
and translation
(up to a scaling) between the two camera's coordinate systems. In these derivations
is seen as a projective element rather than having a well-determined scaling.
Finding one solution
The following method for determining
and
is based on performing a
SVD of
, see Hartley & Zisserman's book.
It is also possible to determine
and
without an SVD, for example, following Longuet-Higgins' paper.
An SVD of
gives
:
where
and
are orthogonal
matrices and
is a
diagonal matrix with
:
The diagonal entries of
are the singular values of
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
In physics and mathematics, an ansatz (; , meaning: "initial placement of a tool at a work piece", plural Ansätze ; ) is an educated guess or an additional assumption made to help solve a problem, and which may later be verified to be part of th ...
:
:
Since
may not completely fulfill the constraints when dealing with real world data (f.e. camera images), the alternative
:
with
may help.
Proof
First, these expressions for
and
do satisfy the defining equation for the essential matrix
:
Second, it must be shown that this
is a matrix representation of the cross product for some
. Since
:
it is the case that
is skew-symmetric, i.e.,
. This is also the case for our
, since
:
According to the general properties of the
matrix representation of the cross product it then follows that
must be the cross product operator of exactly one vector
.
Third, it must also need to be shown that the above expression for
is a rotation matrix. It is the product of three matrices which all are orthogonal which means that
, too, is orthogonal or
. To be a proper rotation matrix it must also satisfy
. Since, in this case,
is seen as a projective element this can be accomplished by reversing the sign of
if necessary.
Finding all solutions
So far one possible solution for
and
has been established given
. It is, however, not the only possible solution and it may not even be a valid solution from a practical point of view. To begin with, since the scaling of
is undefined, the scaling of
is also undefined. It must lie in the
null space
In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map between two vector spaces and , the kern ...
of
since
:
For the subsequent analysis of the solutions, however, the exact scaling of
is not so important as its "sign", i.e., in which direction it points. Let
be normalized vector in the null space of
. It is then the case that both
and
are valid translation vectors relative
. It is also possible to change
into
in the derivations of
and
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.
To summarize, given
there are two opposite directions which are possible for
and two different rotations which are compatible with this essential matrix. In total this gives four classes of solutions for the rotation and translation between the two camera coordinate systems. On top of that, there is also an unknown scaling
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.
The above determination of
and
assumes that
satisfy the
internal constraints of the essential matrix. If this is not the case which, for example, typically is the case if
has been estimated from real (and noisy) image data, it has to be assumed that it approximately satisfy the internal constraints. The vector
is then chosen as right singular vector of
corresponding to the smallest singular value.
3D points from corresponding image points
Many methods exist for computing
given corresponding normalized image coordinates
and
, if the essential matrix is known and the corresponding rotation and translation transformations have been determined.
See also
*
Bundle adjustment
In photogrammetry and computer stereo vision, bundle adjustment is simultaneous refining of the 3D Coordinate system, coordinates describing the scene geometry, the parameters of the relative motion, and the optical characteristics of the camera(s ...
*
Epipolar geometry
Epipolar geometry is the geometry of stereo vision. When two cameras view a 3D scene from two distinct positions, there are a number of geometric relations between the 3D points and their projections onto the 2D images that lead to constraints b ...
*
Fundamental matrix
*
Geometric camera calibration
Camera resectioning is the process of estimating the parameters of a pinhole camera model approximating the camera that produced a given photograph or video; it determines which incoming light ray is associated with each pixel on the resulting imag ...
*
Triangulation (computer vision)
In computer vision, triangulation refers to the process of determining a point in 3D space given its projections onto two, or more, images. In order to solve this problem it is necessary to know the parameters of the camera projection function f ...
*
Trifocal tensor
In computer vision, the trifocal tensor (also tritensor) is a 3×3×3 array of numbers (i.e., a tensor) that incorporates all projective geometric relationships among three views. It relates the coordinates of corresponding points or lines in thr ...
Toolboxes
Essential Matrix Estimationin MATLAB (Manolis Lourakis).
External links
An Investigation of the Essential Matrixby R.I. Hartley
References
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{{refend
Geometry in computer vision