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In computer vision, the fundamental matrix \mathbf is a 3×3 matrix which relates corresponding points in stereo images. In epipolar geometry, with homogeneous image coordinates, x and x′, of corresponding points in a stereo image pair, Fx describes a line (an epipolar line) on which the corresponding point x′ on the other image must lie. That means, for all pairs of corresponding points holds : \mathbf'^ \mathbf = 0. Being of rank two and determined only up to scale, the fundamental matrix can be estimated given at least seven point correspondences. Its seven parameters represent the only geometric information about cameras that can be obtained through point correspondences alone. The term "fundamental matrix" was coined by
QT Luong QT Luong (born 1964) is a French-Vietnamese born American photographer known for his work in the U.S. National Parks, as well as for work in the theory of computer vision. In 2022, Luong received the Ansel Adams Award for Conservation Photography ...
in his influential PhD thesis. It is sometimes also referred to as the "bifocal tensor". As a tensor it is a
two-point tensor Two-point tensors, or double vectors, are tensor-like quantities which transform as Euclidean vectors with respect to each of their indices. They are used in continuum mechanics to transform between reference ("material") and present ("configurat ...
in that it is a bilinear form relating points in distinct coordinate systems. The above relation which defines the fundamental matrix was published in 1992 by both Olivier Faugeras and Richard Hartley. Although H. Christopher Longuet-Higgins' essential matrix satisfies a similar relationship, the essential matrix is a metric object pertaining to calibrated cameras, while the fundamental matrix describes the correspondence in more general and fundamental terms of projective geometry. This is captured mathematically by the relationship between a fundamental matrix \mathbf and its corresponding essential matrix \mathbf, which is : \mathbf = ()^ \; \mathbf \; \mathbf \mathbf and \mathbf' being the intrinsic calibration matrices of the two images involved.


Introduction

The fundamental matrix is a relationship between any two images of the same scene that constrains where the projection of points from the scene can occur in both images. Given the projection of a scene point into one of the images the corresponding point in the other image is constrained to a line, helping the search, and allowing for the detection of wrong correspondences. The relation between corresponding points, which the fundamental matrix represents, is referred to as ''epipolar constraint'', ''matching constraint'', ''discrete matching constraint'', or ''incidence relation''.


Projective reconstruction theorem

The fundamental matrix can be determined by a set of point correspondences. Additionally, these corresponding image points may be ''triangulated'' to world points with the help of camera matrices derived directly from this fundamental matrix. The scene composed of these world points is within a
projective transformation In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, ...
of the true scene.


Proof

Say that the image point correspondence \mathbf \leftrightarrow \mathbf derives from the world point \textbf under the camera matrices \left ( \textbf, \textbf' \right ) as : \begin \mathbf & = \textbf \textbf \\ \mathbf & = \textbf' \textbf \end Say we transform space by a general homography matrix \textbf_ such that \textbf_0 = \textbf \textbf. The cameras then transform as : \begin \textbf_0 & = \textbf \textbf^ \\ \textbf_0' & = \textbf' \textbf^ \end :\textbf_0 \textbf_0 = \textbf \textbf^ \textbf \textbf = \textbf \textbf = \mathbf and likewise with \textbf_0' still get us the same image points.


Derivation of the fundamental matrix using coplanarity condition

The fundamental matrix can also be derived using the coplanarity condition.


For satellite images

The fundamental matrix expresses the epipolar geometry in stereo images. The epipolar geometry in images taken with perspective cameras appears as straight lines. However, in
satellite images Satellite images (also Earth observation imagery, spaceborne photography, or simply satellite photo) are images of Earth collected by imaging satellites operated by governments and businesses around the world. Satellite imaging companies sell ima ...
, the image is formed during the sensor movement along its orbit ( pushbroom sensor). Therefore, there are multiple projection centers for one image scene and the epipolar line is formed as an epipolar curve. However, in special conditions such as small image tiles, the satellite images could be rectified using the fundamental matrix.


Properties

The fundamental matrix is of
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...
2. Its
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learn ...
defines the epipole.


See also

* Epipolar geometry * Essential matrix * Trifocal tensor * Eight-point algorithm


Notes


References

* * * * * * * * * * * * * * *


Toolboxes


fundest
is a GPL C/
C++ C++ (pronounced "C plus plus") is a high-level general-purpose programming language created by Danish computer scientist Bjarne Stroustrup as an extension of the C programming language, or "C with Classes". The language has expanded significan ...
library for
robust Robustness is the property of being strong and healthy in constitution. When it is transposed into a system, it refers to the ability of tolerating perturbations that might affect the system’s functional body. In the same line ''robustness'' ca ...
, non-linear (based on the
Levenberg–Marquardt algorithm In mathematics and computing, the Levenberg–Marquardt algorithm (LMA or just LM), also known as the damped least-squares (DLS) method, is used to solve non-linear least squares problems. These minimization problems arise especially in least sq ...
) fundamental matrix estimation from matched point pairs and various objective functions (Manolis Lourakis).
Structure and Motion Toolkit in MATLAB (Philip H. S. Torr)



The Epipolar Geometry Toolbox (EGT)


External links


Epipolar Geometry and the Fundamental Matrix (chapter from Hartley & Zisserman)


* ttps://web.archive.org/web/20091120063117/http://www2.informatik.hu-berlin.de/~blaschek/diplvortrag/learn_epi/EpipolarGeo.html Visualization of epipolar geometry(originally by Sylvain Bougnoux of
INRIA The National Institute for Research in Digital Science and Technology (Inria) () is a French national research institution focusing on computer science and applied mathematics. It was created under the name ''Institut de recherche en informatiq ...
Robotvis, requires
Java Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's mos ...
)
The Fundamental Matrix Song
Video demonstrating laws of epipolar geometry. {{Matrix classes Geometry in computer vision