The collinearity equations are a set of two equations, used 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 ...
and
computer stereo vision
Computer stereo vision is the extraction of 3D information from digital images, such as those obtained by a CCD camera. By comparing information about a scene from two vantage points, 3D information can be extracted by examining the relative positi ...
, to relate
coordinates
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
in a
sensor plane (in two dimensions) to object coordinates (in three dimensions). The equations originate from the
central projection
In mathematics, a projection is a mapping of a set (or other mathematical structure) into a subset (or sub-structure), which is equal to its square for mapping composition, i.e., which is idempotent. The restriction to a subspace of a projec ...
of a point of the
object
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ...
through the
optical centre
In Gaussian optics, the cardinal points consist of three pairs of points located on the optical axis of a rotationally symmetric, focal, optical system. These are the '' focal points'', the principal points, and the nodal points. For ''ideal'' s ...
of the
camera to the image on the sensor plane.
T. Schenk, ''Introduction to Photogrammetry''
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Definition
Let x,y, and z refer to a coordinate system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
with the x- and y-axis in the sensor plane. Denote the coordinates of the point P on the object by , the coordinates of the image point of P on the sensor plane by ''x'' and ''y'' and the coordinates of the projection (optical) centre by . As a consequence of the projection method there is the same fixed ratio
In mathematics, a ratio shows how many times one number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in lan ...
between and , and , and the distance of the projection centre to the sensor plane and . Hence:
:
:
:
Solving for in the last equation and entering it in the others yields:
:
:
The point P is normally given in some coordinate system "outside" the camera by the coordinates ''X'', ''Y'' and ''Z'', and the projection centre by . These coordinates may be transformed through a rotation and a translation to the system on the camera. The translation doesn't influence the differences of the coordinates, and the rotation, often called camera transform, is given by a 3×3-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 ...
''R'', transforming into:
:
:
and
:
Substitution of these expressions, leads to a set of two equations, known as the ''collinearity equations'':
:
:
The most obvious use of these equations is for images recorded by a camera. In this case the equation describes transformations from object space (X, Y, Z) to image coordinates (x, y). It forms the basis for the equations used in 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 ...
.
They indicate that the image point (on the sensor plate of the camera), the observed point (on the object) and the projection center of the camera were aligned when the picture was taken.
See also
*3D projection
A 3D projection (or graphical projection) is a design technique used to display a three-dimensional (3D) object on a two-dimensional (2D) surface. These projections rely on visual perspective and aspect analysis to project a complex object fo ...
*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 ...
* Pinhole camera model
References
{{reflist
Photogrammetry
Geometry in computer vision