The collinearity equations are a set of two equations, used in
photogrammetry 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 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 proje ...
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'' ...
of the
camera
A camera is an optical instrument that can capture an image. Most cameras can capture 2D images, with some more advanced models being able to capture 3D images. At a basic level, most cameras consist of sealed boxes (the camera body), with a ...
to the image on the sensor plane.
T. Schenk, ''Introduction to Photogrammetry''
/ref>
Definition
Let x,y, and z refer to a coordinate system 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 contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
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
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
to the system on the camera. The translation doesn't influence the differences of the coordinates, and the rotation, often called camera transform
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 ...
, 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'' (franchis ...
''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 coordinates describing the scene geometry, the parameters of the relative motion, and the optical characteristics of the camera(s) employed to acqui ...
.
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
The pinhole camera model describes the mathematical relationship between the coordinates of a point in three-dimensional space and its projection onto the image plane of an ''ideal'' pinhole camera, where the camera aperture is described as a po ...
References
{{reflist
Photogrammetry
Geometry in computer vision