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 and standardize the Position (geometry), position of the Point (geometry), points or other geometric elements on a manifold such as ...

in a

sensor A sensor is often defined as a device that receives and responds to a signal or stimulus. The stimulus is the quantity, property, or condition that is sensed and converted into electrical signal. In the broadest definition, a sensor is a devi ...

plane (in two dimensions) to object coordinates (in three dimensions). The equations originate from the

central projection In mathematics, a projection is an idempotent mapping of a set (or other mathematical structure) into a subset (or sub-structure). In this case, idempotent means that projecting twice is the same as projecting once. The restriction to a subspa ...

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 a ...

through the

optical centre In Gaussian optics, the cardinal points consist of three pairs of Point (geometry), points located on the optical axis of a Rotational symmetry, rotationally symmetric, focal, optical system. These are the ''Focus (optics), focal points'', the p ...

of the

camera A camera is an instrument used to capture and store images and videos, either digitally via an electronic image sensor, or chemically via a light-sensitive material such as photographic film. As a pivotal technology in the fields of photograp ...

to the image on the sensor plane.T. Schenk, ''Introduction to Photogrammetry''
/ref> NYW-CentralProjection02

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 and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...

with the x- and y-axis in the sensor plane. Denote the coordinates of the point P on the object by

x_P,y_P,z_P

, 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

x_0,y_0,z_0

. 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 ...

\lambda

between

x-x_0

and

x_0-x_P

y-y_0

and

y_0-y_P

, and the distance of the projection centre to the sensor plane

z_0=c

and

z_P-z_0

. Hence: :

x-x_0=-\lambda (x_P-x_0)

y-y_0=-\lambda (y_P-y_0)

c=\lambda (z_P-z_0),

Solving for

\lambda

in the last equation and entering it in the others yields: :

x-x_0=-c\ \frac

y-y_0=-c\ \frac

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

X_0,Y_0,Z_0

. These coordinates may be transformed through a

rotation Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...

and a

translation Translation is the communication of the semantics, meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The English la ...

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 (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...

''R'', transforming

(X-X_0,Y-Y_0,Z-Z_0)

into: :

x_P-x_0=R_ (X-X_0)+ R_(Y-Y_0) + R_ (Z-Z_0)

y_P-y_0=R_ (X-X_0)+ R_(Y-Y_0) + R_ (Z-Z_0)

and :

z_P-z_0=R_(X-X_0) + R_ (Y-Y_0) + R_ (Z-Z_0)

Substitution of these expressions, leads to a set of two equations, known as the ''collinearity equations'': :

x - x_0=  -c\ \frac

y - y_0=  -c\ \frac

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 acq ...

. 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.

References

{{reflist Photogrammetry Geometry in computer vision

Definition

See also

References