Calibration of camera

Image processing, the operation of calibration of a camera reverts modelling the formation process of the Image S, i.e. to find the relation between the space coordinates of a point of space with the point associated in the image taken by the Caméra.

Modeling

Several models describing the formation process of the images exist. Simplest is the model of the Sténopé or model pine-hole in the Anglo-Saxon literature. This last is usually used in image processing.

Notations

It is advisable to specify the orthonormés reference marks in which we will work to model the operation of the camera: The associated reference marks are:

$(R_O, \ vec {X_O}, \ vec {Y_O}, \ vec {Z_O})$ : who is the reference mark associated with the workspace. $O$ is the origin of this reference mark,
$(R_C, \ vec {X_C}, \ vec {Y_C}, \ vec {Z_C})$ : who is the reference mark associated with the camera. $C$ is the optical center of the camera,
$(R_i, \ vec {U}, \ vec {v})$ : who is the reference mark associated with the visualized image.

An additional plan was not represented here which is in the reference mark associated with the camera. It is the image plan which is at a distance

f

C

according to the direction given by

\ vec {Z_C}

f

is the focal Distance of the camera.

The model of the pinhole

Also called model pine-hole in the Anglo-Saxon literature, it is about a simple and linear modeling of the formation process of the images within a camera. This model supposes that the optical system of the camera, i.e. its lens respects the Conditions of Gauss. If one uses the matric notation of the homogeneous Coordonnées, it is possible to describe in a simple way this process. It is enough to express the relations of passage of the reference mark world to the reference mark camera, to express the projection of the reference mark camera in the image plan and to apply the transformation closely connected which leads to the coordinates of the image. The relation is the following one for a point

M

coordinates

(X, Y, Z, 1)

in the space (reference mark world) and whose image is coordinates

(known, sv, S)

in the image plan:

\ left (
\begin{matrix}
known \ \ sv \ \ S \ \
\end{matrix}
\ right)

\ left \begin{matrix} k_u & s_ {UV} & c_u \ \ 0 & k_v & c_v \ \ 0 & 0 & 1 \end{matrix} \ right] \ left \begin{matrix} F & 0 & 0 & 0 \ \ 0 & F & 0 & 0 \ \ 0 & 0 & 1 & 0 \ \ \end{matrix} \ right] \ left \begin{matrix} & & & t_x \ \ & R_ {3 \ times3} & & t_y \ \ & & & t_z \ \ 0&0&0&1 \end{matrix} \ right] \ left ( \begin{matrix} X \ \ Y \ \ Z \ \ 1 \ \ \end{matrix} \ right)

The parameters employed in this model are usually divided into two categories: the intrinsic parameters which are interns with the camera, and the extrinsic parameters which can vary according to the position of the camera in the workspace. Among the intrinsic parameters we count:

$f$ : the Focal distance ,
$k_u \ mbox {and} k_v$ : factors of enlarging of the image,
$c_u \ mbox {and} c_v$ : coordinates of the projection of the optical center of the camera on the image plan,
$s_ {UV}$ : who translates the potential not-orthogonality of the lines and the electronic columns of cells Photosensible S which compose the sensor of the camera. Most of the time, this parameter is neglected and takes thus a zero value.

The extrinsic parameters are:

$R_ {3 \ times3}$ : who is the matrix of Rotation allowing to pass from the reference mark related to the workspace the reference mark related to the camera,
$t_x, t_y \ mbox {and} t_z$ : who are the components of the vector of translation allowing to pass from the reference mark related to the workspace the reference mark related to the camera.

On the whole, that made 18 parameters be estimated (the matrix of rotation contains 9 of them).

Étalonner the camera consists in determining the numerical value of the parameters of this model. It is however possible to gather them in a different way, according to the form in which this model must then be exploited. This led to various possible alternatives of the model.

Alternatives of the model of the pinhole

Certain alternatives of the model of the Sténopé gather the intrinsic parameters in the following way:

$\ left
\begin{matrix}
\ alpha_u & S_ {UV} & u_0 & 0 \ \
0 & \ alpha_v & v_0 & 0 \ \
0 & 0 & 1 & 0
\end{matrix}
\ right]$

\ left \begin{matrix} k_u & s_ {UV} & c_u \ \ 0 & k_v & c_v \ \ 0 & 0 & 1 \end{matrix} \ right] \ left \begin{matrix} F & 0 & 0 & 0 \ \ 0 & F & 0 & 0 \ \ 0 & 0 & 1 & 0 \ \ \end{matrix} \ right]

Knowing that the expression $f.s_ {UV}$ is often regarded as being null, we obtain a simplified new handset of intrinsic parameters:

$\ alpha_u \ mbox {and} \ alpha_v$ (sometimes also called $f_x \ mbox {and} f_y$ in the literature) which corresponds to the distance Focale expressed in widths and heights of Pixel S (the latter are not necessarily square),
$u_0 \ mbox {and} v_0$ which is the coordinates of the projection of the optical center of the camera on the image plan.

This alternative contains 17 parameters to be estimated. It has the advantage of separating in two matrices the intrinsic and extrinsic parameters and thus of isolating the latter which change value with each time the camera is moved in the workspace.

Another alternative consists in using only one total matrix of parameters to be estimated:

$\ left
\begin{matrix}
m_ {11} & m_ {12} & m_ {13} & m_ {14} \ \
m_ {21} & m_ {22} & m_ {23} & m_ {24} \ \
m_ {31} & m_ {32} & m_ {33} & m_ {34}
\end{matrix}
\ right]$

The latter presents only 12 parameters to be estimated but it is necessary to reappraise them with each time the camera is moved.

However, in the case of the wide-angle lenses, the optical conditions of Gauss are not respected any more, which optical deformations in the final image of which should then be held account induces.

Distortions optics

Methods of calibration

See too

OpenCV Library

Image processing

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