CN104346813A - Method for calibrating camera parameters in flame emission tomography system - Google Patents

Abstract

The invention discloses a method for calibrating camera parameters in a flame emission tomography system. Spatial location parameters and an image distance of a camera are calibrated by virtue of a three-dimensional camera calibration template; a plurality of sampling points and character graphics for indicating focusing control of the camera are arranged on the surface of the three-dimensional camera calibration template; the method specifically comprises the following steps: (1) building a relationship between a world coordinate system and a camera coordinate system and a relationship between the work coordinate system and a camera imaging surface coordinate system; (2) collecting an image of the calibration template by the camera, and calculating the coordinate of the sampling point in the imaging surface; (3) calculating the spatial location parameters of the camera by virtue of the coordinate relationship in the step (1) according to the world coordinate of the collecting point and the imaging surface coordinate in the step (2); (4) wherein imaging of the focus point in the camera meets the lens imaging formula, solving the image distance of the camera by virtue of the world coordinate and the imaging plane coordinate of the focus point. The method is simple and convenient to operate; the calibration result is relatively accurate; the camera with a relatively wide distribution range is calibrated by virtue of the three-dimensional camera calibration template; the camera calibration range is wide.

Description

The scaling method of camera parameter in a kind of flame emission tomographic system

Technical field

The present invention relates to a kind of camera parameter scaling method, particularly the scaling method of camera parameter in a kind of flame emission tomographic system.

Background technology

Flame emission chromatographic technique in combustion field diagnosis research, is become the important means of reaction mechanism of the transient buildup disclosed in combustion process, research combustion field by increasingly extensive.In conjunction with three-dimensional chromatography reconstruction technique, the optical radiation information of the optical radiation image of combustion field self in multiple directions to different-waveband in combustion field can be utilized to carry out whole audience three-dimensional quantitative measurement, break through the limitation of space single-point and plane monitoring-network.One of research core of flame emission chromatographic technique builds multidirectional projection acquisition device.Along with the development of industrial CCD camera, flame emission tomographic system can utilize multiple CCD camera to realize.But crucial technical matters is demarcation and the measurement of multiple CCD camera locus and respective inner parameter in this system, to realize accurate projection matching and three-dimensional reconstruction.Current camera calibration method is based on pinhole camera model, document 1 (Roger Y.Tsai, " A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses, " IEEE Journal of Robotics and Automation RA-3 (4), 323-344 (1987)) disclose a kind of method utilizing chessboard calibration template to measure the locus of camera and inner parameter, the method have ignored the focus issues of camera lens, although high to the space position calibration precision of camera, but camera image distance (distance between camera lens and CCD target surface) can not be measured well.

Summary of the invention

The object of the present invention is to provide the scaling method of camera parameter in a kind of flame emission tomographic system, improve the degree of accuracy of camera locus and inner parameter measurement result.

The technical scheme realizing the object of the invention is: the scaling method of camera parameter in a kind of flame emission tomographic system, three-dimensional camera calibrating template is utilized to demarcate the space position parameter of camera and image distance, three-dimensional camera calibrating template surface is provided with multiple sampled point and is used to indicate the character graphics of camera focus adjustment, and the method comprises the following steps:

Step 1, set up world coordinate system, camera coordinates system and camera imaging areal coordinate system, determine the transformational relation between three coordinate systems;

The image of step 2, collection three-dimensional camera calibrating template, determines the coordinate of sampled point in camera imaging face;

Step 3, the camera imaging areal coordinate obtained according to world coordinates and the step 2 of sampled point, utilize the space position parameter of the transformational relation determination camera between the coordinate in step 1;

The imaging in the camera of step 4, focus point meets lens imaging formula, based on the space position parameter that step 3 obtains, utilizes the world coordinates of focus point and the image distance of imaging plane coordinate determination camera.

Compared with prior art, its remarkable advantage is in the present invention:

1) between coordinate system of the present invention, the foundation of relation considers T in flame emission tomographic system _zthe feature of ≠ 0, is applicable to flame emission tomographic system;

2) contemplated by the invention the focusing effect of camera, compared with traditional pinhole camera model, measure camera image distance and there is higher precision;

3) camera that the present invention utilizes three-dimensional camera calibrating template wider to distribution range is demarcated, the sampled point that template surface has a series of coordinate clear and definite, and introduce character graphics at model surface to be used for indicating the focus adjustment of camera, camera calibration is wider.

Below in conjunction with accompanying drawing, the present invention will be further described.

Accompanying drawing illustrates:

Fig. 1 is the process flow diagram of the scaling method of camera parameter in flame emission tomographic system of the present invention.

Fig. 2 is coordinate system schematic diagram in flame emission tomographic system of the present invention.

Embodiment:

Composition graphs 1, the scaling method of camera parameter in a kind of flame emission tomographic system, three-dimensional camera calibrating template is utilized to demarcate the space position parameter of camera and image distance, three-dimensional camera calibrating template surface is provided with multiple sampled point and is used to indicate the character graphics of camera focus adjustment, and the method comprises the following steps:

Step 1, set up world coordinate system, camera coordinates system and camera imaging areal coordinate system, determine the transformational relation between three coordinate systems; Be specially:

World coordinate system (x _w, y _w, z _w), set as the case may be;

Camera imaging face (x', y') is CCD target surface, and x' axle is CCD target surface long limit opposite direction, and y' axle is CCD target surface minor face opposite direction, and true origin is defined as the central pixel point of camera CCD;

Camera coordinates system (x, y, z), x-y plane is parallel to camera imaging face, and x-axis is parallel to x' axle and contrary with x' direction of principal axis, and y-axis is parallel to y' axle and contrary with y' direction of principal axis, and true origin is the intersection point of camera optical axis and camera lens;

By rotation matrix R and translation vector T, world coordinate system is transformed into camera coordinates system, its mathematical relation is:

$\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=R\left[\begin{array}{c}{x}_w\\ y_w\\ z_w\end{array}\right]+T---\left(1\right)$

Wherein rotation matrix and translation vector are respectively

$R=\left[\begin{array}{ccc}{r}_1& r_2& r_3\\ r_4& r_5& r_6\\ r_7& r_8& r_9\end{array}\right]$

$T=\left[\begin{array}{c}{T}_x\\ T_y\\ T_z\end{array}\right]$

A bit (x, y, z) in camera coordinates system and its picture point (x', y') on camera imaging face meet relation

$\left\{\begin{array}{c}{x}^{\′}=Z_0\frac{x}{z}\\ y^{\′}=Z_0\frac{y}{z}\end{array}\right.---\left(2\right)$

Wherein, Z ₀for the image distance of camera;

Through type (1) and (2) can obtain

$\left\{\begin{array}{c}{Z}_0(r_1{x}_w+r_2{y}_w+r_3{z}_w+T_x)-x^{\′}(r_7{x}_w+r_8{y}_w+r_9{z}_w+T_z)=0\\ Z_0(r_4{x}_w+r_5{y}_w+r_6{z}_w+T_y)-y^{\′}(r_7{x}_w+r_8{y}_w+r_9{z}_w+T_z)=0\end{array}\right.---\left(3\right)$

Work as T _z≠ 0, equation (3) is expressed as

$\left\{\begin{array}{c}\frac{Z_0{r}_1}{T_z}{x}_w+\frac{Z_0{r}_2}{T_z}{y}_w+\frac{Z_0{r}_3}{T_z}{z}_w+\frac{Z_0{T}_x}{T_z}-\frac{r_7}{T_z}{x}^{\′}{x}_w-\frac{r_8}{T_z}{x}^{\′}{y}_w-\frac{r_9}{T_z}{x}^{\′}{z}_w=x^{\′}\\ \frac{Z_0{r}_4}{T_z}{x}_w+\frac{Z_0{r}_5}{T_z}{y}_w+\frac{Z_0{r}_6}{T_z}{z}_w+\frac{Z_0{T}_y}{T_z}-\frac{r_7}{T_z}{y}^{\′}{x}_w-\frac{r_8}{T_z}{y}^{\′}{y}_w-\frac{r_9}{T_z}{y}^{\′}{z}_w=y^{\′}\end{array}\right.---\left(4\right)$

Position coordinates (the x of object point in world coordinate system _w, y _w, z _w) meet equation (4) described relation with its imaging point coordinate (x', y') on camera imaging face;

The image of step 2, collection three-dimensional camera calibrating template, determines the coordinate of sampled point in camera imaging face; Be specially:

The image of collected by camera calibrating template, chooses N number of sampled point not coplanar on calibrating template, N>=6, the position (x of sampled point in world coordinate system _wi, y _wi, z _wi), i=1,2 ... N, by asking the coordinate (x' of round dot gravity model appoach determination sampled point in camera imaging face _i, y' _i), i=1,2 ... N;

Step 3, the camera imaging areal coordinate obtained according to world coordinates and the step 2 of sampled point, utilize the space position parameter of the transformational relation determination camera between the coordinate in step 1; Detailed process is:

For N number of sampled point, obtain 2N linear equation according to formula (4), system of equations matrix representation is

B _2N×11y _11×1＝b _2N×1(5)

Wherein

$B=\left[\begin{array}{ccccccccccc}{x}_{w1}& y_{w1}& z_{w1}& 1& 0& 0& 0& 0& -{x^{\′}}_1{x}_{w1}& -{x^{\′}}_1{y}_{w1}& -{x^{\′}}_1{z}_{w1}\\ 0& 0& 0& 0& x_{w1}& y_{w1}& z_{w1}& 1& -{y^{\′}}_1{x}_{w1}& {-y^{\′}}_1{y}_{w1}& -{y^{\′}}_1{z}_{w1}\\ & & & & & \·& & & & & \\ & & & & & \·& & & & & \\ & & & & & \·& & & & & \\ x_{\mathrm{wN}}& y_{\mathrm{wN}}& z_{\mathrm{wN}}& 1& 0& 0& 0& 0& -{x^{\′}}_N{x}_{\mathrm{wN}}& -{x^{\′}}_N{y}_{\mathrm{wN}}& -{x^{\′}}_N{z}_{\mathrm{wN}}\\ 0& 0& 0& 0& x_{\mathrm{wN}}& y_{\mathrm{wN}}& z_{\mathrm{wN}}& 1& -{y^{\′}}_N{x}_{\mathrm{wN}}& -{y^{\′}}_N{y}_{\mathrm{wN}}& -{y^{\′}}_N{z}_{\mathrm{wN}}\end{array}\right]$

$y={\left[\begin{array}{ccccccccccc}\frac{Z_0{r}_1}{T_z}& \frac{Z_0{r}_2}{T_z}& \frac{Z_0{r}_3}{T_z}& \frac{Z_0{T}_x}{T_z}& \frac{Z_0{r}_4}{T_z}& \frac{Z_0{r}_5}{T_z}& \frac{Z_0{r}_6}{T_z}& \frac{Z_0{T}_y}{T_z}& \frac{r_7}{T_z}& \frac{r_8}{T_z}& \frac{r_9}{T_z}\end{array}\right]}^T$

b＝[x' ₁y' ₁… x' _Ny' _N] ^T

Least square method solving equation group is utilized to obtain y; Order according to the internal relations of rotation matrix $\sqrt{{(r_1+r_5)}^2+{(r_2+r_4)}^2}+\sqrt{{(r_1+r_5)}^2+{(r_2+r_4)}^2}=2$ Obtain:

$a=\frac{T_z}{Z_0}=\frac{2}{\sqrt{{(y\left(1\right)+y\left(6\right))}^2+{(y\left(2\right)+y\left(5\right))}^2}+\sqrt{{(y\left(1\right)-y\left(6\right))}^2+{(y\left(2\right)+y\left(5\right))}^2}}---\left(6\right)$

Then r can be obtained ₁=y (1) a, r ₂=y (2) a, r ₃=y (3) a, r ₄=y (5) a, r ₅=y (6) a, r ₆=y (7) a, T _x=y (4) a, T _y=y (8) a; Dextrorotation orthogonal property according to rotation matrix obtains r ₇, r ₈and r ₉;

The imaging in the camera of step 4, focus point meets lens imaging formula, based on the space position parameter that step 3 obtains, utilizes the world coordinates of focus point and the image distance of imaging plane coordinate determination camera; Be specially:

Camera is adjusted in advance focuses on a bit (x _wf, y _wf, z _wf), the position of this focus point picture point on camera imaging face is (x' _f, y' _f), the picture point that focus point is corresponding with it in camera imaging system meets lens imaging equation

1/z _f+1/Z ₀＝1/f _lens(7)

$\frac{{x^{\′}}_f}{x_f}=\frac{Z_0}{z_f}---\left(8\right)$

$\frac{{y^{\′}}_f}{y_f}=\frac{Z_0}{z_f}---\left(9\right)$

Wherein, f _lensfor the focal length of camera lens; According to formula (1), (7) and (8), the image distance of camera

$Z_{0x}=f_{\mathrm{lens}}+f_{\mathrm{lens}}\frac{{x^{\′}}_f}{r_1{x}_{\mathrm{wf}}+r_2{y}_{\mathrm{wf}}+r_3{z}_{\mathrm{wf}}+T_x}---\left(10\right)$

According to formula (1), (7) and (9), the image distance of camera

$Z_{0y}=f_{\mathrm{lens}}+f_{\mathrm{lens}}\frac{{y^{\′}}_f}{r_4{x}_{\mathrm{wf}}+r_5{y}_{\mathrm{wf}}+r_6{z}_{\mathrm{wf}}+T_y}---\left(11\right)$

Then image distance Z ₀=(Z _0x+ Z _0y)/2; Camera coordinates system translational movement T in a z-direction _z=aZ ₀.

Below in conjunction with specific embodiment, the present invention will be further described.

Embodiment 1

Composition graphs 2, sets up the flame emission tomographic system in 12 projection acquisition directions, and in this system, camera is to be equidistantly distributed on a semicircle.In system, camera CCD pixel count is 964*1292, and Pixel Dimensions is 3.75 μm, and the focal length of camera lens is 12mm; Three-dimensional camera calibrating template is placed on system centre, gathers the calibrating template image that each camera is corresponding.

Camera is conditioned the focusing indication point focused on below scaling board front, regulation world coordinate system x _w-y _wplane is parallel to calibrating template surface, x _waxle is horizontal square to, y _waxle is vertical positive dirction, and initial point is positioned at calibrating template center.Select 8, scaling board front sampled point, the world coordinate system coordinate of these 8 sampled points is respectively: (-10,30,0), (10,30,0), (-10,10,0), (10,10,0), (-10 ,-10,30), (10,-10,30), (-10 ,-30,30) and (10,-30,30) (unit: mm), the world coordinate system coordinate of focus point is: (0,-20,30) (unit: mm).

Utilization is asked round dot gravity model appoach to solve the coordinate of 8 sampled points in image coordinate system and is respectively: (-0.5653,0.5489), (-0.2072,0.5387), (-0.5586,0.0940), (-0.1999,0.0938), (-0.1361 ,-0.3621), (0.2296 ,-0.3523), (-0.1309,-0.8379) and (0.2342 ,-0.8165) (unit: mm).

Sampled point world coordinate system and image coordinate system are substituted into formula (5), utilizes least square method solving equation group, and calculate according to formula (6) r ₁=0.7782, r ₂=-0.0122, r ₃=0.62,3, r ₄=0.0041, r ₅=0.9999, r ₆=0.0224, r ₇=-0.6215, r ₈=-0.0149, r ₉=0.7782, T _x=-16.7026, T _y=-5.8190.

The position of picture in image coordinate system of focus point is (0.0517 ,-0.5919), is obtained the image distance Z of camera by the mean value of formula (10) and (11) ₀=12.2837, camera coordinates system translational movement T in a z-direction _z=547.9324.

Although illustrate and describe embodiments of the invention above, but can not be interpreted as limitation of the present invention, those of ordinary skill in the art can change above-described embodiment within the scope of the invention when not departing from principle of the present invention and aim, revising, replacing and modification.

Claims (5)

1. the scaling method of camera parameter in a flame emission tomographic system, it is characterized in that, three-dimensional camera calibrating template is utilized to demarcate the space position parameter of camera and image distance, three-dimensional camera calibrating template surface is provided with multiple sampled point and is used to indicate the character graphics of camera focus adjustment, and the method comprises the following steps:

Step 1, set up world coordinate system, camera coordinates system and camera imaging areal coordinate system, determine the transformational relation between three coordinate systems;

The image of step 2, collection three-dimensional camera calibrating template, determines the coordinate of sampled point in camera imaging face;

Step 3, the camera imaging areal coordinate obtained according to world coordinates and the step 2 of sampled point, utilize the space position parameter of the transformational relation determination camera between the coordinate in step 1;

The imaging in the camera of step 4, focus point meets lens imaging formula, based on the space position parameter that step 3 obtains, utilizes the world coordinates of focus point and the image distance of camera imaging areal coordinate determination camera.

2. the camera calibration method in multi-direction flame emission tomographic system according to claim 1, set up world coordinate system, camera coordinates system and camera imaging areal coordinate system described in step 1, determine that the transformational relation between three coordinate systems is specially:

World coordinate system (x _w, y _w, z _w);

Camera imaging face (x', y') is CCD target surface, and x' axle is CCD target surface long limit opposite direction, and y' axle is CCD target surface minor face opposite direction, and true origin is the central pixel point of camera CCD;

Camera coordinates system (x, y, z), x-y plane is parallel to camera imaging face, and x-axis is parallel to x' axle and contrary with x' direction of principal axis, and y-axis is parallel to y' axle and contrary with y' direction of principal axis, and true origin is the intersection point of camera optical axis and camera lens;

By rotation matrix R and translation vector T, world coordinate system is transformed into camera coordinates system, its transformational relation is:

$\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=R\left[\begin{array}{c}{x}_w\\ y_w\\ z_w\end{array}\right]+T---\left(1\right)$

Wherein rotation matrix and translation vector are respectively

$R=\left[\begin{array}{ccc}{r}_1& r_2& r_3\\ r_4& r_5& r_6\\ r_7& r_8& r_9\end{array}\right]$

$T=\left[\begin{array}{c}{T}_x\\ T_y\\ T_z\end{array}\right]$

A bit (x, y, z) in camera coordinates system and its picture point (x', y') on camera imaging face meet relation

$\left\{\begin{array}{c}{x}^{\′}=Z_0\frac{x}{z}\\ y^{\′}=Z_0\frac{y}{z}\end{array}\right.---\left(2\right)$

Wherein, Z ₀for the image distance of camera;

Through type (1) and (2) can obtain

$\left\{\begin{array}{c}{Z}_0(r_1{x}_w+r_2{y}_w+r_3{z}_w+T_x)-x^{\′}(r_7{x}_w+r_8{y}_w+r_9{z}_w+T_z)=0\\ Z_0(r_4{x}_w+r_5{y}_w+r_6{z}_w+T_y)-y^{\′}(r_7{x}_w+r_8{y}_w+r_9{z}_w+T_z)=0\end{array}\right.---\left(3\right)$

Work as T _z≠ 0, equation (3) is expressed as

$\left\{\begin{array}{c}\frac{Z_0{r}_1}{T_z}{x}_w+\frac{Z_0{r}_2}{T_z}{y}_w+\frac{Z_0{r}_3}{T_z}{z}_w+\frac{Z_0{T}_x}{T_z}-\frac{r_7}{T_z}{x}^{\′}{x}_w-\frac{r_8}{T_z}{x}^{\′}{y}_w-\frac{r_9}{T_z}{x}^{\′}{z}_w=x^{\′}\\ \frac{Z_0{r}_4}{T_z}{x}_w+\frac{Z_0{r}_5}{T_z}{y}_w+\frac{Z_0{r}_6}{T_z}{z}_w+\frac{Z_0{T}_y}{T_z}-\frac{r_7}{T_z}{y}^{\′}{x}_w-\frac{r_8}{T_z}{y}^{\′}{y}_w-\frac{r_9}{T_z}{y}^{\′}{z}_w=y^{\′}\end{array}\right.---\left(4\right)$

Position coordinates (the x of object point in world coordinate system _w, y _w, z _w) meet equation (4) described relation with its imaging point coordinate (x', y') on camera imaging face.

3. the camera calibration method in multi-direction flame emission tomographic system according to claim 2, gathers the image of three-dimensional camera calibrating template, determines that sampled point is specially at the coordinate in camera imaging face described in step 2:

The image of collected by camera calibrating template, chooses N number of sampled point not coplanar on calibrating template, N>=6, the position (x of sampled point in world coordinate system _wi, y _wi, z _wi), i=1,2 ... N, by asking the coordinate (x' of round dot gravity model appoach determination sampled point in camera imaging face _i, y' _i), i=1,2 ... N.

4. the camera calibration method in multi-direction flame emission tomographic system according to claim 3, determine described in step 3 that the space position parameter detailed process of camera is:

For N number of sampled point, obtain 2N linear equation according to formula (4), system of equations matrix representation is

B _2N×11y _11×1＝b _2N×1(5)

Wherein

$B=\left[\begin{array}{ccccccccccc}{x}_{w1}& y_{w1}& z_{w1}& 1& 0& 0& 0& 0& -{x^{\′}}_1{x}_{w1}& -{x^{\′}}_1{y}_{w1}& -{x^{\′}}_1{z}_{w1}\\ 0& 0& 0& 0& x_{w1}& y_{w1}& z_{w1}& 1& -{y^{\′}}_1{x}_{w1}& -{y^{\′}}_1{y}_{w1}& -{y^{\′}}_1{z}_{w1}\\ & & & & & .& & & & & \\ & & & & & .& & & & & \\ & & & & & .& & & & & \\ x_{\mathrm{wN}}& y_{\mathrm{wN}}& z_{\mathrm{wN}}& 1& 0& 0& 0& 0& -{x^{\′}}_N{x}_{\mathrm{wN}}& -{x^{\′}}_N{y}_{\mathrm{wN}}& -{x^{\′}}_N{z}_{\mathrm{wN}}\\ 0& 0& 0& 0& x_{\mathrm{wN}}& y_{\mathrm{wN}}& z_{\mathrm{wN}}& 1& -{y^{\′}}_N{x}_{\mathrm{wN}}& -{y^{\′}}_N{y}_{\mathrm{wN}}& -{y^{\′}}_N{z}_{\mathrm{wN}}\end{array}\right]$

$y={\left[\begin{array}{ccccccccccc}\frac{Z_0{r}_1}{T_z}& \frac{Z_0{r}_2}{T_z}& \frac{Z_0{r}_3}{T_z}& \frac{Z_0{T}_x}{T_z}& \frac{Z_0{r}_4}{T_z}& \frac{Z_0{r}_5}{T_z}& \frac{Z_0{r}_6}{T_z}& \frac{Z_0{T}_y}{T_z}& \frac{r_7}{T_z}& \frac{r_8}{T_z}& \frac{r_9}{T_z}\end{array}\right]}^T$

b＝[x' ₁y' ₁… x' _Ny' _N] ^T

Least square method solving equation group is utilized to obtain y; Order according to the internal relations of rotation matrix $\sqrt{{(r_1+r_5)}^2+{(r_2-r_4)}^2}+\sqrt{{(r_1-r_5)}^2+{(r_2+r_4)}^2}=2$ Obtain:

$a=\frac{T_z}{Z_0}=\frac{2}{\sqrt{{(y\left(1\right)+y\left(6\right))}^2+{(y\left(2\right)-y\left(5\right))}^2}+\sqrt{{(y\left(1\right)-y\left(6\right))}^2+{(y\left(2\right)+y\left(5\right))}^2}}---\left(6\right)$

Then r can be obtained ₁=y (1) a, r ₂=y (2) a, r ₃=y (3) a, r ₄=y (5) a, r ₅=y (6) a, r ₆=y (7) a, T _x=y (4) a, T _y=y (8) a; Dextrorotation orthogonal property according to rotation matrix obtains r ₇, r ₈and r ₉.

5. the camera calibration method in multi-direction flame emission tomographic system according to claim 4, utilizes the image distance detailed process of the world coordinates of focus point and camera imaging areal coordinate determination camera to be described in step 4:

Camera is adjusted in advance focuses on a bit (x _wf, y _wf, z _wf), the position of this focus point picture point on camera imaging face is (x' _f, y' _f), the picture point that focus point is corresponding with it in camera imaging system meets lens imaging equation

1/z _f+1/Z ₀＝1/f _lens(7)

$\frac{{x^{\′}}_f}{x_f}=\frac{Z_0}{z_f}---\left(8\right)$

$\frac{{y^{\′}}_f}{y_f}=\frac{Z_0}{z_f}---\left(9\right)$

Wherein, f _lensfor the focal length of camera lens; According to formula (1), (7) and (8), the image distance of camera

$Z_{0x}=f_{\mathrm{lens}}+f_{\mathrm{lens}}\frac{{x^{\′}}_f}{r_1{x}_{\mathrm{wf}}+r_2{y}_{\mathrm{wf}}+r_3{z}_{\mathrm{wf}}+T_x}---\left(10\right)$

According to formula (1), (7) and (9), the image distance of camera

$Z_{0y}=f_{\mathrm{lens}}+f_{\mathrm{lens}}\frac{{y^{\′}}_f}{r_4{x}_{\mathrm{wf}}+r_5{y}_{\mathrm{wf}}+r_6{z}_{\mathrm{wf}}+T_y}---\left(11\right)$

Then image distance Z ₀=(Z _0x+ Z _0y)/2; Camera coordinates system translational movement T in a z-direction _z=aZ ₀.