CN102402785B - Camera self-calibration method based on quadratic curves - Google Patents

Abstract

The invention relates to a camera self-calibration method based on quadratic curves. A planar rectangle is used as a calibration panel, three (or more than three) images of the calibration panel are shot from different directions, image characteristic points are extracted, straight lines are fitted, and straight line intersection points are solved to obtain four vertex coordinates of the rectangle and vanishing point coordinates on two groups of parallel lines. Midpoint coordinates of each edge and circumscribed circle center coordinates are obtained according to projective invariance and harmonic conjugate relation, and vanishing point coordinates on diagonal lines are solved. Image coordinates of circular ring points on a plane are solved according to properties of second-order curves in the quadratic curves, the constraint equation of circular ring point images in relation to camera intrinsic parameters is built, and the camera intrinsic parameters are solved linearly. All intrinsic parameters of a camera can be solved by the aid of the planar panel without complicated image matching, calibration process is simplified, and calibration precision is improved.

Description

A kind of camera self-calibration method based on quafric curve

Technical field

The invention belongs to the computer vision research field, is a kind of camera self-calibration method based on quafric curve, utilizes planar rectangular to be calibrating template.

Background technology

Three-dimensional reconstruction is one of important topic of computer vision research, refers to recover from the image of three-dimensional body the spatial geometric shape of three-dimensional body.Traditional Three-dimensional Gravity found a capital be the image that obtains of the video camera demarcated be to rebuild foundation, therefore in this process, the camera calibration technology is very important, the precision that the accuracy affects later stage of the parameter of camera calibration rebuilds.The camera calibration technology is one of research basic task in computer vision field

What usually adopt in the camera calibration process is classical pin-hole imaging model, document " Computer Vision:AModern Approach " (David A.Forsyth for example, Jean Ponce Faugeras work, forestry Yan, Wang Hong etc. translate. the Electronic Industry Press, 2004) provide model in, the general simple following formula that is described as of this model:

$\mathrm{\λ}\left(\begin{array}{c}u\\ v\\ 1\end{array}\right)=\left(\begin{array}{ccc}{f}_u& s& u_0\\ 0& f_v& v_0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{cc}R& T\end{array}\right)\left(\begin{array}{c}{x}_w\\ y_w\\ z_w\\ 1\end{array}\right)=K\left(\begin{array}{cc}R& T\end{array}\right)\left(\begin{array}{c}{x}_w\\ y_w\\ z_w\\ 1\end{array}\right).---\left(1\right)$

Wherein: in the space, the homogeneous coordinates of any point P in world coordinate system are P _w(x _wy _wz _w1) ^T, the homogeneous coordinates in image coordinate system are p (u v 1) ^Tλ is any one scale factor; K is the camera intrinsic parameter matrix, and wherein s is the pattern distortion factor, f _u, f _vBe the physical coordinates of image picture point on u direction and the v direction scale-up factor to the image pixel coordinate, i.e. effective focal length, (u ₀, v ₀) be the image coordinate of primary optical axis and plane of delineation intersection point.R is the rotation matrix of 3 * 3 unit quadratures; T is a translation vector; Simultaneously, (R, T) is that camera coordinate system is with respect to the position of world coordinate system.

The camera calibration process is exactly to determine the process of above-mentioned Parameters in Formula, and in general, camera marking method can be divided into three major types: traditional scaling method, based on camera marking method and the self-calibrating method of active vision.Wherein traditional scaling method is mainly world coordinates and its corresponding point coordinate on image that utilizes the point on known scenery, document " A Cameracalibration with one-dimensional objects " (Zhang Zhengyou.IEEE Transactions onPattern Analysis and Machine Intelligence for example, 2004,7 (26): 892-899); This kind method can be used in the video camera of arbitrary model, and stated accuracy is high, but its calibration process is complicated, needs high-precision known structure information, can't use calibrating block in a lot of situations in actual applications; Camera marking method based on active vision is mainly some movable information that utilizes known video camera or template, for example document " based on the active vision camera marking method " (Hu Zhanyi, Wu Fuchao. Chinese journal of computers, 2002:1149-1156); The method usually can linear solution and robustness higher, but can not use and uncontrollable occasion unknown at camera motion; Camera self-calibration method is the corresponding relation that only relies between multiple image, do not need to make high-precision calibrating template, document " A New Easy Camere Calibration Technique Based onCircle Points " (MENG Xiao-qiao for example, HU Zhan-yi.Journal of Software, 2000,13 (5): 957-965); The method dirigibility is stronger, and range of application is wider, but generally is nonlinear calibration, thereby computation process complexity and robustness are not too high.Thereby the research of linear self-calibrating method is become focus in current camera calibration.Such as document " a kind of improvement is based on the camera self-calibration method of annulus point " (Hu Peicheng etc. photoelectric project, 2007,34 (12): what propose 54-60) utilizes two pairs of mutual perpendicular diameter to obtain the annulus point coordinate to demarcate.

Summary of the invention

The invention provides a kind of planar rectangular that utilizes and be calibrating template, find the solution according to the character of quafric curve the method that annulus dot image coordinate carries out camera self-calibration.The method only needs video camera to take 3 width from different orientation or the image more than 3 width just can linearity solve camera intrinsic parameter.The method takes full advantage of the character of geometric properties and the quafric curve of rectangle, and calibration process is simple, and precision is higher.

Technical solution of the present invention

1, a kind of camera self-calibration method based on quafric curve, calibrate the method for the whole intrinsic parameters of video camera as calibrating template take a planar rectangular, it is characterized in that simple rectangle template of its employing, according to projective invariance, and taking full advantage of the character of geometric properties and the quafric curve of rectangle, linearity solves camera intrinsic parameter.Concrete steps comprise: each place, limit straight line of fitted rectangle, find the solution summit and vanishing point image coordinate, and find the solution each limit mid point and circumscribed circle center of circle image coordinate, find the solution planar circular dot image coordinate, find the solution the camera intrinsic parameter matrix.

(1) each place, limit straight line of fitted rectangle

If planar rectangular ABCD, some A, B, C, the picture of D are a, b, c, d extracts the characteristics of image point coordinate, utilizes least square fitting limit ab, bc, cd, da place straight line l ₁, l ₂, l ₃, l ₄

(2) find the solution summit and vanishing point image coordinate

If two groups of parallel lines l ₁, l ₃And l ₂, l ₄Upper vanishing point is p ₁, p ₂, utilize the projective transformation invariance, have:

$\left\{\begin{array}{c}a=l_1\×l_4,b=l_1\×l_2\\ c=l_2{\×l}_3,d=l_3{\×l}_4\end{array}\right.$ With $\left\{\begin{array}{c}{p}_1=l_1\×l_3\\ p_2=l_2\×l_4\end{array}\right..$

(3) find the solution each limit mid point and circumscribed circle center of circle image coordinate

If ab, bc, cd, the mid point of da are e, f, g, h, the circumscribed circle center of circle is o, according to projective invariance and harmonic conjugates relation, has:

$\left\{\begin{array}{c}(\mathrm{ab},{\mathrm{ep}}_1)=-1,(\mathrm{bc},{\mathrm{fp}}_2)=-1\\ (\mathrm{cd},{\mathrm{gp}}_1)=-1,(\mathrm{da},{\mathrm{hp}}_2)=-1\end{array}\right..$

If e, g place straight line is l _eg, f, h place straight line is l _fh, can obtain rectangle circumscribed circle center of circle image coordinate is o=l _eg* l _fh

(4) find the solution planar circular dot image coordinate

If rectangle diagonal line ac, on the straight line of bd place, the vanishing point coordinate is p ₃, p ₄, have:

If rectangle ABCD annulus point in the plane be I, J, respective image coordinate are m _i(x _r+ x _iI, y _r+ y _iI, 1) ^T, m _j(x _r-x _iI, y _r-y _iI, 1) ^TAccording to the character of quafric curve, if I, J, C, D regard four fixed points on circle as, and (l is arranged _AIl _AJ, l _ACl _AD)=(l _BIl _BJ, l _BCl _BD), can get according to the projective properties of double ratio: (m _im _j, p ₃p ₂)=(m _im _j, p ₂p ₄), that is:

$({2u}_{p2}-u_{p3}-u_{p4})({x_r}^2+{x_i}^2)+2(u_{p2}^2-u_{p3}{u}_{p4})x_r=2u_{p2}{u}_{p3}{u}_{p4}-u_{p2}^2(u_{p3}+u_{p4}).$

If I, J, B, D regard four fixed points on circle as, and (l is arranged _AIl _AJ, l _ABl _AD)=(l _CIl _CJ, l _CBl _CD), can get according to the projective properties of double ratio: (m _im _j, p ₁p ₂)=(m _im _j, p ₂p ₁), that is:

x _r ²+x _i ²-(u _p1+u _p2)x _r＝-u _p1u _p2。

Above two equations of simultaneous solve x _r, x _iValue, in like manner can solve y _r, y _i, namely obtain annulus point the picture m _i, m _jCoordinate

(5) utilize annulus point character, set up annulus dot image coordinate about the equation of constraint of camera intrinsic parameter, utilize the least square method linearity to solve intrinsic parameter.

Advantage of the present invention is:

(1) the present invention mainly is applicable to contain in scene circle, perhaps contains the condition of n (n 〉=4) the limit shape template that has circumscribed circle in photographed scene, belongs to contactless measurement, directly extracts in image unique point coordinate on curve.

(2) method of the present invention can calibrate the Intrinsic Matrix of 5 parameters of video camera, has comprised all parameters in the optical imagery, mainly contains the demarcation of optical imagery center, inclination factor and effective focal length.

(3) method that adopts in the present invention is to utilize the cross ratio invariability of geometric properties, harmonic conjugates relation and the projective transformation of n limit shape to find the solution apex coordinate and circumscribed circle central coordinate of circle, character direct solution annulus dot image coordinate according to quafric curve, do not need to carry out ellipse fitting, simplified calibration process.

Description of drawings

Fig. 1 the present invention is based on quafric curve, utilizes rectangle template to find the solution the process flow diagram of camera intrinsic parameter method.

The planar rectangular template that Fig. 2 the present invention adopts and its circumscribed circle structural representation.

Fig. 3 the present invention adopts the rectangle template imaging and finds the solution the principle schematic of annulus dot image coordinate.

Embodiment

The below is that the present invention is described in further detail.A kind of camera self-calibration method based on quafric curve has been proposed, calibrate the method for the whole intrinsic parameters of video camera as calibrating template take a planar rectangular, it is characterized in that simple rectangle template of its employing, according to projective invariance, and taking full advantage of the character of geometric properties and the quafric curve of rectangle, linearity solves camera intrinsic parameter.Concrete steps comprise: each place, limit straight line of fitted rectangle, find the solution summit and vanishing point image coordinate, and find the solution each limit mid point and circumscribed circle center of circle image coordinate, find the solution planar circular dot image coordinate, find the solution the camera intrinsic parameter matrix.

(1) each place, limit straight line of fitted rectangle

If planar rectangular ABCD, some A, B, C, the picture of D are a, b, c, d.Input picture extracts unique point coordinate on four edges with the function cvGoodFeaturesToTrack in OpenCV, utilizes least square fitting limit ab, bc, cd, da place straight line l ₁, l ₂, l ₃, l ₄

(2) find the solution summit and vanishing point image coordinate

If two groups of parallel lines l ₁, l ₃And l ₂, l ₄Upper vanishing point is p ₁, p ₂, utilize the projective transformation invariance, have:

$\left\{\begin{array}{c}a=l_1\×l_4,b=l_1\×l_2\\ c=l_2{\×l}_3,d=l_3{\×l}_4\end{array}\right.$ With $\left\{\begin{array}{c}{p}_1=l_1\×l_3\\ p_2=l_2\×l_4\end{array}\right..$

Separate the picture a that top two system of equations obtain the rectangle summit, b, c, d and two groups of opposite side ab, cd and bc, the picture p of the upper vanishing point infinity point of da ₁, p ₂Coordinate.

(3) find the solution each limit mid point and circumscribed circle center of circle image coordinate

If ab, bc, cd, the mid point of da are e, f, g, h, the circumscribed circle center of circle is o, according to projective invariance and harmonic conjugates relation, has:

$\left\{\begin{array}{c}(\mathrm{ab},{\mathrm{ep}}_1)=-1,(\mathrm{bc},{\mathrm{fp}}_2)=-1\\ (\mathrm{cd},{\mathrm{gp}}_1)=-1,(\mathrm{da},{\mathrm{hp}}_2)=-1\end{array}\right..$

Separate following formula and can obtain an e, f, g, the coordinate of h.

If e, g place straight line is l _eg, f, h place straight line is l _fh, can obtain rectangle circumscribed circle center of circle image coordinate is o=l _eg* l _fh

(4) find the solution planar circular dot image coordinate

If rectangle diagonal line ac, on the straight line of bd place, the vanishing point coordinate is p ₃, p ₄, have:

Solve an equation and both can obtain a p ₃, p ₄Coordinate.

If rectangle ABCD annulus point in the plane be I, J, respective image coordinate are m _i(x _r+ x _iI, y _r+ y _iI, 1) ^T, m _j(x _r-x _iI, y _r-y _iI, 1) ^TAccording to the character of quafric curve, if I, J, C, D regard four fixed points on circle as, and (l is arranged _AIl _AJ, l _ACl _AD)=(l _BIl _BJ, l _BCl _BD), can get according to the projective properties of double ratio: (m _im _j, p ₃p ₂)=(m _im _j, p ₂p ₄), that is:

$({2u}_{p2}-u_{p3}-u_{p4})({x_r}^2+{x_i}^2)+2(u_{p2}^2-u_{p3}{u}_{p4})x_r=2u_{p2}{u}_{p3}{u}_{p4}-u_{p2}^2(u_{p3}+u_{p4}).$

If I, J, B, D regard four fixed points on circle as, and (l is arranged _AIl _AJ, l _ABl _AD)=(l _CIl _CJ, l _CBl _CD), can get according to the projective properties of double ratio: (m _im _j, p ₁p ₂)=(m _im _j, p ₂p ₁), that is:

x _r ²+x _i ²-(u _p1+u _p2)x _r＝-u _p1u _p2。

Above two equations of simultaneous solve x _r, x _iValue, in like manner can solve y _r, y _i, namely obtain annulus point the picture m _i, m _jCoordinate

(5) utilize annulus point character, set up annulus dot image coordinate about the equation of constraint of camera intrinsic parameter, utilize the least square method linearity to solve intrinsic parameter.

Embodiment

The present invention proposes a kind of based on quafric curve, utilize the planar rectangular template to find the solution the camera intrinsic parameter method, calculation process as shown in Figure 1, the circumscribed circle structural representation of planar rectangular template and it as shown in Figure 2, rectangle template imaging and the principle schematic of finding the solution annulus dot image coordinate are as shown in Figure 3.

The below makes more detailed description with an example to embodiment of the present invention:

The calibrating template that adopts based on the shooting Camera Self-Calibration method of quafric curve is a planar rectangular ABCD arbitrarily, as shown in Figure 2.The rectangle template that adopts in example of the present invention is set to long 30cm, wide 20cm, i.e. and AB=CD=20cm, it is coordinate origin that AD=BC=30cm. gets a B, and BC is the x axle, and BA is that the y axle is set up plane right-angle coordinate B-xy.Estimate each summit A, B, C, the homogeneous coordinates of D (as Fig. 2) are respectively: A (0,20,1) ^T, B (0,0,1) ^T, C (30,0,1) ^T, D (30,20,1) ^T

Utilize the method in the present invention that camera intrinsic parameter is demarcated, concrete steps are as follows:

(1) extract minutiae

The image resolution ratio that adopts in the present invention is 1280 * 960, and input picture is chosen m (m 〉=3) width clearly, and the image that is evenly distributed of unique point, utilizes function in Opencv to extract the coordinate of image characteristic point.

(2) each place, limit straight line of fitted rectangle

If planar rectangular ABCD, some A, B, C, the picture of D are a, b, c, d.Input picture extracts unique point coordinate on four edges with the function cvGoodFeaturesToTrack in OpenCV, utilizes least square fitting limit ab, bc, cd, da place straight line l ₁, l ₂, l ₃, l ₄

The below is to find the solution straight line l in piece image ₁, l ₂, l ₃, l ₄Be example, provide solution procedure.

In the present invention, establish straight line l ₁, l ₂, l ₃, l ₄Equation is k _ix+b _iY=1 (i=1,2,3,4), l _iLine coordinates be (k _i, b _i,-1) ^TChoose piece image, extract upper different 4 points of limit AB of rectangle, utilize least square method to calculate l ₁=(0.00052871139 ,-0.00762418744 ,-1) ^T, l ₂=(0.00786602122 ,-0.00969718712 ,-1) ^T, l ₃=(0.00082612813,0.00040698276 ,-1) ^T, l ₄=(0.00126051181,0.00125035 ,-1) ^T

(3) find the solution summit and vanishing point image coordinate

The straight line l of the below to solve in previous step ₁, l ₂, l ₃, l ₄Coordinate is example, provides the process of finding the solution summit and vanishing point image coordinate.

If two groups of parallel lines l ₁, l ₃And l ₂, l ₄Upper vanishing point is p ₁, p ₂, according to a=l ₁* l ₄, b=l ₁* l ₂, c=l ₂* l ₃, d=l ₃* l ₄And p ₁=l ₁* l ₃, p ₂=l ₂* l ₄The inhomogeneous coordinate that obtains each summit imaging of rectangle is: a=(864.0002 ,-71.2460) ^T, b=(901.1559,627.8637) ^T, c=(526.5804 ,-261.8013) ^T, d=(108.8.8594,1155.4719) ^T, p ₁=(1320.18245 ,-222.7117) ^T, p ₂=(4584.1284 ,-3821.6083) ^T

(4) find the solution each limit mid point and circumscribed circle center of circle image coordinate

If ab, bc, cd, the mid point of da are e, f, g, h, the circumscribed circle center of circle is o.According to (ab, ep ₁)=-1, (bc, fp ₂)=-1 and (cd, gp ₁)=-1, (da, hp ₂)=-1, the inhomogeneous coordinate that obtains each limit mid point is: e=(350.0086 ,-10.7419) ^T, f=(391.2946 ,-201.2505) ^T, g=(776.3181,462.3985) ^T, h=(271.4846-594.8949) ^T

If e, g place straight line is l _eg, f, h place straight line is l _fh, by o=l _eg* l _fhThe imaging inhomogeneous coordinate that obtains the circumscribed circle center of circle is: o=(3278.2253,1513.3843) ^T

(5) find the solution planar circular dot image coordinate

If rectangle diagonal line ac, on the straight line of bd place, the vanishing point coordinate is p ₃, p ₄, according to (ac, op ₃)=-1 and (bd, op ₄)=-1 obtains p ₃=(257.5436-131.0042) ^T, p ₃=(257.5436-131.0042) ^T

If rectangle ABCD annulus point in the plane be I, J, respective image coordinate are m _i(x _r+ x _iI, y _r+ y _iI, 1) ^T, m _j(x _r-x _iI, y _r-y _iI, 1) ^TAccording to following system of equations:

$\left\{\begin{array}{c}({2u}_{p2}-u_{p3}-u_{p4})({x_r}^2+{x_i}^2)+2(u_{p2}^2-u_{p3}{u}_{p4})x_r=2u_{p2}{u}_{p3}{u}_{p4}-u_{p2}^2(u_{p3}+u_{p4})\\ ({2v}_{p2}-v_{p3}-v_{p4})({y_r}^2+{y_i}^2)+2(v_{p2}^2-v_{p3}{v}_{p4})y_r=2v_{p2}{v}_{p3}{v}_{p4}-v_{p2}^2(v_{p3}+v_{p4})\end{array}\right.$ With

$\left\{\begin{array}{c}{x_r}^2+{x_i}^2-(u_{p1}+u_{p2})x_r=-u_{p1}{u}_{p2}\\ {y_r}^2+{y_i}^2-(v_{p1}+v_{p2})y_r=-v_{p1}{v}_{p2}\end{array}\right.,$ Can obtain: $\left\{\begin{array}{c}{x}_r=-338.9546\\ x_i=-2197.8776\end{array}\right.,$ $\left\{\begin{array}{c}{y}_r=-820.8064\\ y_i=1339.6880\end{array}\right.,$

Namely obtain the picture m of annulus point _i, m _jCoordinate be m _i=(338.9546-2197.8776i ,-820.8064+1339.6880i, 1) ^T, m _i=(338.9546+2197.8776i ,-820.8064-1339.6880i, 1) ^T

(6) find the solution camera intrinsic parameter

Choose m (m 〉=3) image, repeat above 5 steps, obtain m (m 〉=3) to the picture coordinate of annulus point, the picture coordinate of setting up annulus point is as follows about the equation of camera intrinsic parameter:

$\left(\begin{array}{cccccc}{x_r}^2-{x_i}^2& 2(x_r{y}_r-x_i{y}_i)& 2x_r& {y_r}^2-{y_i}^2& 2y_r& 1\\ x_r{x}_i& x_r{y}_i+x_i{y}_r& x_i& y_r{y}_i& y_i& 0\end{array}\right)C=0.$

Utilize m (m 〉=3) width image to obtain m group system of equations as above, their combination is obtained, AC=0, wherein A is the matrix of 2m * 6, and C is the column vector of 6 * 1, and it is by symmetric matrix ω=K ^-TK ^-1Determine.According to least square method, differ under a constant factor can unique C=of set matrix really (0.00000021552010 ,-0.00000000002155,-0.00013792250932,0.00000021552009 ,-0.00010343583915,0.99999998513916) ^T

Obtain $\mathrm{\ω}=\left(\begin{array}{ccc}0.00000021552010& -0.00000000002155& -0.00013792250932\\ -0.00000000856906& 0.00000021552009& -0.00010343583915\\ -0.00013792250932& -0.00010343583915& 0.99999998513916\end{array}\right),$

After utilizing the Cholesky decomposition method that ω is decomposed again finding the inverse matrix obtain K, then with last element normalized of K, the Intrinsic Matrix that namely obtains video camera is

Whole process is carried out according to process flow diagram shown in Figure 1, extracts successively characteristics of image point coordinate, each place, limit straight line of fitted rectangle, finds the solution summit and vanishing point image coordinate, finds the solution each limit mid point and circumscribed circle center of circle image coordinate, finds the solution planar circular dot image coordinate, finds the solution camera intrinsic parameter.

Claims (1)

1. the camera self-calibration method based on quafric curve, is characterized in that the method adopts a simple planar rectangular template, takes full advantage of the character of geometric properties and the quafric curve of rectangle, and according to projective invariance, linearity solves camera intrinsic parameter; Concrete steps comprise:

(1) each place, limit straight line of fitted rectangle

If planar rectangular ABCD, some A, B, C, the picture of D are a, b, c, d extract the characteristics of image point coordinate, utilize least square fitting limit ab, bc, cd, da place straight line 1 ₁, 1 ₂, 1 ₃, 1 ₄

(2) find the solution summit and vanishing point image coordinate

If two groups of parallel lines 1 ₁, 1 ₃With 1 ₂, 1 ₄Upper vanishing point is p ₁, p ₂, utilize the projective transformation invariance, have:

With

(3) find the solution each limit mid point and circumscribed circle center of circle image coordinate

If ab, bc, cd, the mid point of da are e, f, g, h, the circumscribed circle center of circle is o, according to projective invariance and harmonic conjugates relation, has:

If e, g place straight line is 1 _eg, f, h place straight line is 1 _fh, can obtain rectangle circumscribed circle center of circle image coordinate is o=1 _eg* 1 _fh

(4) find the solution planar circular dot image coordinate

If rectangle diagonal line ac, on the straight line of bd place, the vanishing point coordinate is p ₃, p ₄, have:

If rectangle ABCD annulus point in the plane be I, J, respective image coordinate are m _i(x _r+ x _iI, y _r+ y _iI, 1) ^T, m _j(x _r-x _iI, y _r-y _iI, 1) ^TAccording to the character of the curve of order 2 in quafric curve, if I, J, C, D regard four fixed points on circle as, have (1 _AI1 _AJ, 1 _AC1 _AD)=(1 _BI1 _BJ, 1 _BC1 _BD), can get according to the projective properties of double ratio: (m _im _j, P ₃P ₂)=(m _im _j, P ₂P ₄), that is:

If I, J, B, D regard four fixed points on circle as, have (1 _AI1 _AJ, 1 _AB1 _AD)=(1 _CI1 _CJ, 1 _CB1 _CD), can get according to the projective properties of double ratio: (m _im _j, P _iP ₂)=(m _im _j, P ₂P ₁), that is:

x _r ²+x _i ²-(u _p1+u _p2)x _r＝-u _p1u _p2 (3)

y _r ²+y _i ²-(v _p1+v _p2)x _r＝-v _p1v _p2 (4)

The above equation of simultaneous (1) and equation (3) solve x _r, x _iValue; The above equation of simultaneous (2) and equation (4) solve y _r, y _iValue; Namely obtain the picture m of annulus point _i, m _jCoordinate;

(5) utilize annulus point character, set up annulus dot image coordinate about the equation of constraint of camera intrinsic parameter, utilize the least square method linearity to solve intrinsic parameter.

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