CN102402785A - 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

The three-dimensional reconstruction technology is one of important topic of computer vision research, is meant the spatial geometric shape that recovers three-dimensional body from the image of three-dimensional body.Traditional three-dimensional reconstruction all is to serve as to rebuild foundation with the image that the video camera of having demarcated obtains, and 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 the computer vision field

What adopt usually 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: the homogeneous coordinates of any point P in world coordinate system are P in the space _w(x _wy _wz _w1) ^T, the homogeneous coordinates in image coordinate system are p (u v 1) ^Tλ is any 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 v direction scale-up factor, i.e. effective focal length, (u to the image pixel coordinate ₀, v ₀) be the image coordinate of primary optical axis and plane of delineation intersection point.R is the rotation matrix of one 3 * 3 unit quadrature; T is a translation vector; Simultaneously, (R T) is the position of camera coordinate system with respect to world coordinate system.

The camera calibration process is exactly to confirm the process of parameter in the above-mentioned formula, and in general, camera marking method can be divided into three major types: traditional scaling method, based on the camera marking method and the self-calibrating method of active vision.Wherein traditional scaling method mainly is the world coordinates and its pairing point coordinate on image that utilizes the point on the 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, in practical application, can't use calibrating block under a lot of situation; Camera marking method based on active vision mainly is 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); This method usually can linear solution and robustness higher, but can not use unknown and uncontrollable occasion at camera motion; Camera self-calibration method is the corresponding relation that only relies between the multiple image; Need not 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); This flexible method property is stronger, and range of application is wider, but generally is nonlinear calibration, thereby computation process complicacy and robustness are not too high.Thereby the research to linear self-calibrating method becomes the focus in the current camera calibration.For example 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 the method that annulus dot image coordinate carries out camera self-calibration according to the character of quafric curve.This method only needs video camera to take 3 width of cloth or the image more than 3 width of cloth from different orientation just can linearity solve camera intrinsic parameter.This method has made full use of the character of the 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; With a planar rectangular is the method that calibrating template calibrates the whole intrinsic parameters of video camera; It is characterized in that simple rectangle template of its employing; According to projective invariance, and made full use of the character of the 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 ₄Last vanishing point is p ₁, p ₂, utilize the projective transformation invariance, then have:

$\left\{\begin{array}{c}a=l_1\&times;l_4,b=l_1\&times;l_2\\ c=l_2{\&times;\mathrm{<figure-callout\; id="3"\; label="l"\; filenames="HSA00000258757300012.png,HSA00000258757300021.png"\; state="\{\{state\}\}">l</figure-callout>}}_3,d={\mathrm{<figure-callout\; id="3"\; label="l"\; filenames="HSA00000258757300012.png,HSA00000258757300021.png"\; state="\{\{state\}\}">l</figure-callout>}}_3{\&times;l}_4\end{array}\right.$ With $\left\{\begin{array}{c}{p}_1=l_1\&times;l_3\\ p_2=l_2\&times;l_4\end{array}\right..$

(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, then 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, then 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, the vanishing point coordinate is p on the straight line of bd place ₃, p ₄, then have:

If the annulus point on the plane, rectangle ABCD place is 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 the circle as, and (l is then 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:

$({2u}_{p2}-u_{p3}-u_{p4})({x_r}^2+{x_i}^2)+2(u_{p2}^2-u_{p3}{u}_{p4})x_r=2u_{p2}{u}_{p3}{u}_{p4}-u_{p2}^2({\mathrm{<figure-callout\; id="3"\; label="u\; p"\; filenames="HSA00000258757300012.png,HSA00000258757300021.png"\; state="\{\{state\}\}">u</figure-callout>}}_p+u_{p4}).$

If I, J, B, D regard four fixed points on the circle as, and (l is then 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, promptly obtain annulus point the picture m _i, m _jCoordinate

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

Advantage of the present invention is:

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

(2) method of the present invention can calibrate the intrinsic parameter 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 among the present invention is to utilize the double ratio unchangeability of the geometric properties of n limit shape, harmonic conjugates relation and projective transformation to find the solution apex coordinate and circumscribed circle central coordinate of circle; Directly find the solution annulus dot image coordinate according to the character of quafric curve; Need not carry out ellipse fitting, simplify 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.

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

Be that the present invention is done further detailed description below.A kind of camera self-calibration method based on quafric curve has been proposed; With a planar rectangular is the method that calibrating template calibrates the whole intrinsic parameters of video camera; It is characterized in that simple rectangle template of its employing; According to projective invariance, and made full use of the character of the 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 the four edges with the function cvGoodFeaturesToTrack among the 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 ₄Last vanishing point is p ₁, p ₂, utilize the projective transformation invariance, then have:

$\left\{\begin{array}{c}a=l_1\&times;l_4,b=l_1\&times;l_2\\ c=l_2{\&times;\mathrm{<figure-callout\; id="3"\; label="l"\; filenames="HSA00000258757300012.png,HSA00000258757300021.png"\; state="\{\{state\}\}">l</figure-callout>}}_3,d={\mathrm{<figure-callout\; id="3"\; label="l"\; filenames="HSA00000258757300012.png,HSA00000258757300021.png"\; state="\{\{state\}\}">l</figure-callout>}}_3{\&times;l}_4\end{array}\right.$ With $\left\{\begin{array}{c}{p}_1=l_1\&times;l_3\\ p_2=l_2\&times;l_4\end{array}\right..$

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 last 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, then 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, then 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, the vanishing point coordinate is p on the straight line of bd place ₃, p ₄, then have:

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

If the annulus point on the plane, rectangle ABCD place is 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 the circle as, and (l is then 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:

$({2u}_{p2}-u_{p3}-u_{p4})({x_r}^2+{x_i}^2)+2(u_{p2}^2-u_{p3}{u}_{p4})x_r=2u_{p2}{u}_{p3}{u}_{p4}-u_{p2}^2({\mathrm{<figure-callout\; id="3"\; label="u\; p"\; filenames="HSA00000258757300012.png,HSA00000258757300021.png"\; state="\{\{state\}\}">u</figure-callout>}}_p+u_{p4}).$

If I, J, B, D regard four fixed points on the circle as, and (l is then 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, promptly obtain annulus point the picture m _i, m _jCoordinate

(5) utilize annulus point character, set up the equation of constraint of annulus dot image coordinate, utilize the least square method linearity to solve intrinsic parameter about camera 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 is as shown in Figure 1; The planar rectangular template is as shown in Figure 2 with its circumscribed circle structural representation, rectangle template imaging and to find the solution the principle schematic of annulus dot image coordinate as shown in Figure 3.

With an instance embodiment of the present invention are made more detailed description below:

The calibrating template that adopts based on the shooting camera self-calibrating method of quafric curve is a planar rectangular ABCD arbitrarily, and is as shown in Figure 2.The rectangle template that in instance of the present invention, adopts 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 sets up plane right-angle coordinate B-xy for the y axle.Then estimate each summit A, B, C, the homogeneous coordinates of D (like 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 among the present invention that camera intrinsic parameter is demarcated, concrete steps are following:

(1) extract minutiae

The image resolution ratio that adopts among the present invention is 1280 * 960, and input picture is chosen m (m >=3) width of cloth clearly, and the image that is evenly distributed of unique point, utilizes function among the 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 the four edges with the function cvGoodFeaturesToTrack among the OpenCV, utilizes least square fitting limit ab, bc, cd, da place straight line l ₁, l ₂, l ₃, l ₄

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

Among 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), then l _iLine coordinates be (k _i, b _i,-1) ^TChoose piece image, the limit AB that extracts rectangle goes up different 4 points, utilizes 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

Below the straight line l that solves in an above step ₁, l ₂, l ₃, l ₄Coordinate is an 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 ₄Last 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, then 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, the vanishing point coordinate is p on the straight line of bd place 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 the annulus point on the plane, rectangle ABCD place is 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 equation group:

$\left\{\begin{array}{c}({2u}_{p2}-u_{p3}-u_{p4})({x_r}^2+{x_i}^2)+2(u_{p2}^2-u_{p3}{u}_{p4})x_r=2u_{p2}{u}_{p3}{u}_{p4}-u_{p2}^2({\mathrm{<figure-callout\; id="3"\; label="u\; p"\; filenames="HSA00000258757300012.png,HSA00000258757300021.png"\; state="\{\{state\}\}">u</figure-callout>}}_p+u_{p4})\\ ({2v}_{p2}-v_{p3}-v_{p4})({y_r}^2+{y_i}^2)+2(v_{p2}^2-v_{p3}{v}_{p4})y_r=2v_{p2}{v}_{p3}{v}_{p4}-v_{p2}^2({\mathrm{<figure-callout\; id="3"\; label="v\; p"\; filenames="HSA00000258757300012.png,HSA00000258757300021.png"\; state="\{\{state\}\}">v</figure-callout>}}_p+v_{p4})\end{array}\right.$ 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: $\left\{\begin{array}{c}{x}_r=-338.9546\\ x_i=-2197.8776\end{array}\right.,$ $\left\{\begin{array}{c}{y}_r=-820.8064\\ y_i=1339.6880\end{array}\right.,$

Promptly 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 the picture coordinate of m (m >=3) to annulus point, the picture coordinate of setting up annulus point is following 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 of cloth 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 one 6 * 1 a column vector, and it is by symmetric matrix ω=K ^-TK ^-1Confirm.According to least square method, in that differ under the constant factor can unique C=of set matrix really (0.00000021552010 ,-0.00000000002155;-0.00013792250932; 0.00000021552009 ,-0.00010343583915,0.99999998513916) ^T

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

After utilizing the Cholesky decomposition method that ω is decomposed again finding the inverse matrix obtain K; Last element normalization of K is handled, the intrinsic parameter matrix that promptly obtains video camera is for again

Whole process is carried out according to process flow diagram shown in Figure 1, extracts characteristics of image point coordinate, each place, limit straight line of fitted rectangle successively, 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 simple plane rectangle template of its employing, makes full use of the character of the geometric properties and the quafric curve of rectangle, and according to projective invariance, 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 ₄Last vanishing point is p ₁, p ₂, utilize the projective transformation invariance, then have:

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

(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, then 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, then 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, the vanishing point coordinate is p on the straight line of bd place ₃, p ₄, then have:

If the annulus point on the plane, rectangle ABCD place is 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 the quafric curve, if I, J, C, D regard four fixed points on the circle as, and (l is then 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 the circle as, and (l is then 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, promptly obtain annulus point the picture m _i, m _jCoordinate.

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

Images