CN101025824A - Calibrating method based on fixed parameters and variable parameters for three-dimensional scanning system - Google Patents

Abstract

A calibration method based on fixed and variable parameters in three-dimensional scanning system relates to the camera model parameter calibration in the visual system including the following steps: (1) fixed parameters calibration is primarily to acquire coordinates of sampling points on circular contour and heart in image coordinates through image processing techniques, and then use n sampling points to get aspect ratio through least squares method, (2) after obtaining the fixed parameters, it uses the image coordinates of image midpoint and the corresponding space coordinates of world midpoint to demarcate the initial value of variable parameters for the camera, including the equivalent focal length, main point coordinates, axis tilt factor, distortion factor and external parameters, (3) after demarcation, it takes fixed parameters as a constraint for nonlinear optimization objective function to establish a binding target function and get the optimal solution of variable parameters through the nonlinear optimization method.

Description

Scaling method in the 3 D scanning system based on preset parameter and variable element

Technical field

The present invention relates to the demarcation problem of the camera model parameter in the vision system, relate in particular to the scaling method in a kind of 3 D scanning system based on preset parameter and variable element.

Background technology

One of basic task of computer vision is to take the image that obtains from video camera, calculates the three-dimensional information of object in the visual field, comes thus three-dimensional body is rebuild and discerned.The three-dimensional geometric information of body surface point and its mutual relationship between the corresponding point on the image are that in fact the process of setting up this geometric model is exactly the solution procedure of camera parameters by the decision of the imaging model of video camera.Therefore, the demarcation to camera parameters is the prerequisite and the key of this modeling process.Solution procedure to camera parameters is called camera calibration.

Document " Image Processing; Analysis; and Machine Vision " (M.Sonka, V.Hlavac, R.Boyle, International Thomson Publishing, 1998) set forth a kind of comparatively general video camera imaging model in, this imaging model can be described with following formula:

$\left[\begin{array}{c}u\\ v\\ 1\end{array}\right]=\mathrm{\λA}\left[\begin{array}{cc}R& T\end{array}\right]\left[\begin{array}{c}{X}_w\\ Y_w\\ Z_w\\ 1\end{array}\right]$

Wherein, X _w, Y _w, Z _wBe the world coordinate system coordinate of demarcating thing, u, v are the two-dimensional coordinates in image coordinate system, and λ is a scalar, and R, T are the external parameter matrix of video camera, have defined video camera respectively in three-dimensional attitude and position, $A=\left[\begin{array}{ccc}{f}_x& s& u_0\\ 0& f_y& v_0\\ 0& 0& 1\end{array}\right]$ Be intrinsic parameters of the camera matrix, wherein f _x, f _yBe illustrated respectively in the scale-up factor of the physical coordinates of picture point on x direction and the y direction to image coordinate, or be called the equivalent focal length on x direction and the y direction, s represents between centers inclination factor, (u ₀, v ₀) the principal point coordinate of presentation video.

Camera calibration is exactly a process of calculating the camera model parameter.The camera calibration technology roughly can be divided into two classes: traditional camera marking method and camera self-calibration method.

In recent years, video camera obtained very big progress from calibration algorithm, delivered a considerable amount of documents, the some of them algorithm has obtained comparatively widely to use.But because poor with respect to traditional calibration algorithm precision, be not suitable for accuracy of detection being required very high occasion such as 3-D scanning etc. from calibration algorithm.

Traditional calibration algorithm has also obtained using comparatively widely, has also obtained effect preferably simultaneously.Document " A flexible new technique for camera calibration " (Zhang Z Y.IEEETransaction on Pattern Analysis and Machine Intelligence for example, 2000,22 (11): the constraint condition that discloses a kind of corresponding relation that utilizes unique point on the plane template and intrinsic parameters of the camera 1330-1334) is found the solution the scaling method of camera parameters.This method does not need to know the size of marker, only utilize the corresponding relation of the unique point on the plane template, plane template any rotation before video camera can be finished demarcation more than twice or twice, need not to understand the kinematic parameter of template, calibration process is become be more prone to convenient.Shortcoming is only to have considered radial distortion in this method, and is inapplicable for the occasion that tangential distortion is bigger.What adopt when finding the solution initial value in addition is the unique point of entire image, rather than adopts near the less unique point of distortion the picture centre, and initial value is found the solution improper meeting and caused the convergence of nonlinear optimization and the reduction of stated accuracy like this.

Existing traditional cameras scaling method generally is divided into camera parameters inner parameter and external parameter, and inner parameter is usually demarcated in the laboratory and finished, and the demarcation of external parameter is then finished at the scene the 3-D scanning process.This sorting technique is not perfectly in the engineering practical application, usually can bring bigger systematic error.If this is because think that when each the measurement inner parameter is constant and do not demarcate, thereby but in fact some the easy variable element in the inner parameter can change and influence system accuracy; If all demarcate inner parameter at every turn, but be limited to the condition of measure field, can be bigger for the calibrated error of some fixed value in the inner parameter, thus cause the increase of systematic error.

Summary of the invention

The invention provides the scaling method in a kind of 3 D scanning system, have the advantage of stated accuracy height and good stability based on preset parameter and variable element.

Technical scheme of the present invention is as follows: the scaling method based on preset parameter and variable element in the 3 D scanning system of the present invention comprises: at first demarcate preset parameter, also promptly demarcate aspect ratio; Utilize the plane template method to demarcate the initial value of variable element then, promptly demarcate external parameter, principal point coordinate, focal length, between centers inclination factor, distortion factor; At last preset parameter as a constraint, set up the optimization aim function of belt restraining, with the Levenberg-Marquardt method variable element is carried out nonlinear optimization and tries to achieve optimum solution.

The operation steps of this method is:

Step 1: at first vertically take the standard form circle, extract the circle contour in the image coordinate system, utilize least square method that the circle contour that extracts is carried out curve fitting, obtain the central coordinate of circle under the image coordinate system, then, utilization is as the corresponding relation of the plane and the plane of delineation, will be as circle contour equation on the plane:

(x _l-x _c) ²+(y _i-y _c) ²＝r ²，i＝1，2，3，…，n

Be transformed to the circle contour equation of representing with image coordinate:

[μ _x(u _i-u _c)] ²+[μ _y(v _l-v _c)] ²＝r ²，i＝1，2，3，…，n

(x wherein _i, y _l), i=1,2,3 ..., n is as the coordinate of putting on the flat circle profile, (x _c, y _c) be central coordinate of circle, u _i, v _i, u _c, v _cHorizontal ordinate, ordinate, center of circle horizontal ordinate and the ordinate of match circle contour point in the difference presentation video, r is a match radius of a circle on the plane of delineation, μ _x, μ _yFor the physical size of each pixel on u axle and v direction of principal axis, make τ _Xy=μ _x/ μ _y, by μ _x, μ _yDefinition know τ _XyBe exactly aspect ratio to be calibrated, then can obtain:

${\left[{\mathrm{\τ}}_{\mathrm{xy}}(u_i-u_c)\right]}^2+{(v_i-v_c)}^2=\frac{r^2}{{\mathrm{\μ}}_y^2},i=\mathrm{1,2,3},...,n$

At last, utilize the sampled point of the n on the circle contour in the image coordinate system, utilize least square method to obtain aspect ratio τ again _Xy

Step 2: try to achieve the preset parameter τ of video camera _XyAfter, utilizing image coordinate and its corresponding world coordinate of image coordinate system mid point is the volume coordinate of mid point, find the solution homography matrix, according to the initial value of homography matrix calibrating camera variable element, this initial value comprises equivalent focal length, principal point coordinate, between centers inclination factor, distortion factor and external parameter then;

Step 3: after calibrating the initial value of variable element, be a constraint of nonlinear optimization objective function, set up the objective function of belt restraining with the preset parameter:

Find the solution the optimum solution of variable element by nonlinear optimization method.

Compared with prior art, the present invention has following advantage:

(1) among the present invention camera parameters is divided into preset parameter and variable element, agrees with the engineering practical application more.

(2) the present invention has more intactly demarcated the preset parameter and the variable element of video camera, comprises aspect ratio, equivalent focal length, principal point coordinate, between centers inclination factor, attitude parameter and translation parameters, has demarcated the distortion factor of camera lens simultaneously.

(3) adopt nonlinear optimization algorithm among the present invention, with the constraint of the preset parameter demarcated as the nonlinear optimization objective function, set up the objective function of belt restraining, by the demarcation of simple calibration facility realization to variable element, make on-the-spot calibration process simple and efficient, keep higher precision again.

(3) used plane reference object to demarcate among the present invention, plane reference object is three-dimensional to be demarcated thing and has and make simply, and the precision advantages of higher has reduced in the calibration process the dependence of high-precision calibrating thing, has simplified calibration process.

Description of drawings

Fig. 1 is scaling board figure.

Fig. 2 is the standard form circle diagram.

Fig. 3 demarcates the preset parameter process flow diagram.

Fig. 4 demarcates variable element initial value process flow diagram.

Fig. 5 is the 3 D scanning system structural drawing.

Fig. 6 is a camera model parameter calibration process flow diagram.

Embodiment

Demarcation based on preset parameter and variable element in the 3 D scanning system comprises: earlier preset parameter is demarcated; Associated ideal pin hole perspective model solves homography matrix then, utilizes the initial value of plane template method calibrating camera variable element; The constraint of preset parameter, objective function is carried out the optimum solution that nonlinear optimization obtains variable element at last as the nonlinear optimization objective function.

With reference to the accompanying drawings, specific embodiments of the present invention are made more detailed description:

The present invention utilizes plane reference object---and scaling board carries out camera parameters to be demarcated, and array distribution has round monumented point on the scaling board, as shown in Figure 1.Handle by video camera is taken the image that obtains, obtain the world coordinate system coordinate and the image coordinate system coordinate of monumented point.Calibration algorithm among the present invention utilizes these center of circle data, and the camera model parameter is demarcated.

Concrete steps are as follows:

(1) demarcation of preset parameter

Utilize image processing techniques to carry out the demarcation of aspect ratio.At first vertically take the standard form circle, as shown in Figure 2, extract circle contour then, utilize least square method that the circle contour that extracts is carried out curve fitting at last, obtain the central coordinate of circle under the image coordinate system.If be as circle contour equation on the plane:

(x _l-x _c) ²+(y _l-y _c) ²＝r ²，i＝1，2，3，…，n

(x wherein _l, y _i), i=1,2,3 ..., n is as the coordinate of putting on the flat circle profile, (x _c, y _c) be central coordinate of circle, r is a radius.If the physical size of each pixel on u axle and v direction of principal axis is μ _x, μ _y, by camera model as can be known, under the situation of not considering the between centers inclination factor, as planar point (x, y) with plane of delineation corresponding point (u, pass v) is:

$\left\{\begin{array}{c}u=u_0+\frac{x}{{\mathrm{\μ}}_x}\\ v=v_0+\frac{y}{{\mathrm{\μ}}_y}\end{array}\right.$

Following formula is rewritten as:

$\left\{\begin{array}{c}x={\mathrm{\μ}}_x(u-u_0)\\ y={\mathrm{\μ}}_y(v-v_0)\end{array}\right.$

The circle contour equation that can obtain thus representing with image coordinate is:

[μ _x(u _l-u _c)] ²+[μ _y(v _i-v _c)] ²＝r ²，i＝1，2，3，…，n

U wherein _i, v _i, u _c, v _cHorizontal ordinate, ordinate, center of circle horizontal ordinate and the ordinate of match circle contour point on the difference presentation video plane, r is a match radius of a circle on the plane of delineation.

Make τ _Xy=μ _x/ μ _y, by μ _x, μ _yDefinition know τ _XyBe exactly aspect ratio to be calibrated, then can obtain:

${\left[{\mathrm{\τ}}_{\mathrm{xy}}(u_i-u_c)\right]}^2+{(v_i-v_c)}^2=\frac{r^2}{{\mathrm{\μ}}_y^2},i=\mathrm{1,2,3},...,n$

Obviously this can find the solution with linear least-squares, and corresponding optimization aim function is:

$F({\mathrm{\τ}}_{\mathrm{xy}},{\mathrm{\μ}}_y)=\underset{{\mathrm{\τ}}_{\mathrm{xy}},{\mathrm{\μ}}_y}{\min }\left({\underset{i}{\mathrm{\Σ}}\{{\left[{\mathrm{\τ}}_{\mathrm{xy}}(u_i-u_c)\right]}^2+{(v_l-v_c)}^2-\frac{r^2}{{\mathrm{\μ}}_y^2}\}}^2\right)$

Utilize the sampled point of the n on the circle contour in the image coordinate system, can be write the optimization aim function as matrix form:

$\left[\begin{array}{cc}{(u_1-u_c)}^2& -r^2\\ {(u_2-u_c)}^2& -r^2\\ .& .\\ .& .\\ .& .\\ {(u_n-u_c)}^2& -r^2\end{array}\right]\left[\begin{array}{c}{\mathrm{\τ}}_{\mathrm{xy}}^2\\ \frac{1}{{\mathrm{\μ}}_y^2}\end{array}\right]=\left[\begin{array}{c}-{(v_1-v_c)}^2\\ -{(v_2-v_c)}^2\\ .\\ .\\ .\\ -{(v_n-v_c)}^2\end{array}\right]$

Utilize least square method to find the solution following formula again and just can obtain aspect ratio τ _Xy, concrete solution procedure is seen accompanying drawing 3.

(2) the first demarcation of video camera variable element

Try to achieve the preset parameter τ of video camera _XyAfter, utilizing image coordinate and its corresponding world coordinate of image coordinate system mid point is the volume coordinate of mid point, find the solution homography matrix, find the solution the initial value of video camera variable element then according to homography matrix, mainly comprise equivalent focal length, principal point coordinate, between centers inclination factor, distortion factor and external parameter.Document " A Flexible New Technique for Camera Calibration " (ZhangZ Y, IEEE Transactions on Pattern Analysis and Machine Intelligence, 2000,20 (11): 1330-1334) propose a kind of calibration algorithm that is called as the plane template method, it mainly is the inside and outside parameter except that distortion factor of at first coming the linear solution video camera with desirable pin hole perspective model, utilize the actual imaging model to find the solution distortion factor then, utilize the nonlinear optimization algorithm to optimize all camera interior and exterior parameters at last.Adopted first step algorithm in the document among the present invention, but difference is specifically to have provided among the present invention the step to image coordinate and world coordinates normalized, has considered tangential distortion in addition when finding the solution distortion factor.

Make the Z of world coordinate system _wComponent is 0, and then desirable pin hole perspective model can be expressed as:

Wherein

H=λ A[r ₁r ₂T]=[h ₁h ₂h ₃], r _i(i=1～3) and h _iThe column vector of (i=1～3) expression rotation matrix R and matrix H, matrix A is the intrinsic parameter matrix, w, λ are scalar;

The H matrix is called homography matrix, and it has represented the homography between the calibrating template plane and the plane of delineation, and degree of freedom is 8, therefore needs at least 4 above non-colinear corresponding point, just can obtain the H matrix of band scale factor.Finding the solution of H matrix is a non-linear least square problem, can carry out nonlinear optimization with the Levenberg-Marquardt method and solve.Find the solution with nonlinear optimization method and to need a suitable initial value and come iteration, the initial value preparation method is as follows:

Suppose h _iI the row vector of representing matrix H can be rewritten into the desirable pin hole perspective model of video camera:

Order $X={\left[\begin{array}{ccc}{h}_1^T& h_2^T& h_3^T\end{array}\right]}^T,$ Thereby following formula is expressed as:

If n is arranged to given point, we just can obtain n above-mentioned system of equations, and this n of simultaneous system of equations just can obtain the matrix equation of a shape such as LX=0, and wherein L is the matrix of 2n * 9, and separating of X is matrix L ^TThe minimal eigenvalue characteristic of correspondence vector of L.In the actual solution procedure, the formation of observation L matrix as can be known, its element has plenty of the image coordinate system coordinate, has plenty of the world coordinate system coordinate, what have then is the product of these two, each the number of elements value that promptly constitutes matrix L differs greatly, thus L be one very morbid state equation, need transform it.The present invention has carried out normalized to improve the ill performance of matrix L to image coordinate and world coordinates.If each picture point coordinate is (u _i, v _l) (i=1,2 ..., n), the world coordinate system coordinate of each spatial point is (X _Wi, Y _Wi) (i=1,2 ..., n), then whole steps is summarized as follows:

1.) ask u respectively, the mean value m of each coordinate on the v diaxon _u, m _vFor:

$m_u=\frac{\underset{i=0}{\overset{n}{\mathrm{\Σ}}}{u}_i}{n},$ $m_v=\frac{\underset{i=0}{\overset{n}{\mathrm{\Σ}}}{v}_i}{n}$

Then each picture point is with respect to the relative value Δ u of mean point _i, Δ v _iFor:

Δu _i＝u _i-m _u，Δv _i＝v _i-m _v

Equally the world coordinate system coordinate is handled, can be got its mean value and relative value and be:

$m_x=\frac{\underset{i=0}{\overset{n}{\mathrm{\Σ}}}{X}_{\mathrm{wl}}}{n},$ $m_y=\frac{\underset{i=0}{\overset{n}{\mathrm{\Σ}}}{Y}_{\mathrm{wl}}}{n}$

ΔX _wl＝X _wl-m _x，ΔY _wl＝Y _wi-m _y

2.) ask u respectively, reach X on the v diaxon _w, Y _wAll relative coordinates are to the mean value d of barycenter on the diaxon _Uv, d _XyFor:

$d_{\mathrm{uv}}=\frac{\underset{l=0}{\overset{n}{\mathrm{\Σ}}}{(\mathrm{\Δ}{u}_l^2+\mathrm{\Δ}{v}_i^2)}^{\frac{1}{2}}}{n},$ $d_{\mathrm{xy}}=\frac{\underset{i=0}{\overset{n}{\mathrm{\Σ}}}{(\mathrm{\Δ}{X}_{\mathrm{wi}}^2+\mathrm{\Δ}{Y}_{\mathrm{wi}}^2)}^{\frac{1}{2}}}{n}$

3.) ask u respectively, reach X on the v diaxon _w, Y _wScale factor s on the diaxon _Uv, s _XyFor:

$s_{\mathrm{uv}}=\frac{\sqrt{2}}{d_{\mathrm{uv}}},$ $s_{\mathrm{xy}}=\frac{\sqrt{2}}{d_{\mathrm{xy}}}$

4.) u reaches X on the v diaxon _w, Y _wTransformation relation on the diaxon can be used matrix T _Uv, T _XyBe expressed as:

$T_{\mathrm{uv}}=\left[\begin{array}{ccc}{s}_{\mathrm{uv}}& 0& -s_{\mathrm{uv}}{m}_u\\ 0& s_{\mathrm{uv}}& -s_{\mathrm{uv}}{m}_v\\ 0& 0& 1\end{array}\right],$ $T_{\mathrm{xy}}=\left[\begin{array}{ccc}{s}_{\mathrm{xy}}& 0& -s_{\mathrm{xy}}{m}_x\\ 0& s_{\mathrm{xy}}& -s_{\mathrm{xy}}{m}_y\\ 0& 0& 1\end{array}\right]$

5.) image coordinate and the world coordinate system coordinate of establishing through normalized is respectively

With

Then have following relationship to set up:

If the homography matrix with image coordinate after the normalized and world coordinate system coordinate is

Homography matrix in the then former coordinate system between the plane of delineation and the space coordinates is:

$H={T_{\mathrm{uv}}}^{-1}\tilde{H}{T}_{\mathrm{xy}}$

Because rotation matrix R is the unit orthogonal matrix, so r ₁, r ₂Be the unit orthogonal vector, can get two constraints thus:

$\left\{\begin{array}{c}{h}_1^T{A}^{-T}{A}^{-1}{h}_2=0\\ h_1^T{A}^{-T}{A}^{-1}{h}_1=h_2^T{A}^{-T}{A}^{-1}{h}_2\end{array}\right.$

Order $B=A^{-T}{A}^{-1}=\left[\begin{array}{ccc}{b}_1& b_2& b_4\\ b_2& b_3& b_5\\ b_4& b_5& b_6\end{array}\right],$ Define vectorial b=[b ₁b ₂b ₃b ₄b ₅b ₆] ^T, then following formula can be written as again:

$\left[\begin{array}{c}{v}_{12}^T\\ {(v_{11}-v_{22})}^T\end{array}\right]b=\mathrm{Vb}=0$

V wherein _Ij=[h _I1h _J1, h _I1h _J2+ h _L2h _J1, h _L2h _J2, h _I3h _J1+ h _I1h _J3, h _L3h _J2+ h _I2h _J3, h _L3h _J3] ^T, h _LJFor the i of matrix H is capable, the j column element.The image of every bat n (n 〉=3) width of cloth plane reference plate just can obtain n above-mentioned system of equations, thereby solves matrix B.After solving B, just can calibrate the intrinsic parameter matrix A by the Qiao Lisiji decomposition, comprise equivalent focal length, between centers inclination factor and principal point coordinate.After calibrating the intrinsic parameter matrix A, just can determine the external parameter of every width of cloth image, promptly the video camera external parameter is:

$\left\{\begin{array}{c}\mathrm{\ρ}=1/\left|\right|A^{-1}{h}_1\left|\right|=1/\left|\right|A^{-1}{h}_2\left|\right|\\ r_1=\mathrm{\ρ}{A}^{-1}{h}_1,r_2=\mathrm{\ρ}{A}^{-1}{h}_2,r_3=r_1\×r_2\\ T=\mathrm{\ρ}{A}^{-1}{h}_3\end{array}\right.$

The rotation matrix R that is determined by following formula does not satisfy orthogonality, therefore adopts singular value decomposition method with its orthogonalization, obtains the external parameter of video camera.

Because the manufacturing issue of imaging lens itself, for the camera lens that non-precision measurement is used, the influence of distortion is very big.The present invention has considered the radial distortion and the tangential distortion of camera lens, has set up a complete distortion of camera model.If (u v) is desirable picture point coordinate,

It is the picture point coordinate that actual observation arrives.The imaging model of ideal point is deferred to desirable pin hole perspective model.Similarly, establish (X _n, Y _n) be the normalized image coordinate of desirable spatial point,

It is the normalized image coordinate of the spatial point of actual observation.Can get:

${\tilde{X}}_n=X_n+X_n[k_1(X_n^2+Y_n^2)+k_2{(X_n^2+Y_n^2)}^2]+p_1(3X_n^2+Y_n^2)+2p_2{X}_n{Y}_n$

${\tilde{Y}}_n=Y_n+Y_n[k_1(X_n^2+Y_n^2)+k_2{(X_n^2+Y_n^2)}^2]+{2p}_1{X}_n{Y}_n+p_2(X_n^2+{3Y}_n^2)$

K wherein ₁, k ₂Be single order and second order coefficient of radial distortion, p ₁, p ₂It is the tangential distortion coefficient.The transformational relation of image coordinate system coordinate and normalized image coordinate is:

$\left\{\begin{array}{c}u=u_0+f_x{X}_n+s{Y}_n\\ v=v_0+f_y{Y}_n\end{array}\right.$

Above-mentioned two formulas of simultaneous can get:

$+p_2[s(X_n^2+Y_n^2)+{2Y}_n(u-u_0)]$

Following formula is rewritten into matrix form to be had:

Wherein $A_1=(u-u_0)(X_n^2+Y_n^2),B_1=(u-u_0){(X_n^2+Y_n^2)}^2,$

$C_1=f_x(X_n^2+Y_n^2)+2X_n(u-u_0),D_1=s(X_n^2+Y_n^2)+2Y_n(u-u_0),$

$A_2=(v-v_0)(X_n^2+Y_n^2),B_2=(v-v_0){(X_n^2+Y_n^2)}^2,$

$C_2=2X_n(v-v_0),D_2=f_y(X_n^2+3Y_n^2)$

After the data of calibration point that has provided n width of cloth image, these equatioies are piled up the equation that must arrive shape such as Dk=d, wherein k=[k ₁, k ₂, p ₁, p ₂] ^T, solve with linear least square:

k＝(D ^TD) ^-1D ^Td

So far, we have obtained the initial value of all variable elements of video camera, and algorithm flow is seen accompanying drawing 4.

(3) nonlinear optimization of variable element

After calibrating the initial value of variable element, in order to obtain more high-precision variable element, the constraint of preset parameter as the nonlinear optimization objective function, the objective function of setting up belt restraining is:

Wherein n is an amount of images, m _jThe reference mark number of representing every width of cloth Image Acquisition,

Be a M _jSubpoint coordinate on i width of cloth image, m _LJ=(u _LJ, v _IJ) ^TImage coordinate for reality.Variable element to be optimized comprises: intrinsic parameter matrix A, distortion factor matrix k _c=[k ₁, k ₂, p ₁, p ₂] ^T, the outer parameter matrix R of i width of cloth image _i, T _i

If R _iCan be expressed as R _i=[r _1lr _2ir _3l], r wherein _Jl(j=1,2,3) are R _lColumn vector, establish M _CJBe and a M _jCorresponding camera coordinate system coordinate.Then have:

M _cj＝[r _1i r _2i T _i]M _j

Make M _CJ=(X _Cj, Y _Cj, Z _Cj) ^T, establish M _NJ=(X _NJ, Y _Nj) ^TBe M _CjThe normalization coordinate, promptly have:

$\left\{\begin{array}{c}{X}_{\mathrm{nj}}=X_{\mathrm{cj}}/Z_{\mathrm{cj}}\\ Y_{\mathrm{nj}}=Y_{\mathrm{cj}}/Z_{\mathrm{cj}}\end{array}\right.$

Further can establish and M _NjCorresponding homogeneous coordinates are ${\tilde{M}}_{\mathrm{nj}}={(X_{\mathrm{nj}},Y_{\mathrm{nj}},1)}^T,$ M (A, R _l, T _l, M _j)=(u, v) ^TFor with a M _jCorresponding ideal image coordinate then has:

$m(A,R_i,T_i,M_j)=A{\tilde{M}}_{\mathrm{nj}}$

And then obtain easily

In sum, the Levenberg-Marquardt method optimization step of finding the solution the optimum solution of variable element can reduce:

1.) establish k _MaxThe expression iterations, x represents variable element column vector to be optimized, f (x) represents departure function, x ₀The initial value of expression variable element column vector, ε ₁, ε ₂(ε ₁＞0, ε ₂＞0) the expression iteration stops real number, J (x) expression Jacobian matrix.Make k=0, v=2, x=x ₀, B=J ^T(x) J (x), g (x)=J ^T(x) f (x).Make b=(‖ g (x) ‖ _∞≤ ε ₁), b is a Boolean variable.Calculate ratio of damping μ=τ max{b _Ii, b wherein _IiBe diagonal entry in the matrix B;

2.) if b is vacation and k＜k _Max, then turn to 3.), otherwise finish to optimize;

3.) k=k+1 makes that iterative increment is Δ x, calculates Δ x=(J ^T(x) J (x)+μ I) ^-1J ^T(x) f (x);

4.) if ‖ Δ x is ‖≤ε ₂(‖ x ‖+ε ₂), then establish b for true, otherwise turn to 5.);

5.) establish x _New=x+ Δ x, F ' (x)=g (x), η is a gain coefficient, then

$\mathrm{\η}=\frac{F\left(x\right)-F\left(x_{\mathrm{new}}\right)}{L\left(0\right)-L\left(\mathrm{\Δx}\right)}$

Wherein launch definition by Taylor (Taylor) formula:

$L\left(0\right)-L\left(\mathrm{\Δx}\right)=-\mathrm{\Δ}{x}^T{J}^T\left(x\right)f\left(x\right)-\frac{1}{2}\mathrm{\Δ}{x}^T{J}^T\left(x\right)J\left(x\right)\mathrm{\Δx}$

$=-\frac{1}{2}\mathrm{\Δ}{x}^T[2g\left(x\right)+(J^T\left(x\right)J\left(x\right)+\mathrm{\μI}-\mathrm{\μI})\mathrm{\Δx}$

$=\frac{1}{2}\mathrm{\Δ}{x}^T(\mathrm{\μ\Δx}-g\left(x\right))$

Then gain coefficient can be write as:

$\mathrm{\η}=\frac{2(F\left(x\right)-F\left(x_{\mathrm{new}}\right))}{\mathrm{\Δ}{x}^T(\mathrm{\μ\Δx}-g\left(x\right))};$

6.) if η≤0, μ=μ * v then, v=2*v, otherwise turn to 7.);

7.) make x=x _New, calculate B=J ^T(x) J (x), g (x)=J ^T(x) f (x), b=(‖ g (x) ‖≤ε ₁), order $\mathrm{\μ}=\mathrm{\μ}*\max \{\frac{1}{3},1-{(2\mathrm{\η}-1)}^3\},$ Turn to 2.).

From the process of whole optimization, the end condition of iteration has:

1)‖g(x)‖ _∞≤ε ₁(ε ₁＞0)

2)‖Δx‖≤ε ₂(‖x‖+ε ₂)(ε ₂＞0)

3)k≥k _max

One that satisfies in above three conditions just can be finished to optimize.

In the actual 3 D scanning system two ccd video cameras are fixed on the top of the shelf by shown in Figure 5.The left and right sides mirror that requires according to calibration algorithm is respectively taken 5 original scaling board image informations, and each the demarcation selects wherein any 4 width of cloth to demarcate.Handle by above-mentioned steps,, only list left mirror camera model parameter based on length.Being fixed parameter is: τ _Xy=1.00036, the calibration result of variable element sees Table 1, table 2.

Table 1 video camera variable inner parameter calibration result

Picture numbers	(1234)	(1235)	(1245)	(1345)	(2345)	Average	Standard deviation
								f x	2482.81	2480.26	2480.02	2479.15	2478.99	2480.24	1.533
f y	2483.70	2481.12	2480.96	2479.93	2479.90	2481.12	1.548
								u 0	675.11	677.72	675.09	676.37	677.85	676.43	1.343
v 0	521.11	521.77	520.54	519.04	519.05	520.30	1.227
								s	-0.53	-0.71	-0.53	-0.16	-0.75	-0.54	0.233
k 1	-0.0957	-0.1031	-0.0936	-0.0949	-0.0948	-0.0964	0.00381
								k 2	0.52743	0.55823	0.50729	0.50603	0.50312	0.52042	0.02322
p 1	-0.0010	-0.0010	-0.0011	-0.0011	-0.0009	-0.00107	0.00007
								p 2	0.00069	0.00089	0.00093	0.00061	0.00064	0.00075	0.00015

Table 2 video camera variable external parameter calibration result

Whole calibrating procedure is carried out according to the flow process in the accompanying drawing 6, demarcate preset parameter successively, calibrating camera variable element initial value comprises equivalent focal length, principal point coordinate, between centers inclination factor, distortion factor, external parameter, at last, nonlinear optimization is found the solution the optimum solution of variable element.

The present invention is directed to the existing shortcoming of existing traditional cameras calibration technique and restriction, propose the video camera parameter is divided into preset parameter and variable element. Preset parameter is main relevant with the self-characteristic of lens group and CCD, and its character is stable in a long time, such as the aspect ratio between pixel among the CCD. And variable element also comprises equivalent focal length, principal point coordinate, between centers obliquity factor and the distortion factor etc. of some variable inner parameters such as CCD except comprising external parameter. By such classification, the demarcation of preset parameter just can be independent of on-the-spot 3-D scanning process, namely can after using the long period at Laboratory Calibration once, needn't demarcate in measure field at every turn. And for variable element, when carrying out the three-dimensional measurement at every turn, then be placed on measure field and demarcate. The object of the invention is to design the camera calibration method based on preset parameter and variable element in a kind of 3-D scanning system that can more agree with practical implementation. The method has calibrating template makes simply, and field calibration is simple and efficient, precision is high, fireballing advantage.

Claims (1)

1, the scaling method based on preset parameter and variable element in a kind of 3 D scanning system is characterized in that:

Step 1: at first vertically take the standard form circle, extract the circle contour in the image coordinate system, utilize least square method that the circle contour that extracts is carried out curve fitting, obtain the central coordinate of circle under the image coordinate system, then, utilization is as the corresponding relation of the plane and the plane of delineation, will be as circle contour equation on the plane:

(x _i-x _c) ²+(y _i-y _c) ²＝r ²，i＝1，2，3，…，n

Be transformed to the circle contour equation of representing with image coordinate:

[μ _x(u _i-u _c)] ²+[μ _y(v _i-v _c)] ²＝r ²，i＝1，2，3，…，n

(x wherein _i, y _i), i=1,2,3 ..., n is as the coordinate of putting on the flat circle profile, (x _c, y _c) be central coordinate of circle, r is a radius, u _i, v _i, u _c, v _cHorizontal ordinate, ordinate, center of circle horizontal ordinate and the ordinate of match circle contour point in the difference presentation video, r is a match radius of a circle on the plane of delineation, μ _x, μ _yFor the physical size of each pixel on u axle and v direction of principal axis, make τ _Xy=μ _x/ μ _y, by μ _x, μ _yDefinition know τ _XyBe exactly aspect ratio to be calibrated, then can obtain:

${\left[{\mathrm{\τ}}_{\mathrm{xy}}(u_i-u_c)\right]}^2+{(v_i-v_c)}^2=\frac{r^2}{{\mathrm{\μ}}_y^2},i=\mathrm{1,2,3},\·\·\·,n$

At last, utilize the sampled point of the n on the circle contour in the image coordinate system, utilize least square method to obtain aspect ratio τ again _Xy

Step 2: try to achieve the preset parameter τ of video camera _XyAfter, utilizing image coordinate and its corresponding world coordinate of image coordinate system mid point is the volume coordinate of mid point, find the solution homography matrix, according to the initial value of homography matrix calibrating camera variable element, this initial value comprises equivalent focal length, principal point coordinate, between centers inclination factor, distortion factor and external parameter then;

Step 3: after calibrating the initial value of variable element, be a constraint of nonlinear optimization objective function, set up the objective function of belt restraining with the preset parameter:

Find the solution the optimum solution of variable element by nonlinear optimization method.

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