CN106204625A - A kind of variable focal length flexibility pose vision measuring method - Google Patents

Abstract

One variable focal length flexibility pose vision measuring method of the present invention belongs to computer vision measurement technical field, relates to flexibility variable focal length monocular pose measuring method easily.The method completes the motion model position relative to camera coordinate system and the resolving of attitude by the characteristic point of known spatial local coordinate system coordinate, during camera calibration and object pose are measured, video camera keeps constant relative to the position of world coordinate system and camera intrinsic parameter matrix, directly complete the conversion to three-dimensional coordinate of the two dimensional image feature by punctuate parameter after having demarcated, solve the pose of objective.The method need not demarcate measurement system, according to the labelling point of the known relativeness in testee surface, it is achieved to object space, the real-time measurement of attitude information in the case of zoom, not only increases system flexibility, too increases the facilitation of system.

Description

A kind of variable focal length flexibility pose vision measuring method

Technical field

The invention belongs to computer vision measurement technical field, relate to flexibility variable focal length monocular pose measurement easily Method.

Background technology

In multiple fields, many bodies are thrown in the application of technology simultaneously and are got more and more.Due to the continuous progress of modern technologies, right Many bodies are thrown in the measurement requirement of pose and are the most constantly improved.Especially in the many bodies of high speed are thrown in, safety, efficient, high-precision obtains The pose parameter that must throw in each monomer is particularly significant.For this measurement demand, vision measurement becomes due to its high frequency sound, high accuracy For the first-selection in non-contact measurement.But owing to needing often the focal length of camera to be adjusted in a lot of occasions, and camera Being accomplished by recalibrating after carrying out Focussing, calibration process is complex loaded down with trivial details, and vision measurement system every time The adjustment of focal length cannot be completed during measuring.

Dalian University of Technology Liu Wei et al. 2015 permed " the pair based on coloud coding of table in aviation journal the 5th phase 36 Fuel tank wind tunnel model pose measuring method " in propose auxiliary fuel tank pose measuring method based on coloud coding image, utilize coloured silk The self-luminous sign point of color coding carries out images match, solves under low light conditions target label luminance shortage and due to object The labelling point blanking phenomenon that rolling causes.But the method needs when camera focus changes to recalibrate, and calibration steps, is System precision can decrease.Patent of invention Publication No. ZL 201310139656.7 of Dalian University of Technology Liu Wei et al. application, " a kind of high speed rolling posture measuring method " proposes position and attitude vision measuring method in wind-tunnel, can survey in wind-tunnel The sextuple position and attitude information of amount moving object, but directly can not measure in the case of changing camera focus, After demarcation, camera parameter cannot change.

Summary of the invention

The technical barrier that the invention solves the problems that is the defect overcoming prior art, a kind of variable focal length flexibility pose of invention Vision measuring method, solves the difficult problem that after existing vision measurement system is demarcated, parameter cannot change, uses 4 known coordinates Labelling point is as constraint, and the pose measuring method that can not change camera focus after traditional demarcation is expanded to focal length can be at any time The pose measuring method changed, improves the flexibility of system, and changing focal length during solving measurement needs to re-scale phase The problem of machine.

The technical solution used in the present invention is a kind of variable focal length flexibility pose vision measuring method, it is characterized in that, surveys Metering method completes the motion model position relative to camera coordinate system by the characteristic point of known spatial local coordinate system coordinate With the resolving of attitude, during camera calibration and object pose are measured, video camera is relative to the position of world coordinate system Keep constant with camera intrinsic parameter matrix, directly complete two dimensional image feature by punctuate parameter after having demarcated and sit to three-dimensional Target is changed, and solves the pose of objective.Comprising the following steps that of method is shown:

The first step: initially set up the mathematical model between measured target and camera

Setting up local coordinate system with model barycenter on tested model for initial point, the axis of rotation of Definition Model outline is Y-axis, is x-axis by the axle that model barycenter is vertical with y-axis, then determines the direction of z-axis according to the right-hand rule, and model surface owns The coordinate that labelling point is fastened in model local coordinate is ensured by the setting accuracy on work of labelling point,

Video camera pin-hole model describes three dimensions point:

λ_iu_i=PX_i (1)

Wherein, λ_iFor the scale factor relevant with ith feature point, P is video camera projection matrix, U_i=(u_i v_i 1)^TFor The homogeneous coordinates of characteristic point, X on two dimensional image_i=(x_i y_i z_i 1)^TFor the homogeneous coordinates of labelling point on model local coordinate system, The form that video camera projection matrix P is written as:

P=K R | t] (2)

Wherein, K is camera intrinsic parameter matrix, describes three-dimensional to two-dimentional projection relation:

$K=\left[\begin{array}{ccc}f& s& u_0\\ 0& rf& v_0\\ 0& 0& 1\end{array}\right]---\left(3\right)$

F is focal length of camera, u₀For the abscissa of principal point for camera, v₀For the vertical coordinate of principal point for camera, S is that coordinate axes tilts Parameter, r represents the length-width ratio of chip pixel unit, R=(r_ij)³, i, j=1 and T=(t_x t_y t_z)^TRepresent that video camera is sat respectively Spin matrix between mark system and model local coordinate system and translation vector, S is coefficient of skewness, and video camera principal point is positioned at X-Y scheme The center of picture, if: camera chip is square, therefore r=1；Principal point is positioned at the center of image；Coefficient of skewness S=0, then interior Parameter matrix K can be reduced to the diagonal matrix [f f 1] being made up of focal distance f^T, make w=1/f, then K=[1 1 w]^T, by this hypothesis Substitute into formula (1), by video camera projection model be:

${\mathrm{\λ}}_i{U}_i=\left[\begin{array}{cccc}{r}_{11}& r_{12}& r_{13}& t_x\\ r_{21}& r_{22}& r_{23}& t_y\\ {\mathrm{wr}}_{31}& {\mathrm{wr}}_{32}& {\mathrm{wr}}_{33}& {\mathrm{wt}}_z\end{array}\right]X_i---\left(4\right)$

Radial distortion is incorporated in projection model, and sets up pattern distortion model according to ordinary circumstance and be:

$P_u~P_d/(1+{\mathrm{kr}}_d^2)---\left(5\right)$

Wherein, k is distortion factor, P_u=[u_u v_u 1]^TAnd P_d=[u_d v_d 1]^TImage before and after being respectively distorted Point coordinates, r_dFor a P_dTo the distance of center of distortion, i.e. distort radius, it is assumed that center of distortion is positioned at the center of two dimensional image, then r_d ²=u_d ²+v_d ², then on two dimensional image, the coordinate of actual imaging point is:

$U_i={\[u_i,v_i,1+k(u_d^2+v_d^2)\]}^T---\left(6\right)$

Above-mentioned lens distortion model is joined in projection equation, and according to [u_i]_xu_i=0, by projection formula (4) both sides It is multiplied by [u simultaneously_i] can be by proportionality factors lambda after x_iEliminate, obtain projection equation as follows:

$\left[\begin{array}{ccc}0& -1-k(u_i^2+v_i^2)& v_i\\ 1+k(u_i^2+v_i^2)& 0& u_i\\ -v_i& u_i& 0\end{array}\right]\left[\begin{array}{cccc}{r}_{11}& r_{12}& r_{13}& t_x\\ r_{21}& r_{22}& r_{23}& t_y\\ {\mathrm{wr}}_{31}& {\mathrm{wr}}_{32}& {\mathrm{wr}}_{33}& {\mathrm{wt}}_z\end{array}\right]\left[\begin{array}{c}{x}_i\\ y_i\\ z_i\\ 1\end{array}\right]=0---\left(7\right)$

Wherein, this projection equation describes three-dimensional feature point X_iTo X-Y scheme picture point U comprising distortion_iProjection relation, side The unknown number of journey is the transition matrix [P] between Two coordinate system_3×4All elements and distortion coefficients of camera lens k, obtain root after P The Intrinsic Matrix of video camera and outer parameter matrix, therefore, the spin matrix between the amount of solving coordinate system is decomposited according to formula (2) The process solving video camera projection matrix P it is with the process of translation vector,

Second step: the coordinate system of video camera and tested model is changed and asked for

In defined formula (7), each element of video camera projection matrix P is p_ij, p_ijThe ith row and jth column of representing matrix P Element, the third line of matrix equation (7) is individually listed, obtains following form:

-v_i(p₁₁x_i+p₁₂y_i+p₁₃z_i+p₁₄)+u_i(p₁₂x_i+p₂₂y_i+p₂₃z_i+p₂₄)=0 (8)

The equation is for having 8 unknown number (p₁₁,p₁₂,p₁₃,p₁₄,p₁₂,p₂₂,p₂₃,p₂₄) its sublinear equation, as long as Obtain the accurate coordinates of 4 three-dimensional point and in two dimensional image, find the image coordinate of they correspondences, just can utilize this 4 correspondences Some equation is write as the form of matrix:

Mv=0 (9)

Wherein M is the coefficient matrix of 4 × 8, unknown number vector v=[p₁₁,p₁₂,p₁₃,p₁₄,p₁₂,p₂₂,p₂₃,p₂₄]^T, v With the orthonormal basis n of above-mentioned matrix equation_iLinear combination represent, formula is as follows:

$v=\underset{i=1}{\overset{4}{\Σ}}{\mathrm{\α}}_i{n}_i---\left(10\right)$

Wherein α_iFor the new unknown number obtained after v is carried out parametrization, make α₄=1, then front two row of projection matrix P can To be expressed as three unknown number α₁, α₂And α₃Linear combination, only require and solve α₁, α₂And α₃, just two row elements before P can be asked Going out, the 2nd row of matrix equation can be with the form being written as:

$(1+{\mathrm{kr}}_i^2)(p_{11}{x}_i+p_{12}{y}_i+p_{13}{z}_i+p_{14})-u_i(p_{31}{x}_i+p_{32}{y}_i+p_{33}{z}_i+p_{34})=0---\left(11\right)$

In conjunction with formula (10), the equation can be write as:

A[p₃₁,p₃₂,p₃₃,p₃₄]^T=B [α₁,α₂,α₃,kα₁,kα₂,kα₃,k,1]^T (12)

Wherein A, B are respectively the coefficient matrix of 4 × 4 and 4 × 8, work as X₁, X₂, X₃, X₄Time non-coplanar, matrix A is reversible, equation (12) write as:

[p₃₁,p₃₂,p₃₃,p₃₄]^T=A^-1B[α₁,α₂,α₃,kα₁,kα₂,kα₃,k,1]^T (13)

By formula (13), by the 3rd row α of P₁, α₂,α₃Linear combination with K represents, i.e. completes projection matrix P's Parametrization, owing to before P matrix, 3 are classified as the operation result after K with R is multiplied, and K=[1 1 w]^T, therefore front 3 row of P have The character identical with spin matrix R, utilizes the property of orthogonality of spin matrix to set up the Constrained equations about P matrix all elements As follows:

$\left\{\begin{array}{c}{p}_{11}{p}_{21}+p_{12}{p}_{22}+p_{13}{p}_{23}=0,\\ p_{31}{p}_{11}+p_{32}{p}_{12}+p_{33}{p}_{13}=0,\\ p_{31}{p}_{21}+p_{32}{p}_{22}+p_{33}{p}_{23}=0,\\ p_{11}^2+p_{12}^2+p_{13}^2-p_{21}^2-p_{22}^2-p_{23}^2=0.\end{array}\right.---\left(14\right)$

Equation group (14) has 4 equations, solves 4 unknown number (α₁,α₂,α₃, k), i.e. obtain projection matrix P, Jin Erfen Solve the model spin matrix R relative to camera coordinate system₀With translation vector T₀,

3rd step: the coordinate system transformational relation of video camera and world coordinate system is asked for

For coplanar characteristic point X_i, it is assumed that a little coordinate figure in Z-direction be 0, i.e. z_i, now with 3 row of formula (7) Corresponding formula (8) is write as:

-v_i(p₁₁x_i+p₁₂y_i+p₁₄)+u_i(p₁₂x_i+p₂₂y_i+p₂₄)=0 (15)

The form of matrix of being write as by the equation of 4 characteristic points can obtain Mv=0, and M is the coefficient matrix of 4 × 6, v= [p₁₁,p₁₂,p₁₄,p₁₂,p₂₂,p₃₄]^TFor unknown number vector, v two orthonormal basis vector n of matrix M₁And n₂It is expressed as:

V=β₁n₁+n₂ (16)

Wherein, β₁For the unknown number of new definition, front two provisional capitals of projection matrix P are with unknown number and n₁And n₂Replace, when When 4 characteristic points used are coplanar, the 3rd row of formula (7) is promising:

$(1+{\mathrm{kr}}_i^2)(p_{11}{x}_i+p_{12}{y}_i+p_{14})-u_i(p_{31}{x}_i+p_{32}{y}_i+p_{34})=0---\left(17\right)$

Utilizing this formula by the 3rd traveling line parameter of projection matrix P, process is as follows:

C[p₃₁,p₃₂,p₃₄]^T=D [β₁,kβ₁,k,1]^T (18)

Wherein C and D is respectively the coefficient matrix of 3 × 3 and 3 × 4, works as X₁, X₂, X₃During conllinear, Matrix C is not reversible, equation (18) can be write as:

[p₃₁,p₃₂,p₃₄]^T=C^-1D[β₁,kβ₁,k,1]^T (19)

Utilize above-mentioned formula β₁Replace all elements in P with two unknown numbers of k, i.e. complete the parametrization of P, due to P Front 3 row to have any two row of the character identical with spin matrix R, i.e. spin matrix mutually orthogonal and have identical mould, profit The equation about P matrix all elements is listed by these character, as follows:

wp₁₁wp₁₂+wp₂₁wp₂₂+p₃₁wp₃₂=0, (20)

$w^2{p}_{11}^2+w^2{p}_{21}^2+p_{31}^2-w^2{p}_{12}^2-w^2{p}_{22}^2-p_{32}^2=0.---\left(21\right)$

In conjunction with the 4th the projection relation of 2 peacekeeping 3-dimensional corresponding point can be obtained equation:

$(1+{\mathrm{kr}}_4^2)(p_{11}{x}_4+p_{12}{y}_4+p_{14})-u_4(p_{31}{x}_4+p_{32}{y}_4+p_{34})=0---\left(22\right)$

Aggregative formula (20), (21), the unknown number β in (22) solving equation₁, k and w=1/f, in conjunction with the parametrization of P All elements in formula reverse P matrix, and then solve the camera transition matrix R relative to world coordinate system_CAnd T_C；

4th step: model sport pose solves

Initially set up model local coordinate system, using model barycenter as local coordinate system initial point O_t, along model center axis Direction model caudal directions is O_tY_tAxle positive direction, took initial point O_tWith O_tY_tAxle vertical and with same group of red-label point of head and the tail The direction that line intersects is O_tX_tAxle positive direction, defines local coordinate system O according to the right-hand rule_tZ_tDirection of principal axis, surveys at model pose First the video camera initial position transformational relation relative to world coordinate system, i.e. spin matrix R is obtained before amount_CWith translation vector T_C, Then model each moment model local coordinate system in motor process is obtained by zoom pose measurement process relative Translation vector T in camera coordinates system_OWith spin matrix R_O, utilize below equation computation model thing local coordinate system to sit with wind-tunnel The transition matrix of mark system:

$\left[\begin{array}{c}{x}_t\\ y_t\\ z_t\end{array}\right]=R_o^{-1}{R}_c\left[\begin{array}{c}{x}_w\\ y_w\\ z_w\end{array}\right]+R_o^{-1}(T_c-T_o)---\left(23\right)$

Wherein, (x_t,y_t,z_t) it is labelling point coordinate, (x under model local coordinate system_w,y_w,z_w) it is that labelling point is at wind-tunnel Coordinate under coordinate system, and:

$R=R_o^{-1}{R}_o---\left(24\right)$

$T=R_o^{-1}(T_c-T_o)={\left[\begin{array}{ccc}{t}_x& t_y& t_z\end{array}\right]}^T---\left(25\right)$

R and T is respectively model relative to the spin matrix of world coordinate system and translation matrix.

The invention has the beneficial effects as follows and need not measurement system is demarcated, according to known relative pass, testee surface The labelling point of system, it is achieved to object space, the real-time measurement of attitude information in the case of zoom, not only increase system flexibility Change, too increase the facilitation of system.

Accompanying drawing explanation

Fig. 1 is variable focal length flexibility pose vision measuring method schematic diagram.Wherein, 1-testee coordinate system 0_tX_tY_tZ_t, 2-camera coordinates system 0XYZ, 3-world coordinate system 0_wX_wY_wZ_w, 4-labelling point, X₁It it is some office of first, space table Portion's coordinate, X₂Be second, space table some local coordinate, X₃Be the 3rd, space table some local coordinate, X₄It it is space 4th table some local coordinate, U₁It is X₁Image coordinate on camera, U₂It is X₂Image coordinate on camera, U₃It is X₃ Image coordinate on camera, U₄It is X₄Image coordinate on camera.

Fig. 2 is zoom flexible posture measuring method flow chart.

Detailed description of the invention

The detailed description of the invention of the present invention is described in detail below in conjunction with technical scheme and accompanying drawing.

Accompanying drawing 1 is variable focal length flexibility pose vision measuring method schematic diagram, and Fig. 2 is zoom flexible posture measurement side Method flow chart.According to some the labelling points in known local, testee surface relative coordinate, mark tally amount is more than or equal to 4. First according to local coordinate information known to labelling point, solve measured target relative to the position of camera coordinates system, angular relationship, Then further according to the relation between camera coordinates system and world coordinate system, the position of testee and angle information are transformed into generation Under boundary's coordinate system, finally in the case of unknown camera focus information, complete the position of testee and the measurement of attitude information.

The coordinate system of step 1 video camera and tested model is changed and is asked for

First the relative coordinate of testee surface markers point is determined, as follows:

$X_1=\left[\begin{array}{c}6.325\\ 2.125\\ 10.56\end{array}\right],X_2=\left[\begin{array}{c}8.265\\ 4.256\\ 2.36\end{array}\right],X_3=\left[\begin{array}{c}5.632\\ 3.687\\ 3.25\end{array}\right],X_4=\left[\begin{array}{c}9.167\\ 8.648\\ 0.52\end{array}\right]$

Then according to formula (14), wherein the length-width ratio of vision sensor chip is 1, and for camera, S is 0, calculates Obtain the testee posture information spin matrix R relative to camera_oWith translation vector T_o:

$R_o=\left[\begin{array}{ccc}2.34641& 2.12451& 3.34641\\ 0.46341& 2.34641& 2.4101\\ 1.4687& 0.1654& 1.34641\end{array}\right]$

$T_o=\left[\begin{array}{c}20.65\\ 40.35\\ 120.36\end{array}\right]$

The coordinate system transformational relation of step 2 video camera and world coordinate system is asked for

Relation R that camera coordinates system is relative with world coordinate system can be calculated according to formula (22)_cAnd T_c, as follows:

$R_c=\left[\begin{array}{ccc}2.34641& 2.12451& 3.34641\\ 0.46341& 2.34641& 2.4101\\ 1.4687& 0.1654& 1.34641\end{array}\right]$

$T_c=\left[\begin{array}{c}80.21\\ 120.36\\ 30.12\end{array}\right]$

Step 3 model sport pose solves

Testee can be obtained relative to the position of world coordinate system, attitude information according to formula (23), (24), (25), As shown in the table:

Testee pose parameter

The present invention is according to the labelling point of the known relativeness in testee surface, it is achieved to object position in the case of zoom Put, attitude information obtains and measures in real time, and the method need not demarcate measurement system, not only increases system flexibility, also Add the facilitation of system.In addition for situations such as the camera focus changes during measuring, there is more preferable effect, Can be to the most accurately pose measurement at depth of field direction displacement larger object.

Claims (1)

1. a variable focal length flexibility pose vision measuring method, is characterized in that, measuring method is sat by known spatial local The characteristic point of mark system coordinate completes the motion model position relative to camera coordinate system and the resolving of attitude, at camera calibration During measuring with object pose, video camera keeps not relative to position and the camera intrinsic parameter matrix of world coordinate system Become, directly complete the conversion to three-dimensional coordinate of the two dimensional image feature by punctuate parameter after having demarcated, solve objective Pose；Specifically comprising the following steps that of method

The first step: initially set up the mathematical model between measured target and camera

Setting up local coordinate system with model barycenter on tested model for initial point, the axis of rotation of Definition Model outline is y-axis, It is x-axis by the axle that model barycenter is vertical with y-axis, then determines the direction of z-axis according to the right-hand rule, all labellings of model surface The coordinate that point is fastened in model local coordinate is ensured by the setting accuracy on work of labelling point,

Video camera pin-hole model describes three dimensions point:

λ_iU_i=PX_i (1)

Wherein, λ_iFor the scale factor relevant with ith feature point, P is video camera projection matrix, U_i=(u_i v_i 1)^TFor two dimension The homogeneous coordinates of characteristic point, X on image_i=(x_i y_i z_i 1)^TFor the homogeneous coordinates of labelling point on model local coordinate system, shooting The form that machine projection matrix P is written as:

P=K [R | t] (2)

Wherein, K is camera intrinsic parameter matrix, describes three-dimensional to two-dimentional projection relation:

$K=\left[\begin{array}{ccc}f& s& u_0\\ 0& rf& v_0\\ 0& 0& 1\end{array}\right]---\left(3\right)$

F is focal length of camera, u₀For the abscissa of principal point for camera, v₀For the vertical coordinate of principal point for camera, S is coordinate axes tilt parameters, R represents the length-width ratio of chip pixel unit, R=(r_ij)³, i, j=1 and T=(t_x t_y t_z)^TRepresent camera coordinate system respectively And the spin matrix between model local coordinate system and translation vector, S is coefficient of skewness, and video camera principal point is positioned at two dimensional image Center, if: camera chip is square, therefore r=1；Principal point is positioned at the center of image；Coefficient of skewness S=0, then intrinsic parameter Matrix K can be reduced to the diagonal matrix [f f 1] being made up of focal distance f^T, make w=1/f, then K=[1 1 w]^T, this hypothesis is substituted into Formula (1), by video camera projection model be:

${\mathrm{\λ}}_i{U}_i=\left[\begin{array}{cccc}{r}_{11}& r_{12}& r_{13}& t_x\\ r_{21}& r_{22}& r_{23}& t_y\\ {\mathrm{wr}}_{31}& {\mathrm{wr}}_{32}& {\mathrm{wr}}_{33}& {\mathrm{wt}}_z\end{array}\right]X_i---\left(4\right)$

Radial distortion is incorporated in projection model, and sets up pattern distortion model according to ordinary circumstance and be:

$P_u~P_d/(1+{\mathrm{kr}}_d^2)---\left(5\right)$

Wherein, k is distortion factor, P_u=[u_u v_u 1]^TAnd P_d=[u_d v_d 1]^TPicture point before and after being respectively distorted is sat Mark, r_dFor a P_dTo the distance of center of distortion, i.e. distort radius, it is assumed that center of distortion is positioned at the center of two dimensional image, then r_d ²= u_d ²+v_d ², then on two dimensional image, the coordinate of actual imaging point is:

$U_i={\[u_i,v_i,1+k(u_d^2+v_d^2)\]}^T---\left(6\right)$

Above-mentioned lens distortion model is joined in projection equation, and according to [u_i]_xu_i=0, by projection formula (4) both sides simultaneously It is multiplied by [u_i] can be by proportionality factors lambda after x_iEliminate, obtain projection equation as follows:

$\left[\begin{array}{ccc}0& -1-k(u_i^2+v_i^2)& v_i\\ 1+k(u_i^2+v_i^2)& 0& u_i\\ -v_i& u_i& 0\end{array}\right]\left[\begin{array}{cccc}{r}_{11}& r_{12}& r_{13}& t_x\\ r_{21}& r_{22}& r_{23}& t_y\\ {\mathrm{wr}}_{31}& {\mathrm{wr}}_{32}& {\mathrm{wr}}_{33}& {\mathrm{wt}}_z\end{array}\right]\left[\begin{array}{c}{x}_i\\ y_i\\ z_i\\ 1\end{array}\right]=0---\left(7\right)$

Wherein, this projection equation describes three-dimensional feature point X_iTo X-Y scheme picture point U comprising distortion_iProjection relation, equation Unknown number is the transition matrix [P] between Two coordinate system_3×4All elements and distortion coefficients of camera lens k, obtain after P according to public affairs Formula (2) decomposites the Intrinsic Matrix of video camera and outer parameter matrix, and therefore, the spin matrix between the amount of solving coordinate system is with flat The process of the amount of shifting to is the process solving video camera projection matrix P,

Second step: the coordinate system of video camera and tested model is changed and asked for

In defined formula (7), each element of video camera projection matrix P is p_ij, p_ijThe unit of the ith row and jth column of representing matrix P Element, individually lists the third line of matrix equation (7), obtains following form:

-v_i(p₁₁x_i+p₁₂y_i+p₁₃z_i+p₁₄)+u_i(p₁₂x_i+p₂₂y_i+p₂₃z_i+p₂₄)=0 (8)

The equation is for having 8 unknown number (p₁₁,p₁₂,p₁₃,p₁₄,p₁₂,p₂₂,p₂₃,p₂₄) its sublinear equation, as long as obtaining 4 The accurate coordinates of individual three-dimensional point also finds the image coordinate of they correspondences in two dimensional image, just can utilize these 4 corresponding points Equation is write as the form of matrix:

Mv=0 (9)

Wherein M is the coefficient matrix of 4 × 8, unknown number vector v=[p₁₁,p₁₂,p₁₃,p₁₄,p₁₂,p₂₂,p₂₃,p₂₄]^T, v uses State the orthonormal basis n of matrix equation_iLinear combination represent, formula is as follows:

$v=\underset{i=1}{\overset{4}{\Σ}}{\mathrm{\α}}_i{n}_i---\left(10\right)$

Wherein α_iFor the new unknown number obtained after v is carried out parametrization, make α₄=1, then front two row of projection matrix P can be with table It is shown as three unknown number α₁, α₂And α₃Linear combination, only require and solve α₁, α₂And α₃, just two row elements before P can be obtained, 2nd row of matrix equation can be with the form being written as:

$(1+{\mathrm{kr}}_i^2)(p_{11}{x}_i+p_{12}{y}_i+p_{13}{z}_i+p_{14})-u_i(p_{31}{x}_i+p_{32}{y}_i+p_{33}{z}_i+p_{34})=0---\left(11\right)$

In conjunction with formula (10), the equation can be write as:

A[p₃₁,p₃₂,p₃₃,p₃₄]^T=B [α₁,α₂,α₃,kα₁,kα₂,kα₃,k,1]^T (12)

Wherein A, B are respectively the coefficient matrix of 4 × 4 and 4 × 8, work as X₁, X₂, X₃, X₄Time non-coplanar, matrix A is reversible, and equation (12) is write Become:

[p₃₁,p₃₂,p₃₃,p₃₄]^T=A^-1B[α₁,α₂,α₃,kα₁,kα₂,kα₃,k,1]^T (13)

By formula (13), by the 3rd row α of P₁, α₂,α₃Linear combination with K represents, i.e. completes the parameter of projection matrix P Change, owing to before P matrix, 3 are classified as the operation result after K with R is multiplied, and K=[1 1 w]^T, therefore front 3 row of P have and revolve The character that torque battle array R is identical, utilizes the property of orthogonality of spin matrix to set up the Constrained equations about P matrix all elements such as Under:

$\left\{\begin{array}{c}{p}_{11}{p}_{21}+p_{12}{p}_{22}+p_{13}{p}_{23}=0,\\ p_{31}{p}_{11}+p_{32}{p}_{12}+p_{33}{p}_{13}=0,\\ p_{31}{p}_{21}+p_{32}{p}_{22}+p_{33}{p}_{23}=0,\\ p_{11}^2+p_{12}^2+p_{13}^2-p_{21}^2-p_{22}^2-p_{23}^2=0.\end{array}\right.---\left(14\right)$

Equation group (14) has 4 equations, solves 4 unknown number (α₁,α₂,α₃, k), i.e. obtain projection matrix P, and then decomposite Model is relative to the spin matrix R of camera coordinate system₀With translation vector T₀,

3rd step: the coordinate system transformational relation of video camera and world coordinate system is asked for

For coplanar characteristic point X_i, it is assumed that a little coordinate figure in Z-direction be 0, i.e. z_i, now relative with 3 row of formula (7) The formula (8) answered is write as:

-v_i(p₁₁x_i+p₁₂y_i+p₁₄)+u_i(p₁₂x_i+p₂₂y_i+p₂₄)=0 (15)

The form of matrix of being write as by the equation of 4 characteristic points can obtain Mv=0, and M is the coefficient matrix of 4 × 6, v=[p₁₁, p₁₂,p₁₄,p₁₂,p₂₂,p₃₄]^TFor unknown number vector, v two orthonormal basis vector n of matrix M₁And n₂It is expressed as:

V=β₁n₁+n₂ (16)

Wherein, β₁For the unknown number of new definition, front two provisional capitals of projection matrix P are with unknown number and n₁And n₂Replace, when used 4 characteristic points coplanar time, the 3rd row of formula (7) is promising:

(1+kr_i ²)(p₁₁x_i+p₁₂y_i+p₁₄)-u_i(p₃₁x_i+p₃₂y_i+p₃₄)=0 (17)

Utilizing this formula by the 3rd traveling line parameter of projection matrix P, process is as follows:

C[p₃₁,p₃₂,p₃₄]^T=D [β₁,kβ₁,k,1]^T (18)

Wherein C and D is respectively the coefficient matrix of 3 × 3 and 3 × 4, works as X₁, X₂, X₃During conllinear, Matrix C is not reversible, and equation (18) can To be write as:

[p₃₁,p₃₂,p₃₄]^T=C^-1D[β₁,kβ₁,k,1]^T (19)

Utilize above-mentioned formula β₁Replace all elements in P with two unknown numbers of k, i.e. complete the parametrization of P, due to before P 3 Row have the character identical with spin matrix R, i.e. spin matrix any two and arrange mutually orthogonal and have identical mould, utilize these Character lists the equation about P matrix all elements, as follows:

wp₁₁wp₁₂+wp₂₁wp₂₂+p₃₁wp₃₂=0, (20)

$w^2{p}_{11}^2+w^2{p}_{21}^2+p_{31}^2-w^2{p}_{12}^2-w^2{p}_{22}^2-p_{32}^2=0.---\left(21\right)$

In conjunction with the 4th the projection relation of 2 peacekeeping 3-dimensional corresponding point can be obtained equation:

$(1+{\mathrm{kr}}_4^2)(p_{11}{x}_4+p_{12}{y}_4+p_{14})-u_4(p_{31}{x}_4+p_{32}{y}_4+p_{34})=0---\left(22\right)$

Aggregative formula (20), (21), unknown number β in (22) solving equation₁, k and w=1/f, in conjunction with the parameterized Equation reverse of P All elements in P matrix, and then solve the camera transition matrix R relative to world coordinate system_CAnd T_C；

4th step: model sport pose solves

Initially set up model local coordinate system, using model barycenter as local coordinate system initial point O_t, point to mould along model center axis Type caudal directions is O_tY_tAxle positive direction, took initial point O_tWith O_tY_tAxle is vertical and line phase with same group of red-label point of head and the tail The direction handed over is O_tX_tAxle positive direction, defines local coordinate system O according to the right-hand rule_tZ_tDirection of principal axis, first before model pose measurement First obtain the video camera initial position transformational relation relative to world coordinate system, i.e. spin matrix R_CWith translation vector T_C, then lead to Cross zoom pose measurement process and obtain model each moment model local coordinate system in motor process relative to camera The translation vector T of coordinate system_OWith spin matrix R_O, utilize below equation computation model thing local coordinate system and wind tunnel axis system Transition matrix:

$\left[\begin{array}{c}{x}_t\\ y_t\\ z_t\end{array}\right]=R_o^{-1}{R}_c\left[\begin{array}{c}{x}_w\\ y_w\\ z_w\end{array}\right]+R_o^{-1}(T_c-T_o)---\left(23\right)$

Wherein, (x_t,y_t,z_t) it is labelling point coordinate, (x under model local coordinate system_w,y_w,z_w) it is that labelling point is at wind-tunnel coordinate The lower coordinate of system, and:

$R=R_o^{-1}{R}_o---\left(24\right)$

$T=R_o^{-1}(T_c-T_o)={\left[\begin{array}{ccc}{t}_x& t_y& t_z\end{array}\right]}^T---\left(25\right)$

R and T is respectively model relative to the spin matrix of world coordinate system and translation matrix.