CN103247048A - Camera mixing calibration method based on quadratic curve and straight lines - Google Patents

Abstract

A camera mixing calibration method based on a quadratic curve and straight lines aims to enable a camera to exactly complete calibration in a small field of view environment so as to accurately perform subsequent three-dimensional measurement. According to the invention, a brand-new standard circle calibration template is designed; the template is shot from any three different angles; an equation of a circle and linear equations of the radius are detected out; then anharmonic ratio invariance of projective transformation is utilized to accurately obtain two points outside the circle; accordingly, a tangential equation passing through the two points outside the circle is obtained; combining the equation of the circle and the three linear equations, so as to obtain a homography matrix; establishing a corresponding relation of the world coordinate system and an image coordinate system; calculating inside parameters and outside parameters of the camera by using unit orthogonality of a rotation matrix; and finally considering a second-order radial distortion to optimize a calibration result, so as to complete the whole calibration process of the camera.

Description

A kind of video camera mixed calibration method based on quafric curve and straight line

Technical field

The invention belongs to the technical field of three-dimensional information reconstruct, is a kind of in the small items three-dimension measuring system, utilizes the characteristic of quafric curve and straight line, by demarcating Justify calibrating template, improve the technology of small items three-dimensional measurement precision.

Background technology

The three-dimensional measurement technology is a branch of computer image processing technology, be that computer vision and computer graphic image are handled a research direction that combines, it all has a wide range of applications in fields such as the production automation, robot vision, CAD, virtual reality and the diagnosis of medical science reflection.Camera calibration is requisite prior step in the three-dimensional measurement.The precision of demarcating affects the precision of whole three-dimension measuring system to a great extent.

Camera marking method can be divided into two big classes: traditional camera marking method and camera self-calibration method.Traditional camera marking method has direct linear transformation's method, the two-step approach that Tsai proposes, Zhang Zhengyou scaling method and be the various methods based on plane template of representative with it.Camera self-calibration does not rely on specific demarcation masterplate, only utilizes video camera to finish demarcation at the image of different angles shooting and the corresponding relation between the image.Common camera self-calibration method has based on the self-calibrating method of Kruppa equation, based on quadric self-calibrating method, based on self-calibrating method of active vision etc.Because camera self-calibration method do not need scaling board, with respect to traditional camera marking method, demarcate certainly on dirigibility and practicality and all be greatly improved.But on precision, self-calibrating method and traditional camera marking method also have bigger gap.Therefore, in the high precision three-dimensional measurement system, the application of traditional camera marking method is more extensive.

In the traditional cameras scaling method, be considered at present comprehensive various factors based on the camera marking method of plane template and consider best camera marking method.Common calibrating template has gridiron pattern and circular index point.Adopt tessellated method to have some inherent shortcomings: to be right-angled intersection point on the gridiron pattern based on the monumented point of the scaling method utilization of gridiron pattern template, relation conversion on the multiple image between the world coordinates of the image coordinate of right-angled intersection point and this right-angled intersection point needs the process of a logos point coupling, is to be difficult to accomplish very stable images match process and only utilize the perspective projection relation between the right-angled intersection point.Comparatively speaking, circle marker thing template has more superiority.The first, circular target is identified easily and threshold value is cut apart insensitive, and this will simplify the process of demarcating and improve the final precision of demarcating.The second, when the circle marker point template is set, the relation that several big roundlets retrain image coordinate and world coordinates can be set, this will simplify the matching process of the image coordinate of circular index point greatly, for calibration process facilitates.Therefore, the scaling method based on the circle marker thing has obtained approval widely.

Circle marker thing template is segmented, can be divided into the composition template of circular dot matrix mark template, circular target and other geometric configuratioies and the gang form of minority circular target.Camera marking method stated accuracy height based on circular dot matrix mark, noiseproof feature is good, but arranging of circular dot matrix mark is loaded down with trivial details relatively, and in calibration process each circular dot matrix is carried out image coordinate coupling, has brought many inconvenience to calibration process.In recent years, many scholars are from the practicality of demarcating and the angle of convenience, and the camera marking method based on single or two circle marker thing has been carried out broad research.Wherein, the camera marking method based on concentric circles, coaxial circle, circle and straight line combination is applied.Though these methods have been simplified the manufacture difficulty of calibrating template to a certain extent, in calibration process, the template that they adopt is often not really common or difficult extraction in practice.In the template of circle and straight line combination, apart from influences such as far and near and illumination powers, often cause the fracture of continuous straight line during owing to shooting especially, extract to straight line and brought very burden, directly influenced asking for of camera parameters.

Summary of the invention

Goal of the invention: at above-mentioned prior art, the invention provides a kind of video camera mixed calibration method based on quafric curve and straight line, to reduce the parameter error of camera calibration under the small field of view environment, improve measuring accuracy.

Technical scheme:

A kind of video camera mixed calibration method based on quafric curve and straight line comprises the steps:

Step 1: adopt video camera from three different angles to standard

Circular shuttering is taken, and obtains three width of cloth images;

Step 2: piece image is wherein handled, obtained 3 * 3 homography matrix H of this width of cloth image correspondence:

Step 2.1: utilize the Canny edge detection algorithm to extract in the image

Oval edge;

Step 2.2: utilize least square fitting

The equation of oval equation and oval two axle place straight lines; Wherein, oval corresponding quadratic form equation Q is:

$Q=\left[\begin{array}{ccc}a& b/2& d/2\\ b/2& c& e/2\\ d/2& e/2& f\end{array}\right]$ (1) formula

Article two, elliptical shaft place straight-line equation is respectively l ₁, l ₂:

l ₁：k ₁x+m ₁y+n ₁＝0

(2) formula

l ₂：k ₂x+m ₂y+n ₂＝0

Namely

l ₁＝[k ₁，m ₁，n ₁] ^T

(3) formula

l ₂＝[k ₂，m ₂，n ₂] ^T

Wherein, a, b, c, d, e, f are the parameters of oval general equation, k ₁, m ₁, n ₁, k ₂, m ₂, n ₂It is the parameter of straight line general equation;

Step 2.3: the elliptic equation that obtains according to described step 2.2 and elliptical shaft straight-line equation wherein, try to achieve intersecting point coordinate A (x oval and the elliptical shaft straight line ₁, y ₁) and B (x ₂, y ₂);

Step 2.4: according to two elliptical shaft straight-line equations, try to achieve oval central coordinate of circle O (x ₀, y ₀),

Step 2.5: choose double ratio

Try to achieve ray

Last 1 C (x ₃, y ₃) coordinate; Choose double ratio

Try to achieve ray

Last 1 D (x ₄, y ₄) coordinate;

Step 2.6: obtain the oval outer some C (x that tries to achieve by described step 2.5 respectively ₃, y ₃) and D (x ₄, y ₄) with

Oval tangent tangent line, two tangent lines that order is asked for are l ₃, l ₄:

l ₃＝[k ₃，m ₃，n ₃] ^T

(4) formula

l ₄＝[k ₄，m ₄，n ₄] ^T

Wherein, k ₃, m ₃, n ₃, k ₄, m ₄, n ₄It is the parameter of straight line general equation;

Step 2.7: in standard

Before the perspective projection of video camera of circle template process, standard The equation of circle template under world coordinate system is formula (5):

x ²+ y ²=R ²(5) formula

Wherein, R is standard

The radius of circle template, x, y are the parameter of circle general equation; Its quadratic form equation P is:

$P=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& {-R}^2\end{array}\right]$ (6) formula

Article two, the straight line L at radius place ₁, L ₂Equation is:

L ₁＝[0，1，0] ^T

(7) formula

L ₂＝[1，0，0] ^T

According to the double ratio value

Try to achieve straight line l ₃And l ₄Be respectively L through the equation before the perspective projection of video camera ₃, L ₄:

$L_3=[\frac{\sqrt{3}}{3},-1,\frac{2\sqrt{3}}{3}R]$

(8) formula

$L_4=[\frac{\sqrt{3}}{3},1,-\frac{2\sqrt{3}}{3}R]$

Wherein, R is standard

The radius of circle template;

Step 2.8: the quadratic form equation P and the L that try to achieve according to described step 2.7 ₁, L ₂, L ₃, L ₄, according to the image-forming principle of video camera, following relational expression is arranged:

H ^TQH＝P

H ^Tl ₁=s ₁L ₁(9) formula

H ^Tl ₃＝s ₂L ₃

H ^Tl ₄＝s ₃L ₄

That is: H ^T[l ₁, l ₃, l ₄]=[L ₁, L ₃, L ₄] S (10) formula

Wherein, s ₁, s ₂, s ₃Be scale factor, S=diag{s ₁, s ₂, s ₃; Trying to achieve homography matrix H is:

H＝[l ₁，L ₂，L ₃] ^-TS[L ₁，L ₂，L ₃] ^T；

Step 3: calibrating camera inner parameter K:

Step 3.1: other two width of cloth images taken according to described step 2, are obtained corresponding homography matrix H respectively ₂, H ₃

Step 3.2: the homography matrix H that asks for can be expressed as:

λ H=λ [h ₁, h ₂, h ₃]=K[r ₁, r ₂, T] and (11) formula

Wherein, λ is scale factor, h ₁, h ₂, h ₃Be three column vectors of homography matrix H, K is 3 * 3 confidential reference items matrixes, r ₁And r ₂Be two rotating vectors of rotation matrix R in 3 * 3 external parameters, T is 3 * 1 translation matrix in the external parameter;

There is relational expression in unit orthogonality according to rotation matrix:

$\left\{\begin{array}{c}{h}_1^T{K}^{-T}{K}^{-1}{h}_2=0\\ h_1^T{K}^{-T}{K}^{-1}{h}_1=h_2^T{K}^{-T}{K}^{-1}{h}_2\end{array}\right.$ (12) formula

The confidential reference items matrix K can be expressed as:

$K=\left[\begin{array}{ccc}{f}_x& w& u_0\\ 0& f_y& v_0\\ 0& 0& 1\end{array}\right]$

Wherein comprise the equivalent focal length (f on x axle and the y direction of principal axis _x, f _y), isoboles inconocenter point coordinate (u ₀, v ₀) and horizontal ordinate angle coefficient w totally 5 unknown quantitys, three homography matrix H ₁, H ₂, H ₃Can try to achieve one group respectively suc as formula the system of equations of (12), according to three groups of system of equations that draw, try to achieve the confidential reference items matrix K accordingly;

Step 4: calibrating camera external parameter [R, T]:

Video camera external parameter [R, T] bag 4 is drawn together 3 * 3 rotation matrix R=[r ₁, r ₂, r ₃] and 3 * 1 translation matrix T; Wherein, r ₁, r ₂, r ₃Be three column vectors of rotation matrix R, obtain can trying to achieve outer ginseng matrix R and T according to formula (13) after the confidential reference items matrix K according to described step 3:

r ₁＝δK ^-1h ₁

r ₂=δ K ^-1h ₂(13) formula

r ₃＝r ₁×r ₂

T＝δK ^-1h ₃

Wherein, δ=1/||K ^-1h ₁||=1/||K ^-1h ₂|| be scale factor;

Step 5: ask for the distortion of camera coefficient:

According to the relation of ideal image point with actual imaging point, try to achieve distortion factor; Described ideal image point with the pass of actual imaging point is:

$\hat{x}=x+x[j_1(x^2+y^2)+j_2{(x^2+y^2)}^2]$

(14) formula

$\hat{y}=y+y[j_1(x^2+y^2)+j_2{(x^2+y^2)}^2]$

Wherein, j ₁And j ₂Be the distortion factor of asking, Be the picture point coordinate that actual detected arrives, (x y) is the ideal image point coordinate; Solve video camera confidential reference items matrix K, join matrix R and T and distortion parameter j outward ₁, j ₂After, namely finished camera calibration.

Beneficial effect: the present invention is mainly used in the small field of view environment camera calibration in following time.Compared with prior art, the present invention has the following advantages: at first, traditional scaling method need be put customization before video camera length and width are tens of centimetres accurate scaling board, but in the small items measuring system, the visual field of video camera only is 4cm*3cm, can't from traditional scaling board, extract abundant information and finish demarcation, the brand-new standard of the present invention's redesign

Circular shuttering utilizes

Circumference place equation and tangential equation are demarcated in the circle, have solved in small field of view environment following time to use traditional scaling board can't finish the problem of demarcation; Secondly, the mixed calibration method that the present invention uses utilizes the double ratio unchangeability of projective transformation to try to achieve outer several tangential equations of circle, in conjunction with the quadratic form equation with circle, can accurately try to achieve homography matrix, and then draw camera interior and exterior parameter accurately; At last, a double ratio value is set obtains two tangential equations, in conjunction with diameter, three straight-line equations and a curvilinear equation can obtain minimal solution altogether, a plurality of different double ratio values are set can obtain many group tangential equations, thereby the result is optimized, obtains the higher result of precision.Therefore, the present invention not only can finish the camera calibration under the small field of view environment, more is enhanced than existing traditional cameras calibration technique on dirigibility and robustness.

Description of drawings

Fig. 1 is the process flow diagram of whole calibrating procedure of the present invention;

Fig. 2 is the standard that the present invention designs

Circular masterplate;

Fig. 3 is the calibrating template figure that takes from three different angles;

Fig. 4 is standard

The synoptic diagram of circular shuttering before the process video camera imaging;

Fig. 5 is standard

The synoptic diagram of circular shuttering after the process video camera imaging.

Embodiment

Below in conjunction with accompanying drawing the present invention is done further explanation.

As shown in Figure 1, use the present invention and realize that the concrete steps of camera calibration are as follows:

Step 1: adopt video camera from three different angles to as shown in Figure 2 standard

Circular shuttering is taken, and obtains three width of cloth images as shown in Figure 3;

Step 2: piece image is wherein handled, obtained 3 * 3 homography matrix H of this width of cloth image correspondence:

Step 2.1: utilize the Canny edge detection algorithm to extract in the image

Oval edge;

Step 2.2: utilize least square fitting The equation of oval equation and oval two axle place straight lines, as shown in Figure 4; Wherein, oval corresponding quadratic form equation Q is:

$Q=\left[\begin{array}{ccc}a& b/2& d/2\\ b/2& c& e/2\\ d/2& e/2& f\end{array}\right]$ (1) formula

Article two, elliptical shaft place straight-line equation is respectively l ₁, l ₂:

l ₁：k ₁x+m ₁y+n ₁＝0

(2) formula

l ₂：k ₂x+m ₂y+n ₂＝0

Namely

l ₁＝[k ₁，m ₁，n ₁] ^T

(3) formula

l ₂＝[k ₂，m ₂，n ₂]T

Wherein, a, b, c, d, e, f are the parameters of oval general equation, k ₁, m ₁, n ₁, k ₂, m ₂, n ₂It is the parameter of straight line general equation;

Step 2.3: the elliptic equation that obtains according to described step 2.2 and elliptical shaft straight-line equation wherein, try to achieve intersecting point coordinate A (x oval and the elliptical shaft straight line ₁, y ₁) and B (x ₂, y ₂);

Step 2.4: according to two elliptical shaft straight-line equations, try to achieve oval central coordinate of circle O (x ₀, y ₀),

Step 2.5: choose double ratio

Try to achieve ray

Last 1 C (x ₃, y ₃) coordinate; Choose double ratio

Utilize the double ratio unchangeability of photography conversion, try to achieve ray

Last 1 D (x ₄, y ₄) coordinate;

Step 2.6: obtain the oval outer some C (x that tries to achieve by described step 2.5 respectively ₃, y ₃) and D (x ₄, y ₄) with

Oval tangent tangent line, two tangent lines that order is asked for are l ₃, l ₄:

l ₃＝[k ₃，m ₃，n ₃] ^T

(4) formula

l ₄＝[k ₄，m ₄，n ₄] ^T

Wherein, k ₃, m ₃, n ₃, k ₄, m ₄, n ₄It is the parameter of straight line general equation;

Step 2.7: in standard

Before the perspective projection of video camera of circle template process, standard

The equation of circle template under world coordinate system is formula (5):

x ²+ y ²=R ²(5) formula

Wherein, R is standard The radius of circle template, x, y are the parameter of circle general equation; Its quadratic form equation P is:

$P=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& {-R}^2\end{array}\right]$ (6) formula

As shown in Figure 5, the straight line L at two radius places ₁, L ₂Equation is:

L ₁＝[0，1，0] ^T

(7) formula

L ₂＝[1，0，0] ^T

According to the double ratio value

Try to achieve straight line l ₃And l ₄Be respectively L through the equation before the perspective projection of video camera ₃, L ₄:

$L_3=[\frac{\sqrt{3}}{3},-1,\frac{2\sqrt{3}}{3}R]$

(8) formula

$L_4=[\frac{\sqrt{3}}{3},1,-\frac{2\sqrt{3}}{3}R]$

Wherein, R is standard

The radius of circle template;

Step 2.8: the quadratic form equation P and the L that try to achieve according to described step 2.7 ₁, L ₂, L ₃, L ₄, according to the image-forming principle of video camera, following relational expression is arranged:

H ^TQH＝P

H ^Tl ₁=s ₁L ₁(9) formula

H ^TL ₃＝s ₂L ₃

H ^Tl ₄＝s ₃L ₄

That is: H ^T[l ₁, l ₃, l ₄]=[L ₁, L ₃, L ₄] S (10) formula

Wherein, s ₁, s ₂, s ₃Be scale factor, S=diag{s ₁, s ₂, s ₃; Trying to achieve homography matrix H is:

H＝[l ₁，l ₂，l ₃] ^-1S[L ₁，L ₂，L ₃] ^T；

Wherein, H is 3 * 3 homography matrixes of the step 2 piece image correspondence of trying to achieve, l ₁, l ₂Be two elliptical shaft place straight lines that step 2.2 is tried to achieve, l ₃, l ₄For step 2.6 is tried to achieve Oval tangent tangent line;

Step 3: calibrating camera inner parameter K:

Step 3.1: other two width of cloth images taken according to described step 2, are obtained corresponding homography matrix H respectively ₂, H ₃

Step 3.2: the homography matrix H that asks for can be expressed as:

λ H=λ [h ₁, h ₂, h ₃]=K[r ₁, r ₂, T] and (11) formula

Wherein, λ is scale factor, h ₁, h ₂, h ₃Be three column vectors of homography matrix H, K is 3 * 3 confidential reference items matrixes, r ₁And r ₂Be two rotating vectors of rotation matrix R in 3 * 3 external parameters, T is 3 * 1 translation matrix in the external parameter;

There is relational expression in unit orthogonality according to rotation matrix:

$\left\{\begin{array}{c}{h}_1^T{K}^{-T}{K}^{-1}{h}_2=0\\ h_1^T{K}^{-T}{K}^{-1}{h}_1=h_2^T{K}^{-T}{K}^{-1}{h}_2\end{array}\right.$ (12) formula

The confidential reference items matrix K can be expressed as:

$K=\left[\begin{array}{ccc}{f}_x& w& u_0\\ 0& f_y& v_0\\ 0& 0& 1\end{array}\right]$

Wherein comprise the equivalent focal length (f on x axle and the y direction of principal axis _x, f _y), isoboles inconocenter point coordinate (u ₀, v ₀) and horizontal ordinate angle coefficient w totally 5 unknown quantitys, three homography matrix H ₁, H ₂, H ₃Can try to achieve one group respectively suc as formula the system of equations of (12), according to three groups of system of equations that draw, try to achieve the confidential reference items matrix K accordingly;

Step 4: calibrating camera external parameter [R, T]:

Video camera external parameter [R, T] bag 4 is drawn together 3 * 3 rotation matrix R=[r ₁, r ₂, r ₃] and 3 * 1 translation matrix T; Wherein, r ₁, r ₂, r ₃Be three column vectors of rotation matrix R, obtain can trying to achieve outer ginseng matrix R and T according to formula (13) after the confidential reference items matrix K according to described step 3:

r ₁＝δK ^-1h ₁

r ₂=δ K ^-1h ₂(13) formula

r ₃＝r ₁×r ₂

T＝δK ^-1h ₃

Wherein, δ=1/||K ^-1h ₁||=1/||K ^-1h ₂|| be scale factor;

Step 5: ask for the distortion of camera coefficient:

Camera lens distortion is that the manufacturing process by camera lens causes, and according to the relation of ideal image point and actual imaging point, tries to achieve distortion factor; Described ideal image point with the pass of actual imaging point is:

$\hat{x}=x+x[j_1(x^2+y^2)+j_2{(x^2+y^2)}^2]$

(14) formula

$\hat{y}=y+y[j_1(x^2+y^2)+j_2{(x^2+y^2)}^2]$

Wherein, j ₁And j ₂Be the distortion factor of asking,

Be the picture point coordinate that actual detected arrives, (x y) is the ideal image point coordinate; Solve video camera confidential reference items matrix K, join matrix R and T and distortion parameter j outward ₁, j ₂After, namely finished camera calibration.

The above only is preferred implementation of the present invention; should be pointed out that for those skilled in the art, under the prerequisite that does not break away from the principle of the invention; can also make some improvements and modifications, these improvements and modifications also should be considered as protection scope of the present invention.

Claims (1)

1. the video camera mixed calibration method based on quafric curve and straight line is characterized in that: comprise the steps:

Step 1: adopt video camera from three different angles to standard

Circular shuttering is taken, and obtains three width of cloth images;

Step 2: piece image is wherein handled, obtained 3 * 3 homography matrix H of this width of cloth image correspondence:

Step 2.1: utilize the Canny edge detection algorithm to extract in the image

Oval edge;

Step 2.2: utilize least square fitting The equation of oval equation and oval two axle place straight lines; Wherein, oval corresponding quadratic form equation Q is:

$Q=\left[\begin{array}{ccc}a& b/2& d/2\\ b/2& c& e/2\\ d/2& e/2& f\end{array}\right]$ (1) formula

Article two, elliptical shaft place straight-line equation is respectively l ₁, l ₂:

l ₁：k ₁x+m ₁y+n ₁＝0

(2) formula

l ₂：k ₂x+m ₂y+n ₂＝0

Namely

l ₁＝[k ₁，m ₁，n ₁] ^T

(3) formula

l ₂＝[k ₂，m ₂，n ₂] ^T

Wherein, a, b, c, d, e, f are the parameters of oval general equation, k ₁, m ₁, n ₁, k ₂, m ₂, n ₂It is the parameter of straight line general equation;

Step 2.3: the elliptic equation that obtains according to described step 2.2 and elliptical shaft straight-line equation wherein, try to achieve intersecting point coordinate A (x oval and the elliptical shaft straight line ₁, y ₁) and B (x ₂, y ₂);

Step 2.4: according to two elliptical shaft straight-line equations, try to achieve oval central coordinate of circle O (x ₀, y ₀),

Step 2.5: choose double ratio

Try to achieve ray Last 1 C (x ₃, y ₃) coordinate; Choose double ratio

Try to achieve ray

Last 1 D (x ₄, y ₄) coordinate;

Step 2.6: obtain the oval outer some C (x that tries to achieve by described step 2.5 respectively ₃, y ₃) and D (x ₄, y ₄) with

Oval tangent tangent line, two tangent lines that order is asked for are l ₃, l ₄:

l ₃＝[k ₃，m ₃，n ₃] ^T

(4) formula

l ₄＝[k ₄，m ₄，n ₄] ^T

Wherein, k ₃, m ₃, n ₃, k ₄, m ₄, n ₄It is the parameter of straight line general equation;

Step 2.7: in standard Before the perspective projection of video camera of circle template process, standard

The equation of circle template under world coordinate system is formula (5):

x ²+ y ²=R ²(5) formula

Wherein, R is standard

The radius of circle template, x, y are the parameter of circle general equation; Its quadratic form equation P is:

$P=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& {-R}^2\end{array}\right]$ (6) formula

Article two, the straight line L at radius place ₁, L ₂Equation is:

L ₁＝[0，1，0] ^T

(7) formula

L ₂＝[1，0，0] ^T

According to the double ratio value

Try to achieve straight line l ₃And l ₄Be respectively L through the equation before the perspective projection of video camera ₃, L ₄:

$L_3=[\frac{\sqrt{3}}{3},-1,\frac{2\sqrt{3}}{3}R]$

(8) formula

$L_4=[\frac{\sqrt{3}}{3},1,-\frac{2\sqrt{3}}{3}R]$

Wherein, R is standard

The radius of circle template;

Step 2.8: the quadratic form equation P and the L that try to achieve according to described step 2.7 ₁, L ₂, L ₃, L ₄, according to the image-forming principle of video camera, following relational expression is arranged:

H ^TQH＝P

H ^Tl ₁=s ₁L ₁(9) formula

H ^Tl ₃＝s ₂L ₃

H ^Tl ₄＝s ₃L ₄

That is: H ^T[l ₁, l ₃, l ₄]=[l ₁, L ₃, L ₄] S (10) formula

Wherein, s ₁, s ₂, s ₃Be scale factor, S=diag{s ₁, s ₂, s ₃; Trying to achieve homography matrix H is:

H＝[l ₁，l ₂，l ₃] ^-TS[L ₁，L ₂，L ₃] ^T；

Step 3: calibrating camera inner parameter K:

Step 3.1: other two width of cloth images taken according to described step 2, are obtained corresponding homography matrix H respectively ₂, H ₃

Step 3.2: the homography matrix H that asks for can be expressed as:

λ H=λ [h ₁, h ₂, h ₃]=K[r ₁, r ₁, T] and (11) formula

Wherein, λ is scale factor, h ₁, h ₂, h ₃Be three column vectors of homography matrix H, K is 3 * 3 confidential reference items matrixes, r ₁And r ₂Be two rotating vectors of rotation matrix R in 3 * 3 external parameters, T is 3 * 1 translation matrix in the external parameter;

There is relational expression in unit orthogonality according to rotation matrix:

$\left\{\begin{array}{c}{h}_1^T{K}^{-T}{K}^{-1}{h}_2=0\\ h_1^T{K}^{-T}{K}^{-1}{h}_1=h_2^T{K}^{-T}{K}^{-1}{h}_2\end{array}\right.$ (12) formula

The confidential reference items matrix K can be expressed as:

$K=\left[\begin{array}{ccc}{f}_x& w& u_0\\ 0& f_y& v_0\\ 0& 0& 1\end{array}\right]$

Wherein comprise the equivalent focal length (f on x axle and the y direction of principal axis _x, f _y), isoboles inconocenter point coordinate (u ₀, v ₀) and horizontal ordinate angle coefficient w totally 5 unknown quantitys, three homography matrix H ₁, H ₂, H ₃Can try to achieve one group respectively suc as formula the system of equations of (12), according to three groups of system of equations that draw, try to achieve the confidential reference items matrix K accordingly;

Step 4: calibrating camera external parameter [R, T]:

Video camera external parameter [R, T] bag 4 is drawn together 3 * 3 rotation matrix R=[r ₁, r ₂, r ₃] and 3 * 1 translation matrix T; Wherein, r ₁, r ₂, r ₃Be three column vectors of rotation matrix R, obtain can trying to achieve outer ginseng matrix R and T according to formula (13) after the confidential reference items matrix K according to described step 3:

r ₁＝δK ^-1h ₁

r ₂=δ K ^-1h ₂(13) formula

r ₃＝r ₁×r ₂

T＝δK ^-1h ₃

Wherein, δ=1/||K ^-1h ₁||=1/||K ^-1h ₂|| be scale factor;

Step 5: ask for the distortion of camera coefficient:

According to the relation of ideal image point with actual imaging point, try to achieve distortion factor; Described ideal image point with the pass of actual imaging point is:

$\hat{x}=x+x[j_1(x^2+y^2)+j_2{(x^2+y^2)}^2]$

(14) formula

$\hat{y}=y+y[j_1(x^2+y^2)+j_2{(x^2+y^2)}^2]$

Wherein, j ₁And j ₂Be the distortion factor of asking,

Be the picture point coordinate that actual detected arrives, (x y) is the ideal image point coordinate; Solve video camera confidential reference items matrix K, join matrix R and T and distortion parameter j outward ₁, j ₂After, namely finished camera calibration.

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