CN102800084B - Method for measuring image principal point coordinates and distortion coefficient of linear target - Google Patents

Abstract

The invention discloses a method for measuring the image principal point coordinates and distortion coefficient of a linear target. The method comprises the following steps of: 10, imaging a symmetry axis linear equation of a curve target: fitting the measured object, and calculating the slope of the symmetry axis of the measured object; converting the image points of the measured object to the coordinates of the curve coordinate system; reckoning the coefficient of a parabolic equation; converting the vertex of the parabola to the coordinates of the image coordinate system; and establishing a linear equation of the symmetry axis of the measured object; 20, reckoning the principal point coordinates of the image; and 30, measuring the image distortion coefficient: selecting an image point from the measured object, and obtaining two-dimensional coordinates of the image point in the image coordinate system; establishing a distortion coefficient equation; recovering the distortion so that the measured object becomes a line from a curve; giving the normal equation of the measured object; and establishing the normal equation of the measured object and reckoning the distortion coefficient of the image. The measuring method can be used for measuring the image distortion coefficient and principal point coordinates of the imaged picture.

Description

A kind of image slices principal point coordinate of straight-line target and the assay method of distortion factor

Technical field

The invention belongs to image measurement field, specifically, relate to a kind of image slices principal point coordinate of straight-line target and the assay method of distortion factor.

Background technology

Computer vision field usually requires, by image, photographic is carried out to space three-dimensional measurement, and wherein, pattern distortion is very large on the impact of measurement result, requires by obtaining the distortion factor of image, the coordinate of picture point to be carried out to distortion correction in measuring process.At present, the computing method of pattern distortion coefficient have two kinds: the one, and chamber is demarcated by experiment, utilizes special-purpose instrument and equipment to identify in laboratory, offers user; The 2nd, the reference mark of laying some before photography in camera coverage, goes out distortion factor by calculated with mathematical model.Yet these two kinds of methods cannot be to the conventional images calculating that distorts, particularly, the coordinate of image slices principal point also can affect the demarcation of pattern distortion coefficient.

Summary of the invention

Technical matters: the technical problem to be solved in the present invention is: a kind of image slices principal point coordinate of straight-line target and the assay method of distortion factor are provided, this assay method carries out curve fitting by the point on a plurality of symmetrical curves on picture, determine the equation of curve symmetric axle, and carry out matching according to a plurality of axis of symmetry equations, and then calculate its intersecting point coordinate, be image slices principal point coordinate, then by the method that a plurality of symmetrical curves are recovered, determine the distortion factor of image; This assay method is for the picture of imaging, completes the mensuration to pattern distortion coefficient and image slices principal point coordinate, need to before photography, to photographic equipment, not demarcate or lay sign in camera coverage.

Technical scheme: for solving the problems of the technologies described above, the technical solution used in the present invention is:

The image slices principal point coordinate of straight-line target and an assay method for distortion factor, this assay method comprises the following steps:

10. become the axis of symmetry straight-line equation of image curve target:

101. utilize picture pick-up device to take pictures to straight-line target, obtain image, and straight-line target is shown as symmetrical curve on image; Foundation be take picture centre as initial point, and horizontal ordinate is x axle, and ordinate is the image coordinate system o-xy of y axle;

102. select m bar symmetrical curve as determination object from image, and m is integer, and m >=2, select a symmetrical curve to carry out step 103 to the operation of step 110;

103. choose n picture point from determination object, and n is integer, and n>=5, obtain the two-dimensional coordinate (x of n picture point in image coordinate system o-xy _i, y _i), i=1 wherein, 2 ... n;

104. use quadratic polynomial Ax _i ²+ Bx _iy _i+ Cy _i ²+ Dx _i+ Ey _i+ 1=0 carries out matching to the determination object in step 103, and wherein, A, B, C, D and E are quadratic polynomial coefficient, x _iand y _ithe two-dimensional coordinate of picture point on expression determination object in image coordinate system o-xy, utilizes formula (1) to calculate quadratic polynomial coefficient A, B, C, D and E,

$\left[\begin{array}{ccccc}\mathrm{\Σ}{x}_i^4& \mathrm{\Σ}{x}_i^3{y}_i& \mathrm{\Σ}{x}_i^2{y}_i^2& \mathrm{\Σ}{x}_i^3& \mathrm{\Σ}{x}_i^2{y}_i\\ \mathrm{\Σ}{x}_i^3{y}_i& \mathrm{\Σ}{x}_i^2{y}_i^2& \mathrm{\Σ}{x}_i{y}_i^3& \mathrm{\Σ}{x}_i^2{y}_i& \mathrm{\Σ}{x}_i{y}_i^2\\ \mathrm{\Σ}{x}_i^2{y}_i^2& \mathrm{\Σ}{x}_i{y}_i^3& \mathrm{\Σ}{y}_i^4& \mathrm{\Σ}{x}_i{y}_i^2& \mathrm{\Σ}{y}_i^3\\ \mathrm{\Σ}{x}_i^3& \mathrm{\Σ}{x}_i^2{y}_i& \mathrm{\Σ}{x}_i{y}_i^2& \mathrm{\Σ}{x}_i^2& \mathrm{\Σ}{x}_i{y}_i\\ \mathrm{\Σ}{x}_i^2{y}_i& \mathrm{\Σ}{x}_i{y}_i^2& \mathrm{\Σ}{y}_i^3& \mathrm{\Σ}{x}_i{y}_i& \mathrm{\Σ}{y}_i^2\end{array}\right]\·\left[\begin{array}{c}A\\ B\\ C\\ D\\ E\end{array}\right]=-\left[\begin{array}{c}\mathrm{\Σ}{x}_i^2\\ \mathrm{\Σ}{x}_i{y}_i\\ \mathrm{\Σ}{y}_i^2\\ \mathrm{\Σ}{x}_i\\ \mathrm{\Σ}{y}_i\end{array}\right]$ Formula (1)

In formula, i=1,2 ... n;

105. quadratic polynomial coefficient A, B, C, D and the E that calculate according to step 104, and formula (2), calculate the slope u of the axis of symmetry of determination object:

$u=\±\sqrt{\frac{A}{C}}$ Formula (2)

The symbol of u according to axis of symmetry the direction in image coordinate system o-xy relevant, axis of symmetry is got "+" number when first quartile or third quadrant, at the second quadrant or fourth quadrant, gets "-" number;

106. set up curvilinear coordinate system o-x ' y ': take the initial point of image coordinate system o-xy as initial point, take and be parallel to the right-handed coordinate system that the direction of determination object axis of symmetry is y ';

107. coordinate conversion: first according to the indexing α between formula (3) measuring and calculating image coordinate system o-xy and curvilinear coordinate system o-x ' y ',

$\tan \mathrm{\α}=\frac{1}{u}$ Formula (3);

Then utilize the coordinate conversion matrix R between formula (4) measuring and calculating image coordinate system o-xy and curvilinear coordinate system o-x ' y ':

$R=\left[\begin{array}{cc}\cos \mathrm{\α}& \sin \mathrm{\α}\\ -\sin \mathrm{\α}& \cos \mathrm{\α}\end{array}\right]$ Formula (4);

Coordinate (the x of the picture point of finally utilizing the determination object that formula (5) selects step 103 in image coordinate system o-xy _i, y _i), be transformed into coordinate in curvilinear coordinate system o-x ' y ' (x ' _i, y ' _i);

$\left[\begin{array}{c}{x}_i^{\′}\\ y_i^{\′}\end{array}\right]=R\·\left[\begin{array}{c}{x}_i\\ y_i\end{array}\right]=\left[\begin{array}{cc}\cos \mathrm{\α}& \sin \mathrm{\α}\\ -\sin \mathrm{\α}& \cos \mathrm{\α}\end{array}\right]\·\left[\begin{array}{c}{x}_i\\ y_i\end{array}\right]$ Formula (5);

108. utilize para-curve determination object is carried out in curvilinear coordinate system o-x ' y ' to matching, utilizes formula (6) to calculate parabolic equation coefficient a, b, c:

$\left[\begin{array}{ccc}\mathrm{\Σ}{x_i^{\′}}^4& \mathrm{\Σ}{x_i^{\′}}^3& \mathrm{\Σ}{x_i^{\′}}^2\\ \mathrm{\Σ}{x_i^{\′}}^3& \mathrm{\Σ}{x_i^{\′}}^2& \mathrm{\Σ}{x}_i^{\′}\\ \mathrm{\Σ}{x_i^{\′}}^2& \mathrm{\Σ}{x}_i^{\′}& n\end{array}\right]\·\left[\begin{array}{c}a\\ b\\ c\end{array}\right]=\left[\begin{array}{c}\mathrm{\Σ}{x_i^{\′}}^2{y}_y^{\′}\\ \mathrm{\Σ}{x}_i^{\′}{y}_y^{\′}\\ \mathrm{\Σ}{y}_y^{\′}\end{array}\right]$ Formula (6)

I=1 in formula, 2 ... n;

109. utilize formula (7) to calculate para-curve the coordinate of summit q in curvilinear coordinate system o-x ' y ' (x ' _q, y ' _q), and utilize formula (8) will (x ' _q, y ' _q) be converted to the coordinate (x of image coordinate system o-xy _q, y _q):

$\left\{\begin{array}{c}{x}_q^{\′}=-\frac{b}{2a}\\ y_q^{\′}=-\frac{b^2}{4a}+c\end{array}\right.$ Formula (7);

$\left[\begin{array}{c}{x}_q\\ y_q\end{array}\right]=R^T\left[\begin{array}{c}{x}_q^{\′}\\ y_q^{\′}\end{array}\right]$ Formula (8);

Coordinate (the x of the slope u of the axis of symmetry of 110. determination objects of measuring according to step 105 and the para-curve summit q that step 109 is measured _q, y _q), set up the straight-line equation of determination object axis of symmetry, shown in (9):

Y=S ₁x+T ₁formula (9)

Wherein, S ₁=u, T ₁=y _q-ux _q;

M-1 bar symmetrical curve remaining in 111. pairs of images carries out respectively step 103 to the operation of step 110, sets up the straight-line equation of each symmetrical curve axis of symmetry, and the straight-line equation of m bar symmetrical curve axis of symmetry forms suc as formula system of equations shown in (10):

Y=S _jx+T _j(j=1,2 ... m) formula (10);

The measuring and calculating of 20. image slices principal point coordinates:

M the equation in formula (10), carries out indirect adjustment with principle of least square method, and the normal equation obtaining, suc as formula shown in (11), calculates the least square intersection point of m axis of symmetry, and this least square intersection point is exactly image slices principal point O'(x ₀, y ₀),

$\left[\begin{array}{cc}\mathrm{\Σ}{S_j}^2& -\mathrm{\Σ}{S}_j\\ -\mathrm{\Σ}{S}_j& m\end{array}\right]\left[\begin{array}{c}{x}_0\\ y_0\end{array}\right]=\left[\begin{array}{c}-\mathrm{\Σ}{S}_j{T}_j\\ \mathrm{\Σ}{T}_j\end{array}\right]$ Formula (11)

J=1 in formula, 2 ... m;

30. measure pattern distortion coefficient:

In 301. m bar symmetrical curves from image, select a symmetrical curve as determination object;

302. choose n picture point from determination object, and n is integer, and n>=5, obtain the two-dimensional coordinate (x of n picture point in image coordinate system o-xy _i, y _i), i=1 wherein, 2 ... n; N picture point two-dimensional coordinate in curvilinear coordinate system o-x ' y ' be (x ' _i, y ' _i);

303. set up distortion factor equation:

First according to formula (12) by image slices principal point O'(x ₀, y ₀) coordinate (x in image coordinate system o-xy ₀, y ₀), be converted to coordinate in curvilinear coordinate system o-x ' y ' (x ' ₀, y ' ₀):

$\left[\begin{array}{c}{x}_0^{\′}\\ y_0^{\′}\end{array}\right]=R\·\left[\begin{array}{c}{x}_0\\ y_0\end{array}\right]=\left[\begin{array}{cc}\cos \mathrm{\α}& \sin \mathrm{\α}\\ -\sin \mathrm{\α}& \cos \mathrm{\α}\end{array}\right]\·\left[\begin{array}{c}{x}_0\\ y_0\end{array}\right]$ Formula (12);

Then set up distortion factor equation, shown in (13):

Δ y ' _i=k ₁(y ' _i-y ' ₀) [((x ' _i-x ' ₀) ²+ (y ' _i-y ' ₀) ²)] formula (13);

In formula (13), Δ y ' _irepresent on determination object 1 p (x ' _i, y ' _i) at the distortion correction value of y ' direction, k ₁represent distortion factor;

304. recover after distortion, and determination object becomes straight line from curve, and on determination object, n picture point recovered should meet formula (14) after distortion:

(y ' _i-y ' ₀)+k ₁(y ' _i-y ' ₀) r ²=C ₁formula (14)

Wherein, C ₁for constant, r presentation graphs picture point (x ' _i, y ' _i) to principal point (x ' ₀, y ' ₀) distance, $r=\sqrt{{{(x_i^{\′}-x_0^{\′})}_i}^2+{{(y_i^{\′}-y^{\′})}_i}^2};$

305. utilize formula (14) can list the normal equation of determination object, shown in (15):

$\left[\begin{array}{cc}\mathrm{\Σ}{\left[(y_i^{\′}-y_0^i)r^2\right]}^2& -\mathrm{\Σ}(y_i^{\′}-y_0^{\′})r^2\\ -\mathrm{\Σ}(y_i^{\′}-y_0^{\′})r^2& n\end{array}\right]\·\left[\begin{array}{c}{k}_1\\ C_1\end{array}\right]=\left[\begin{array}{c}-\mathrm{\Σ}{(y_i^{\′}-y_0^{\′})}^2{r}^2\\ \mathrm{\Σ}(y_i^{\′}-y_0^{\′})\end{array}\right]$ Formula (15)

Formula (15) is expressed as formula (16) with general formula:

$\left[\begin{array}{cc}{N_{11}}^{\left(1\right)}& {N_{12}}^{\left(1\right)}\\ {N_{12}}^{\left(1\right)}& {N_{22}}^{\left(1\right)}\end{array}\right]\·\left[\begin{array}{c}{k}_1\\ C_1\end{array}\right]=\left[\begin{array}{c}{W_1}^{\left(1\right)}\\ {W_2}^{\left(1\right)}\end{array}\right]$ Formula (16)

In formula (16), $N_{11}^{\left(1\right)}=\mathrm{\Σ}{\left[(y_i^{\′}-y_0^{\′})r^2\right]}^2,N_{12}^{\left(1\right)}=-\mathrm{\Σ}(y_i^{\′}-y_0^{\′})r^2,N_{22}^{\left(1\right)}=n,$ W ₁ ⁽¹⁾＝-Σ(y′ _i-y′ ₀) ²r ²，W ₂ ⁽¹⁾＝Σ(y′ _i-y′ ₀)；

Remaining m-1 bar symmetrical curve in 306. pairs of images, respectively as determination object, according to step 302-305, is set up the normal equation of determination object, with general formula, the overall normal equation of m bar symmetrical curve represented, shown in (17):

$\left[\begin{array}{cc}{N_{11}}^{\left(j\right)}& {N_{12}}^{\left(j\right)}\\ {N_{12}}^{\left(j\right)}& {N_{22}}^{\left(j\right)}\end{array}\right]\·\left[\begin{array}{c}{k}_1\\ C_j\end{array}\right]=\left[\begin{array}{c}{W_1}^{\left(j\right)}\\ {W_2}^{\left(j\right)}\end{array}\right]$ Formula (17)

Wherein, j=1,2 ... m;

307. organize formula (17) constitutive equation by m, shown in (18), can calculate the distortion factor k of image ₁:

$\left[\begin{array}{cccccc}\mathrm{\Σ}{N}_{11}^{\left(j\right)}& N_{12}^{\left(1\right)}& ...& N_{12}^{\left(j\right)}& ...& N_{12}^{\left(m\right)}\\ N_{12}^{\left(1\right)}& N_{22}^{\left(1\right)}& ...& 0& ...& 0\\ ...& ...& ...& ...& ...& ...\\ N_{12}^{\left(j\right)}& 0& ...& N_{22}^{\left(j\right)}& ...& 0\\ ...& ...& ...& ...& ...& ...\\ N_{12}^{\left(m\right)}& 0& ...& 0& ...& N_{22}^{\left(m\right)}\end{array}\right]\·\left[\begin{array}{c}{k}_1\\ C_1\\ ...\\ C_j\\ ...\\ C_m\end{array}\right]=\left[\begin{array}{c}\mathrm{\Σ}{W_1}^{\left(j\right)}\\ {W_2}^{\left(1\right)}\\ ...\\ {W_2}^{\left(j\right)}\\ ...\\ {W_2}^{\left(m\right)}\end{array}\right]$ Formula (18)

J=1 wherein, 2 ... m.

Beneficial effect: compared with prior art, the present invention has following beneficial effect:

1. without photography and vedio recording equipment is demarcated in advance, can carry out to the picture of the imaging in any source the mensuration of image slices principal point coordinate and pattern distortion coefficient.The technical program is for the picture of imaging, to carry out the mensuration of image slices principal point coordinate and pattern distortion coefficient, need in photography and vedio recording process, equipment not demarcated in advance, or lay sign in camera coverage.To the picture of imaging, utilize the technical program can carry out the mensuration of image slices principal point coordinate and pattern distortion coefficient, do not need photography and vedio recording again.Like this, for the target of not reproducible photography and vedio recording, utilize the technical program can carry out to the picture of imaging the mensuration of image slices principal point coordinate and pattern distortion coefficient; For can repeat photography the target of shooting, utilize the technical program, can save repeat photography shooting, directly utilize the picture of imaging to carry out the mensuration of image slices principal point coordinate and pattern distortion coefficient.

2. it is convenient to implement, and applicability is strong.Distortion makes straight line be imaged as symmetrical curve, conversely, imageable target curve is reverted to straight line with certain distortion factor, just can determine the distortion factor of image.The assay method of the technical program recovers with certain distortion factor the discrete point on imageable target curve, makes it revert to straight line, and then calculates distortion factor.This assay method is applicable to all pictures that straight-line target is taken, and applicability is strong.Meanwhile, this assay method is implemented convenient, need in photography and vedio recording process, to equipment, not demarcate in advance, or lay sign in camera coverage.

3. the mensuration of pattern distortion coefficient is more accurate.The selection of image slices principal point coordinate can have influence on the mensuration of pattern distortion coefficient.And image slices principal point coordinate conventionally can produce and depart from image center.Assay method provided by the invention, measured image slices principal point coordinate before this, and then according to image slices principal point coordinate, carried out the mensuration of pattern distortion coefficient, thereby made the mensuration of pattern distortion coefficient more accurate.

Accompanying drawing explanation

Fig. 1 is the schematic diagram that the present invention measures image slices principal point, the initial point of o presentation video coordinate system wherein, the position of o ' presentation video principal point.

Fig. 2 is the schematic diagram that the present invention measures pattern distortion coefficient, wherein p is expressed as any point on image curve, t represents p point to carry out the position after y ' direction corrects, p0 represents p point to carry out the position after the distortion correction of x ', y ' direction simultaneously, line between 3 of t, p, p0 forms a right-angle triangle, and the coordinate of y ' direction that t point is ordered with p0 is simultaneously identical.

Embodiment

Below in conjunction with accompanying drawing, technical scheme of the present invention is described in detail.

As shown in Figure 1, the image slices principal point coordinate of a kind of straight-line target of the present invention and the assay method of distortion factor, comprise the following steps:

10. become the axis of symmetry straight-line equation of image curve target:

101. utilize picture pick-up device to take pictures to straight-line target, obtain image, and straight-line target is shown as symmetrical curve on image; Foundation be take picture centre as initial point, and horizontal ordinate is x axle, and ordinate is the image coordinate system o-xy of y axle;

102. select m bar symmetrical curve as determination object from image, and m is integer, and m >=2, select a symmetrical curve to carry out step 103 to the operation of step 110;

103. choose n picture point from determination object, and n is integer, and n>=5, obtain the two-dimensional coordinate (x of n picture point in image coordinate system o-xy _i, y _i), i=1 wherein, 2 ... n;

104. use quadratic polynomial Ax _i ²+ Bx _iy _i+ Cy _i ²+ Dx _i+ Ey _i+ 1=0 carries out matching to the determination object in step 103, and wherein, A, B, C, D and E are quadratic polynomial coefficient, x _iand y _ithe two-dimensional coordinate of picture point on expression determination object in image coordinate system o-xy, utilizes formula (1) to calculate quadratic polynomial coefficient A, B, C, D and E,

$\left[\begin{array}{ccccc}\mathrm{\Σ}{x}_i^4& \mathrm{\Σ}{x}_i^3{y}_i& \mathrm{\Σ}{x}_i^2{y}_i^2& \mathrm{\Σ}{x}_i^3& \mathrm{\Σ}{x}_i^2{y}_i\\ \mathrm{\Σ}{x}_i^3{y}_i& \mathrm{\Σ}{x}_i^2{y}_i^2& \mathrm{\Σ}{x}_i{y}_i^3& \mathrm{\Σ}{x}_i^2{y}_i& \mathrm{\Σ}{x}_i{y}_i^2\\ \mathrm{\Σ}{x}_i^2{y}_i^2& \mathrm{\Σ}{x}_i{y}_i^3& \mathrm{\Σ}{y}_i^4& \mathrm{\Σ}{x}_i{y}_i^2& \mathrm{\Σ}{y}_i^3\\ \mathrm{\Σ}{x}_i^3& \mathrm{\Σ}{x}_i^2{y}_i& \mathrm{\Σ}{x}_i{y}_i^2& \mathrm{\Σ}{x}_i^2& \mathrm{\Σ}{x}_i{y}_i\\ \mathrm{\Σ}{x}_i^2{y}_i& \mathrm{\Σ}{x}_i{y}_i^2& \mathrm{\Σ}{y}_i^3& \mathrm{\Σ}{x}_i{y}_i& \mathrm{\Σ}{y}_i^2\end{array}\right]\·\left[\begin{array}{c}A\\ B\\ C\\ D\\ E\end{array}\right]=-\left[\begin{array}{c}\mathrm{\Σ}{x}_i^2\\ \mathrm{\Σ}{x}_i{y}_i\\ \mathrm{\Σ}{y}_i^2\\ \mathrm{\Σ}{x}_i\\ \mathrm{\Σ}{y}_i\end{array}\right]$ Formula (1)

In formula, i=1,2 ... n;

105. quadratic polynomial coefficient A, B, C, D and the E that calculate according to step 104, and formula (2), calculate the slope u of the axis of symmetry of determination object:

$u=\±\sqrt{\frac{A}{C}}$ Formula (2)

The symbol of u according to axis of symmetry the direction in image coordinate system o-xy relevant, axis of symmetry is got "+" number when first quartile or third quadrant, at the second quadrant or fourth quadrant, gets "-" number;

106. set up curvilinear coordinate system o-x ' y ': take the initial point of image coordinate system o-xy as initial point, take and be parallel to the right-handed coordinate system that the direction of determination object axis of symmetry is y ';

107. coordinate conversion: first according to the indexing α between formula (3) measuring and calculating image coordinate system o-xy and curvilinear coordinate system o-x ' y ',

$\tan \mathrm{\α}=\frac{1}{u}$ Formula (3);

Then utilize the coordinate conversion matrix R between formula (4) measuring and calculating image coordinate system o-xy and curvilinear coordinate system o-x ' y ':

$R=\left[\begin{array}{cc}\cos \mathrm{\α}& \sin \mathrm{\α}\\ -\sin \mathrm{\α}& \cos \mathrm{\α}\end{array}\right]$ Formula (4);

Coordinate (the x of the picture point of finally utilizing the determination object that formula (5) selects step 103 in image coordinate system o-xy _i, y _i), be transformed into coordinate in curvilinear coordinate system o-x ' y ' (x ' _i, y ' _i):

$\left[\begin{array}{c}{x}_i^{\′}\\ y_i^{\′}\end{array}\right]=R\·\left[\begin{array}{c}{x}_i\\ y_i\end{array}\right]=\left[\begin{array}{cc}\cos \mathrm{\α}& \sin \mathrm{\α}\\ -\sin \mathrm{\α}& \cos \mathrm{\α}\end{array}\right]\·\left[\begin{array}{c}{x}_i\\ y_i\end{array}\right]$ Formula (5);

108. utilize para-curve determination object is carried out in curvilinear coordinate system o-x ' y ' to matching, utilizes formula (6) to calculate parabolic equation coefficient a, b, c:

$\left[\begin{array}{ccc}\mathrm{\Σ}{x_i^{\′}}^4& \mathrm{\Σ}{x_i^{\′}}^3& \mathrm{\Σ}{x_i^{\′}}^2\\ \mathrm{\Σ}{x_i^{\′}}^3& \mathrm{\Σ}{x_i^{\′}}^2& \mathrm{\Σ}{x}_i^{\′}\\ \mathrm{\Σ}{x_i^{\′}}^2& \mathrm{\Σ}{x}_i^{\′}& n\end{array}\right]\·\left[\begin{array}{c}a\\ b\\ c\end{array}\right]=\left[\begin{array}{c}\mathrm{\Σ}{x_i^{\′}}^2{y}_y^{\′}\\ \mathrm{\Σ}{x}_i^{\′}{y}_y^{\′}\\ \mathrm{\Σ}{y}_y^{\′}\end{array}\right]$ Formula (6)

I=1 in formula, 2 ... n;

109. utilize formula (7) to calculate para-curve the coordinate of summit q in curvilinear coordinate system o-x ' y ' (x ' _q, y ' _q), and utilize formula (8) will (x ' _q, y ' _q) be converted to the coordinate (x of image coordinate system o-xy _q, y _q):

$\left\{\begin{array}{c}{x}_q^{\′}=-\frac{b}{2a}\\ y_q^{\′}=-\frac{b^2}{4a}+c\end{array}\right.$ Formula (7);

$\left[\begin{array}{c}{x}_q\\ y_q\end{array}\right]=R^T\left[\begin{array}{c}{x}_q^{\′}\\ y_q^{\′}\end{array}\right]$ Formula (8);

Coordinate (the x of the slope u of the axis of symmetry of 110. determination objects of measuring according to step 105 and the para-curve summit q that step 109 is measured _q, y _q), set up the straight-line equation of determination object axis of symmetry, shown in (9):

Y=S ₁x+T ₁formula (9)

Wherein, S ₁=u, T ₁=y _q-ux _q;

M-1 bar symmetrical curve remaining in 111. pairs of images carries out respectively step 103 to the operation of step 110, sets up the straight-line equation of each symmetrical curve axis of symmetry, and the straight-line equation of m bar symmetrical curve axis of symmetry forms suc as formula system of equations shown in (10):

Y=S _jx+T _j(j=1,2 ... m) formula (10);

The measuring and calculating of 20. image slices principal point coordinates:

M the equation in formula (10), carries out indirect adjustment with principle of least square method, and the normal equation obtaining, suc as formula shown in (11), calculates the least square intersection point of m axis of symmetry, and this least square intersection point is exactly image slices principal point O'(x ₀, y ₀),

$\left[\begin{array}{cc}\mathrm{\Σ}{S_j}^2& -\mathrm{\Σ}{S}_j\\ -\mathrm{\Σ}{S}_j& m\end{array}\right]\left[\begin{array}{c}{x}_0\\ y_0\end{array}\right]=\left[\begin{array}{c}-\mathrm{\Σ}{S}_j{T}_j\\ \mathrm{\Σ}{T}_j\end{array}\right]$ Formula (11)

J=1 in formula, 2 ... m;

30. measure pattern distortion coefficient:

In 301. m bar symmetrical curves from image, select a symmetrical curve as determination object;

302. choose n picture point from determination object, and n is integer, and n>=5, obtain the two-dimensional coordinate (x of n picture point in image coordinate system o-xy _i, y _i), i=1 wherein, 2 ... n; N picture point two-dimensional coordinate in curvilinear coordinate system o-x ' y ' be (x ' _i, y ' _i);

303. set up distortion factor equation:

First according to formula (12) by image slices principal point O'(x ₀, y ₀) coordinate (x in image coordinate system o-xy ₀, y ₀), be converted to coordinate in curvilinear coordinate system o-x ' y ' (x ' ₀, y ' ₀):

$\left[\begin{array}{c}{x}_0^{\′}\\ y_0^{\′}\end{array}\right]=R\·\left[\begin{array}{c}{x}_0\\ y_0\end{array}\right]=\left[\begin{array}{cc}\cos \mathrm{\α}& \sin \mathrm{\α}\\ -\sin \mathrm{\α}& \cos \mathrm{\α}\end{array}\right]\·\left[\begin{array}{c}{x}_0\\ y_0\end{array}\right]$ Formula (12);

Then set up distortion factor equation, shown in (13):

Δ y ' _i=k ₁(y ' _i-y ' ₀) [((x ' _i-x ' ₀) ²+ (y ' _i-y ' ₀) ²)] formula (13);

In formula (13), Δ y ' _irepresent on determination object 1 p (x ' _i, y ' _i) at the distortion correction value of y ' direction, k ₁represent distortion factor;

304. recover after distortion, and determination object becomes straight line from curve, and on determination object, n picture point recovered should meet formula (14) after distortion:

(y ' _i-y ' ₀)+k ₁(y ' _i-y ' ₀) r ²=C ₁formula (14)

Wherein, C ₁for constant, r presentation graphs picture point (x ' _i, y ' _i) to principal point (x ' ₀, y ' ₀) distance, $r=\sqrt{{{(x_i^{\′}-x_0^{\′})}_i}^2+{{(y_i^{\′}-y^{\′})}_i}^2};$

305. utilize formula (14) can list the normal equation of determination object, shown in (15):

$\left[\begin{array}{cc}\mathrm{\Σ}{\left[(y_i^{\′}-y_0^i)r^2\right]}^2& -\mathrm{\Σ}(y_i^{\′}-y_0^{\′})r^2\\ -\mathrm{\Σ}(y_i^{\′}-y_0^{\′})r^2& n\end{array}\right]\·\left[\begin{array}{c}{k}_1\\ C_1\end{array}\right]=\left[\begin{array}{c}-\mathrm{\Σ}{(y_i^{\′}-y_0^{\′})}^2{r}^2\\ \mathrm{\Σ}(y_i^{\′}-y_0^{\′})\end{array}\right]$ Formula (15)

Formula (15) is expressed as formula (16) with general formula:

$\left[\begin{array}{cc}{N_{11}}^{\left(1\right)}& {N_{12}}^{\left(1\right)}\\ {N_{12}}^{\left(1\right)}& {N_{22}}^{\left(1\right)}\end{array}\right]\·\left[\begin{array}{c}{k}_1\\ C_1\end{array}\right]=\left[\begin{array}{c}{W_1}^{\left(1\right)}\\ {W_2}^{\left(1\right)}\end{array}\right]$ Formula (16)

In formula (16), $N_{11}^{\left(1\right)}=\mathrm{\Σ}{\left[(y_i^{\′}-y_0^{\′})r^2\right]}^2,N_{12}^{\left(1\right)}=-\mathrm{\Σ}(y_i^{\′}-y_0^{\′})r^2,N_{22}^{\left(1\right)}=n,$ W ₁ ⁽¹⁾＝-Σ(y′ _i-y′ ₀) ²r ²，W ₂ ⁽¹⁾＝Σ(y′ _i-y′ ₀)；

Remaining m-1 bar symmetrical curve in 306. pairs of images, respectively as determination object, according to step 302-305, is set up the normal equation of determination object, with general formula, the overall normal equation of m bar symmetrical curve represented, shown in (17):

$\left[\begin{array}{cc}{N_{11}}^{\left(j\right)}& {N_{12}}^{\left(j\right)}\\ {N_{12}}^{\left(j\right)}& {N_{22}}^{\left(j\right)}\end{array}\right]\·\left[\begin{array}{c}{k}_1\\ C_j\end{array}\right]=\left[\begin{array}{c}{W_1}^{\left(j\right)}\\ {W_2}^{\left(j\right)}\end{array}\right]$ Formula (17)

Wherein, j=1,2 ... m;

307. organize formula (17) constitutive equation by m, shown in (18), can calculate the distortion factor k of image ₁:

$\left[\begin{array}{cccccc}\mathrm{\Σ}{N}_{11}^{\left(j\right)}& N_{12}^{\left(1\right)}& ...& N_{12}^{\left(j\right)}& ...& N_{12}^{\left(m\right)}\\ N_{12}^{\left(1\right)}& N_{22}^{\left(1\right)}& ...& 0& ...& 0\\ ...& ...& ...& ...& ...& ...\\ N_{12}^{\left(j\right)}& 0& ...& N_{22}^{\left(j\right)}& ...& 0\\ ...& ...& ...& ...& ...& ...\\ N_{12}^{\left(m\right)}& 0& ...& 0& ...& N_{22}^{\left(m\right)}\end{array}\right]\·\left[\begin{array}{c}{k}_1\\ C_1\\ ...\\ C_j\\ ...\\ C_m\end{array}\right]=\left[\begin{array}{c}\mathrm{\Σ}{W_1}^{\left(j\right)}\\ {W_2}^{\left(1\right)}\\ ...\\ {W_2}^{\left(j\right)}\\ ...\\ {W_2}^{\left(m\right)}\end{array}\right]$ Formula (18)

J=1 wherein, 2 ... m.

Be subject to the impact of pattern distortion, the imaging of straight-line target shows as symmetrical curve.The image slices principal point coordinate of a kind of straight-line target of the present invention and the assay method of distortion factor, the present invention is according to this principle, extract the symmetrical curve in image, point on a plurality of symmetrical curves is carried out curve fitting, calculate the axis of symmetry equation of symmetrical curve, and according to many axis of symmetry equations, carry out matching and obtain its intersecting point coordinate, be image slices principal point coordinate, then by the method that a plurality of symmetrical curves are recovered, calculate the distortion factor of image.This assay method is widely used in computer vision field.The present invention is symmetrical curve after can utilizing many straight-line target imagings, and the principal point coordinate of image and distortion factor are carried out to simultaneous determination.

Below by an embodiment, specifically measure principal point coordinate and the distortion factor of image.

First, choose 3 straight-line targets, it is imaged as symmetrical curve, usings these three symmetrical curves as determination object, selects 7 picture point on each symmetrical curve, and its coordinate is in Table the original image coordinate system in 1:

The coordinate of table 1 picture point in image coordinate system and curvilinear coordinate system

Then, use quadratic polynomial Ax _i ²+ Bx _iy _i+ Cy _i ²+ Dx _i+ Ey _i+ 1=0 carries out matching to above-mentioned three determination objects, calculates quadratic polynomial coefficient A, B, C, D and E, then calculates the axis of symmetry of three determination objects and the angle of x axle is respectively according to formula (2):

Wherein, α ₁axis of symmetry after 1 imaging of expression straight-line target and the angle of x axle, α ₂axis of symmetry after 2 imagings of expression straight-line target and the angle of x axle, α ₃axis of symmetry after 3 imagings of expression straight-line target and the angle of x axle; Curvilinear coordinate system o-x ' the y ' setting up respectively according to three determination objects, by the picture point coordinate conversion of determination object, to the coordinate in curvilinear coordinate system o-x ' y ' (x ', y '), transformation result is in Table 2.Utilize parabolic equation that formula (6), formula (7), formula (8) and formula (9) obtain picture point coordinate (x ', the y ') matching after conversion and apex coordinate in Table 2.

The parabolic equation of table 2 symmetrical curve and apex coordinate

Then, utilize the axis of symmetry slope of para-curve apex coordinate and determination object, calculate the straight-line equation of the axis of symmetry of three determination objects, be respectively:

$\left\{\begin{array}{c}y=0.0237928x+12.1\\ y=-0.0208769x-1.2\\ y=-23.2253533x-1274.4\end{array}\right.$

Subsequently, according to the straight-line equation of the axis of symmetry of above-mentioned three determination objects and formula (11), the image slices principal point coordinate calculating is

$\left\{\begin{array}{c}{x}_0=-55.1\\ y_0=5.4\end{array}\right.$

Finally, utilize the normal equation general formula of formula (18), calculate distortion factor:

k ₁＝2.65×10 ^-8。

Claims (1)

1. the image slices principal point coordinate of straight-line target and an assay method for distortion factor, is characterized in that, this assay method comprises the following steps:

10. become the axis of symmetry straight-line equation of image curve target:

101. utilize picture pick-up device to take pictures to straight-line target, obtain image, and straight-line target is shown as symmetrical curve on image; Foundation be take picture centre as initial point, and horizontal ordinate is x axle, and ordinate is the image coordinate system o-xy of y axle;

102. select m bar symmetrical curve as determination object from image, and m is integer, and m >=2, select a symmetrical curve to carry out step 103 to the operation of step 110;

103. choose n picture point from determination object, and n is integer, and n>=5, obtain the two-dimensional coordinate (x of n picture point in image coordinate system o-xy _i, y _i), i=1 wherein, 2 ... n;

104. use quadratic polynomial Ax _i ²+ Bx _iy _i+ Cy _i ²+ Dx _i+ Ey _i+ 1=0 carries out matching to the determination object in step 103, and wherein, A, B, C, D and E are quadratic polynomial coefficient, x _iand y _ithe two-dimensional coordinate of picture point on expression determination object in image coordinate system o-xy, utilizes formula (1) to calculate quadratic polynomial coefficient A, B, C, D and E,

$\left[\begin{array}{ccccc}\mathrm{\Σ}{x}_i^4& \mathrm{\Σ}{x}_i^3{y}_i& \mathrm{\Σ}{x}_i^2{y}_i^2& \mathrm{\Σ}{x}_i^3& \mathrm{\Σ}{x}_i^2{y}_i\\ \mathrm{\Σ}{x}_i^3{y}_i& \mathrm{\Σ}{x}_i^2{y}_i^2& \mathrm{\Σ}{x}_i{y}_i^3& \mathrm{\Σ}{x}_i^2{y}_i& \mathrm{\Σ}{x}_i{y}_i^2\\ \mathrm{\Σ}{x}_i^2{y}_i^2& \mathrm{\Σ}{x}_i{y}_i^3& \mathrm{\Σ}{y}_i^4& \mathrm{\Σ}{x}_i{y}_i^2& \mathrm{\Σ}{y}_i^3\\ \mathrm{\Σ}{x}_i^3& \mathrm{\Σ}{x}_i^2{y}_i& \mathrm{\Σ}{x}_i{y}_i^2& \mathrm{\Σ}{x}_i^2& \mathrm{\Σ}{x}_i{y}_i\\ \mathrm{\Σ}{x}_i^2{y}_i& \mathrm{\Σ}{x}_i{y}_i^2& \mathrm{\Σ}{y}_i^3& \mathrm{\Σ}{x}_i{y}_i& \mathrm{\Σ}{y}_i^2\end{array}\right]\·\left[\begin{array}{c}A\\ B\\ C\\ D\\ E\end{array}\right]=-\left[\begin{array}{c}\mathrm{\Σ}{x}_i^2\\ \mathrm{\Σ}{x}_i{y}_i\\ \mathrm{\Σ}{y}_i^2\\ \mathrm{\Σ}{x}_i\\ \mathrm{\Σ}{y}_i\end{array}\right]$ Formula (1)

In formula, i=1,2 ... n;

105. quadratic polynomial coefficient A, B, C, D and the E that calculate according to step 104, and formula (2), calculate the slope u of the axis of symmetry of determination object:

$u=\±\sqrt{\frac{A}{C}}$ Formula (2)

The symbol of u according to axis of symmetry the direction in image coordinate system o-xy relevant, axis of symmetry is got "+" number when first quartile or third quadrant, at the second quadrant or fourth quadrant, gets "-" number;

106. set up curvilinear coordinate system o-x ' y ': take the initial point of image coordinate system o-xy as initial point, take and be parallel to the right-handed coordinate system that the direction of determination object axis of symmetry is y ';

107. coordinate conversion: first according to the indexing α between formula (3) measuring and calculating image coordinate system o-xy and curvilinear coordinate system o-x ' y ',

$\tan \mathrm{\α}=\frac{1}{u}$ Formula (3);

Then utilize the coordinate conversion matrix R between formula (4) measuring and calculating image coordinate system o-xy and curvilinear coordinate system o-x ' y ':

$R=\left[\begin{array}{cc}\cos \mathrm{\α}& \sin \mathrm{\α}\\ -\sin \mathrm{\α}& \cos \mathrm{\α}\end{array}\right]$ Formula (4);

Coordinate (the x of the picture point of finally utilizing the determination object that formula (5) selects step 103 in image coordinate system o-xy _i, y _i), be transformed into coordinate in curvilinear coordinate system o-x ' y ' (x ' _i, y ' _i);

$\left[\begin{array}{c}{x}_i^{\′}\\ y_i^{\′}\end{array}\right]=R\·\left[\begin{array}{c}{x}_i\\ y_i\end{array}\right]=\left[\begin{array}{cc}\cos \mathrm{\α}& \sin \mathrm{\α}\\ -\sin \mathrm{\α}& \cos \mathrm{\α}\end{array}\right]\·\left[\begin{array}{c}{x}_i\\ y_i\end{array}\right]$ Formula (5);

108. utilize para-curve determination object is carried out in curvilinear coordinate system o-x ' y ' to matching, utilizes formula (6) to calculate parabolic equation coefficient a, b, c:

$\left[\begin{array}{ccc}\mathrm{\Σ}{x_i^{\′}}^4& \mathrm{\Σ}{x_i^{\′}}^3& \mathrm{\Σ}{x_i^{\′}}^2\\ \mathrm{\Σ}{x_i^{\′}}^3& \mathrm{\Σ}{x_i^{\′}}^2& \mathrm{\Σ}{x}_i^{\′}\\ \mathrm{\Σ}{x_i^{\′}}^2& \mathrm{\Σ}{x}_i^{\′}& n\end{array}\right]\·\left[\begin{array}{c}a\\ b\\ c\end{array}\right]=\left[\begin{array}{c}\mathrm{\Σ}{x_i^{\′}}^2{y}_y^{\′}\\ \mathrm{\Σ}{x}_i^{\′}{y}_y^{\′}\\ \mathrm{\Σ}{y}_y^{\′}\end{array}\right]$ Formula (6)

I=1 in formula, 2 ... n;

109. utilize formula (7) to calculate para-curve the coordinate of summit q in curvilinear coordinate system o-x ' y ' (x ' _q, y ' _q), and utilize formula (8) will (x ' _q, y ' _q) be converted to the coordinate (x of image coordinate system o-xy _q, y _q):

$\left\{\begin{array}{c}{x}_q^{\′}=-\frac{b}{2a}\\ y_q^{\′}=-\frac{b^2}{4a}+c\end{array}\right.$ Formula (7);

$\left[\begin{array}{c}{x}_q\\ y_q\end{array}\right]=R^T\left[\begin{array}{c}{x}_q^{\′}\\ y_q^{\′}\end{array}\right]$ Formula (8);

Coordinate (the x of the slope u of the axis of symmetry of 110. determination objects of measuring according to step 105 and the para-curve summit q that step 109 is measured _q, y _q), set up the straight-line equation of determination object axis of symmetry, shown in (9):

Y=S ₁x+T ₁formula (9)

Wherein, S ₁=u, T ₁=y _q-ux _q;

M-1 bar symmetrical curve remaining in 111. pairs of images carries out respectively step 103 to the operation of step 110, sets up the straight-line equation of each symmetrical curve axis of symmetry, and the straight-line equation of m bar symmetrical curve axis of symmetry forms suc as formula system of equations shown in (10):

Y=S _jx+T _jformula (10);

In formula (10), j=1,2 ... m;

The measuring and calculating of 20. image slices principal point coordinates:

M the equation in formula (10), carries out indirect adjustment with principle of least square method, and the normal equation obtaining, suc as formula shown in (11), calculates the least square intersection point of m axis of symmetry, and this least square intersection point is exactly image slices principal point O'(x ₀, y ₀),

$\left[\begin{array}{cc}\mathrm{\Σ}{S_j}^2& -\mathrm{\Σ}{S}_j\\ -\mathrm{\Σ}{S}_j& m\end{array}\right]\left[\begin{array}{c}{x}_0\\ y_0\end{array}\right]=\left[\begin{array}{c}-\mathrm{\Σ}{S}_j{T}_j\\ \mathrm{\Σ}{T}_j\end{array}\right]$ Formula (11)

J=1 in formula, 2 ... m;

30. measure pattern distortion coefficient:

In 301. m bar symmetrical curves from image, select a symmetrical curve as determination object;

302. choose n picture point from determination object, and n is integer, and n>=5, obtain the two-dimensional coordinate (x of n picture point in image coordinate system o-xy _i, y _i), i=1 wherein, 2 ... n; N picture point two-dimensional coordinate in curvilinear coordinate system o-x ' y ' be (x ' _i, y ' _i);

303. set up distortion factor equation:

First according to formula (12) by image slices principal point O'(x ₀, y ₀) coordinate (x in image coordinate system o-xy ₀, y ₀), be converted to coordinate in curvilinear coordinate system o-x ' y ' (x ' ₀, y ' ₀):

$\left[\begin{array}{c}{x}_0^{\′}\\ y_0^{\′}\end{array}\right]=R\·\left[\begin{array}{c}{x}_0\\ y_0\end{array}\right]=\left[\begin{array}{cc}\cos \mathrm{\α}& \sin \mathrm{\α}\\ -\sin \mathrm{\α}& \cos \mathrm{\α}\end{array}\right]\·\left[\begin{array}{c}{x}_0\\ y_0\end{array}\right]$ Formula (12);

Then set up distortion factor equation, shown in (13):

Δ y ' _i=k ₁(y ' _i-y ' ₀) [((x ' _i-x ' ₀) ²+ (y ' _i-y ' ₀) ²)] formula (13);

In formula (13), Δ y ' _irepresent on determination object 1 p (x ' _i, y ' _i) at the distortion correction value of y ' direction, k ₁represent distortion factor;

304. recover after distortion, and determination object becomes straight line from curve, and on determination object, n picture point recovered should meet formula (14) after distortion:

(y ' _i-y ' ₀)+k ₁(y ' _i-y ' ₀) r ²=C ₁formula (14)

Wherein, C ₁for constant, r presentation graphs picture point (x ' _i, y ' _i) to principal point (x ' ₀, y ' ₀) distance, $r=\sqrt{{{(x_i^{\′}-x_0^{\′})}_i}^2+{{(y_i^{\′}-y^{\′})}_i}^2};$

305. utilize formula (14) can list the normal equation of determination object, shown in (15):

$\left[\begin{array}{cc}\mathrm{\Σ}{\left[(y_i^{\′}-y_0^i)r^2\right]}^2& -\mathrm{\Σ}(y_i^{\′}-y_0^{\′})r^2\\ -\mathrm{\Σ}(y_i^{\′}-y_0^{\′})r^2& n\end{array}\right]\·\left[\begin{array}{c}{k}_1\\ C_1\end{array}\right]=\left[\begin{array}{c}-\mathrm{\Σ}{(y_i^{\′}-y_0^{\′})}^2{r}^2\\ \mathrm{\Σ}(y_i^{\′}-y_0^{\′})\end{array}\right]$ Formula (15)

Formula (15) is expressed as formula (16) with general formula:

$\left[\begin{array}{cc}{N_{11}}^{\left(1\right)}& {N_{12}}^{\left(1\right)}\\ {N_{12}}^{\left(1\right)}& {N_{22}}^{\left(1\right)}\end{array}\right]\·\left[\begin{array}{c}{k}_1\\ C_1\end{array}\right]=\left[\begin{array}{c}{W_1}^{\left(1\right)}\\ {W_2}^{\left(1\right)}\end{array}\right]$ Formula (16)

In formula (16), $N_{11}^{\left(1\right)}=\mathrm{\Σ}{\left[(y_i^{\′}-y_0^{\′})r^2\right]}^2,N_{12}^{\left(1\right)}=-\mathrm{\Σ}(y_i^{\′}-y_0^{\′})r^2,N_{22}^{\left(1\right)}=n,$ W ₁ ⁽¹⁾＝-Σ(y′ _i-y′ ₀) ²r ²，W ₂ ⁽¹⁾＝Σ(y′ _i-y′ ₀)；

Remaining m-1 bar symmetrical curve in 306. pairs of images, respectively as determination object, according to step 302-305, is set up the normal equation of determination object, with general formula, the overall normal equation of m bar symmetrical curve represented, shown in (17):

$\left[\begin{array}{cc}{N_{11}}^{\left(j\right)}& {N_{12}}^{\left(j\right)}\\ {N_{12}}^{\left(j\right)}& {N_{22}}^{\left(j\right)}\end{array}\right]\·\left[\begin{array}{c}{k}_1\\ C_j\end{array}\right]=\left[\begin{array}{c}{W_1}^{\left(j\right)}\\ {W_2}^{\left(j\right)}\end{array}\right]$ Formula (17)

Wherein, j=1,2 ... m;

307. organize formula (17) constitutive equation by m, shown in (18), can calculate the distortion factor k of image ₁:

$\left[\begin{array}{cccccc}\mathrm{\Σ}{N}_{11}^{\left(j\right)}& N_{12}^{\left(1\right)}& ...& N_{12}^{\left(j\right)}& ...& N_{12}^{\left(m\right)}\\ N_{12}^{\left(1\right)}& N_{22}^{\left(1\right)}& ...& 0& ...& 0\\ ...& ...& ...& ...& ...& ...\\ N_{12}^{\left(j\right)}& 0& ...& N_{22}^{\left(j\right)}& ...& 0\\ ...& ...& ...& ...& ...& ...\\ N_{12}^{\left(m\right)}& 0& ...& 0& ...& N_{22}^{\left(m\right)}\end{array}\right]\·\left[\begin{array}{c}{k}_1\\ C_1\\ ...\\ C_j\\ ...\\ C_m\end{array}\right]=\left[\begin{array}{c}\mathrm{\Σ}{W_1}^{\left(j\right)}\\ {W_2}^{\left(1\right)}\\ ...\\ {W_2}^{\left(j\right)}\\ ...\\ {W_2}^{\left(m\right)}\end{array}\right]$ Formula (18)

J=1 wherein, 2 ... m.