CN102800084A - 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 the 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 through image photographic to be carried out space three-dimensional and measures, and wherein, pattern distortion is very big to the influence of measurement result, requires in measuring process through obtaining the distortion in images coefficient coordinate of picture point to be carried out distortion correction.At present, the computing method of pattern distortion coefficient have two kinds: the one, demarcate through the laboratory, and promptly utilize special-purpose instrument and equipment to identify in the laboratory, offer the user; The 2nd, the reference mark of before photography, in camera coverage, laying some goes out distortion factor through calculated with mathematical model.Yet these two kinds of methods can't be to the conventional images calculating that distorts, and particularly, the coordinate of image slices principal point also can influence the demarcation of pattern distortion coefficient.

Summary of the invention

Technical matters: the technical matters that the present invention will solve 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 determines the equation of curve axis of symmetry through the point on a plurality of symmetrical curves on the picture is carried out curve fitting, and carries out match according to a plurality of axis of symmetry equations; And then calculate its intersecting point coordinate; Be image slices principal point coordinate, the method through a plurality of symmetrical curves are recovered determines the distortion in images coefficient then; This assay method is to the picture that has formed images, accomplishes the mensuration to pattern distortion coefficient and image slices principal point coordinate, need before photography, not demarcate or in camera coverage, lay sign to photographic equipment.

Technical scheme: for solving the problems of the technologies described above, the technical scheme that the present invention adopts is:

A kind of image slices principal point coordinate of straight-line target and the assay method of distortion factor, this assay method may further comprise the steps:

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

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

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

103. choose n picture point from determination object, n is an integer, and n>=5, obtains 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 match to the determination object in the step 103, and wherein, A, B, C, D and E are quadratic polynomial coefficient, x _iAnd y _iThe two-dimensional coordinate of picture point in image coordinate system o-xy on the expression determination object 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 the formula, i=1,2 ... N;

105. the 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 is relevant according to the direction of axis of symmetry in image coordinate system o-xy, and axis of symmetry is got "+" number when first quartile or third quadrant, gets "-" number at second quadrant or four-quadrant;

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

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

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

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

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

Utilize picture point the coordinate (x among image coordinate system o-xys of formula (5) at last with the determination object of step 103 selection _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

that determination object is carried out match in curvilinear coordinate system o-x ' y ', utilize 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 the formula, 2 ... N;

109. the formula of utilization (7) calculates para-curve

Summit q, the coordinate 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);

110. the slope u of the axis of symmetry of the determination object of measuring according to step 105 and the coordinate (x of 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

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

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

20. the measuring and calculating of image slices principal point coordinate:

M equation in the formula (10) carries out indirect adjustment with principle of least square method, and normal equation calculates the least square intersection point of m axis of symmetry suc as formula shown in (11), 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 the formula, 2 ... M;

30. measure the pattern distortion coefficient:

301. select a symmetrical curve in the m bar symmetrical curve from image as determination object;

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

303. set up the distortion factor equation:

At first according to formula (12) with image slices principal point O' (x ₀, y ₀) coordinate (x in image coordinate system o-xy ₀, y ₀), the conversion in curvilinear coordinate system o-x ' y ' coordinate (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);

Set up the distortion factor equation then, 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 the formula (13), Δ y ' _i1 p on the expression determination object (x ' _i, y ' _i) at the distortion correction value of y ' direction, k ₁The expression distortion factor;

304. after recovering distortion, determination object becomes straight line by curve, n picture point recovered should satisfy formula (14) after the distortion on the determination object:

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

Wherein, C ₁Be 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. the formula of utilization (14) can be listed the normal equation of determination object, shown in (15):

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

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

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

In the formula (16), $N_{11}^{\left(1\right)}=\mathrm{\Σ}{\left[(y_i^{\′}-y_0^{\′})r^2\right]}^2,$ $N_{12}^{\left(1\right)}=-\mathrm{\Σ}(y_i^{\′}-y_0^{\′})r^2,$ $N_{22}^{\left(1\right)}=n,$ ${W_1}^{\left(1\right)}=-\mathrm{\Σ}{(y_i^{\′}-y_0^{\′})}^2{r}^2,$ ${W_2}^{\left(1\right)}=\mathrm{\Σ}(y_i^{\′}-y_0^{\′});$

306. the m-1 bar symmetrical curve of the remainder in the image 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 is 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. m is organized formula (17) constitutive equation, shown in (18), can calculate the distortion in images coefficient k ₁:

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

J=1 wherein, 2 ... M.

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

1. need not photography and vedio recording equipment is demarcated in advance, can carry out the mensuration of image slices principal point coordinate and pattern distortion coefficient the picture that has formed images in any source.The present technique scheme is the mensuration of carrying out image slices principal point coordinate and pattern distortion coefficient to the picture that has formed images, need in the photography and vedio recording process, equipment not demarcated in advance, or in camera coverage, lays sign.To the picture that has formed images, utilize the present technique scheme 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 present technique scheme can carry out the mensuration of image slices principal point coordinate and pattern distortion coefficient to the picture of imaging; But for the target of repeat photography shooting, utilize the present technique scheme, can save the repeat photography shooting, directly utilize the picture that forms images 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, the imageable target curve is reverted to straight line with certain distortion factor, just can determine the distortion in images coefficient.The assay method of present technique scheme recovers with certain distortion factor the discrete point on the imageable target curve, makes it revert to straight line, and then calculates distortion factor.This assay method carries out picture shot applicable to all to straight-line target, and applicability is strong.Simultaneously, this assay method is implemented convenient, need in the photography and vedio recording process, not demarcate in advance equipment, or in camera coverage, lay sign.

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 can produce usually and departs from image center.Assay method provided by the invention was 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 that the mensuration of pattern distortion coefficient is more accurate.

Description of drawings

Fig. 1 is the synoptic diagram that the present invention measures the 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 synoptic diagram that the present invention measures the pattern distortion coefficient; Wherein p be expressed as on the image curve more arbitrarily; T representes the p point is carried out the position after y ' direction corrects; P0 representes the p point is carried out the position after the distortion correction of x ', y ' direction simultaneously, and the line between 3 of the t, p, p0 constitutes a right-angle triangle, and the coordinate of t point and the p0 y ' direction of ordering is identical simultaneously.

Embodiment

Below in conjunction with accompanying drawing, technical scheme of the present invention is carried out detailed explanation.

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 may further comprise the steps:

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

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

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

103. choose n picture point from determination object, n is an integer, and n>=5, obtains 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 match to the determination object in the step 103, and wherein, A, B, C, D and E are quadratic polynomial coefficient, x _iAnd y _iThe two-dimensional coordinate of picture point in image coordinate system o-xy on the expression determination object 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 the formula, i=1,2 ... N;

105. the 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 is relevant according to the direction of axis of symmetry in image coordinate system o-xy, and axis of symmetry is got "+" number when first quartile or third quadrant, gets "-" number at second quadrant or four-quadrant;

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

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

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

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

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

Utilize picture point the coordinate (x among image coordinate system o-xys of formula (5) at last with the determination object of step 103 selection _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

that determination object is carried out match in curvilinear coordinate system o-x ' y ', utilize 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 the formula, 2 ... N;

109. the formula of utilization (7) calculates para-curve

Summit q, the coordinate 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);

110. the slope u of the axis of symmetry of the determination object of measuring according to step 105 and the coordinate (x of 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

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

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

20. the measuring and calculating of image slices principal point coordinate:

M equation in the formula (10) carries out indirect adjustment with principle of least square method, and normal equation calculates the least square intersection point of m axis of symmetry suc as formula shown in (11), 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 the formula, 2 ... M;

30. measure the pattern distortion coefficient:

301. select a symmetrical curve in the m bar symmetrical curve from image as determination object;

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

303. set up the distortion factor equation:

At first according to formula (12) with image slices principal point O' (x ₀, y ₀) coordinate (x in image coordinate system o-xy ₀, y ₀), the conversion in curvilinear coordinate system o-x ' y ' coordinate (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);

Set up the distortion factor equation then, 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 the formula (13), Δ y ' _i1 p on the expression determination object (x ' _i, y ' _i) at the distortion correction value of y ' direction, k ₁The expression distortion factor;

304. after recovering distortion, determination object becomes straight line by curve, n picture point recovered should satisfy formula (14) after the distortion on the determination object:

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

Wherein, C ₁Be 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. the formula of utilization (14) can be listed the normal equation of determination object, shown in (15):

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

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

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

In the formula (16), $N_{11}^{\left(1\right)}=\mathrm{\Σ}{\left[(y_i^{\′}-y_0^{\′})r^2\right]}^2,$ $N_{12}^{\left(1\right)}=-\mathrm{\Σ}(y_i^{\′}-y_0^{\′})r^2,$ $N_{22}^{\left(1\right)}=n,$ ${W_1}^{\left(1\right)}=-\mathrm{\Σ}{(y_i^{\′}-y_0^{\′})}^2{r}^2,$ ${W_2}^{\left(1\right)}=\mathrm{\Σ}(y_i^{\′}-y_0^{\′});$

306. the m-1 bar symmetrical curve of the remainder in the image 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 is 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. m is organized formula (17) constitutive equation, shown in (18), can calculate the distortion in images coefficient k ₁:

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

J=1 wherein, 2 ... M.

Receive the influence 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 extracts the symmetrical curve in the image according to this principle; Point on a plurality of symmetrical curves carries out curve fitting; Calculate the axis of symmetry equation of symmetrical curve, and carry out match according to many axis of symmetry equations and obtain its intersecting point coordinate, be image slices principal point coordinate; Method through a plurality of symmetrical curves are recovered calculates the distortion in images coefficient then.This assay method is widely used in computer vision field.The present invention is symmetrical curve after can utilizing many straight-line target imagings, comes the principal point coordinate and the distortion factor of image are carried out simultaneous determination.

Through an embodiment, come concrete principal point coordinate and the distortion factor of measuring image below.

At first, choose 3 straight-line targets, it is imaged as symmetrical curve, as determination object, on each symmetrical curve, selects 7 picture point with these three symmetrical curves, and its coordinate is seen the original image coordinate system in the table 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 match to above-mentioned three determination objects, calculates quadratic polynomial coefficient A, B, C, D and E, calculates the axis of symmetry of three determination objects according to formula (2) again and the angle of x axle is respectively:

Wherein, α ₁The axis of symmetry after 1 imaging of expression straight-line target and the angle of x axle, α ₂The axis of symmetry after 2 imagings of expression straight-line target and the angle of x axle, α ₃The axis of symmetry after 3 imagings of expression straight-line target and the angle of x axle; Curvilinear coordinate system o-x ' the y ' that sets up respectively according to three determination objects, with coordinate to curvilinear coordinate system o-x ' y ' of the picture point coordinate conversion of determination object (x ', y '), transformation result is seen table 2.The parabolic equation and the apex coordinate that utilize picture point coordinate after formula (6), formula (7), formula (8) and formula (9) will be changed (x ', y ') match to obtain are seen 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 and the formula (11) of the axis of symmetry of above-mentioned three determination objects, the image slices principal point coordinate that calculates does

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

At last, 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 a straight-line target and the assay method of distortion factor is characterized in that, this assay method may further comprise the steps:

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

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

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

103. choose n picture point from determination object, n is an integer, and n>=5, obtains 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 match to the determination object in the step 103, and wherein, A, B, C, D and E are quadratic polynomial coefficient, x _iAnd y _iThe two-dimensional coordinate of picture point in image coordinate system o-xy on the expression determination object 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 the formula, i=1,2 ... N;

105. the 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 is relevant according to the direction of axis of symmetry in image coordinate system o-xy, and axis of symmetry is got "+" number when first quartile or third quadrant, gets "-" number at second quadrant or four-quadrant;

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

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

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

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

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

Utilize picture point the coordinate (x among image coordinate system o-xys of formula (5) at last with the determination object of step 103 selection _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

that determination object is carried out match in curvilinear coordinate system o-x ' y ', utilize 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 the formula, 2 ... N;

109. the formula of utilization (7) calculates para-curve

Summit q, the coordinate 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);

110. the slope u of the axis of symmetry of the determination object of measuring according to step 105 and the coordinate (x of 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

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

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

20. the measuring and calculating of image slices principal point coordinate:

M equation in the formula (10) carries out indirect adjustment with principle of least square method, and normal equation calculates the least square intersection point of m axis of symmetry suc as formula shown in (11), 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 the formula, 2 ... M;

30. measure the pattern distortion coefficient:

301. select a symmetrical curve in the m bar symmetrical curve from image as determination object;

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

303. set up the distortion factor equation:

At first according to formula (12) with image slices principal point O' (x ₀, y ₀) coordinate (x in image coordinate system o-xy ₀, y ₀), the conversion in curvilinear coordinate system o-x ' y ' coordinate (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);

Set up the distortion factor equation then, 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 the formula (13), Δ y ' _i1 p on the expression determination object (x ' _i, y ' _i) at the distortion correction value of y ' direction, k ₁The expression distortion factor;

304. after recovering distortion, determination object becomes straight line by curve, n picture point recovered should satisfy formula (14) after the distortion on the determination object:

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

Wherein, C ₁Be 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. the formula of utilization (14) can be listed the normal equation of determination object, shown in (15):

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

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

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

In the formula (16), $N_{11}^{\left(1\right)}=\mathrm{\Σ}{\left[(y_i^{\′}-y_0^{\′})r^2\right]}^2,$ $N_{12}^{\left(1\right)}=-\mathrm{\Σ}(y_i^{\′}-y_0^{\′})r^2,$ $N_{22}^{\left(1\right)}=n,$ ${W_1}^{\left(1\right)}=-\mathrm{\Σ}{(y_i^{\′}-y_0^{\′})}^2{r}^2,$ ${W_2}^{\left(1\right)}=\mathrm{\Σ}(y_i^{\′}-y_0^{\′});$

306. the m-1 bar symmetrical curve of the remainder in the image 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 is 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. m is organized formula (17) constitutive equation, shown in (18), can calculate the distortion in images coefficient k ₁:

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

J=1 wherein, 2 ... M.

Images