CN103925912B - Interior visual field optical segmentation type large CCD images geometry joining method - Google Patents

Abstract

The present invention relates to a kind of interior visual field optical segmentation type large area array CCD boat camera sub-image geometry splicing synthetic method, the large format digitized video utilizing the method to generate can meet photogrammetric measurement to high photographic efficiency and the requirement of high aerial survey mapping precision.Basic ideas are: first high precision measures each sub-image overlapping region same place image coordinate, and obtain the position control coordinate of sub-image benchmark picture point on actual focal plane provided by camera manufacturer; Secondly, observation equation group is set up based on splicing relational model; Then, least squares adjustment technology is adopted to solve the splicing parameter of each sub-image to final resultant image; Finally utilize splicing parameter to convert and resampling sub-image, thus complete the splicing of each sub-image to final resultant image.Experiment shows that the present invention has good reliability and good splicing precision, for interior visual field optical segmentation type digital camera, the present invention can seamless spliced generation high precision without the large format aerial stereo images of geometric dislocation.

Description

Interior visual field optical segmentation type large CCD images geometry joining method

Technical field

The present invention relates to the sub-image geometry splicing synthetic method of a kind of interior visual field optical segmentation type large area array CCD boat camera, belong to photogrammetry and remote-sensing technique field.

Background technology

Up to the present, along with the rise of digital mapping camera, the film type simulation boat camera used in the past is fully substituted.But owing to being subject to the restriction of CCD manufacturing process, the film size scale of individual Digital Aerial Photography image cannot reach the image planes size of original analogue camera all the time, and this largely have impact on the embody rule of the digital mapping camera such as photographic efficiency and plotting accuracy.

But under vigorous application demand promotes, there is multiple compound large battle array boat camera both at home and abroad, to solve Digital Aerial Photography image film size dimension-limited in the problem of single CCD face battle array size, these compound cameras can be divided into outer visual field Splittable, the large class of interior visual field Splittable two.Its China and foreign countries' visual field Splittable camera utilizes the medium and small area array CCD camera binding of multiple separate unit to form, the SWDC camera etc. that the DMC type camera that Typical Representative has American I ntergraph company to produce, national Mapping remote sensing technology institute manufacture; Interior visual field Splittable camera comprises the UltraCam camera of Microsofot/Vexcel production, the DMZ camera etc. of Nanjing point's Remote Sensing company limited manufacture, in these two kinds also there is constructional difference in visual field Splittable camera, the former is the interior visual field optical segmentation utilizing many camera lenses timesharing photograph mode to realize, and the latter directly utilizes the disposable entirety of carrying out of optics half-reflection and half-transmission prism to split in the posterior nodal point visual field of single camera lens.The large class camera in inside and outside visual field two respectively has relative merits, but it should be noted that herein, the clear superiority of interior visual field optical segmentation type camera is that after having good splicing precision and splicing, image has good measurement performance, and this is most important to meeting later stage plotting accuracy requirement.

The application is just for interior visual field this class camera of optical segmentation type, a kind of interior large CCD images joining method of visual field optical segmentation is proposed, to solve the splicing of high precision sub-image, the high large format image composition generation problem measuring performance, same or analogous large battle array image synthesis method has no appearance with it both at home and abroad.

Summary of the invention

The technical issues that need to address of the present invention are to provide one and can be used in interior visual field Splittable boat camera large format image high-precision composition generation method, and the large format digitized video after utilizing the method to generate can meet the application demand of photogrammetric measurement to photographic efficiency and aerial survey mapping precision.

The basic ideas of the interior visual field Splittable boat camera large format image high-precision composition generation method that the present invention proposes are: first high precision measures the same place image coordinate of each sub-image overlapping region, and obtain provided by camera manufacturer certain/the position control coordinate of a little image benchmark picture point on actual focal plane; Secondly, based on certain splicing relational model---as the fractal transforms such as two dimension or affined transformation, set up observation equation group; Then, least squares adjustment technology is adopted to solve the splicing parameter of each sub-image to final resultant image; Finally utilize splicing parameter to convert and resampling sub-image, thus complete the splicing of each sub-image to final resultant image.

Technical solution of the present invention is:

Interior visual field optical segmentation type large CCD images geometry joining method, is characterized in that the method comprises following step:

(1) measure corresponding image points image coordinate in each adjacent sub-image overlap area, obtain the position of focal plane coordinate of the sub-image reference point that camera manufacturer provides; Being numbered of definition sub-image, corresponding image points and reference point: sub-image label coding adopts Greek alphabet (I, II, III ...), corresponding image points label coding adopts arabic numeral (1,2,3 ...), reference point label coding adopts English capitalization (A, B, C ...);

(2) observation equation group is set up based on splicing relational model

If establishing splicing relational model is the fractal transforms such as two dimension, formula is

X＝ax-by+T _X

(1)

Y＝ay+bx+T _Y

Wherein, x, y are sub-image coordinate system coordinate, and X, Y are resultant image coordinate system coordinate after splicing.A, b, T _x, T _ybe 4 conversion parameters, with this, splicing relational design carried out to sub-image reference image point and adjacent sub-image corresponding image points

The first splices relational design---about sub-image benchmark picture point

For the benchmark picture point C on VI work song image for having

$\begin{array}{c}{X}_C=a_{VI}{x}_{C_{VI}}-b_{VI}{y}_{C_{VI}}+T_{X_{VI}}\\ Y_C=a_{VI}{y}_{C_{VI}}+b_{VI}{x}_{C_{VI}}+T_{Y_{VI}}\end{array}---\left(2\right)$

Wherein, for the picpointed coordinate of reference point C on sub-image VI under its sub-image coordinate system, X _c, Y _cbe then image coordinate after the synthesis of reference point C position, reference mark on focal plane, a _vI, b _vI, for the splicing parameter of sub-image VI, one group of corresponding splicing relation equation group can be write out by above formula equally for A, B, D, E, F, G reference point on other I, IV, VII, IX, X, XII sub-image.

The second splicing relational design---about adjacent sub-image corresponding image points

After utilizing the synthesis on adjacent sub-image corresponding to corresponding image points, image picpointed coordinate this condition equal lists splicing relation equation, for the corresponding image points 1 on adjacent sub-image I and sub-image II for having:

$\begin{array}{c}{a}_I{x}_{1_I}-b_I{y}_{1_I}+T_{X_I}-(a_{II}{x}_{1_{II}}-b_{II}{y}_{1_{II}}+T_{X_{II}})=0\\ a_I{y}_{1_I}-b_I{x}_{1_I}+T_{Y_I}-(a_{II}{y}_{1_{II}}-b_{II}{x}_{1_{II}}+T_{Y_{II}})=0\end{array}---\left(3\right)$

For other often pair adjacent sub-image, each corresponding image points occurred in overlay region can write out one group of corresponding splicing relation equation group by above formula.

Like this, corresponding splicing relation observation equation group listed by all reference points and corresponding image points combined, the observation equation group matrix form finally obtained is

_mA ⁿ _nX ¹＝ _mL ¹+ _mV ¹(4)

Wherein, coefficient matrices A is set up according to two kinds of splicing relations, and X is the splicing parameter of all sub-images, L is the scalar matrix be made up of element 0 and reference point position of focal plane coordinate, V is residual vector, and m size is the twice of corresponding image points and reference point number sum, and n size is four times of sub-image quantity;

(3) utilize least squares adjustment technology to solve the splicing parameter of each sub-image to final resultant image, and provide calculation accuracy

All observation equations often open splicing parameter (a, b, T that sub-image is corresponding after building _x, T _y) ask by least square adjustment solution, after these parameters determine all sub-images to synthesis, the geometry of image splices relation

_mA ⁿ _nX ¹＝ _mL ¹+ _mV ¹，X＝(A ^TA) ^-1(A ^TL)

The internal accuracy of splicing parameter X is calculated as follows:

V＝AX-L(5)

Weight unit standard deviation is:

$S_0=\sqrt{\frac{V^{T}V}{r}},r=m-n$

for (A ^ta) ^-1the i-th row, the i-th column element,

$S_{x_i}=S_0\sqrt{Q_{x_i{x}_i}}$

Be the internal accuracy of i-th splicing parameter in X;

(4) finally utilize splicing parameter to convert and resampling sub-image, thus complete the splicing of each sub-image to final resultant image

These should be spliced parameter " oppositely " application in practical operation, the matrix form of formula (1) is as follows:

$\left[\begin{array}{c}X\\ Y\end{array}\right]=\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]+\left[\begin{array}{c}{T}_X\\ T_Y\end{array}\right]---\left(6\right)$

Oppositely solve,

$\left[\begin{array}{c}x\\ y\end{array}\right]={\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]}^{-1}\left[\begin{array}{c}X-T_X\\ Y-T_Y\end{array}\right]=\frac{1}{a^2+b^2}\left[\begin{array}{cc}a& b\\ -b& a\end{array}\right]\left[\begin{array}{c}X-T_X\\ Y-T_Y\end{array}\right]---\left(7\right)$

Image pixel coordinate (X after the splicing that use not yet generates, Y) and corresponding sub-image conversion parameter, corresponding location of pixels on sub-image corresponding to this pixel can be calculated by formula (7), then can sample and obtain half-tone information, for image after the splicing corresponding to overlapping region, can gray-scale value be obtained at the anyon image up-sampling comprising overlay region or carry out the weighted mean of gray-scale value.

The present invention can solve the splicing parameter of interior visual field each sub-image of Splittable digital mapping camera, thus complete the splicing of large format image high-precision and generate, large format resultant image can meet the demand of photogrammetric measurement to high photographic efficiency and high image measurement performance.The present invention is expected to strengthen visual field Splittable digital mapping camera sub-image high-precision joining in domestic optics and synthesizes this technology short slab, for the later stage aerophotogrammetric field work of the type boat camera provides the necessary technical support.

Accompanying drawing explanation

Fig. 1 is that in 1 point 12, visual field Splittable DMZ camera I ~ XII CCD machine back of the body profile is settled

Fig. 2 is that in 1 point 12, in visual field Splittable DMZ camera, visual field dispenser and I ~ XII CCD distribute

Fig. 3 is corresponding image points and the datum point distribution of corresponding sub-image topological relation and supposition under Fig. 2

Fig. 4 is the element filling effect of coefficient matrices A

Fig. 5 is the pixel after splicing, image being positioned at dissimilar region

Embodiment

For the DMZ of Nanjing point's Remote Sensing company limited boat camera, Fig. 1 is I ~ No. XII and amounts to the arrangement relation of 12 CCD at camera machine back, corresponding interior visual field optical segmentation device and I ~ XII CCD install signal as Fig. 2, sub-image topological relation corresponding to I ~ XII CCD and the corresponding image points of supposition and datum point distribution are as Fig. 3, sub-image label coding is similarly: I ~ XII, corresponding image points label coding is: 1 ~ 33, reference point label coding is: A ~ G, illustrate the embodiment of the interior visual field optical segmentation type large CCD images geometry joining method that the present invention proposes below.

(1) each sub-image overlapping region corresponding image points image coordinate is measured, obtain the position of focal plane coordinate of the sub-image reference point that camera manufacturer provides, being numbered of definition sub-image, corresponding image points and reference point: sub-image label coding adopts Greek alphabet, as I, II, III ... corresponding image points label coding adopts arabic numeral, as 1,2,3 ..., reference point label coding adopts English capitalization, as A, B, C

About the measurement of corresponding image points coordinate, the High-precision image coupling that current commercial full digital photogrammetric picture point measuring tool Eardas, PCI, ENVI and Ermaper etc. can be utilized to carry out picture point measures, according to each corresponding image points in table 1 and adjacent sub-image relation corresponding with it, measure and can obtain coordinate points string corresponding to each adjacent sub-image:

$\begin{array}{c}I,II:(x_{1_I},y_{1_I}),(x_{1_{II}},y_{1_{II}});(x_{4_I},y_{4_I}),(x_{4_{II}},y_{4_{II}});(x_{9_I},y_{9_I}),(x_{9_{II}},y_{9_{II}})\\ ...\\ XI,XII:(x_{{25}_{XI}},y_{{25}_{XI}}),(x_{{25}_{XII}},y_{{25}_{XII}});(x_{{30}_{XI}},y_{{30}_{XI}}),(x_{{30}_{XII}},y_{{30}_{XII}});(x_{{33}_{XI}},y_{{33}_{XI}}),(x_{{33}_{XII}},y_{{33}_{XII}})\end{array}$

About reference point position of focal plane coordinate, it is provided by camera manufacturer, and A ~ G reference point coordinate string form is:

(X _A,Y _A),(X _B,Y _B)…(X _G,Y _G)

Each corresponding image points of table 1 and adjacent sub-image relation corresponding with it

(2) based on the first, the second splicing relation set up observation equation group

To splice relational model formula

X＝ax-by+T _X

(1)

Y＝ay+bx+T _Y

Based on, design sets up two kinds of splicing relations, for the benchmark picture point C on VI work song image for having

$\begin{array}{c}{X}_C=a_{VI}{x}_{C_{VI}}-b_{VI}{y}_{C_{VI}}+T_{X_{VI}}\\ Y_C=a_{VI}{y}_{C_{VI}}+b_{VI}{x}_{C_{VI}}+T_{Y_{VI}}\end{array}---\left(2\right)$

For the corresponding image points 1 on adjacent sub-image I and sub-image II for having

$\begin{array}{c}{a}_I{x}_{1_I}-b_I{y}_{1_I}+T_{X_I}-(a_{II}{x}_{1_{II}}-b_{II}{y}_{1_{II}}+T_{X_{II}})=0\\ a_I{y}_{1_I}-b_I{x}_{1_I}+T_{Y_I}-(a_{II}{y}_{1_{II}}-b_{II}{x}_{1_{II}}+T_{Y_{II}})=0\end{array}---\left(3\right)$

And associative list 1 sets up the observation equation group about all about corresponding image points and reference point, matrix of coefficients ₈₀a ⁴⁸, unknown number vector ₄₈x ¹and constant term ₈₀l ¹structure as follows:

Element in matrix A is with 1,4 corresponding image points of I, No. II adjacent sub-image overlapping region, 30,33 corresponding image points and reference point A, G of XI, No. XII adjacent sub-image overlapping region fill for representing sample, other corresponding image points and reference point can with reference to carrying out, then remaining most elements is 0, the element filling effect of this sparse matrix is shown in Fig. 4, the size of each filling block is that 2 row × 4 arrange represent corresponding image points filling block, look represents reference point filling block.

${X_{48}}^1=\left[\begin{array}{c}{a}_I\\ b_I\\ T_{X_I}\\ T_{Y_I}\\ .\\ .\\ .\\ a_i\\ b_i\\ T_{X_i}\\ T_{Y_i}\\ .\\ .\\ .\\ a_{XII}\\ b_{XII}\\ T_{X_{XII}}\\ T_{Y_{XII}}\end{array}\right],{L_{80}}^1=\left[\begin{array}{c}0\\ .\\ .\\ .\\ 0\\ X_A\\ Y_A\\ .\\ .\\ .\\ X_G\\ Y_G\end{array}\right]$

So far, the matrix form of observation equation group _ma ⁿ _nx ¹= _ml ¹+ _mv ¹set up complete, wherein, m=(33+7) × 2=80, n=12 × 4=48.

(3) utilize least squares adjustment technology to solve the splicing parameter of each sub-image to final resultant image, and provide calculation accuracy

All observation equations often open splicing parameter (a, b, T that sub-image is corresponding after building _x, T _y) ask by least square adjustment solution, after these parameters determine all sub-images to synthesis, the geometry of image splices relation

_mA ⁿ _nX ¹＝ _mL ¹+ _mV ¹，X＝(A ^TA) ^-1(A ^TL)

The internal accuracy of splicing parameter X is calculated as follows:

V＝AX-L(5)

Weight unit standard deviation is:

$S_0=\sqrt{\frac{V^{T}V}{r}},r=m-n$

for (A ^ta) ^-1the i-th row, the i-th column element,

$S_{x_i}=S_0\sqrt{Q_{x_i{x}_i}}$

Be the internal accuracy of i-th splicing parameter in X;

(4) finally utilize splicing parameter to convert and resampling sub-image, thus complete the splicing of each sub-image to final resultant image

These should be spliced parameter " oppositely " application in practical operation, the matrix form of formula (1) is as follows:

$\left[\begin{array}{c}X\\ Y\end{array}\right]=\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]+\left[\begin{array}{c}{T}_X\\ T_Y\end{array}\right]---\left(6\right)$

Oppositely solve,

$\left[\begin{array}{c}x\\ y\end{array}\right]={\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]}^{-1}\left[\begin{array}{c}X-T_X\\ Y-T_Y\end{array}\right]=\frac{1}{a^2+b^2}\left[\begin{array}{cc}a& b\\ -b& a\end{array}\right]\left[\begin{array}{c}X-T_X\\ Y-T_Y\end{array}\right]---\left(7\right)$

Image pixel coordinate (X after the splicing that use not yet generates, Y) and corresponding sub-image conversion parameter, corresponding location of pixels on sub-image corresponding to this pixel can be calculated by formula (7), then can sample and obtain half-tone information, certainly the concrete method of sampling can take the interpolating method such as neighbor or bilinearity, for image after the splicing of overlapping region, can gray-scale value be obtained at any one the sub-image up-sampling comprising overlay region or carry out the weighted mean of multiple gray-scale value, in Fig. 5, namely outermost rectangle is image after splicing, wherein illustrate the pixel being positioned at three kinds of dissimilar regions, pixel L is positioned at " dead space ", pixel J can only find corresponding picture point on I work song image, pixel K is positioned at No. I, in the overlapping region of No. 2 adjacent sub-images.

For pixel L: use any one sub-image to splice parameter calculating gained image point position and all can not drop in effective picture coordinate range of corresponding sub-image, namely be invalid, at this moment the gray-scale value splicing image pixel point L place can be filled with the specific a certain numerical value representing invalid gray scale, such as numerical value 0;

For pixel J: I work song image joint parameter can only be used to calculate and obtain being positioned at the effective picture point under I work song coordinate systems in image, and the gray-scale value at this moment splicing image pixel point J place directly can use I work song image coordinate the sampling gray-scale value at place is filled

$\left[\begin{array}{c}{x}_I^J\\ y_I^J\end{array}\right]=\frac{1}{a_I^2+b_I^2}\left[\begin{array}{cc}{a}_I^2& b_I^2\\ -b_I^2& a_I^2\end{array}\right]\left[\begin{array}{c}{X}_J-T_{X_I}\\ Y_J-T_{Y_I}\end{array}\right]$

For pixel K: can use No. I, 2 work song image joint parameters calculate and obtain correspondence respectively and be positioned at No. I, two effective picture points under 2 work song coordinate systems in image, the gray-scale value at this moment splicing image pixel point K place can use I work song image coordinate with II work song image coordinate fill after the gray scale sampled value at two places is weighted average treatment

$\left[\begin{array}{c}{x}_I^K\\ y_I^K\end{array}\right]=\frac{1}{a_I^2+b_I^2}\left[\begin{array}{cc}{a}_I^2& b_I^2\\ -b_I^2& a_I^2\end{array}\right]\left[\begin{array}{c}{X}_K-T_{X_I}\\ Y_K-T_{Y_I}\end{array}\right],$

$\left[\begin{array}{c}{x}_{II}^K\\ y_{II}^K\end{array}\right]=\frac{1}{a_{II}^2+b_{II}^2}\left[\begin{array}{cc}{a}_{II}^2& b_{II}^2\\ -b_{II}^2& a_{II}^2\end{array}\right]\left[\begin{array}{c}{X}_K-T_{X_{II}}\\ Y_K-T_{Y_{II}}\end{array}\right]$

So far, for each pixel of image after splicing, it must belong to a L, a certain pixel in J or K, method according to above-mentioned correspondence carries out sampling and the filling of grey scale pixel value, until the generation of final splicing image, as can be seen from abundant experimental results, for the sub-image of interior visual field optical segmentation type boat camera, use the interior visual field optical segmentation type large CCD images geometry joining method that the present invention proposes, can seamless spliced generation high precision without the large format aerial stereo images of geometric dislocation, thus meet aerophotogrammetric field work measures performance practical application request to digital mapping camera height photographic efficiency and high geometry.

Claims (1)

1. a visual field optical segmentation type large CCD images geometry joining method in, is characterized in that: the method comprises following step:

(1) corresponding image points image coordinate in each adjacent sub-image overlap area is measured, obtain the position of focal plane coordinate of the sub-image benchmark picture point that camera manufacturer provides, being numbered of definition sub-image, corresponding image points and benchmark picture point: sub-image label coding adopts Greek alphabet (I, II, III ...), corresponding image points label coding adopts arabic numeral (1,2,3 ...), reference image piont mark coding adopts English capitalization (A, B, C ...);

(2) observation equation group is set up based on splicing relational model

Establishing splicing relational model is the fractal transforms such as two dimension, and formula is

X＝ax-by+T _X(1)

Y＝ay+bx+T _Y

Wherein, x, y are sub-image coordinate system coordinate, and X, Y are resultant image coordinate system coordinate after splicing, a, b, T _x, T _ybe 4 conversion parameters, with this, splicing relational design carried out to sub-image reference image point and adjacent sub-image corresponding image points:

The first splices relational design---about sub-image benchmark picture point

Benchmark picture point C on VI work song image is had:

$X_C=a_{VI}{x}_{C_{VI}}-b_{VI}{y}_{C_{VI}}+T_{X_{VI}}$

$Y_C=a_{VI}{y}_{C_{VI}}+b_{VI}{x}_{C_{VI}}+T_{Y_{VI}}---\left(2\right)$

Wherein, for the picpointed coordinate of benchmark picture point C on sub-image VI under its sub-image coordinate system, X _c, Y _cbe then image coordinate after the synthesis of benchmark picture point C position, reference mark on focal plane, a _vI, b _vI, for the splicing parameter of sub-image VI, equally all one group of corresponding splicing relation equation group to be write out by above formula for the benchmark picture point on other sub-image;

The second splicing relational design---about adjacent sub-image corresponding image points

After utilizing the synthesis on adjacent sub-image corresponding to corresponding image points, image picpointed coordinate this condition equal lists splicing relation equation, for the corresponding image points 1 on adjacent sub-image I and sub-image II for having:

$a_I{x}_{1_I}-b_I{y}_{1_I}+T_{X_I}-(a_{II}{x}_{1_{II}}-b_{II}{y}_{1_{II}}+T_{X_{II}})=0$

$a_I{y}_{1_I}+b_I{x}_{1_I}+T_{Y_I}-(a_{II}{y}_{1_{II}}+b_{II}{x}_{1_{II}}+T_{Y_{II}})=0---\left(3\right)$

For other often pair adjacent sub-image, the each corresponding image points occurred in overlay region can write out one group of corresponding splicing relation equation group by above formula, like this, corresponding splicing relation observation equation group listed by all benchmark picture points and corresponding image points combined, the observation equation group matrix form finally obtained is

_mA ⁿ _nX ¹＝ _mL ¹+ _mV ¹(4)

Wherein, coefficient matrices A is set up according to two kinds of splicing relations, X is the splicing parameter of all sub-images, L is the scalar matrix be made up of element 0 and benchmark picture point position of focal plane coordinate, V is residual vector, m size is the twice of corresponding image points and benchmark number of dots sum, and n size is four times of sub-image quantity;

(3) utilize least squares adjustment technology to solve the splicing parameter of each sub-image to final resultant image, and provide calculation accuracy

All observation equations often open splicing parameter (a, b, T that sub-image is corresponding after building _x, T _y) ask by least square adjustment solution, after these parameters determine all sub-images to synthesis, the geometry of image splices relation

_mA ⁿ _nX ¹＝ _mL ¹+ _mV ¹，X＝(A ^TA) ^-1(A ^TL)

The internal accuracy of splicing parameter X is calculated as follows:

V＝AX-L(5)

Weight unit standard deviation is:

$S_0=\sqrt{\frac{V^{T}V}{r}},r=m-n$

for (A ^ta) ^-1the i-th row, the i-th column element,

$S_{x_i}=S_0\sqrt{Q_{x_i{x}_i}}$

Be the internal accuracy of i-th splicing parameter in X;

(4) finally utilize splicing parameter to convert and resampling sub-image, thus complete the splicing of each sub-image to final resultant image

These should be spliced parameter " oppositely " application in practical operation, the matrix form of formula (1) is as follows:

$\left[\begin{array}{c}X\\ Y\end{array}\right]=\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]+\left[\begin{array}{c}{T}_X\\ T_Y\end{array}\right]---\left(6\right)$

Oppositely solve,

$\left[\begin{array}{c}x\\ y\end{array}\right]={\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]}^{-1}\left[\begin{array}{c}X-T_X\\ Y-T_Y\end{array}\right]=\frac{1}{a^2+b^2}\left[\begin{array}{cc}a& b\\ -b& a\end{array}\right]\left[\begin{array}{c}X-T_X\\ Y-T_Y\end{array}\right]---\left(7\right)$

Image pixel coordinate (X after the splicing that use not yet generates, Y) and corresponding sub-image conversion parameter, corresponding location of pixels on sub-image corresponding to this pixel can be calculated by formula (7), then can sample and obtain half-tone information, for image after the splicing corresponding to overlapping region, can gray-scale value be obtained at the anyon image up-sampling comprising overlay region or carry out the weighted mean of gray-scale value.