CN109982049B - Panorama fusion display control system - Google Patents
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- H—ELECTRICITY
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- H04N23/80—Camera processing pipelines; Components thereof
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- H—ELECTRICITY
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- H04N23/00—Cameras or camera modules comprising electronic image sensors; Control thereof
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- H04N5/00—Details of television systems
- H04N5/222—Studio circuitry; Studio devices; Studio equipment
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- H—ELECTRICITY
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Abstract
The invention discloses a panoramic fusion display and control system, and relates to the field of computer vision. The system comprises a tunnel display controller, a road display controller, a flight display controller and panoramic fusion display control software, wherein the panoramic fusion display control software comprises a parameter setting module, a cooperative control module, a server and an image processing module, and the image processing module converts an original panoramic image into a corrected panoramic image by adopting a panoramic fusion display control method; the panoramic fusion display and control method mainly adopts a quadrilateral projection transformation algorithm to obtain a transformation formula, and projects an original panoramic image to a new view plane. The system can subvert the current monitoring mode of the application places such as the current expressway, improve the monitoring effect and the coverage range and provide better support for disaster reduction, accident treatment and safety guarantee of the expressway.
Description
Technical Field
The invention relates to the field of computer vision, in particular to a panoramic fusion display and control system.
Background
The panoramic cooperative control camera is mainly used in the security industry and is suitable for monitoring long and narrow open scenes, such as highway roads and tunnels. With the increasing number of vehicles running on the expressway, people put higher requirements on monitoring safety, and schedule is gradually increased in the whole-course full-coverage monitoring mode.
At present, a tripod head omnibearing camera is used for monitoring the whole course on an expressway, but only one direction can be monitored at the same time, and a large blind area exists; fixed gunlocks are adopted in the highway tunnels mostly, the fixed gunlocks are arranged at intervals of not more than 150 meters, but due to the fact that most of tunnels are identical in scene, the specific physical positions of the displayed images cannot be distinguished, logical positions cannot be displayed on a monitoring screen, when safety problems occur, long time is consumed to determine the physical positions, and the time for rapidly handling accidents is delayed.
In the traditional monitoring, a large number of cameras and required vertical rods are required to be deployed for realizing the whole-course full coverage, and the requirement of the non-blind-area full coverage is difficult to realize. The panoramic fusion display and control system is required to meet the requirement of full-process monitoring and full coverage of a highway scene, and can fuse, splice and display images of the panoramic camera in sequence besides the panoramic camera meeting the requirement.
Disclosure of Invention
The invention aims to: the panoramic fusion display and control system solves the problems that a large number of cameras and required vertical rods are required to be deployed when the whole-course full coverage is required to be realized in the traditional monitoring, and the requirement of non-blind-area full coverage is difficult to realize. The system comprises a tunnel display controller, a road display controller, a flight display controller conforming to the characteristics of the expressway and panoramic fusion display control software, realizes the fusion display and control of multi-path panoramic images, subverts the current monitoring mode of the application places such as the expressway and the like, improves the monitoring effect and the coverage range, and provides better support for disaster reduction, accident treatment and safety guarantee of the expressway.
The technical scheme adopted by the invention is as follows:
a panoramic fusion display and control system comprises a tunnel display and control device, a road display and control device, a flight display and control device and panoramic fusion display and control software,
the tunnel display controller adopts panoramic fusion display control software to decode, cut, correct distortion and fuse the original panoramic image collected in the tunnel, and finally displays a set of tunnel full-coverage images displayed according to the complete logical physical position sequence on the main screen of the monitoring center screen;
the road display controller adopts panoramic fusion display control software to decode, cut, correct distortion and fuse the meta-panoramic image collected on the road, and finally displays a set of road full-coverage image displayed according to the complete logical physical position sequence on the main screen of the monitoring center screen;
the flight display controller can display a gradually moving corrected panoramic image, namely a tracking image, on a main screen of a monitor center screen according to system time or set flight speed, and visually form a flight monitoring effect;
the panoramic fusion display and control software comprises a parameter setting module, a cooperative control module, a server and an image processing module.
Each tunnel hole is provided with a tunnel display controller, and each 8 kilometers is provided with a road display controller. The flight display controller sets parameters of the whole road section (tunnel and road) fusion display in advance, sets according to system time or flight speed, displays a section of corrected panoramic image which gradually moves on a main screen of a fixed number of monitoring center screens, and visually forms a flight monitoring effect. And the panoramic image at the position can be moved to the auxiliary screen of the monitoring center for magnifying and watching by double clicking the panorama at the position, and the system can decode and display the panoramic and PTZ image, and realize the operations of image control, manual tracking selection, PTZ control and the like.
Furthermore, a parameter setting module in the panoramic fusion display and control software is used for setting parameters of cutting and distortion correction of original panoramic images, setting parameters of fusion width and up-down position between the original panoramic images, setting parameters of flight monitoring direction, flight monitoring speed and the like in the flight display and control device, setting the number of panoramic groups according to the number of main screens of a screen of a field monitoring center, and controlling different panoramic groups to be fused and displayed on the main screens;
the cooperative control module is used for cooperatively controlling a main screen and an auxiliary screen of a screen of the on-site monitoring center, and when the cooperative control module receives the alarm information transmitted by the server, the corrected panoramic image and the corrected tracking image corresponding to the alarm position are moved from the main screen to the auxiliary screen for display;
the server is used for receiving alarm information and transmitting the information to the cooperative control module, and is also used for receiving original panoramic images collected by the tunnel display controller and the road display controller, arranging the original panoramic images according to a logical and physical position sequence, and transmitting the arranged images to the image processing module;
the image processing module converts the original panoramic image into a corrected panoramic image by adopting a panoramic fusion display control method, and transmits the corrected panoramic image to the cooperative control module for display.
Further, the panoramic fusion display and control method comprises the following steps,
step 1: collecting a plurality of original panoramic images by a panoramic camera, wherein the edges of the original panoramic images are partially overlapped;
step 2: carrying out hexagonal point selection on the original panoramic image to form a hexagonal pattern;
and step 3: splitting the hexagonal graph into any two quadrangles, wherein one side of each quadrangle is overlapped;
and 4, step 4: distortion correction, adopting any quadrilateral projection transformation algorithm to perform projection transformation on the two quadrilaterals, and converting the two quadrilaterals into a regular quadrilateral;
and 5: and (4) fusing and splicing the two square in the step (4) to form a corrected panoramic image.
Further, the method for hexagonal point selection in step 2 includes,
step 41: setting the middle point of the upper side of the original panoramic image as a point A, setting the middle point of the lower side of the original panoramic image as a point B, and setting the middle point of an AB line segment as a point H;
step 42: setting an overlapped line of the original panoramic image and another original panoramic image adjacent to the right side of the original panoramic image as a CD line segment, enabling the CD line segment to be vertical to the road, and ensuring that the center point of the road width is overlapped with the midpoint I of the CD line segment;
step 43: and setting an overlapped line of the original panoramic image and another original panoramic image adjacent to the left side of the original panoramic image as an EF line segment, enabling the EF line segment to be vertical to the road, and ensuring that the center point of the road width is overlapped with the midpoint G of the EF line segment.
Step 44: and connecting the adjacent points to form the hexagonal ACDBFE.
In the hexagonal point selection, a point A is the middle point of the upper edge of the panoramic image, a point B is the middle point of the lower edge of the panoramic image, and a point H is the middle point of an AB line segment; C. and D, selecting two points according to the adjacent panoramic images, ensuring that the CD line segment is an overlapped line of the two panoramic images, namely ensuring that the adjacent panoramic images are completely covered, keeping the center point of the road width and the center point I of the CD line segment to be overlapped, and enabling the CD line segment to be vertical to the road. Therefore, the road of the corrected panoramic image is positioned in the middle of the image, the phenomenon that the highway is broken at the junction of the two panoramic images can be solved, and the overlapping area or blind area of the highway and the right panoramic image can be reduced. The selection of two points E, F is the same as C, D.
Further, the method for splitting the hexagonal figure in step 3 is to split the hexagonal acdffe into a quadrilateral ABDC and a quadrilateral abde with the AB line segment as an axis.
The existing related image processing does not have a hexagonal distortion correction algorithm, but has an arbitrary quadrilateral projection transformation algorithm, the distortion correction of the panoramic hexagon which needs to be processed by the method can be divided into 2 projection transformations of arbitrary quadrilaterals, and only two quadrilaterals have one edge which is coincident. Namely, taking the AB line segment as an axis, splitting the hexagonal ACDBFE into a quadrilateral ABDC and a quadrilateral ABFE.
Further, the quadrilateral projective transformation algorithm comprises the specific steps of,
step 61: let u, v be the original panoramic image coordinates and x, y be the rectified panoramic image coordinates, i.e.
Step 62: the transformation formula of the rewrite step 61 is available,
and step 63: the auxiliary variables are defined such that,
Δx1＝x1-x2，Δx2＝x3-x2，Δx3＝x0-x1+x2-x3，
Δy1＝y1-y2，Δy2＝y3-y2，Δy3＝y0-y1+y2-y3；
step 64: when Δ x3,Δy3When the value is other than 0, the compound is obtained,
a11＝x1-x0+a12x1，a21＝x3-x0+a12x2，a31＝x0，
a12＝y1-y0+a13y1，a22＝y3-y0+a23y3，a32＝y0，
step 65: when Δ x3,Δy3When the ratio is 0, the compound is obtained,
a11＝x1-x0，a21＝x2-x1，a31＝x0，
a12＝y1-y0，a22＝y2-y1，a32＝y0，
a13＝0，a12＝0；
and step 66: according to the transformation formula in step 62,
x0＝a31，x1＝a11+a31-a13x1，x2＝a11+a21+a31-a13x2-a23x2，x3＝a21+a31-a23x3
y0＝a32，y1＝a12+a32-a13y1，y2＝a12+a22+a32-a23y2-a23y2，y3＝a22+a32-a23y3；
step 67: the four sets of corresponding points after transformation are represented as,
(0,0)→(x0,y0),(1,0)→(x1,y1),(1,1)→(x2,y2),(0,1)→(x3,y3)。
the quadrilateral projection transformation mainly adopts an OPEN CV related OPEN source algorithm, geometrical image transformations-Perspective transformation, namely, a picture is projected to a new visual plane, which is also called projection mapping. The general transformation formula is:
u，vand x and y are coordinates of the original panoramic image, wherein x is x '/w', and y is y '/w'. Transformation matrix
Can be disassembled into 4 parts, and the two parts are combined,
representing a linear transformation, [ a ]31a32]For translation, [ a ]31a32]TA perspective transformation is generated and the picture after the perspective transformation is usually not a parallelogram unless there is a case where the mapped view plane is parallel to the original plane. The transformation formula before rewriting can result in:
therefore, the transformation formula can be obtained by knowing several points corresponding to the transformation.
If the proportion of the panoramic image after the distortion correction is still larger than 16 of the normal display: 9 or 4: and 3, displaying the corrected panoramic image on a single display to have transverse compression deformation, and in order to solve the problem of the transverse deformation, transversely stretching the corrected panoramic image, namely displaying 1 panoramic image on 2 displays to remarkably solve the problem of the transverse compression deformation.
In summary, due to the adoption of the technical scheme, the invention has the beneficial effects that:
1. the invention relates to a panoramic fusion display and control system which comprises a tunnel display and control device, a road display and control device, a flight display and control device conforming to the characteristics of a highway and panoramic fusion display and control software, realizes the fusion display and control of multi-path panoramic images, overturns the current monitoring mode of application places such as the current highway and the like, improves the monitoring effect and coverage range and provides better support for disaster reduction, accident handling and safety guarantee of the highway.
2. According to the panorama fusion display and control system, the front-end panorama cooperative control camera and the rear-end panorama fusion display and control system are combined to form a currently feasible narrow and long area such as a highway, the whole-process full-coverage monitoring system is really realized, and the problem that the traditional monitoring system and monitoring equipment cannot realize whole-process full-coverage monitoring from the practical application is solved.
3. The panoramic fusion display and control system is provided with the flight display and control device, provides an innovative flight monitoring mode and better meets the monitoring requirement of a long and narrow monitoring area.
Detailed Description
All of the features disclosed in this specification, or all of the steps in any method or process so disclosed, may be combined in any combination, except combinations of features and/or steps that are mutually exclusive.
It is noted that relational terms such as "first" and "second," and the like, may be used solely to distinguish one entity or action from another entity or action without necessarily requiring or implying any actual such relationship or order between such entities or actions. Also, the terms "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus. Without further limitation, an element defined by the phrase "comprising an … …" does not exclude the presence of other identical elements in a process, method, article, or apparatus that comprises the element.
The present invention will be described in detail below.
Example 1
A panoramic fusion display and control system comprises a tunnel display and control device, a road display and control device, a flight display and control device and panoramic fusion display and control software,
the tunnel display controller adopts panoramic fusion display control software to decode, cut, correct distortion and fuse the original panoramic image collected in the tunnel, and finally displays a set of tunnel full-coverage images displayed according to the complete logical physical position sequence on the main screen of the monitoring center screen;
the road display controller adopts panoramic fusion display control software to decode, cut, correct distortion and fuse the meta-panoramic image collected on the road, and finally displays a set of road full-coverage image displayed according to the complete logical physical position sequence on the main screen of the monitoring center screen;
the flight display controller can display a gradually moving corrected panoramic image, namely a tracking image, on a main screen of a monitor center screen according to system time or set flight speed, and visually form a flight monitoring effect;
the panoramic fusion display and control software comprises a parameter setting module, a cooperative control module, a server and an image processing module.
Each tunnel hole is provided with a tunnel display controller, and each 8 kilometers is provided with a road display controller. The flight display controller sets parameters of the whole road section (tunnel and road) fusion display in advance, sets according to system time or flight speed, displays a section of corrected panoramic image which gradually moves on a main screen of a fixed number of monitoring center screens, and visually forms a flight monitoring effect. And the panoramic image at the position can be moved to the auxiliary screen of the monitoring center for magnifying and watching by double clicking the panorama at the position, and the system can decode and display the panoramic and PTZ image, and realize the operations of image control, manual tracking selection, PTZ control and the like.
Example 2
The parameter setting module in the panoramic fusion display and control software is used for setting parameters of cutting and distortion correction of original panoramic images, setting parameters of fusion width and up-down position between the original panoramic images, setting parameters of flight monitoring direction, flight monitoring speed and the like in the flight display and control device, setting the number of panoramic groups according to the number of main screens of a screen of a field monitoring center, and controlling the fusion display of different panoramic groups on the main screens;
the cooperative control module is used for cooperatively controlling a main screen and an auxiliary screen of a screen of the on-site monitoring center, and when the cooperative control module receives the alarm information transmitted by the server, the corrected panoramic image and the corrected tracking image corresponding to the alarm position are moved from the main screen to the auxiliary screen for display;
the server is used for receiving alarm information and transmitting the information to the cooperative control module, and is also used for receiving original panoramic images collected by the tunnel display controller and the road display controller, arranging the original panoramic images according to a logical and physical position sequence, and transmitting the arranged images to the image processing module;
the image processing module converts the original panoramic image into a corrected panoramic image by adopting a panoramic fusion display control method, and transmits the corrected panoramic image to the cooperative control module for display.
Example 3
The display and control method for the panoramic fusion comprises the following steps,
step 1: collecting a plurality of original panoramic images by a panoramic camera, wherein the edges of the original panoramic images are partially overlapped;
step 2: carrying out hexagonal point selection on the original panoramic image to form a hexagonal pattern;
and step 3: splitting the hexagonal graph into any two quadrangles, wherein one side of each quadrangle is overlapped;
and 4, step 4: distortion correction, adopting any quadrilateral projection transformation algorithm to perform projection transformation on the two quadrilaterals, and converting the two quadrilaterals into a regular quadrilateral;
and 5: and (4) fusing and splicing the two square in the step (4) to form a corrected panoramic image.
Example 4
The method for hexagonal dot selection in step 2 described in example 3 includes,
step 41: setting the middle point of the upper side of the original panoramic image as a point A, setting the middle point of the lower side of the original panoramic image as a point B, and setting the middle point of an AB line segment as a point H;
step 42: setting an overlapped line of the original panoramic image and another original panoramic image adjacent to the right side of the original panoramic image as a CD line segment, enabling the CD line segment to be vertical to the road, and ensuring that the center point of the road width is overlapped with the midpoint I of the CD line segment;
step 43: and setting an overlapped line of the original panoramic image and another original panoramic image adjacent to the left side of the original panoramic image as an EF line segment, enabling the EF line segment to be vertical to the road, and ensuring that the center point of the road width is overlapped with the midpoint G of the EF line segment.
Step 44: and connecting the adjacent points to form the hexagonal ACDBFE.
In the hexagonal point selection, a point A is the middle point of the upper edge of the panoramic image, a point B is the middle point of the lower edge of the panoramic image, and a point H is the middle point of an AB line segment; C. and D, selecting two points according to the adjacent panoramic images, ensuring that the CD line segment is an overlapped line of the two panoramic images, namely ensuring that the adjacent panoramic images are completely covered, keeping the center point of the road width and the center point I of the CD line segment to be overlapped, and enabling the CD line segment to be vertical to the road. Therefore, the road of the corrected panoramic image is positioned in the middle of the image, the phenomenon that the highway is broken at the junction of the two panoramic images can be solved, and the overlapping area or blind area of the highway and the right panoramic image can be reduced. The selection of two points E, F is the same as C, D.
Example 5
The method for splitting the hexagonal figure in step 3 described in embodiment 3 is to split the hexagonal acdffe into a quadrilateral ABDC and a quadrilateral abde with the AB line segment as an axis.
The existing related image processing does not have a hexagonal distortion correction algorithm, but has an arbitrary quadrilateral projection transformation algorithm, the distortion correction of the panoramic hexagon which needs to be processed by the method can be divided into 2 projection transformations of arbitrary quadrilaterals, and only two quadrilaterals have one edge which is coincident. Namely, taking the AB line segment as an axis, splitting the hexagonal ACDBFE into a quadrilateral ABDC and a quadrilateral ABFE.
Example 6
The quadrilateral projective transformation algorithm comprises the specific steps of,
step 61: let u, v be the original panoramic image coordinates and x, y be the rectified panoramic image coordinates, i.e.
Step 62: the transformation formula of the rewrite step 61 is available,
and step 63: the auxiliary variables are defined such that,
Δx1＝x1-x2，Δx2＝x3-x2，Δx3＝x0-x1+x2-x3，
Δy1＝y1-y2，Δy2＝y3-y2，Δy3＝y0-y1+y2-y3；
step 64: when Δ x3,Δy3When the value is other than 0, the compound is obtained,
a11＝x1-x0+a12x1，a21＝x3-x0+a12x2，a31＝x0，
a12＝y1-y0+a13y1，a22＝y3-y0+a23y3，a32＝y0，
step 65: when Δ x3,Δy3When the ratio is 0, the compound is obtained,
a11＝x1-x0，a21＝x2-x1，a31＝x0，
a12＝y1-y0，a22＝y2-y1，a32＝y0，
a13＝0，a12＝0；
and step 66: according to the transformation formula in step 62,
x0＝a31，x1＝a11+a31-a13x1，x2＝a11+a21+a31-a13x2-a23x2，x3＝a21+a31-a23x3
y0＝a32，y1＝a12+a32-a13y1，y2＝a12+a22+a32-a23y2-a23y2，y3＝a22+a32-a23y3；
step 67: the four sets of corresponding points after transformation are represented as,
(0,0)→(x0,y0),(1,0)→(x1,y1),(1,1)→(x2,y2),(0,1)→(x3,y3)。
the above description is only a preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any changes or substitutions that can be made by those skilled in the art without inventive work within the technical scope of the present invention are included in the scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the protection scope defined by the claims.
Claims (4)
1. The utility model provides a panorama fusion shows accuse system which characterized in that: comprises a tunnel display controller, a road display controller, a flight display controller and panoramic fusion display control software,
the tunnel display controller adopts panoramic fusion display control software to decode, cut, correct distortion and fuse the original panoramic image collected in the tunnel, and finally displays a set of tunnel full-coverage images displayed according to the complete logical physical position sequence on the main screen of the monitoring center screen;
the road display controller adopts panoramic fusion display control software to decode, cut, correct distortion and fuse the meta-panoramic image collected on the road, and finally displays a set of road full-coverage image displayed according to the complete logical physical position sequence on the main screen of the monitoring center screen;
the flight display controller can display a gradually moving corrected panoramic image, namely a tracking image, on a main screen of a monitor center screen according to system time or set flight speed, and visually form a flight monitoring effect;
the panoramic fusion display and control software comprises a parameter setting module, a cooperative control module, a server and an image processing module;
the parameter setting module in the panoramic fusion display and control software is used for setting parameters of cutting and distortion correction of original panoramic images, setting parameters of fusion width and up-down position between the original panoramic images, setting parameters of flight monitoring direction, flight monitoring speed and the like in the flight display and control device, setting the number of panoramic groups according to the number of main screens of a screen of a field monitoring center, and controlling the fusion display of different panoramic groups on the main screens;
the cooperative control module is used for cooperatively controlling a main screen and an auxiliary screen of a screen of the on-site monitoring center, and when the cooperative control module receives the alarm information transmitted by the server, the corrected panoramic image and the corrected tracking image corresponding to the alarm position are moved from the main screen to the auxiliary screen for display;
the server is used for receiving alarm information and transmitting the information to the cooperative control module, and is also used for receiving original panoramic images collected by the tunnel display controller and the road display controller, arranging the original panoramic images according to a logical and physical position sequence, and transmitting the arranged images to the image processing module;
the image processing module converts an original panoramic image into a corrected panoramic image by adopting a panoramic fusion display control method and transmits the corrected panoramic image to the cooperative control module for display;
the display and control method for the panoramic fusion comprises the following steps,
step 1: collecting a plurality of original panoramic images by a panoramic camera, wherein the edges of the original panoramic images are partially overlapped;
step 2: carrying out hexagonal point selection on the original panoramic image to form a hexagonal pattern;
and step 3: splitting the hexagonal graph into any two quadrangles, wherein one side of each quadrangle is overlapped;
and 4, step 4: distortion correction, adopting any quadrilateral projection transformation algorithm to perform projection transformation on the two quadrilaterals, and converting the two quadrilaterals into a regular quadrilateral;
and 5: and (4) fusing and splicing the two square in the step (4) to form a corrected panoramic image.
2. The display and control system for panoramic fusion of claim 1, characterized in that: the hexagonal point selection method in the step 2 comprises the following steps,
step 41: setting the middle point of the upper side of the original panoramic image as a point A, setting the middle point of the lower side of the original panoramic image as a point B, and setting the middle point of an AB line segment as a point H;
step 42: setting an overlapped line of the original panoramic image and another original panoramic image adjacent to the right side of the original panoramic image as a CD line segment, enabling the CD line segment to be vertical to the road, and ensuring that the center point of the road width is overlapped with the midpoint I of the CD line segment;
step 43: and setting an overlapped line of the original panoramic image and another original panoramic image adjacent to the left side of the original panoramic image as an EF line segment, enabling the EF line segment to be vertical to the road, and ensuring that the center point of the road width is overlapped with the midpoint G of the EF line segment.
Step 44: and connecting the adjacent points to form the hexagonal ACDBFE.
3. The display and control system for panoramic fusion of claim 1, characterized in that: the method for splitting the hexagonal graph in the step 3 is to split the hexagonal ACDBFE into a quadrilateral ABDC and a quadrilateral ABFE by taking the AB line segment as an axis.
4. The display and control system for panoramic fusion of claim 1, characterized in that: the quadrilateral projective transformation algorithm comprises the specific steps of,
step 61: let u, v be the original panoramic image coordinates and x, y be the rectified panoramic image coordinates, i.e.
Step 62: the transformation formula of the rewrite step 61 is available,
and step 63: the auxiliary variables are defined such that,
Δx1＝x1-x2，Δx2＝x3-x2，Δx3＝x0-x1+x2-x3，
Δy1＝y1-y2，Δy2＝y3-y2，Δy3＝y0-y1+y2-y3；
step 64: when Δ x3,Δy3When the value is other than 0, the compound is obtained,
a11＝x1-x0+a12x1，a21＝x3-x0+a12x2，a31＝x0，
a12＝y1-y0+a13y1，a22＝y3-y0+a23y3，a32＝y0，
step 65: when Δ x3,Δy3When the ratio is 0, the compound is obtained,
a11＝x1-x0，a21＝x2-x1，a31＝x0，
a12＝y1-y0，a22＝y2-y1，a32＝y0，
a13＝0，a12＝0；
and step 66: according to the transformation formula in step 62,
x0＝a31，x1＝a11+a31-a13x1，x2＝a11+a21+a31-a13x2-a23x2，x3＝a21+a31-a23x3，
y0＝a32，y1＝a12+a32-a13y1，y2＝a12+a22+a32-a23y2-a23y2，y3＝a22+a32-a23y3；
step 67: the four sets of corresponding points after transformation are represented as,
(0,0)→(x0,y0),(1,0)→(x1,y1),(1,1)→(x2,y2),(0,1)→(x3,y3)。
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