CN107316325A - A kind of airborne laser point cloud based on image registration and Image registration fusion method - Google Patents
A kind of airborne laser point cloud based on image registration and Image registration fusion method Download PDFInfo
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Abstract
The invention discloses a kind of airborne laser point cloud based on image registration and Image registration fusion method, for the dimensionally form point cloud of airborne LiDAR system acquisitions to be merged with aerial images, RGB three-dimensional point cloud atlas is generated, is comprised the following steps:Point cloud orthographic projection images A is generated using three-dimensional laser point cloud and sets up a cloud pixel index;Taken photo by plane using aerial images generation and just penetrate stitching image B;Image registration operation is carried out to image A and image B, a cloud orthographic projection images A pixel coordinate is transformed into take photo by plane and just penetrated under stitching image B pixel coordinate system;Back projection is carried out using a cloud pixel index, each point corresponding takes photo by plane of cloud is found and just penetrates stitching image B pixel coordinate, pixel color value is assigned to a cloud, RGB point cloud chart is generated after fusion.To cloud data, whether synchronous acquisition is not restricted this method with image data, it might even be possible to using taking photo by plane of generating of third party or remote sensing orthography is used as registering image.
Description
Technical field
The present invention relates to unmanned plane technical field of mapping, and in particular to a kind of airborne laser point based on image registration
Cloud and Image registration fusion method.
Background technology
Laser scanning measurement technology (Light Detection And Ranging), abbreviation LiDAR, also known as " outdoor scene is replicated
Technology ", is newest one kind in three-dimensional data technology and scene modeling technology.Airborne LiDAR Technology uses LiDAR technologies
Earth observation is realized on aircraft, integrates global positioning system, three kinds of technologies of inertial navigation system and laser, can be fast
Speed efficiently obtains the accurate three-dimensional space coordinate of each sampled point in atural object surface.In recent years, as airborne lidar is measured
The fast development of technology, before it has wide application in terms of digital city, disaster monitoring, coastal engineering, Forestry Investigation
Scape.However, the technology can only obtain the discrete three dimensional space coordinate and Reflection intensity information of target object, acquisition thing is not had
The ability of body real-texture information.And with the fast development of digital imaging technology and imaging sensor, obtain high definition visible ray
Image is not problem, and corresponding image procossing and integration technology take photo by plane field acquisition widely in machine vision, remote sensing
Using.The three-dimensional information that the target object texture information that image is carried is obtained with laser scanner technique forms complementation.
In order to generate RGB point cloud chart, it is necessary to will point cloud and the registering fusion of image progress.Three-dimensional point cloud and aerial images
The registration fusion of data is the Registration of Measuring Data fusion problem between dissimilar sensor, belongs to classical 2 d-3 d fusion
Problem.According to the difference of the flow of registration and registration features, the method for registering of current two and three dimensions can be divided into following several:
(1) laser scanner puts standardization with camera with seat in the plane;(2) the 3D-2D method for registering of feature based;(3) it is based on stereogram
3D-3D method for registering.First method needs that camera and scanner is fixedly mounted and synchronous acquisition, and needs to carry out essence
True proving operation, has some limitations;Second method requires there is more obvious feature in point cloud and image
(point, line, plane) is as registering primitive, and the extraction accuracy of registering primitive influences big to registration result;The third method is to clapping
The angle of inclination of photogra and degree of overlapping are required, and generate imaging point cloud algorithm and process is complex, and precision also is difficult to
Ensure, it is impossible to meet the requirement for quickly and efficiently generating RGB point cloud.Taken photo by plane mapping for unmanned aerial vehicle onboard, work as sweep object
During for the few landform of the obvious characteristics such as wasteland, mountain peak or coastline, existing method can not be fully solved the quick of data
Efficiently registering fusion problem.
The content of the invention
The invention aims to solve drawbacks described above of the prior art there is provided a kind of based on the airborne of image registration
Laser point cloud and Image registration fusion method.
The purpose of the present invention can be reached by adopting the following technical scheme that:
A kind of airborne laser point cloud based on image registration and Image registration fusion method, methods described include following step
Suddenly:
S1, orthogonal projection is carried out to three-dimensional laser point cloud, generation point cloud orthographic projection images A simultaneously sets up a cloud-pixel rope
Draw;
S2, the aviation image progress distortion correction shot to camera, are just being penetrated and splicing to the image after correction,
Generation, which is taken photo by plane, just penetrates stitching image B;
S3, will point cloud orthographic projection images A as floating image, take photo by plane and just penetrate stitching image B as reference picture, to point
Cloud orthographic projection images A just penetrates stitching image B progress image registration operations with taking photo by plane, and uses image registration algorithm to solve conversion ginseng
Number, a cloud orthographic projection images A pixel coordinate conversion is just penetrated under stitching image B coordinate system to taking photo by plane;
S4, thrown cloud orthographic projection images A pixel is counter using the point cloud-pixel index set up in projection process
Shadow, finds corresponding take photo by plane of each cloud and just penetrates stitching image B pixel coordinate, so that the face for just penetrating stitching image B of taking photo by plane
Colour is assigned to corresponding cloud, generates RGB point cloud chart.
Further, generation point cloud orthographic projection images A process includes in the step S1:S11, sampling process;
S12, quantizing process;S13, Interpolation Process.
Further, the sampling process includes:
S111, selection projection plane, set up pixel coordinate system and calculate in projection plane outside the minimum of whole subpoints
Bag rectangle, the O-XY planes using east northeast under coordinate system is perspective planes, using the upper left corner of outsourcing rectangle as coordinate origin O, just
South is to for V axles, and due east direction is axle U, sets up pixel coordinate system O-UV;
S112, the excursion for calculating point cloud X-coordinate and Y-coordinate, YmaxAnd YminRepresent respectively Y-axis coordinate maximum and most
Small value, XmaxAnd XminThe maximum and minimum value of X-axis coordinate is represented respectively, by manually setting projected image width W, pixel size
S and picture altitude H calculation formula is:
S113, each point cloud corresponding pixel coordinate under projected image are (u, v), and calculation formula is:
S114, using nearest neighbor algorithm pixel coordinate is rounded, pixel coordinate closest therewith is assigned to (u, v),
The corresponding relation of measuring point cloud coordinate and pixel coordinate, generates point cloud-pixel index simultaneously.
Further, the quantizing process is specially:
Value to each pixel carries out quantization filling, and selection elevation carries out quantifying generation elevation projected image, selection
Laser reflection intensity carries out quantifying generation intensity projection images.
Further, the Interpolation Process is specially:
For the blank pixel in image, the pixel around use carries out interpolation by high order interpolation methods, while also to point
Cloud carries out interpolation, and sets up a cloud-pixel index, and high order interpolation function is as follows:
Wherein, | x | for the distance value of surrounding pixel to (u, v).
Further, the transformation model that image registration is used in the image registration algorithm of the step S3 is affine transformation mould
Type, the homogeneous equation of the affine Transform Model is expressed as follows:
Wherein, (xA,yA) it is a cloud orthographic projection images A pixel coordinate, (xB,yB) just penetrated in stitching image B to take photo by plane
Pixel coordinate, M is the point cloud orthographic projection images A affine transformation relationships of just penetrating stitching image B to taking photo by plane, wherein
It is the composite matrix of rotation, scaling and reversion,It is translation vector, the affine Transform Model has 6 unknown parameters, finds 3 pairs
Not conllinear feature point coordinates of the same name can obtain unknown parameter.
Further, the process that the affine Transform Model is simplified is as follows:
Unknown parameter is restricted, rotation Pan and Zoom, order is only considered
a11=a22=k cos θ, a12=-a21=k sin θs, b1=kc1,b2=kc2, affine Transform Model is simplified to
RST transformation models, are expressed as follows:
Wherein k represents scaling, and θ represents the anglec of rotation between image, [c1,c2]TRepresent the translation vector before scaling, letter
RST transformation models after change have 4 unknown parameters, and result can be calculated by finding 2 pairs of characteristic points of the same name.
The present invention has the following advantages and effect relative to prior art:
1) the inventive method is applied to unmanned plane survey field, has the advantages that stability height, precision are high.
2) the inventive method to laser scanner and camera whether fixed installation and synchronous acquisition are not restricted, or even can
To use third party's aerial images as registering image, flexibility is high.
3) the inventive method can be applied to the scanning area without obvious characteristic, such as wasteland, meadow, coastline very well,
And it is low with amount of calculation, the characteristics of into figure efficiency high and high stability.
Brief description of the drawings
Fig. 1 is the flow of the airborne laser point cloud disclosed by the invention based on image registration and Image registration fusion method
Figure;
Fig. 2 is a cloud orthogonal projection model;
Fig. 3 is a cloud and image coordinate relationship map figure.
Embodiment
To make the purpose, technical scheme and advantage of the embodiment of the present invention clearer, below in conjunction with the embodiment of the present invention
In accompanying drawing, the technical scheme in the embodiment of the present invention is clearly and completely described, it is clear that described embodiment is
A part of embodiment of the present invention, rather than whole embodiments.Based on the embodiment in the present invention, those of ordinary skill in the art
The every other embodiment obtained under the premise of creative work is not made, belongs to the scope of protection of the invention.
Embodiment one
A kind of airborne laser point cloud based on image registration disclosed in the present embodiment and Image registration fusion method, for machine
Laser point cloud fusion registering with aerial images and the generation of RGB point cloud chart are carried, is comprised the following steps:
S1, orthogonal projection is carried out to three-dimensional laser point cloud, generation point cloud orthographic projection images A simultaneously sets up a cloud-pixel rope
Draw, regard point cloud orthographic projection images A as floating image;
Wherein, generation point cloud orthographic projection images A includes sampling, quantified and interpolation, and detailed process is as follows:
S11, sampling process:
S111, selection projection plane, set up pixel coordinate system and calculate in projection plane outside the minimum of whole subpoints
Bag rectangle.O-XY planes with east northeast under (North-East-Down, NED) coordinate system are perspective plane, with a left side for outsourcing rectangle
Upper angle is coordinate origin O, and due south direction is V axles, and due east direction is axle U, sets up pixel coordinate system O-UV.
S112, the excursion for calculating point cloud X-coordinate and Y-coordinate, YmaxAnd YminRepresent respectively Y-axis coordinate maximum and most
Small value, XmaxAnd XminThe maximum and minimum value of X-axis coordinate is represented respectively, by manually setting projected image width W, pixel size
S and picture altitude H calculation formula is:
S113, each point cloud corresponding pixel coordinate under projected image are (u, v), and calculation formula is:
S114, using nearest neighbor algorithm pixel coordinate is rounded, pixel coordinate closest therewith is assigned to (u, v),
The corresponding relation of measuring point cloud coordinate and pixel coordinate, generation point cloud-pixel index, conveniently subsequently carry out back projection's inquiry simultaneously
Use.
S12, quantizing process:Need to carry out the value of each pixel quantization filling after the completion of sampling, selection elevation enters
Row quantifies then generate elevation projected image, and selection laser reflection intensity is quantified, and generates intensity projection images.
S13, Interpolation Process:For the blank pixel in image, the pixel (except blank pixel) around use passes through height
Rank interpolation method carries out interpolation, while also carrying out interpolation to a cloud, and sets up a cloud-pixel index, high order interpolation function is as follows:
Wherein, | x | for the distance value of surrounding pixel to (u, v).
S2, the aviation image progress distortion correction shot to camera, are just being penetrated and splicing to the image after correction,
Generation, which is taken photo by plane, just penetrates stitching image B, is used as reference picture;
S3, will point cloud orthographic projection images A as floating image, take photo by plane and just penetrate stitching image B as reference picture, to point
Cloud orthographic projection images A just penetrates stitching image B progress image registration operations with taking photo by plane, and uses image registration algorithm to solve conversion ginseng
Number, a cloud orthographic projection images A pixel coordinate conversion is just penetrated under stitching image B coordinate system to taking photo by plane;
Wherein, the transformation model that image registration is used in image registration algorithm is affine Transform Model, the affine transformation mould
The homogeneous equation of type is expressed as follows:
Wherein, (xA,yA) it is a cloud orthographic projection images A pixel coordinate, (xB, yB) just penetrated in stitching image B to take photo by plane
Pixel coordinate, M is the point cloud orthographic projection images A affine transformation relationships of just penetrating stitching image B to taking photo by plane, wherein
It is the composite matrix of rotation, scaling and reversion,It is translation vector, the affine Transform Model there are 6 unknown parameters, need to only looks for
Unknown parameter can be obtained to 3 pairs of not conllinear feature point coordinates of the same name.
Affine Transform Model can be simplified.Unknown parameter is restricted, rotation Pan and Zoom is only considered, makes a11
=a22=k cos θ, a12=-a21=k sin θs, b1=kc1,b2=kc2, so that it may so that affine Transform Model is simplified to RST (rotations
Turn-scaling-translation) transformation model, it is expressed as follows:
Wherein k represents scaling, and θ represents the anglec of rotation between image, [c1,c2]TRepresent the translation vector before scaling, letter
RST transformation models after change only have 4 unknown parameters, it is only necessary to which 2 pairs of characteristic points of the same name can just calculate result.
According to the method for registering images of feature based, characteristic point of the same name is at least 2 pairs.
S4, thrown cloud orthographic projection images A pixel is counter using the point cloud-pixel index set up in projection process
Shadow, finds corresponding take photo by plane of each cloud and just penetrates stitching image B pixel coordinate, so that the face for just penetrating stitching image B of taking photo by plane
Colour is assigned to corresponding cloud, generates RGB point cloud chart.
The point cloud orthographic projection images in point cloud-pixel index relation and step S3 in step S1 are just penetrated to taking photo by plane
Splice the coordinate corresponding relation of image, find corresponding take photo by plane of each three-dimensional point cloud coordinate and just penetrate stitching image B pixel coordinate,
Pixel value is assigned to corresponding cloud, RGB point cloud chart is ultimately generated.
Embodiment two
It is specifically real present embodiment discloses a kind of airborne laser point cloud based on image registration and Image registration fusion method
Mode is applied, the algorithm steps based on image registration are as shown in Figure 1.
Small-sized depopulated helicopter is influenceed that flight attitude can be caused not in flight by low air flow and engine vibration
It is stable, so as to influence shooting visual angle to cause flating to distort, external high resolution camera typically use big wide angle camera with
Large-scale topography and geomorphology is shot, this also result in the geometric distortion of aerial images.Due to distortion, the pixel in image
The straight line that point can occur in the skew on geometric position, such as space can become curve in the picture, image be carried out abnormal
Become calibration be exactly enable each pixel geometrical relationship return to it is relatively correct in the state of.According to camera actual imaging mould
Type:
(xi,yi) it is coordinate of the preferred view point under image coordinate system in ideal image model, and (xr,yr) it is actual
Coordinate of the actual subpoint under image coordinate system, (u in imaging model0,v0) represent image coordinate system origin in pixel coordinate system
In coordinate, sx,syIt is the scale factor of image level axle and vertical axis, k1,k2For coefficient of radial distortion,For
The pixel is to the distance of image plane center, p1,p2For tangential distortion coefficient.
In the case where camera focus is constant, the intrinsic parameter and distortion factor that camera calibration is obtained are substituted into above formula, just
The ideal coordinates position of image slices vegetarian refreshments after correction can be tried to achieve, because the pixel coordinate of image in practice is integer, but by upper
It is generally not integer to state formula and calculate obtained pixel coordinate, and this Experimental comparison has selected the correcting distorted figure of bilinear interpolation
Picture.After distortion correction, splice software PhotoScan generation splicing orthographies using unmanned plane later stage aerophotograph.
As shown in Fig. 2 point cloud orthogonal projection illustratons of model, a cloud is just being penetrated orthogonal projection and using elevation as quantify object can
To generate the point positive exit point cloud atlas of cloud level journey.
Accurate geometric position is closed between splicing orthography and point cloud orthographic projection images can express terrain and its features
Affine transformation relationship is met between system, two class images, introduction point cloud orthographic projection images A is as floating image for experiment, takes photo by plane just
Stitching image B is penetrated as reference picture, the affine transformation relationship set up between two images, affine transformation can realize the rotation of image
Turn translation scaling and invert, the pixel coordinate (x in point cloud orthographic projection images AA,yA) obtain taking photo by plane just after affine transformation M
Pixel coordinate (the x penetrated in stitching image BB,yB), affine Transform Model homogeneous form is represented by
Solving equation is
The image registration of feature based is generally divided into feature extraction, characteristic matching, transformation model parameter Estimation and image and matched somebody with somebody
Accurate four steps, the precision of image registration is determined by the extraction accuracy of feature.In feature extracting and matching, according to need to not need
It is artificial to participate in, two kinds of matching and Auto-matching manually can be divided into.For there is the image compared with multiple features, it can use special based on angle point
Two images are first carried out using Sobel operator extractions edge after morphology noise reduction by the semi-automatic method for registering levied, then at edge
Alternative same place is extracted using Harris Corner Detection Algorithms in image, is finally manually extracted in alternative point;And be directed to
The unconspicuous image of feature atural object, then can be used and manually select characteristic point calculating transformation parameter, realize the registration of image.In this reality
Apply manual extraction three in example and, to characteristic point of the same name, transformation parameter is calculated according to formula.
The index of a cloud coordinate and projected image pixel coordinate, image registration are established in the projection process of three-dimensional point cloud
Projected image pixel coordinate is obtained in journey to the mapping relations of Aerial Images pixel coordinate, therefore each three-dimensional point can be obtained
Mapping relations of the cloud coordinate to aerial images pixel coordinate.Finally, the color value of Aerial Images pixel is assigned to corresponding cloud,
I.e. achievable point cloud is merged with image.The coordinate mapping relations of point cloud and aerial images are as shown in Figure 3.
Above-described embodiment is preferably embodiment, but embodiments of the present invention are not by above-described embodiment of the invention
Limitation, other any Spirit Essences without departing from the present invention and the change made under principle, modification, replacement, combine, simplification,
Equivalent substitute mode is should be, is included within protection scope of the present invention.
Claims (7)
1. a kind of airborne laser point cloud based on image registration and Image registration fusion method, it is characterised in that methods described bag
Include the following steps:
S1, orthogonal projection is carried out to three-dimensional laser point cloud, generation point cloud orthographic projection images A simultaneously sets up a cloud-pixel index;
S2, the aviation image progress distortion correction shot to camera, are just being penetrated and splicing to the image after correction, are being generated
Take photo by plane and just penetrate stitching image B;
S3, will point cloud orthographic projection images A as floating image, take photo by plane and just penetrate stitching image B as reference picture, to a cloud just
Penetrate projected image A and take photo by plane and just penetrate stitching image B progress image registration operations, transformation parameter is solved using image registration algorithm,
A cloud orthographic projection images A pixel coordinate conversion is just penetrated under stitching image B coordinate system to taking photo by plane;
S4, using the point cloud-pixel index set up in projection process back projection is carried out to the pixel of a cloud orthographic projection images A,
Find corresponding take photo by plane of each cloud and just penetrate stitching image B pixel coordinate, so that the color value for just penetrating stitching image B of taking photo by plane
Corresponding cloud is assigned to, RGB point cloud chart is generated.
2. a kind of airborne laser point cloud based on image registration according to claim 1 and Image registration fusion method, its
It is characterised by, generation point cloud orthographic projection images A process includes in the step S1:S11, sampling process;S12, quantified
Journey;S13, Interpolation Process.
3. a kind of airborne laser point cloud based on image registration according to claim 2 and Image registration fusion method, its
It is characterised by, the sampling process includes:
S111, selection projection plane, set up pixel coordinate system and calculate the minimum outsourcing square of whole subpoints in projection plane
Shape, the O-XY planes using east northeast under coordinate system is perspective planes, using the upper left corner of outsourcing rectangle as coordinate origin O, Due South
To for V axles, due east direction is axle U, sets up pixel coordinate system O-UV;
S112, the excursion for calculating point cloud X-coordinate and Y-coordinate, YmaxAnd YminThe minimum and maximum of Y-axis coordinate is represented respectively
Value, XmaxAnd XminThe maximum and minimum value of X-axis coordinate is represented respectively, by manually setting projected image width W, pixel size S
Calculation formula with picture altitude H is:
S113, each point cloud corresponding pixel coordinate under projected image are (u, v), and calculation formula is:
S114, using nearest neighbor algorithm pixel coordinate is rounded, pixel coordinate closest therewith be assigned to (u, v), simultaneously
The corresponding relation of measuring point cloud coordinate and pixel coordinate, generation point cloud-pixel index.
4. a kind of airborne laser point cloud based on image registration according to claim 2 and Image registration fusion method, its
It is characterised by, the quantizing process is specially:
Value to each pixel carries out quantization filling, and selection elevation carries out quantifying generation elevation projected image, selects laser
Reflected intensity carries out quantifying generation intensity projection images.
5. a kind of airborne laser point cloud based on image registration according to claim 2 and Image registration fusion method, its
It is characterised by, the Interpolation Process is specially:
For the blank pixel in image, the pixel around use carries out interpolation by high order interpolation methods, while also entering to a cloud
Row interpolation, and a cloud-pixel index is set up, high order interpolation function is as follows:
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6. a kind of airborne laser point cloud based on image registration according to claim 1 and Image registration fusion method, its
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The homogeneous equation for penetrating transformation model is expressed as follows:
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</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>y</mi>
<mi>B</mi>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>=</mo>
<mi>M</mi>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<msub>
<mi>x</mi>
<mi>A</mi>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>y</mi>
<mi>A</mi>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<msub>
<mi>a</mi>
<mn>11</mn>
</msub>
</mtd>
<mtd>
<msub>
<mi>a</mi>
<mn>12</mn>
</msub>
</mtd>
<mtd>
<msub>
<mi>b</mi>
<mn>1</mn>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>a</mi>
<mn>21</mn>
</msub>
</mtd>
<mtd>
<msub>
<mi>a</mi>
<mn>22</mn>
</msub>
</mtd>
<mtd>
<msub>
<mi>b</mi>
<mn>2</mn>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mfenced>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
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<mi>x</mi>
<mi>A</mi>
</msub>
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<mtr>
<mtd>
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<mi>y</mi>
<mi>A</mi>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
Wherein, (xA,yA) it is a cloud orthographic projection images A pixel coordinate, (xB,yB) it is the picture just penetrated in stitching image B of taking photo by plane
Plain coordinate, the affine transformation relationship that M just penetrates stitching image B for a cloud orthographic projection images A to taking photo by plane, whereinIt is rotation
The composite matrix for turning, scaling and inverting,It is translation vector, the affine Transform Model there are 6 unknown parameters, finds 3 pairs not altogether
The feature point coordinates of the same name of line can obtain unknown parameter.
7. a kind of airborne laser point cloud based on image registration according to claim 6 and Image registration fusion method, its
It is characterised by, the process that the affine Transform Model is simplified is as follows:
Unknown parameter is restricted, rotation Pan and Zoom, order is only considered
a11=a22=k cos θ, a12=-a21=k sin θs, b1=kc1,b2=kc2, affine Transform Model is simplified to RST and become
Mold changing type, is expressed as follows:
<mrow>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<msub>
<mi>x</mi>
<mi>B</mi>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
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<mi>y</mi>
<mi>B</mi>
</msub>
</mtd>
</mtr>
<mtr>
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<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>=</mo>
<mi>k</mi>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mrow>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mi>&theta;</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>s</mi>
<mi>i</mi>
<mi>n</mi>
<mi>&theta;</mi>
</mrow>
</mtd>
<mtd>
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<mi>c</mi>
<mn>1</mn>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
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<mo>-</mo>
<mi>s</mi>
<mi>i</mi>
<mi>n</mi>
<mi>&theta;</mi>
</mrow>
</mtd>
<mtd>
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<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mi>&theta;</mi>
</mrow>
</mtd>
<mtd>
<msub>
<mi>c</mi>
<mn>2</mn>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mfenced>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<msub>
<mi>x</mi>
<mi>A</mi>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>y</mi>
<mi>A</mi>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
Wherein k represents scaling, and θ represents the anglec of rotation between image, [c1,c2]TThe translation vector before scaling is represented, after simplifying
RST transformation models have 4 unknown parameters, result can be calculated by finding 2 pairs of characteristic points of the same name.
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