A fundamental matrix solution and calibration method for synchronous cameras based on a moving sphere

Abstract

The present invention provides a method for solving and calibrating the basic matrix of a synchronous camera based on a moving sphere. Use multiple cameras to be calibrated to synchronously shoot a single moving sphere from different angles, obtain multiple synchronous videos, and then use a circle detection algorithm to automatically detect the sphere in each frame image and obtain the pixel coordinates of its center in the image, and then obtain a series of 2D corresponding point coordinates, and then calculate the basic matrix in the epipolar constraint based on this, and combine the camera's internal parameter calculation to obtain the camera's projection matrix. The advantages of this method are: easy to use, high degree of automation, accurate calibration results, and can be used in a variety of outdoor environments.

Description

Technical Field

[0001] The invention belongs to the technical field of camera calibration, and relates to a basic matrix solving and calibration method of a synchronous camera based on a moving sphere. Background Art

[0002] The significance of camera calibration is: if the two-dimensional pixel coordinates of a three-dimensional point (three-dimensional coordinates are unknown) in three-dimensional space in multiple views are obtained, the three-dimensional coordinates of the point can be calculated according to the camera's projection matrix. This is a three-dimensional reconstruction process based on the principle of multi-view geometry.

[0003] The key to multi-view 3D reconstruction is to accurately estimate the projection matrices of multiple synchronized cameras. To do this, it is necessary to find several pairs of corresponding points in the synchronized views. Based on the principle of epipolar geometry, taking two cameras as an example, the eight-point method can be used to calculate the basic matrix, that is: find several pairs (no less than eight pairs) of 2D corresponding points in the two views, and then establish a linear equation system to find the least squares solution of the basic matrix. Then, the corresponding essential matrix is ​​calculated based on the known camera intrinsic parameter matrix, and finally the essential matrix is ​​decomposed to obtain the required two camera projection matrices. The accuracy of the estimated camera projection matrix determines the accuracy of subsequent 3D reconstruction.

[0004] In order to detect several groups of 2D corresponding points in the images of two perspectives for calculating the basic matrix in the epipolar constraint, a common method is to detect SIFT feature points in the image and then match them in the images of the two perspectives so that each group of 2D corresponding points corresponds to the same 3D point in the three-dimensional space. The disadvantage of the method of detecting SIFT feature points in the image is that thousands of SIFT feature points are often automatically detected in a picture, but they contain many mismatched points. It is necessary to use algorithms such as RANSAC to eliminate mismatched noise points, which makes the overall calculation very large and the whole process is relatively time-consuming. In addition, since most of the detected SIFT feature points come from the background in the image, the method of detecting SIFT features is also subject to the background environment. For different actual backgrounds in the image, the results are often not stable enough.

[0005] In addition, Chinese patent ZL200720154429 provides a detachable three-dimensional calibration framework, which provides a total of 8 calibration rods, each of which is fixed with 3 or 4 white balls as marking points, and the diameter of the balls is 34mm. When in use, all calibration rods can be fixed on the same base, and after assembly, they are radially distributed, providing a total of 26 white balls as marking points in three-dimensional space. If there is no overlap between the marking balls in the pictures taken by the two cameras, the three-dimensional calibration framework can provide 26 pairs of 2D corresponding points in one frame of the image. However, this scheme has the following limitations: on the one hand, the 3D calibration framework is often expensive, which hinders relevant practitioners from conveniently conducting relevant experiments and research; on the other hand, the marker balls fixed on the calibration rods in the 3D calibration framework are generally small, and it is difficult to automatically calculate the pixel coordinates of the marker balls in the captured images using relevant algorithms such as circle detection, and it is not easy to accurately match the corresponding marker balls in the two views. This deficiency is particularly prominent when shooting outdoors at a distance from the 3D calibration framework, so it is usually necessary to use manpower to mark the positions of the marker balls in the image, which is time-consuming and laborious; in addition, the 3D calibration framework has high requirements for use conditions, which makes it very limited in outdoor scenes. And the spatial range of the camera calibrated by the 3D calibration framework is limited, and the reconstruction results will have a large deviation when it exceeds its effective range. In the reconstruction of large scenes, especially the human body moving in a large range in the air (such as the aerial movements of skiers), the calibration frame only occupies a small part of the entire space. It is inaccurate to reconstruct the structure of the entire image only through a small concentrated area of ​​points in the image. Therefore, if the object to be reconstructed is far away from the calibration frame, the reconstruction ratio will be out of balance. Summary of the invention

[0006] To solve the above technical problems, the present invention provides a basic matrix solution and calibration method for a synchronous camera based on a moving sphere, which takes the center of the sphere as a reference point and can freely and conveniently change the position of the sphere in the three-dimensional scene space, so that the multiple groups of 2D corresponding points provided by the center of the sphere in multiple views can more fully characterize the three-dimensional space where the shooting scene is located, thereby improving the accuracy of the basic matrix solution and further improving the accuracy of the camera calibration.

[0007] The present invention provides a method for solving and calibrating a fundamental matrix of a synchronous camera based on a moving sphere, comprising:

[0008] Step 1: Fix multiple cameras in the shooting scene and shoot videos synchronously;

[0009] Step 2: Calibrate the camera's intrinsic parameter matrix offline or online;

[0010] Step 3: Use multiple cameras to synchronously shoot a single moving sphere to obtain multiple sets of images at different times;

[0011] Step 4: Use the Daugman circle detection algorithm to automatically detect the coordinates of the center of the circle of the sphere in each set of multiple images to obtain a set of 2D corresponding points. Multiple sets of images at different times obtain multiple sets of 2D corresponding points.

[0012] Step 5: Calculate the basic matrix that satisfies the epipolar constraint based on multiple sets of 2D corresponding points;

[0013] Step 6: Calculate the intrinsic matrix based on the intrinsic parameters of the camera, decompose the intrinsic matrix to get the extrinsic matrix of the two cameras, and calculate the projection matrix of the camera based on the intrinsic parameter matrix of the camera to complete the calibration.

[0014] In the basic matrix solution and calibration method of the synchronous camera based on the moving sphere of the present invention, the step 3 is specifically as follows:

[0015] Step 3.1: When shooting, first choose the color and size of the sphere according to the size of the venue, the distance from the camera, and the focal length;

[0016] Step 3.2: Freely control the movement range and direction of the sphere, and control the balance between the frame rate and movement speed of the video.

[0017] In the basic matrix solving and calibration method of the synchronous camera based on the moving sphere of the present invention, the step 4 is specifically as follows:

[0018] Step 4.1: Use multiple cameras to synchronously shoot the moving sphere, and obtain a set of multiple images of the sphere at each moment;

[0019] Step 4.2: Use the Daugman circle detection algorithm to automatically detect the circle of the sphere in the image and obtain the pixel coordinates or sub-pixel coordinates of the center of the circle;

[0020] Step 4.3: The 2D coordinates of the 3D point of the center of the sphere in the images of multiple cameras are obtained. The 2D coordinates in the images of multiple cameras at the same time constitute a set of 2D corresponding points;

[0021] Step 4.4: Using a series of grouped images at different times, multiple groups of 2D corresponding points are obtained.

[0022] In the basic matrix solving and calibration method of the synchronous camera based on the moving sphere of the present invention, the step 5 is specifically as follows:

[0023] Step 5.1: Establish a set of linear equations corresponding to 2D points according to the epipolar constraints;

[0024] Step 5.2: Establish a linear equation system through multiple groups of 2D points, use SVD decomposition to find the least squares solution of the linear equation system, and obtain the basic matrix.

[0025] In the method for solving and calibrating the basic matrix of a synchronous camera based on a moving sphere of the present invention, step 5.1 is specifically as follows:

[0026] Step 5.1.1: In two-view geometry, the fundamental matrix F represents the transformation from a point in an image to its epipolar line. Suppose the fundamental matrix F is a 3×3 matrix with 9 parameters and 8 degrees of freedom:

[0027] Step 5.1.2: Select two images from each set of images taken by multiple cameras, let x be the 2D corresponding point on the first image, l' be the epipolar line on the second image, then l' = F x; let x' be the 2D corresponding point on the second image, since x' is on the epipolar line l', the epipolar constraint expression is obtained:

[0028] x' T ·F·x=0

[0029] Step 5.1.3: Let the homogeneous coordinates of x be The homogeneous coordinates of the corresponding point x' are Substituting x, x' and the basic matrix F into the epipolar constraint expression and simplifying it, we obtain the following expression:

[0030]

[0031] Where, f = [F 11 ,F 12 ,F 13 ,F 21 ,F 22 ,F 23 ,F 31 ,F 32 ,F 33 ] T .

[0032] In the method for solving and calibrating the basic matrix of a synchronous camera based on a moving sphere of the present invention, step 5.2 is specifically as follows:

[0033] The above epipolar constraint expression can be obtained through two 2D corresponding points in a set of images. Multiple sets of 2D corresponding points can obtain a unique solution when they differ by a scale factor. When there are more than 8 sets of 2D corresponding points, SVD decomposition is used to find the least squares solution. The solution of f is the singular vector corresponding to the minimum singular value of the coefficient matrix, that is, A=UDV -1 The last column vector of the matrix V in .

[0034] In the basic matrix solving and calibration method of the synchronous camera based on the moving sphere of the present invention, the step 6 is specifically as follows:

[0035] Step 6.1: Calculate the intrinsic matrix by combining the camera intrinsic parameters and the basic matrix:

[0036]

[0037] Among them, K1 and K2 are the internal parameter matrices of two cameras selected from multiple cameras, F is the basic matrix, and E is the essential matrix;

[0038] Step 6.2: Decompose the essential matrix to obtain the extrinsic matrix of the two cameras;

[0039] Two of the three singular values ​​of the essential matrix are equal, and the third singular value is 0. The singular value decomposition of the essential matrix yields:

[0040] E=U·diag(1,1,0)·V T

[0041] Without considering the sign, there are two possible solutions:

[0042] S1=UZU T , R1=UWV T

[0043] S2=-UZU T , R1=UW T V T

[0044] in:

[0045]

[0046] Therefore, there are four cases of external parameter matrices: P1 = [R1|S1], P2 = [R1|S2], P3 = [[R2|S1] and P4 = [R2|S2];

[0047] Step 6.3: Calculate the coordinates of the three-dimensional points according to the four situations respectively. The group with the largest number of points with positive depth is the required external parameter matrix.

[0048] The method for solving and calibrating the fundamental matrix of a synchronous camera based on a moving sphere of the present invention has at least the following beneficial effects:

[0049] 1. In the method of the present invention, the center of a sphere is used as a reference point, and the position of the sphere in the three-dimensional scene space can be freely and conveniently changed, so that the multiple groups of 2D corresponding points provided by the center of the sphere in multiple views can more fully characterize the three-dimensional space where the shooting scene is located, thereby improving the accuracy of solving the basic matrix, and further improving the accuracy of camera calibration.

[0050] 2. The technical solution proposed by the present invention is easy to use and widely applicable, and uses circle detection algorithms such as Daugman to automatically detect spheres in images, with a high degree of automation.

[0051] 3. The technical solution proposed in the present invention has a small amount of calculation and does not require additional computing resources to eliminate noise points in feature points and perform matching calculations on 2D corresponding points. At the same time, it is not restricted by the shooting background of the camera and has stronger robustness to the shooting environment.

[0052] 4. The present invention overcomes the defects of the existing SIFT feature point method, such as many noise points, large amount of calculation, and being restricted by the shooting background. It also overcomes the defects of the three-dimensional calibration framework method, such as high cost, the need to manually mark 2D corresponding points, difficulty in matching, and easy damage. Compared with SIFT feature point matching, the obtained matching point pairs are more accurate, require less calculation, and can be applied to a variety of shooting backgrounds. Compared with the method using the three-dimensional calibration framework, the method of the present invention overcomes the limitation of its spatial range, and can obtain corresponding points of any part of the entire image, and is easy to operate, and the automatic detection of corresponding points is more accurate. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 It is a flow chart of a method for solving and calibrating a basic matrix of a synchronous camera based on a moving sphere of the present invention;

[0054] Figure 2 It is a schematic diagram of using multiple cameras to synchronously capture a single moving sphere. DETAILED DESCRIPTION

[0055] The present invention innovatively uses a moving sphere in the image. In practice, a basketball, volleyball, etc. can be selected as a calibration object to provide multiple sets of corresponding points in the views of more than two synchronous cameras, calculate the basic matrix of the multi-view geometry that satisfies the epipolar constraint, and then calculate the camera's projection matrix under the condition that the camera's intrinsic parameters are known to achieve camera calibration.

[0056] like Figure 1 As shown, a method for solving and calibrating a basic matrix of a synchronous camera based on a moving sphere of the present invention comprises:

[0057] Step 1: Fix multiple cameras in the shooting scene and shoot videos synchronously;

[0058] Step 2: Calibrate the camera's intrinsic parameter matrix offline or online;

[0059] Step 3: Use multiple cameras to synchronously shoot a single moving sphere to obtain multiple sets of images at different times, such as Figure 2 As shown, specifically:

[0060] Step 3.1: When shooting, first select the color and size of the sphere according to the shooting conditions such as the size of the venue, the distance from the camera, and the focal length;

[0061] Step 3.2: Freely control the range and direction of the ball's movement, but consider the balance between the frame rate of the video and the speed of the ball's movement. Too low a frame rate or too fast a ball speed will result in blurred imaging.

[0062] Step 4: Use Daugman and other circle detection algorithms to automatically detect the center coordinates of the circle of the sphere in each set of multiple images to obtain a set of 2D corresponding points. Multiple sets of 2D corresponding points are obtained for multiple sets of images at different times.

[0063] Theoretically, the sphere appears as a standard circle in the image. In the multiple sets of images obtained at different times, the center of the sphere is used as the observation point. Although the center of the sphere is invisible inside the sphere in practice, it is still a point in the three-dimensional world coordinate system, and its 2D projection point in the image is the center of the circle in the image. That is, for the same three-dimensional point - the center of the sphere, in multiple images of the synchronized camera, there is a set of 2D corresponding points - the pixel coordinates of the center of the circle of the sphere in the image, which are exactly the 2D corresponding points required to solve the basic matrix in the epipolar constraint.

[0064] In the multiple sets of images obtained at different times, the position of the center of the sphere in the three-dimensional world coordinate system is variable and can be controlled arbitrarily. Therefore, using the multiple sets of images obtained, any multiple sets of 2D corresponding points can be obtained. These 2D corresponding points are generated by the centers of the spheres distributed at different positions in the three-dimensional world coordinate system in the images of the synchronized camera. Based on these 2D corresponding points, not only can the basic matrix in the two-view epipolar constraint be solved, but the position of the sphere can also be conveniently adjusted so that the generated 2D corresponding points can more fully represent the three-dimensional space where the shooting scene is located, which is conducive to improving the accuracy of solving the basic matrix.

[0065] In order to automatically and quickly detect the pixel coordinates of the center of the circle presented by the sphere in the image, a circle detection algorithm similar to Daugman is used in the multiple sets of images obtained by shooting, and the circle presented by the sphere in each image is automatically detected and the pixel (or sub-pixel) coordinates of the center of the circle are obtained. The 2D pixel coordinates corresponding to the 3D point of the center of the sphere in the images of multiple cameras are obtained. Since the camera is fixed during the shooting process, its internal and external parameters remain unchanged. The internal and external parameters of the camera calculated using the 2D corresponding points provided by the images of different frames are consistent, so a set of 2D corresponding points can be detected in each set of multiple images. Therefore, any number of sets of 2D corresponding points can be obtained based on multiple sets of images for the calculation of the basic matrix. Based on the above analysis, step 4 is specifically as follows:

[0066] Step 4.1: Use multiple cameras to synchronously shoot the moving sphere, and obtain a set of multiple images of the sphere at each moment;

[0067] Step 4.2: Use Daugman and other circle detection algorithms to automatically detect the circle of the sphere in the image and obtain the pixel coordinates or sub-pixel coordinates of the center of the circle;

[0068] Step 4.3: The 2D coordinates of the 3D point of the center of the sphere in the images of multiple cameras are obtained. The 2D coordinates in the images of multiple cameras at the same time constitute a set of 2D corresponding points;

[0069] Step 4.4: Using a series of grouped images at different times, multiple groups of 2D corresponding points are obtained.

[0070] Step 5: Calculate the basic matrix that satisfies the epipolar constraint based on multiple sets of 2D corresponding points, specifically:

[0071] Step 5.1: Establish a set of linear equations corresponding to 2D points according to the epipolar constraints;

[0072] Step 5.1.1: In two-view geometry, the fundamental matrix F represents the transformation from a point in an image to its epipolar line. Suppose the fundamental matrix F is a 3×3 matrix with 9 parameters and 8 degrees of freedom:

[0073]

[0074] Step 5.1.2: Select two images from each set of images taken by multiple cameras, let x be the 2D corresponding point on the first image, l' be the epipolar line on the second image, then l' = F x; let x' be the 2D corresponding point on the second image, since x' is on the epipolar line l', the epipolar constraint expression is obtained:

[0075] x' T ·F·x=0

[0076] Step 5.1.3: Let the homogeneous coordinates of x be The homogeneous coordinates of the corresponding point x' are Substituting x, x' and the basic matrix F into the epipolar constraint expression and simplifying it, we obtain the following expression:

[0077]

[0078] Where, f = [F 11 ,F 12 ,F 13 ,F 21 ,F 22 ,F 23 ,F 31 ,F 32 ,F33 ] T .

[0079] Step 5.2: Establish a linear equation system through multiple groups of 2D points, use SVD decomposition to find the least squares solution of the linear equation system, and obtain the basic matrix;

[0080] The above epipolar constraint expression can be obtained through two 2D corresponding points in a set of images. Multiple sets of 2D corresponding points can obtain a unique solution when they differ by a scale factor. When there are more than 8 sets of 2D corresponding points, SVD decomposition is used to find the least squares solution. The solution of f is the singular vector corresponding to the minimum singular value of the coefficient matrix, that is, A=UDV -1 The last column vector of the matrix V in .

[0081] Step 6: Calculate the intrinsic matrix based on the intrinsic parameters of the camera, decompose the intrinsic matrix to get the extrinsic matrix of the two cameras, and calculate the projection matrix of the camera based on the intrinsic matrix of the camera to complete the calibration. Specifically:

[0082] Step 6.1: Calculate the intrinsic matrix by combining the camera intrinsic parameters and the basic matrix:

[0083]

[0084] Among them, K1 and K2 are the internal parameter matrices of two cameras selected from multiple cameras, F is the basic matrix, and E is the essential matrix;

[0085] Step 6.2: Decompose the essential matrix to obtain the extrinsic matrix of the two cameras;

[0086] Two of the three singular values ​​of the essential matrix are equal, and the third singular value is 0. The singular value decomposition of the essential matrix yields:

[0087] E=U·diag(1,1,0)·V T

[0088] Without considering the sign, there are two possible solutions:

[0089] S1=UZU T , R1=UWV T

[0090] S2=-UZU T , R1=UW T V T

[0091] in:

[0092]

[0093] Therefore, there are four cases of external parameter matrices: P1 = [R1|S1], P2 = [R1|S2], P3 = [[R2|S1] and P4 = [R2|S2];

[0094] Step 6.3: Calculate the coordinates of the three-dimensional points according to the four situations respectively. The group with the largest number of points with positive depth is the required external parameter matrix.

[0095] The present invention uses a standard sphere moving in a sequence of images as a calibration object for a synchronous camera, and uses a circle detection algorithm such as Daugman to automatically detect the 2D pixel coordinates corresponding to the center of the sphere in the image, obtain multiple groups of 2D corresponding points provided by the center of the sphere, and then calculate the basic matrix, the essential matrix and the projection matrix of the camera based on the 2D corresponding points, thereby realizing the calibration of the camera. The key point of the present invention is to use a sphere as a calibration object, specifically using the center of the sphere as a reference point, and at the same time being able to freely and conveniently transform the position of the sphere in the three-dimensional scene space, so that the multiple groups of 2D corresponding points provided by the center of the sphere in multiple views can more fully characterize the three-dimensional space where the shooting scene is located, thereby improving the accuracy of solving the basic matrix, and then improving the accuracy of camera calibration.

[0096] The above description is only a preferred embodiment of the present invention and is not intended to limit the concept of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims

1. A method for solving and calibrating the fundamental matrix of a synchronous camera based on a moving sphere, characterized in that: include: Step 1: Fix multiple cameras in the shooting scene and shoot videos synchronously; Step 2: Calibrate the camera's intrinsic parameter matrix offline or online; Step 3: Use multiple cameras to synchronously shoot a single moving sphere to obtain multiple sets of images at different times; Step 4: Use the Daugman circle detection algorithm to automatically detect the coordinates of the center of the circle of the sphere in each set of multiple images to obtain a set of 2D corresponding points. Multiple sets of images at different times obtain multiple sets of 2D corresponding points. Step 5: Calculate the basic matrix that satisfies the epipolar constraint based on multiple sets of 2D corresponding points; Step 6: Calculate the intrinsic matrix based on the intrinsic parameters of the camera, decompose the intrinsic matrix to get the extrinsic matrix of the two cameras, and calculate the projection matrix of the camera based on the intrinsic matrix of the camera to complete the calibration; The step 4 is specifically as follows: Step 4.1: Use multiple cameras to synchronously shoot the moving sphere, and obtain a set of multiple images of the sphere at each moment; Step 4.2: Use the Daugman circle detection algorithm to automatically detect the circle of the sphere in the image and obtain the pixel coordinates or sub-pixel coordinates of the center of the circle; Step 4.3: The 2D coordinates of the 3D point of the center of the sphere in the images of multiple cameras are obtained. The 2D coordinates in the images of multiple cameras at the same time constitute a set of 2D corresponding points; Step 4.4: Using a series of grouped images at different times, obtain multiple groups of 2D corresponding points; The step 5 is specifically as follows: Step 5.1: Establish a set of linear equations corresponding to 2D points according to the epipolar constraints; Step 5.2: Establish a linear equation system through multiple groups of 2D points, use SVD decomposition to find the least squares solution of the linear equation system, and obtain the basic matrix; The step 6 is specifically as follows: Step 6.1: Calculate the intrinsic matrix by combining the camera intrinsic parameters and the basic matrix: Among them, K1 and K2 are the internal parameter matrices of two cameras selected from multiple cameras, F is the basic matrix, and E is the essential matrix; Step 6.2: Decompose the essential matrix to obtain the extrinsic matrix of the two cameras; Two of the three singular values ​​of the essential matrix are equal, and the third singular value is 0. The singular value decomposition of the essential matrix yields: E=U·diag(1,1,0)·V T Without considering the sign, there are two possible solutions: S1=UZU T ,R1=UWV T S2=-UZU T ,R1=UW T V T in: Therefore, there are four cases of external parameter matrices: P1 = [R1|S1], P2 = [R1|S2], P3 = [[R2|S1] and P4 = [R2|S2]; Step 6.3: Calculate the coordinates of the three-dimensional points according to the four situations respectively. The group with the largest number of points with positive depth is the required external parameter matrix.

2. The method for solving and calibrating the fundamental matrix of a synchronous camera based on a moving sphere as claimed in claim 1, characterized in that: The step 3 is specifically as follows: Step 3.1: When shooting, first choose the color and size of the sphere according to the size of the venue, the distance from the camera, and the focal length; Step 3.2: Freely control the movement range and direction of the sphere, and control the balance between the frame rate and movement speed of the video.

3. The method for solving and calibrating the fundamental matrix of a synchronous camera based on a moving sphere as claimed in claim 1, characterized in that: The step 5.1 is specifically as follows: Step 5.1.1: In two-view geometry, the fundamental matrix F represents the transformation from a point in an image to its epipolar line. Suppose the fundamental matrix F is a 3×3 matrix with 9 parameters and 8 degrees of freedom: Step 5.1.2: Select two images from each set of images taken by multiple cameras, let x be the 2D corresponding point on the first image, l' be the epipolar line on the second image, then l' = F x; let x' be the 2D corresponding point on the second image, since x' is on the epipolar line l', the epipolar constraint expression is obtained: x' T ·F·x=0 Step 5.1.3: Let the homogeneous coordinates of x be The homogeneous coordinates of the corresponding point x' are Substituting x, x' and the basic matrix F into the epipolar constraint expression and simplifying it, we obtain the following expression: where f = [F 11 , F 12 , F 13 , F 21 , F 22 , F 23 , F 31 , F 32 , F 33 T .​ 4. The method for solving and calibrating the fundamental matrix of a synchronous camera based on a moving sphere as claimed in claim 1, characterized in that: The step 5.2 is specifically as follows: The above epipolar constraint expression can be obtained through two 2D corresponding points in a set of images. Multiple sets of 2D corresponding points can obtain a unique solution when they differ by a scale factor. When there are more than 8 sets of 2D corresponding points, SVD decomposition is used to find the least squares solution. The solution of f is the singular vector corresponding to the minimum singular value of the coefficient matrix, that is, A=UDV -1 The last column vector of the matrix V in .

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