A calibration method for telecentric linear array cameras for aircraft engine measurement based on enhanced dynamic imaging model

Abstract

A calibration method for a telecentric linear array camera for measuring an aircraft engine based on an enhanced dynamic imaging model belongs to the field of precision measurement and instrument technology. The method comprises the following steps: adjusting a transparent glass checkerboard calibration plate (3) to different positions and collecting images; determining the projection point coordinates of the corner points of the transparent glass checkerboard calibration plate (3) in a pixel coordinate system and the coordinates in a world coordinate system; combining the enhanced dynamic imaging model of the telecentric linear array camera that ignores the inclination of a guide rail (5) and the distortion of a telecentric lens (7), and estimating the initial parameters of the telecentric linear array camera based on direct linear transformation; based on the enhanced dynamic imaging model of the telecentric linear array camera and the lens distortion model when the guide rail (5) is considered to be tilted, a nonlinear optimization algorithm is used to optimize the parameters to obtain the optimal calibration result. The present invention solves the problem of inaccurate nonlinear optimization initial values directly obtained from camera and lens manuals, thereby improving the calibration accuracy of the telecentric linear array camera.

Description

Technical Field

[0001] The present invention belongs to the field of precision measurement and instrument technology, and in particular relates to a calibration method for a telecentric linear array camera for aircraft engine measurement based on an enhanced dynamic imaging model. Background Art

[0002] The development of a high-precision, rapid, and comprehensive measuring instrument for low-axis form and position errors in aircraft engines is a pressing need in the development, production, and quality control of new aircraft engine models. The inherent characteristics of aircraft engine shafting make contact measurement methods, as well as non-contact laser and ultrasonic methods, difficult to meet these requirements. Telecentric line array camera measurement technology overcomes the shortcomings of manual inspection, which is inefficient and subjectivist, while significantly improving inspection efficiency and measurement accuracy while reducing labor requirements. It offers numerous advantages, including speed, high precision, traceability, non-contact operation, and strong anti-interference capabilities. Furthermore, it can measure structures that are difficult to measure with traditional measurement methods, such as fillets and slots.

[0003] Due to the unique properties of line scan cameras and telecentric lenses, telecentric lenses are widely used in conjunction with line scan cameras in many machine vision applications. Theoretically, the height of the image captured by a line scan camera can be infinite, making it suitable for structures with large aspect ratios. Regarding lens selection, telecentric lenses can significantly reduce image distortion and perspective errors, a property that is particularly important when measuring non-planar objects. In telecentric line scan camera measurement, system calibration plays a crucial role in determining the coordinate transformation from the image plane to the world plane and eliminating image distortion. However, line scan cameras adhere to a specific model, making calibration methods designed for area scan cameras unsuitable for determining their parameters. Furthermore, unlike perspective lenses, telecentric lenses achieve parallel projection of objects onto the image plane, requiring different calibration methods. Therefore, developing appropriate calibration methods for telecentric line scan cameras is crucial for achieving precise positioning and measurement in world space.

[0004] The paper (Steger C and Ulrich M. A camera model for line-scan cameras with telecentric lenses [J]. Int. J. Comput. Vis. 129 (2020) 80-99.) proposes an imaging model specifically for line-scan cameras equipped with telecentric lenses, and also proposes an algorithm for calibrating telecentric line-scan cameras using planar calibration objects. This method requires calibrating the velocity in three orthogonal directions in space. However, since the camera scanning speed and line frequency are known, only the inclination angle of the actual motion direction relative to the ideal motion direction needs to be calibrated. In addition, in Steger's paper, the initial values for nonlinear optimization are obtained directly from the camera and lens manuals. The inaccurate initial values lead to low calibration accuracy.

[0005] Patent CN117928401A, "Machine Vision-Based Polymer Monofilament Diameter Detection Device and Method," proposes a machine vision-based polymer monofilament diameter detection device and method. The detection device includes two CCD linear array cameras, two telecentric lenses, two highly uniform strip light sources, a rectangular soft backlight, a grating ruler, a controller, and image processing software. The detection method includes preprocessing the linear array cameras, grayscale conversion of the acquired raw image, radiometric calibration, smoothing filtering, image noise removal, threshold segmentation, contour feature extraction, contour geometry fitting, and extraction of polymer monofilament diameter based on pixel points between contours. This invention enables intelligent online detection of large quantities of polymer monofilaments. However, the linear array cameras equipped with telecentric lenses were not calibrated, resulting in inaccurate measurement parameters for the telecentric lenses and linear array cameras.

[0006] A comprehensive analysis of the above methods reveals that existing telecentric line scan camera calibration methods still have significant limitations. The imaging model of telecentric line scan cameras is complex, and the initial values for the nonlinear optimization of telecentric line scan camera parameters are directly obtained from the camera and lens manuals. These initial values are not accurately calculated, resulting in low calibration accuracy. Summary of the Invention

[0007] In response to the problems existing in the prior art, the present invention discloses a calibration method for a telecentric linear array camera for aircraft engine measurement based on an enhanced dynamic imaging model. The method uses a transparent glass checkerboard calibration plate as a calibration object, uses a six-degree-of-freedom fixture to adjust the transparent glass checkerboard calibration plate to different positions, and collects images. Using a telecentric linear array camera enhanced dynamic imaging model that ignores the guide rail tilt and telecentric lens distortion, the initial parameters of the telecentric lens and linear array camera are estimated based on direct linear transformation. After determining the initial estimated value of each parameter, based on the telecentric linear array camera enhanced dynamic imaging model and lens distortion model considering the guide rail tilt, a nonlinear optimization algorithm is used to optimize the parameters to obtain the optimal calibration result. Based on the telecentric linear array camera enhanced dynamic imaging model, the present invention determines the initial values of the telecentric linear array camera parameters through direct linear transformation, combines the linear transformation method with the subsequent nonlinear refinement stage, and realizes accurate calibration of the telecentric linear array camera.

[0008] The technical solution of the present invention is:

[0009] A calibration method for a telecentric linear array camera for aircraft engine measurement based on an enhanced dynamic imaging model comprises the following steps:

[0010] 1) First, a transparent glass checkerboard calibration plate is fixed to a multi-degree-of-freedom fixture. The multi-degree-of-freedom fixture is adjusted to place the transparent glass checkerboard calibration plate within the imaging range of the line scan camera. A telecentric backlight source projects a light beam onto the surface of the transparent glass checkerboard calibration plate. The telecentric lens forms an image in the line scan camera. The telecentric backlight source, telecentric lens, and line scan camera simultaneously scan up and down along a guide rail to capture an image of the transparent glass checkerboard calibration plate.

[0011] 2) Adjust the multi-degree-of-freedom fixture to place the transparent glass checkerboard calibration plate in different positions, capture the image of the transparent glass checkerboard calibration plate at each position, and use the sub-pixel corner detection method to locate the projection point coordinate p of the corner point P of the transparent glass checkerboard calibration plate in the pixel coordinate system o-uv p (x p ,y p ), use the distance between adjacent corner points of the transparent glass checkerboard calibration plate to determine the corner point P of the transparent glass checkerboard calibration plate in the world coordinate system O w -X w Y w Z w Coordinate P in w (x w ,y w ,z w );

[0012] 3) The initial parameters of the telecentric lens and the linear array camera are estimated based on direct linear transformation, ignoring the tilt of the guide rail and the distortion of the telecentric lens. The corner point P of the transparent glass checkerboard calibration plate is in the world coordinate system O. w -X w Y w Z w Coordinate P in w (x w ,y w ,z w ) and the projection point coordinates p in the pixel coordinate system o-uv p (x p ,y p ), that is, the enhanced dynamic imaging model of the telecentric line scan camera is as follows:

[0013]

[0014] Where m is the magnification of the lens, dx is the pixel size in the u direction of the pixel coordinate system o-uv, dy is the pixel size in the v direction of the pixel coordinate system o-uv, u0 represents the abscissa of the origin of the image coordinate system in the pixel coordinate system o-uv, v0 represents the ordinate of the origin of the image coordinate system in the pixel coordinate system o-uv, and θ represents the motion direction S in the camera coordinate system O c -X c Y c Z c Plane O c -X c Y c The projection direction S′ on the axis Y c The angle between From the world coordinate system O w -X w Y w Z w To the camera coordinate system O c -X c Y c Z c The 3×3 rotation matrix, T=[t x t y t z ] T , is from the world coordinate system O w -X w Y w Z w To the camera coordinate system O c -X c Y c Z c The translation vector of

[0015] Using a transparent glass checkerboard calibration plate for calibration, the simplified telecentric line scan camera enhanced dynamic imaging model is:

[0016]

[0017] Use the homography matrix H′ to express the corner point P of the transparent glass checkerboard calibration plate in the world coordinate system O w -X w Y w Z w Coordinate P in w (x w ,y w ,z w ) and the projection point coordinates p in the pixel coordinate system o-uv p (x p ,y p ) are as follows:

[0018]

[0019] Where, The relationship between each element of the matrix H′ and the camera parameters is as follows:

[0020]

[0021] The homogeneous linear equations for all elements in the homography matrix H′ are as follows:

[0022]

[0023] Where h T =(h 11 ,h 12 ,h 13 ,h 21 ,h 22 ,h 23 ), is a vector containing all the elements of H′, based on the corner point P of the transparent glass checkerboard calibration plate in the world coordinate system O w -X w Y w Z w Coordinate P in w (x w ,y w ,z w ) and the projection point coordinates p in the pixel coordinate system o-uv p (x p ,y p ) Calculate vector h;

[0024] After calculating the vector h, determine the initial value of dy according to the system parameters, and determine the magnification m according to the following expression:

[0025]

[0026] Select a non-negative root as the initial value of m, the initial value of u0 is half the number of pixels of the linear array camera, and the initial value of v0 is 0;

[0027] The rotation matrix R and translation vector T are calculated by the following expression:

[0028]

[0029] Subscript i represents the i-th image of the transparent glass checkerboard calibration plate. By translating the calibration plate, a marker point Q is determined in the world coordinate system O. w -X w Y w Z w The three-dimensional coordinates Q w (x′ w ,y′ w ,z′ w ), according to the marker point Q in the world coordinate system O w -X w Y w Z w The three-dimensional coordinates Q w (x′ w ,y′ w ,z′ w ) and the projection point coordinate q in the pixel coordinate system o-uv p (x p ′,y p ′) coordinate transformation relationship to determine r 13 and r 23 The positive and negative of , combined with the property 1 of the orthogonal matrix: Determine r 13 and r 23 , then use the property 2 of the orthogonal matrix: r3 = r1 × r2 to determine the third row vector r3 of the rotation matrix, and finally restore the rotation matrix R;

[0030] 4) After determining the initial estimated value of each parameter, a nonlinear optimization algorithm is used to optimize the parameters based on the enhanced dynamic imaging model and lens distortion model of the telecentric line scan camera considering the inclination of the guide rail to obtain the optimal calibration result.

[0031] In step 3), the method for determining the initial value of dy according to the system parameters is to determine the camera scanning speed h v and the camera line frequency h l , according to dy=h v / h l Determine the initial value of dy.

[0032] In step 3), according to the marked point Q in the world coordinate system O w -X w Y w Z w The three-dimensional coordinates Q w (x′ w ,y′ w ,z′ w ) and the projection point coordinate q in the pixel coordinate system o-uv p (x p ′,y p ′) coordinate transformation relationship to determine r 13 and r 23 The steps for positive and negative are as follows:

[0033] Mark point Q in world coordinate system O w -X w Y w Z w The three-dimensional coordinates Q w (x′ w ,y′ w ,z′ w ) and the projection point coordinate q in the pixel coordinate system o-uv p (x p ′,y p The coordinate transformation relationship expression of ′) is as follows:

[0034]

[0035] Inferred:

[0036]

[0037] Thus, we can determine r 13 and r 23 positive and negative.

[0038] The enhanced dynamic imaging model of the telecentric line scan camera considering the inclination of the guide rail in step 4) is:

[0039] BRIEF DESCRIPTION OF THE DRAWINGS

[0040] Figure 1 This is a schematic diagram of the calibration device for aero-engine telecentric line scan cameras.

[0041] Figure 2 Schematic diagram of the enhanced dynamic imaging model for a telecentric line scan camera;

[0042] Figure 3 Schematic diagram of the guide rail movement direction;

[0043] Figure 4is the transparent glass checkerboard calibration plate image;

[0044] In the figure: 1. Telecentric backlight source; 2. Light beam; 3. Transparent glass checkerboard calibration plate; 4. Six-degree-of-freedom fixture; 5. Guide rail; 6. Linear array camera; 7. Telecentric lens.

[0045] The present invention has the following characteristics and beneficial effects:

[0046] 1. This invention combines a linear transformation method with a subsequent nonlinear refinement stage. Initial parameter estimation is performed based on a telecentric line array camera enhanced dynamic imaging model. Then, parameter refinement is performed through nonlinear optimization, achieving accurate calibration of the telecentric line array camera. This avoids the problem of inaccurate parameters caused by the inability to calibrate telecentric line array cameras.

[0047] 2. Based on the enhanced dynamic imaging model of telecentric line array cameras, this invention accurately and uniquely determines the initial values for nonlinear optimization of telecentric line array camera parameters through direct linear transformation, thereby improving the accuracy of nonlinear optimization of telecentric line array camera parameters. This avoids the problem of low calibration accuracy caused by inaccurate initial values for nonlinear optimization obtained directly from camera and lens manuals.

[0048] The invention has a wide range of uses, and is particularly suitable for dynamic measurement of aircraft engines based on a telecentric linear array camera. DETAILED DESCRIPTION

[0049] The embodiments of the present invention are described in detail below with reference to the accompanying drawings.

[0050] 1. A calibration method for a telecentric linear array camera for aircraft engine measurement based on an enhanced dynamic imaging model, comprising the following steps:

[0051] 1) Using a transparent glass checkerboard calibration plate with a 1mm spacing between adjacent corner points and 48×48 corner points, the telecentric line array camera for aircraft engine measurement was calibrated. The transparent glass checkerboard calibration plate 3 was fixed to a multi-degree-of-freedom fixture 4. The multi-degree-of-freedom fixture 4 was adjusted to place the transparent glass checkerboard calibration plate 3 within the imaging range of the line array camera 6. A telecentric backlight source 1 projected a light beam 2 onto the surface of the transparent glass checkerboard calibration plate 3. The light beam was formed in the line array camera 6 through a telecentric lens 7. The telecentric backlight source 1, the telecentric lens 7, and the line array camera 6 simultaneously scanned up and down along a guide rail 5 to capture an image of the transparent glass checkerboard calibration plate 3.

[0052] 2) Adjust the multi-degree-of-freedom fixture 4 to place the transparent glass checkerboard calibration plate 3 in different positions. Capture 8 images of the transparent glass checkerboard calibration plate 3 at each position, with 2209 corner points in total. Use the sub-pixel corner point detection method to locate the projection point coordinates p of the corner point P of the transparent glass checkerboard calibration plate 3 in the pixel coordinate system o-uv p (x p,y p ), the distance between adjacent corner points of the transparent glass checkerboard calibration plate 3 is used to determine the corner point P of the transparent glass checkerboard calibration plate 3 in the world coordinate system O w -X w Y w Z w Coordinate P in w (x w ,y w ,z w );

[0053] 3) The initial parameters of the telecentric lens (7) and the linear array camera (6) are estimated based on direct linear transformation, and the tilt of the guide rail (5) and the distortion of the telecentric lens (7) are ignored. The corner point P of the transparent glass checkerboard calibration plate (3) is located in the world coordinate system O. w -X w Y w Z w Coordinate P in w (x w ,y w ,z w ) and the projection point coordinates p in the pixel coordinate system o-uv p (x p ,y p ), that is, the enhanced dynamic imaging model of the telecentric line scan camera is as follows:

[0054]

[0055] Where m is the magnification of the lens, dx is the pixel size in the u direction of the pixel coordinate system o-uv, dy is the pixel size in the v direction of the pixel coordinate system o-uv, u0 represents the abscissa of the origin of the image coordinate system in the pixel coordinate system o-uv, v0 represents the ordinate of the origin of the image coordinate system in the pixel coordinate system o-uv, and θ represents the motion direction S in the camera coordinate system O c -X c Y c Z c Plane O c -X c Y c The projection direction S′ on the axis Y c The angle between From the world coordinate system O w -X w Y w Z w To the camera coordinate system O c -X c Y c Z c The 3×3 rotation matrix, T=[t x ty t z ] T , is from the world coordinate system O w -X w Y w Z w To the camera coordinate system O c -X c Y c Z c The translation vector of

[0056] Using the transparent glass checkerboard calibration plate (3) for calibration, the simplified telecentric line scan camera enhanced dynamic imaging model is:

[0057]

[0058] Use the homography matrix H′ to express the corner point P of the transparent glass checkerboard calibration plate (3) in the world coordinate system O w -X w Y w Z w Coordinate P in w (x w ,y w ,z w ) and the projection point coordinates p in the pixel coordinate system o-uv p (x p ,y p ) are as follows:

[0059]

[0060] Where, The relationship between each element of the matrix H′ and the camera parameters is as follows:

[0061]

[0062] The homogeneous linear equations for all elements in the homography matrix H′ are as follows:

[0063]

[0064] Where h T =(h 11 ,h 12 ,h 13 ,h 21 ,h 22 ,h 23 ), is a vector containing all the elements of H′, according to the corner point P of the transparent glass checkerboard calibration plate (3) in the world coordinate system O w -X w Y w Zw Coordinate P in w (x w ,y w ,z w ) and the projection point coordinates p in the pixel coordinate system o-uv p (x p ,y p ) Calculate vector h;

[0065] After calculating the vector h, the camera scanning speed h is determined v and the camera line frequency h l , according to dy=h v / h l Determine the initial value of dy and determine the magnification m according to the following expression:

[0066]

[0067] The non-negative root is selected as the initial value of m, the initial value of u0 is half of the number of pixels of the linear array camera (6), and the initial value of v0 is 0;

[0068] The rotation matrix R and translation vector T are calculated by the following expression:

[0069]

[0070] The subscript i represents the i-th image of the transparent glass checkerboard calibration plate (3). By translating the calibration plate, a marker point Q is determined in the world coordinate system O. w -X w Y w Z w The three-dimensional coordinates Q w (x′ w ,y′ w ,z′ w ), the marker point Q is in the world coordinate system O w -X w Y w Z w The three-dimensional coordinates Q w (x′ w ,y′ w ,z′ w ) and the projection point coordinate q in the pixel coordinate system o-uv p (x p ′,y p The coordinate transformation relationship expression of ′) is as follows:

[0071]

[0072] Inferred:

[0073]

[0074] Thus, we can determine r 13 and r 23 The positive and negative of , combined with the property 1 of the orthogonal matrix: Determine r 13 and r 23 , then use the property 2 of the orthogonal matrix: r3 = r1 × r2 to determine the third row vector r3 of the rotation matrix, and finally restore the rotation matrix R;

[0075] 4) The enhanced dynamic imaging model of the telecentric line scan camera considering the inclination of the guide rail 5 is:

[0076]

[0077] After determining the initial estimated value of each parameter, a nonlinear optimization algorithm is used to optimize the parameters based on the enhanced dynamic imaging model of the telecentric linear array scanning camera and the lens distortion model when the guide rail (5) is tilted, and the optimal calibration result is obtained. The average reprojection error is 0.7112 pixels and the standard deviation is 0.0161 pixels.

Claims

1. A calibration method for a telecentric linear array camera for aircraft engine measurement based on an enhanced dynamic imaging model, characterized by: The method comprises the following steps: 1) First, a transparent glass checkerboard calibration plate (3) is fixed on a multi-degree-of-freedom fixture (4), and the multi-degree-of-freedom fixture (4) is adjusted to place the transparent glass checkerboard calibration plate (3) within the imaging range of a linear array camera (6). A telecentric backlight source (1) projects a light beam (2) onto the surface of the transparent glass checkerboard calibration plate (3), and an image is formed in the linear array camera (6) through a telecentric lens (7). The telecentric backlight source (1), the telecentric lens (7) and the linear array camera (6) simultaneously scan up and down along a guide rail (5) to achieve image acquisition of the transparent glass checkerboard calibration plate (3); 2) Adjust the multi-degree-of-freedom fixture (4) to place the transparent glass checkerboard calibration plate (3) at different positions, capture the image of the transparent glass checkerboard calibration plate (3) at each position, and use the sub-pixel corner detection method to locate the projection point coordinate p of the corner point P of the transparent glass checkerboard calibration plate (3) in the pixel coordinate system o-uv p (x p ,y p ), using the distance between adjacent corner points of the transparent glass checkerboard calibration plate (3) to determine the corner point P of the transparent glass checkerboard calibration plate (3) in the world coordinate system O w -X w Y w Z w Coordinate P in w (x w ,y w ,z w ); 3) The initial parameters of the telecentric lens (7) and the linear array camera (6) are estimated based on direct linear transformation, and the tilt of the guide rail (5) and the distortion of the telecentric lens (7) are ignored. The corner point P of the transparent glass checkerboard calibration plate (3) is located in the world coordinate system O. w -X w Y w Z w Coordinate P in w (x w ,y w ,z w ) and the projection point coordinates p in the pixel coordinate system o-uv p (x p ,y p ), that is, the enhanced dynamic imaging model of the telecentric line scan camera is as follows: Where m is the magnification of the lens, dx is the pixel size in the u direction of the pixel coordinate system o-uv, dy is the pixel size in the v direction of the pixel coordinate system o-uv, u0 represents the abscissa of the origin of the image coordinate system in the pixel coordinate system o-uv, and v0 represents the ordinate of the origin of the image coordinate system in the pixel coordinate system o-uv. From the world coordinate system O w -X w Y w Z w To the camera coordinate system O c -X c Y c Z c The 3×3 rotation matrix, T=[t x t y t z ] T , is from the world coordinate system O w -X w Y w Z w To the camera coordinate system O c -X c Y c Z c The translation vector of Using the transparent glass checkerboard calibration plate (3) for calibration, the simplified telecentric line scan camera enhanced dynamic imaging model is: Use the homography matrix H′ to express the corner point P of the transparent glass checkerboard calibration plate (3) in the world coordinate system O w -X w Y w Z w Coordinate P in w (x w ,y w ,z w ) and the projection point coordinates p in the pixel coordinate system o-uv p (x p ,y p ) are as follows: Where, The relationship between each element of the matrix H′ and the camera parameters is as follows: The homogeneous linear equations for all elements in the homography matrix H′ are as follows: Where h T =(h 11 ,h 12 ,h 13 ,h 21 ,h 22 ,h 23 ), is a vector containing all the elements of H′, according to the corner point P of the transparent glass checkerboard calibration plate (3) in the world coordinate system O w -X w Y w Z w Coordinate P in w (x w ,y w ,z w ) and the projection point coordinates p in the pixel coordinate system o-uv p (x p ,y p ) Calculate vector h; After calculating the vector h, determine the initial value of dy according to the system parameters, and determine the magnification m according to the following expression: The non-negative root is selected as the initial value of m, the initial value of u0 is half of the number of pixels of the linear array camera (6), and the initial value of v0 is 0; The rotation matrix R and translation vector T are calculated by the following expression: The subscript i represents the i-th image of the transparent glass checkerboard calibration plate (3). By translating the calibration plate, a marker point Q is determined in the world coordinate system O. w -X w Y w Z w The three-dimensional coordinates Q w (x′ w ,y′ w ,z′ w ), according to the marker point Q in the world coordinate system O w -X w Y w Z w The three-dimensional coordinates Q w (x′ w ,y′ w ,z′ w ) and the projection point coordinate q in the pixel coordinate system o-uv p (x p ′,y p ′) coordinate transformation relationship to determine r 13 and r 23 The positive and negative of , combined with the property 1 of the orthogonal matrix: Determine r 13 and r 23 , then use the property 2 of the orthogonal matrix: r3 = r1 × r2 to determine the third row vector r3 of the rotation matrix, and finally restore the rotation matrix R; 4) The enhanced dynamic imaging model of the telecentric linear array scanning camera considering the inclination of the guide rail (5) is: Where θ represents the motion direction S in the camera coordinate system O c -X c Y c Z c Plane O c -X c Y c The projection direction S′ on the axis Y c The angle between After determining the initial estimated value of each parameter, a nonlinear optimization algorithm is used to optimize the parameters based on the enhanced dynamic imaging model of the telecentric linear array scanning camera and the lens distortion model when the guide rail (5) is tilted to obtain the optimal calibration result.

2. The method for calibrating a telecentric linear array camera for aircraft engine measurement based on an enhanced dynamic imaging model according to claim 1, characterized in that: The method for determining the initial value of dy according to the system parameters in step 3) is to determine the camera scanning speed h v and the camera line frequency h l , according to dy=h v / h l Determine the initial value of dy.

3. The method for calibrating a telecentric linear array camera for aircraft engine measurement based on an enhanced dynamic imaging model according to claim 1, characterized in that: In the step 3), the marker point Q is located in the world coordinate system O w -X w Y w Z w The three-dimensional coordinates Q w (x′ w ,y′ w ,z′ w ) and the projection point coordinate q in the pixel coordinate system o-uv p (x p ′,y p ′) coordinate transformation relationship to determine r 13 and r 23 The steps for positive and negative are as follows: Mark point Q in world coordinate system O w -X w Y w Z w The three-dimensional coordinates Q w (x′ w ,y′ w ,z′ w ) and the projection point coordinate q in the pixel coordinate system o-uv p (x p ′,y p The coordinate transformation relationship expression of ′) is as follows: Inferred: Thus, we can determine r 13 and r 23 positive and negative.

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