CN114565679B - Focal length, radial distortion and attitude calibration method based on camera position - Google Patents
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Abstract
The invention provides a method for calibrating focal length, radial distortion and attitude based on a camera position, and belongs to the technical fields of machine vision, photogrammetry, SLAM and the like. Three mark points are distributed, the positions of a camera and the positions of the three mark points are measured, the image of the mark points is shot to obtain the imaging corresponding relation between the camera and the mark points, an iteration solving method for focal length and multi-order distortion is established by utilizing the important geometrical phenomenon that the azimuth between the camera and the main point of the camera is not changed after radial distortion, further, the image is subjected to distortion correction, and finally, the imaging position of the mark points corresponding to the undistorted image is used for calibrating the pose of the camera. Compared with the existing calibration method, the method has the advantages of reducing the number of the mark points, shortening the calibration time, having unique solutions, being capable of adopting various radial distortion models and the like.
Description
Technical Field
The invention relates to the technical fields of machine vision, photogrammetry, SLAM (slit-level image acquisition) and the like, in particular to a camera position-based focal length, radial distortion and attitude calibration method.
Background
In the related fields of machine vision, photogrammetry, SLAM and the like, the internal and external parameters such as distortion, focal length, position, attitude and the like of a camera are required to be calibrated for measurement and estimation. There are many internal and external parameter calibration methods at present, such as An external parameter calibration method [ An accurato (n) solution to the pnp problem ] which is known for the internal parameters of the camera or can adopt ideal internal parameters; an internal and external parameter calibration method [ Camera pose and calibration from 4or 5known 3d points ] for partial camera internal parameters known; an inside and outside parameter calibration method known for camera part pose [ Closed-form solutions to minimal absolute pose problems with known vertical direction ]. With the development of positioning technology, positioning equipment is gradually miniaturized, cheaper and the precision meets engineering requirements, and is widely applied to camera positioning to acquire the position of a camera. In addition, in machine vision, in order to enlarge the field of view of a camera, a short-focus or fisheye lens is often adopted, but the type of lens has the defects of serious distortion and the like, and if the influence of the distortion is not considered, great errors are brought to measurement and estimation. The invention aims at the condition that the position of a camera is known, the focal length is unknown and the lens has serious distortion, and calibrates the camera, including parameters such as the focal length, radial distortion, gesture and the like.
Currently, under the condition of no priori knowledge, the number of mark points required for calibrating parameters such as the focal length, radial distortion, attitude and the like of a camera is required to be more than or equal to 4. Document [ Pose Estimation WITH RADIAL Distortion and Unknown Focal Length ] proposes a focal length, radial distortion and gesture calibration method based on 4 mark points, in order to simplify the calibration process, a division radial distortion model is adopted, and only one radial distortion coefficient can be estimated; the document [ Real-time solution to the absolute pose problem with unknown radial distortion and focal length ] proposes a focal length, radial distortion and gesture calibration method based on 5 mark points, and in order to simplify the calibration process, a division radial distortion model is adopted, and the number of radial distortion coefficient estimation can reach 3; the above method has a polynary phenomenon. It can be seen that, to achieve focal length, radial distortion and pose calibration, not less than 4 marker points are required, and to simplify the calibration process, a division radial distortion model is often adopted, and a multi-solution phenomenon is provided, so that additional constraint conditions are required to obtain a unique solution. The acquisition and maintenance of accurate mark points need to consume a large amount of manpower and material resources, for example, when the attitude of an aircraft is measured in a field strong wind environment, a plurality of mark points need to be set up by calibrating a camera in advance, and the field strong wind environment brings serious challenges to the establishment of the mark points.
Disclosure of Invention
The invention aims to provide a camera position-based focal length, radial distortion and gesture calibration method, which solves the technical problems that only one distortion model is adopted in the existing calibration method, more than or equal to 4 mark points are needed when the focal length, distortion and gesture of a camera are solved at the same time, and part of the methods have multiple solutions.
In order to achieve the above purpose and solve the above technical problems, the technical scheme of the invention is as follows:
A method for calibrating focal length, radial distortion and attitude based on camera position is characterized by comprising the following steps:
Let the camera position point C (x 0 y0 z0) be known, the three marker points P1(x1 y1 z1)、P2(x2 y2 z2)、P3(x3 y3z3) be known, and the outer marker points have the following relationship with the camera imaging position under no distortion: p 1、p2、p3 is the imaging position of P 1、P2、P3 in the camera under no distortion conditions; p c is the main point of the camera and is positioned in the center of the image; oc is the camera position;
Step 1, establishing a nonlinear equation set for iteratively solving intermediate variables
Selecting a division distortion model and a traditional distortion model;
the division distortion model is as follows
The conventional distortion model is as follows
Wherein k i is the radial distortion coefficient,The geometry under radial distortion is as follows: angle P 1Ocp2,∠p2Ocp3,∠p3Ocp1 is denoted as α 1,β1,γ1, and since the positions of P i and Oc are known, the angle sizes of the three angles are calculated by triangle Δp 1OcP2,ΔP2OcP3,ΔP3OcP1; meanwhile, angle p 1pcp2,∠p2pcp3,∠p3pcp1 is denoted as α 2,β2,γ2 respectively; principal point p c and distortion point/>It is known that α 2,β2,γ2 passes through triangle/>Acquiring;
Let p cpi＝xi,Ocpi＝yi be the triangle Δp 1pcp2,Δp1Ocp2 with common edge p 1p2 by cosine law
Where O cpc = f, which is the focal length, is perpendicular to the plane p 1p2p3, and is therefore derived from the triangle Δo cpcpi
By taking formula (4) into formula (3), the following can be obtained
Similarly, two other equations can be obtained
Is provided withThen a system of equations with three unknowns is obtained
Solving a solution f i of the nonlinear equation set by adopting an iteration method;
Step 2, calculating focal length and distortion coefficient
2.1 Distortion model for division
The two parameter models are as follows:
Wherein k j (j=1, 2) is the radial distortion coefficient, Is the distance from the distortion point to the main point, and further obtains
Wherein,Is a known quantity; a system of polynomial equations is obtained:
Considering f, k 1f,k2 f as three unknowns to the above system of equations, then a linear system of equations can be derived as follows:
A1X1＝Y1 (10)
Wherein the method comprises the steps of
The solution of the equation set isThe focal length and distortion coefficients are calculated as follows:
2.2 for the traditional distortion model
The two parameter models are as follows:
Similarly, a system of linear equations is obtained as follows:
A2X2＝Y2 (14)
Wherein the method comprises the steps of
The focal length and distortion coefficients are obtained by linear solution of equation (16):
step 3, calculating the camera gesture
And (3) according to the distortion coefficient and the corresponding distortion model obtained in the step (2), obtaining the position of an imaging point after distortion correction, wherein the corresponding relation of the 2D-3D point after distortion correction is as follows: let O c-XcYcZc be the camera coordinate system and O w-XwYwZw be the world coordinate system; calculating a camera pose, namely a rotation matrix R w_c and a translation vector T w_c, by adopting any two mark points and camera positions;
Defining a new camera coordinate system O c-Xc2Yc2Zc2 and a new world coordinate system O c-Xw2Yw2Zw2; the new camera coordinate system is defined as follows:
wherein, The known quantity is obtained after the first step of solving; by definition, under a new camera coordinate system, the X c2 axis is vector/>The Z c2 axis is perpendicular to the plane O cp1p2,Yc2 axis and is defined according to the right-hand coordinate system; the coordinates of the point P c in the original camera coordinate system O c-XcYcZc are converted to the coordinates P c2 in the new camera coordinate system O c-Xw2Yw2Zw2 by:
The new world coordinate system is defined as follows:
Under the new world coordinate system, O c is the origin of the coordinate system and the X w2 axis is the vector according to the definition The Z w2 axis is perpendicular to the plane O cP1P2,Yw2 axis and is defined according to the right-hand coordinate system; the coordinates of point P w in the original world coordinate system O c-XwYwZw can be converted to coordinates P w2 in the new world coordinate system O c-Xw2Yw2Zw2 by:
A new camera coordinate system and a new world coordinate system, assuming that a point P c under the original camera coordinate system and a point P w under the original world coordinate system are the same point, and obtaining a conversion relation under each coordinate system according to the definition of the new camera coordinate system and the new world coordinate system;
The rotation matrix R w_c and the translation vector T w_c are calculated as follows:
so far, the camera gesture is calculated.
Further, in the step 1, f i is solved by using an LM method; the value of f i under the condition of no distortion is selected as an initial value of iterative calculation, the initial value is given by a traditional PnP algorithm, and the solution f i of the nonlinear equation can be solved by a small number of iterations because the initial value is close to a true value.
Compared with the prior art, the invention has the following effective benefits:
1. The method solves the technical problems that in the existing calibration method, more than or equal to 4 mark points are needed when the focal length, distortion and gesture of a camera are solved simultaneously, and part of the method has multiple solutions and only adopts a certain distortion model.
2. The method provided by the invention can simultaneously solve and obtain the focal length, distortion and gesture of the camera by using 3 mark points and known camera positions, has unique solutions and can be suitable for various distortion models.
3. The method is suitable for application scenes with the advantages of camera position acquisition in advance, serious distortion, unknown radial distortion model and small number of available mark points.
Drawings
FIG. 1 is a schematic diagram of correspondence between undistorted 3D points and camera imaging points;
FIG. 2 is a schematic diagram of camera imaging geometry under radial distortion;
FIG. 3 is a schematic diagram of the correspondence between 2D and 3D points after distortion correction;
FIG. 4 is a diagram illustrating the transformation relationship of different coordinate systems.
Detailed Description
The invention is explained and illustrated in detail below with reference to the drawings and the examples.
The invention provides a camera focal length, distortion and gesture calibration method based on a camera position and three mark points, which only needs to lay the three mark points, measure the camera position and the three mark point positions, shoot mark point images to obtain the imaging corresponding relation between the camera and the mark points, and establish an iterative focal length and 4-order distortion solving method by utilizing the important geometrical phenomenon that the azimuth between the radial distortion and the main point of the camera is not changed, thereby carrying out distortion correction on images, and finally calibrating the camera gesture by utilizing the imaging position of the mark point corresponding to the undistorted images. Compared with the existing calibration method, the method has the advantages that the number of the mark points is reduced, the calibration time is shortened, the unique solution is provided, various radial distortion models can be adopted, and the like.
When the camera position point C (x 0 y0 z0) is known, the three marker points P1(x1 y1 z1)、P2(x2 y2 z2)、P3(x3 y3z3) are known, and the specific relationship between the external marker point and the imaging position of the camera under the distortion-free condition is shown in fig. 1.
Where P 1、p2、p3 is the imaging position of P 1、P2、P3 in the camera under no distortion conditions. p c is the main point of the camera and is positioned in the center of the image; oc is the camera position. In practical application, due to the existence of distortion, only the distorted 2D imaging point can be obtainedWhereas the undistorted 2D imaging point p i is not available. The object of the present invention is to utilize 3D landmark P i (i=1, 2, 3) and distorted 2D imaging point/>To calibrate the focal length, distortion and pose of the camera.
In order to simplify the process of solving the technical problem, the calibration problem is decomposed into two sub-problems to be solved, so that the solution is more efficient: the first problem is to solve for focal length and distortion parameters, and the second problem is to solve for out-of-camera parameters.
For the first problem, an important geometrical phenomenon that the azimuth between the radial distortion and the main point of the camera is not changed is utilized, a method for iteratively solving the focal length and 4-order distortion is established, and in the iterative process, an intermediate variable without considering distortion is utilized as an initial value, and the initial value is relatively close to a true value, so that even if the method needs iteration, the initial value is very good, the global optimal solution can be quickly obtained; in addition, the method can adapt to various distortion models, and the applicability of the method is enlarged. And after the distortion and the focal length are obtained by solving, obtaining a corrected image, and linearly solving the external parameters of the camera by utilizing geometric constraints.
In the second problem, distortion correction is carried out on the image by utilizing the distortion parameters obtained in the first problem to obtain an undistorted image; and then calibrating the camera gesture by using the camera position, the known mark point and the corresponding imaging point.
The specific implementation steps of the invention are as follows:
Step 1, establishing a nonlinear equation set for iteratively solving intermediate variables
Currently, there are many distortion models, and for most digital cameras, the main distortion is radial distortion, with division distortion models and conventional distortion models being the most widely used.
The division distortion model is as follows
The conventional distortion model is as follows
Wherein k i is the radial distortion coefficient,The radial distortion is mainly affected by the first two terms, so the first two terms are analyzed in the present invention. Of course, as the number of marker points increases, the number of distortion coefficients analyzed also increases. It can be seen that in either distortion model, under distortion, the direction between the imaging point and the center of the distortion (i.e., the principal point of the camera) does not change, but only the distance. The invention calculates the focal length and distortion of the camera by utilizing the important and key geometric feature that the direction between an imaging point and the center of the distortion is not changed under radial distortion. According to this feature, the geometry under radial distortion is shown in fig. 2.
In fig. 2, < P 1Ocp2,∠p2Ocp3,∠p3Ocp1 > is denoted as a 1,β1,γ1, respectively, and since the P i and Oc positions are known, the angular magnitudes of these three angles can be calculated from the triangle Δp 1OcP2,ΔP2OcP3,ΔP3OcP1 in fig. 1. Meanwhile, < p 1pcp2,∠p2pcp3,∠p3pcp1 > is denoted as α 2,β2,γ2, respectively. Here, principal point p c and distortion pointThe known, but undistorted point p i is not known. However, since the direction between the imaging point and the principal point does not change before and after distortion, α 2,β2,γ2 can pass through the triangle/> And (5) calculating to obtain the product. In the subsequent calculation process, only the distance between the undistorted point and the distortion center is involved, and no distortion coefficient is involved, so the method does not directly calculate the distortion coefficient.
Let p cpi＝xi,Ocpi＝yi be the result of the cosine law because triangle Δp 1pcp2,Δp1Ocp2 has a common edge p 1p2
In fig. 2, O cpc = f, which is the focal length, perpendicular to the plane p 1p2p3, and therefore, from the triangle Δo cpcpi, one can obtain
By taking formula (4) into formula (3), the following can be obtained
Similarly, two other equations can be obtained
Is provided withThen a system of equations with three unknowns is obtained
The system of nonlinear equations is then solved using an iterative method, such as the LM (Levenberg-Marquardt) method. For a nonlinear equation set, the speed of iterating the initial value of the difference to the global optimal solution is low, even the local optimal solution can be obtained only by solving, the global optimal solution can not be obtained, and the good initial value can improve the calculation speed and can obtain the global optimal solution, so the method needs the good initial value. Therefore, in order to obtain a globally optimal solution, it is important to select the initial value of the system of non-linear equations. Here, an initial value of f i under the condition of no distortion is adopted, which can be given by a traditional PnP algorithm, and since the initial value is close to a true value, the solution f i of the nonlinear equation can be solved through a small number of iterations.
Step 2, calculating focal length and distortion coefficient
After the value of f i is obtained, the focal length and the radial distortion coefficient can be obtained according to different radial distortion models. Because the commonly used radial distortion model is a division distortion model and a traditional distortion model, the method disclosed by the invention is used for respectively expanding and describing the calculation methods under the two models.
2.1 Distortion model for division
The two parameter models are as follows:
Wherein k j (j=1, 2) is the radial distortion coefficient, Is the distance from the distortion point to the main point, and then is obtained
Here the number of the elements is the number,Is a known quantity; thus, a polynomial equation set can be obtained as follows.
To achieve a linear solution, f, k 1f,k2 f is considered as three unknowns of the above equation set. A system of linear equations can then be derived as follows:
A1X1＝Y1 (31)
Wherein the method comprises the steps of
The solution of the equation set isThen, the focal length and distortion coefficient are calculated as follows:
2.2 conventional distortion model. The two parameter models are as follows:
similarly, a system of linear equations can be obtained as follows:
A2X2＝Y2 (35)
Wherein the method comprises the steps of
Finally, the focal length and distortion coefficients can be directly linearly solved:
so far, under two distortion models, the focal length and two distortion coefficients are obtained through three mark points and camera positions.
Step 3, calculating the camera gesture
And (3) obtaining the position of the imaging point after distortion correction according to the distortion coefficient obtained in the step (1) and the corresponding distortion model. The corresponding relationship of the 2D-3D points after distortion correction is shown in figure 3.
In fig. 3, O c-XcYcZc is a camera coordinate system, and O w-XwYwZw is a world coordinate system. Any two landmark points and camera positions are used to calculate the camera pose, i.e. the rotation matrix R w_c and the translation vector T w_c.
To calculate the camera pose, a new camera coordinate system O c-Xc2Yc2Zc2 and a new world coordinate system O c-Xw2Yw2Zw2 are defined; the new camera coordinate system is defined as follows:
wherein, The known quantity is obtained after the first step of solving. By definition, under a new camera coordinate system, the X c2 axis is vector/>The Z c2 axis is defined perpendicular to the plane O cp1p2,Yc2 axis according to the right-hand coordinate system. The coordinates of point P c in the original camera coordinate system O c-XcYcZc can be converted to the coordinates P c2 in the new camera coordinate system O c-Xw2Yw2Zw2 by:
The new world coordinate system is defined as follows:
Under the new world coordinate system, O c is the origin of the coordinate system and the X w2 axis is the vector according to the definition The Z w2 axis is defined perpendicular to the plane O cP1P2,Yw2 axis according to the right-hand coordinate system. The coordinates of point P w in the original world coordinate system O c-XwYwZw can be converted to coordinates P w2 in the new world coordinate system O c-Xw2Yw2Zw2 by:
Obviously, the new camera coordinate system coincides with the new world coordinate system. Assuming that the point P c in the original camera coordinate system and the point P w in the original world coordinate system are the same point, the conversion relationship in each coordinate system can be obtained according to the definition of the new camera coordinate system and the new world coordinate system as shown in fig. 4.
Thus, the rotation matrix R w_c and the translation vector T w_c are calculated as follows:
so far, the camera gesture is calculated.
Example 1
Three landmark coordinates P1 (-9.31, 13.50, 200.00), P2 (22.86,0.06, 200.00), P3 (22.58,6.38, 230.00), camera position Oc (0, 0) are given below. Setting the resolution of the camera to 1280 multiplied by 800, the pixel size to 14 mu m, and adopting a division distortion model, wherein two radial distortion coefficients are respectively-0.01, -0.02 and the focal length is 50mm; the other camera internal parameters are known, and the theoretical external parameter rotation matrix of the camera is set as follows:
To approximate reality, an error of 0.2 pixels is added to the 2D imaging points of each 3D marker point. By adopting the method, the focal length is 50.0385mm, the distortion coefficients are-0.0101 and 0.0201, and the external parameter rotation matrix measurement result is as follows:
from the calculation results, it can be seen that the errors of the focal length, the distortion coefficient and the external parameters are small. In order to better analyze the calculation accuracy of the method, the calculation is carried out on the re-projection errors of each point, and the average re-projection error is 0.12 pixel, which indicates that the method has high calculation accuracy.
Claims (2)
1. A method for calibrating focal length, radial distortion and attitude based on camera position is characterized by comprising the following steps:
Let the camera position point C (x 0 y0 z0) be known, the three marker points P1(x1 y1 z1)、P2(x2 y2 z2)、P3(x3 y3 z3) be known, and the outer marker points have the following relationship with the camera imaging position under no distortion: p 1、p2、p3 is the imaging position of P 1、P2、P3 in the camera under no distortion conditions; p c is the main point of the camera and is positioned in the center of the image; oc is the camera position;
Step 1, establishing a nonlinear equation set for iteratively solving intermediate variables
Selecting a division distortion model and a traditional distortion model;
the division distortion model is as follows
The conventional distortion model is as follows
Wherein k i is the radial distortion coefficient,The geometry under radial distortion is as follows: angle P 1Ocp2,∠p2Ocp3,∠p3Ocp1 is denoted as α 1,β1,γ1, and since the positions of P i and Oc are known, the angle sizes of the three angles are calculated by triangle Δp 1OcP2,ΔP2OcP3,ΔP3OcP1; meanwhile, angle p 1pcp2,∠p2pcp3,∠p3pcp1 is denoted as α 2,β2,γ2 respectively; principal point p c and distortion point/>It is known that α 2,β2,γ2 passes through triangle/>Acquiring;
Let p cpi＝xi,Ocpi＝yi be the triangle Δp 1pcp2,Δp1Ocp2 with common edge p 1p2 by cosine law
Where O cpc = f, which is the focal length, is perpendicular to the plane p 1p2p3, and is therefore derived from the triangle Δo cpcpi
By taking formula (4) into formula (3), the following can be obtained
Similarly, two other equations can be obtained
Is provided withThen a system of equations with three unknowns is obtained
Solving a solution f i of the nonlinear equation set by adopting an iteration method;
Step 2, calculating focal length and distortion coefficient
2.1 Distortion model for division
The two parameter models are as follows:
Wherein k j (j=1, 2) is the radial distortion coefficient, Is the distance from the distortion point to the main point, and further obtains
Wherein,Is a known quantity; a system of polynomial equations is obtained:
Considering f, k 1f,k2 f as three unknowns to the above system of equations, then a linear system of equations can be derived as follows:
A1X1＝Y1 (10)
Wherein the method comprises the steps of
The solution of the equation set isThe focal length and distortion coefficients are calculated as follows:
2.2 for the traditional distortion model
The two parameter models are as follows:
Similarly, a system of linear equations is obtained as follows:
A2X2＝Y2 (14)
Wherein the method comprises the steps of
The focal length and distortion coefficients are obtained by linear solution of equation (16):
step 3, calculating the camera gesture
And (3) according to the distortion coefficient and the corresponding distortion model obtained in the step (2), obtaining the position of an imaging point after distortion correction, wherein the corresponding relation of the 2D-3D point after distortion correction is as follows: let O c-XcYcZc be the camera coordinate system and O w-XwYwZw be the world coordinate system; calculating a camera pose, namely a rotation matrix R w_c and a translation vector T w_c, by adopting any two mark points and camera positions;
Defining a new camera coordinate system O c-Xc2Yc2Zc2 and a new world coordinate system O c-Xw2Yw2Zw2; the new camera coordinate system is defined as follows:
wherein, The known quantity is obtained after the first step of solving; by definition, under a new camera coordinate system, the X c2 axis is vector/>The Z c2 axis is perpendicular to the plane O cp1p2,Yc2 axis and is defined according to the right-hand coordinate system; the coordinates of the point P c in the original camera coordinate system O c-XcYcZc are converted to the coordinates P c2 in the new camera coordinate system O c-Xw2Yw2Zw2 by:
The new world coordinate system is defined as follows:
Under the new world coordinate system, O c is the origin of the coordinate system and the X w2 axis is the vector according to the definition The Z w2 axis is perpendicular to the plane O cP1P2,Yw2 axis and is defined according to the right-hand coordinate system; the coordinates of point P w in the original world coordinate system O c-XwYwZw can be converted to coordinates P w2 in the new world coordinate system O c-Xw2Yw2Zw2 by:
A new camera coordinate system and a new world coordinate system, assuming that a point P c under the original camera coordinate system and a point P w under the original world coordinate system are the same point, and obtaining a conversion relation under each coordinate system according to the definition of the new camera coordinate system and the new world coordinate system;
The rotation matrix R w_c and the translation vector T w_c are calculated as follows:
so far, the camera gesture is calculated.
2. The method for calibrating focal length, radial distortion and pose based on camera position according to claim 1, wherein f i in step 1 is solved by LM method; in order to increase the calculation speed and obtain the global optimal solution, the value of f i under the condition of no distortion is selected as the initial value of iterative calculation, the initial value is given by the traditional PnP algorithm, and the solution f i of the nonlinear equation can be solved by a small number of iterations because the initial value is close to the true value.
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| CN1719477A (en) * | 2005-05-19 | 2006-01-11 | 上海交通大学 | Calibration method of pick up camera or photographic camera geographic distortion |
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| CN111667536A (en) * | 2019-03-09 | 2020-09-15 | 华东交通大学 | Parameter calibration method based on zoom camera depth estimation |
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