CN113920201A - Fisheye camera calibration method with epipolar geometric constraints - Google Patents

Fisheye camera calibration method with epipolar geometric constraints Download PDF

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CN113920201A
CN113920201A CN202110746242.5A CN202110746242A CN113920201A CN 113920201 A CN113920201 A CN 113920201A CN 202110746242 A CN202110746242 A CN 202110746242A CN 113920201 A CN113920201 A CN 113920201A
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周国清
谢永繁
宋汝昊
王庆阳
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Guilin University of Technology
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Abstract

The invention discloses a fisheye camera calibration method based on epipolar geometric constraint, which comprises the following implementation steps: firstly, establishing an indoor three-dimensional calibration field, measuring the three-dimensional coordinates and fisheye image coordinates of artificial mark points, then establishing the relationship between fisheye image points and perspective projection image points through the relationship between fisheye cameras and projection functions of traditional optical cameras, and resolving the optimal projection transformation matrix and the basic matrix of the fisheye image points and the space three-dimensional coordinate points by using Levenberg-Marquardt (LM algorithm) to minimize errors; and then adding the radial distortion and tangential distortion model into an epipolar geometric constraint formula, solving radial and tangential distortion parameters by a Newton method, and correcting fisheye image points. And finally, calculating a projection transformation matrix and a basic matrix of the fisheye image points and the spatial three-dimensional coordinate points, and decomposing the projection transformation matrix to obtain the internal and external orientation elements. The method can calculate the internal parameters of the fisheye camera and improve the solving precision of the internal parameters.

Description

Polar line geometric constraint fisheye camera calibration method
Technical Field
The invention relates to the technical field of photogrammetry, in particular to a fisheye camera calibration method which is suitable for fisheye camera three-dimensional reconstruction work.
Background
In the development process of photogrammetry for more than 150 years, all theoretical bases (such as collinearity equations, adjustment by a beam method, differential orthorectification and the like) are 'ordinary' aerial cameras (such as frame cameras and linear array sensors) based on perspective (central) projection, and relevant aerial image projective measurement processing is relatively mature. However, such cameras are designed based on vision, the observation range is generally limited to 40-50 degrees, the whole information of the surrounding environment cannot be obtained, and partial targets are difficult to find, namely, the targets easily "escape" from the visual field and the observation range. Therefore, international proposals have been made for "omnidirectional vision", i.e., obtaining panoramic information of a large field of view of 180 ° or 360 ° at a time. In recent years, the research on such cameras (such as 180-degree fisheye cameras) is internationally keen, and the cameras are gradually applied to the fields of robot navigation, star exploration, panoramic monitoring, public security, pipeline detection, auxiliary driving, field detection, vehicle-mounted inspection, aircraft guidance, virtual reality and the like.
The existing fisheye camera calibration method can be roughly divided into three types:
(1) fisheye camera calibration based on camera models (e.g., Zhang, y., Zhao, l., & Hu, W. (2013). advance of catadioptic uniformity camera calibration. international Journal of Information Technology and Computer Science (ijtcs), 5(3), 13.). Although the method is reasonable in derivation, the distortion of the camera has various factors, so that a very strict mathematical model cannot be constructed, the relation between a space point and an image point is not established, the method belongs to self calibration, and the actual application precision has uncertainty.
(2) Fisheye camera calibration based on imaging geometry (e.g., Schwalbe, E. (2005, February). Geotric modeling and calibration of fish lens camera systems. In Proc.2nd Panoramic photo metrology works, int. archives of photo metrology and Remote Sensing (Vol.36, No. part 5, p.W8)). The method can calculate calibration parameters more correctly, but only one fisheye image is used for calibration, so that the information of a single image is incomplete, some important view geometric information is easy to miss, and the method is not beneficial to three-dimensional reconstruction.
(3) Fisheye camera calibration based on single view geometric constraints (e.g., Fitzgibbon, A.W. (2001, Decumber). Simultaneous linearity estimation of multiple view geometry and lens translation. in Proceedings of the 2001IEEE Computer Society Conference on Computer Vision and Pattern registration. CVPR2001(Vol.1, pp.I-I). IEEE.). Most of the methods are calibrated by using the geometric constraint of a single image, but the distortion of a fisheye camera is not caused by a single factor, some constraints of the single image are not strictly established, and the methods do not use a spatial three-dimensional coordinate, belong to self-calibration and have uncertainty in the precision of practical application. The fisheye camera calibration of view geometry constraints has great limitations.
Aiming at the limitations of the three fisheye camera calibration methods, the invention discloses a fisheye camera calibration method based on polar line geometric constraint.
Disclosure of Invention
The invention discloses a fisheye camera calibration method based on epipolar line geometric constraint. In order to achieve the above purpose, the present invention adopts the following technical scheme, including the following steps.
Step 1, establishing an indoor three-dimensional calibration field, carrying out coordinate measurement on artificial mark points in the calibration field, then carrying out shooting of multiple angles on the calibration field, and measuring fisheye image point coordinates of the artificial mark points on an image.
And 2, deducing the mathematical relationship between the fisheye image point and the perspective projection image point by using the relationship between the equidistant projection function of the fisheye camera and the perspective projection function of the traditional optical camera, and then deducing the mathematical relationship between the fisheye image point and the spatial three-dimensional coordinate point, wherein the process needs to determine the initial value of the internal orientation element. And finally, solving a projection transformation matrix to obtain a basic matrix.
The method for determining the initial value of the internal orientation element comprises the following steps: since the fisheye camera cannot be manufactured without error, the fisheye image appears as an ellipse of an approximate circle. According to the characteristic of the fisheye camera, the images are fitted into a curve, and noise is removed. The centers and radii of the two images are estimated from the curve equation. Using optical principles, an initial value of the focal length can be calculated.
The projective transformation matrix determining method comprises the following steps: firstly, obtaining a mathematical relation between a fisheye image point and a perspective projection image point and an initial value of an internal orientation element, then using a plurality of spatial three-dimensional coordinate points and fisheye image points corresponding to the spatial three-dimensional coordinate points to calculate an optimal projection transformation matrix based on a Direct Linear Transformation (DLT) method and an LM algorithm, and calculating an initial value of a basic matrix through a polar line geometric theory formula.
And 3, adding the distortion model into the polar line geometric constraint formula, and solving the distortion parameter of the fisheye camera by using a Newton method.
Distortion parameter determination method: after the distortion parameters are added into the polar line geometric constraint formula, the initial value of the basic matrix and the initial value of the internal orientation element are solved, so that the constraint formula only has unknown distortion parameters, and the distortion parameters can be obtained by solving a nonlinear equation set by a Newton method.
And 4, correcting the fisheye image points by using distortion, solving a projection transformation matrix, and decomposing the projection transformation matrix to obtain inner and outer orientation elements.
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FIG. 1 is a flow chart of the present invention
FIG. 2 is an indoor three-dimensional calibration field of the present invention
FIG. 3 is a schematic diagram of the distortion of the fisheye lens of the present invention
FIG. 4 is a schematic diagram of the imaging process of the fisheye camera of the present invention
FIG. 5 is a fisheye image of an indoor calibration field according to the invention
FIG. 6 is a fish-eye image segmentation chart of the present invention
FIG. 7 is a graph of a curve fit of the present invention to determine the internal orientation element
FIG. 8 is a schematic view of epipolar geometry constraint according to the present invention
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, preferred embodiments are described below in detail with reference to the accompanying drawings.
The steps of the present invention will be described in further detail with reference to the accompanying drawings and examples.
For calibration purposes, a coordinate system is first established:
(1)(XW,YW,ZW) Is a space coordinate system OW-XWYWZWCoordinates of any point in the space, OWIs the origin of the spatial coordinate system.
(2)(XS,YS,ZS) Is the camera coordinate system OS-XSYSZSCoordinates of any point in the space, OSIs the origin of the camera coordinate system.
(3) (x, y) fisheye image coordinate system Oxy-any point in xy, OxyIs the origin of the coordinate system of the fisheye image.
(4) (u, v) is the coordinates of any point in the fisheye image pixel coordinate system O-uv, and O is the origin of the fisheye image pixel coordinate system.
(5) (x ', y ') is a virtual perspective projection image coordinate system O 'xy-coordinates of any point in x 'y', Ox'yIs the origin of the virtual perspective projection image coordinate system.
(6) (u ', v') is an arbitrary point in the pixel coordinate system O '-u' v 'of the virtual perspective projection image, and O' is the origin of the pixel coordinate system of the virtual perspective projection image.
(7)Oxy(u0,v0) Is the center of the fisheye image and the virtual perspective projection image,
radial direction
Figure BDA0003144507500000041
(see fig. 3).
With reference to fig. 1, the steps of the fisheye camera calibration method with epipolar geometric constraint are described.
Step 1, establishing an indoor three-dimensional calibration field. And carrying out coordinate measurement on the artificial mark points in the calibration field, then carrying out shooting at a plurality of angles on the calibration field, and measuring the fisheye image point coordinates of the artificial mark points on the image.
The establishment of an indoor three-dimensional calibration field is described in connection with fig. 2. The calibration field is 6.5m long, 1.5m wide and 4.0m high, and has 30 steel columns in total, each steel column is uniformly provided with 10 ground control points (hereinafter abbreviated as GCP), the steel column distribution structure is three layers in front and at the back of uniform distribution, the distance between two adjacent steel columns is 0.75m, and meanwhile, an indium steel ruler is arranged below the calibration field and used as a standard ruler during measurement, the background wall is white, the illumination condition is good, and multi-angle shooting can be performed at different heights and different angles, as shown in fig. 2 (a). In order to observe all GCP as far as possible, two solid measuring stations with the height of 1.15m are respectively arranged at the position 4.8m in front of the left side and 3.2m in front of the right side of the calibration field to serve as measuring stations. The GCP artificial mark is a white square light-reflecting sticker of 40mm multiplied by 40mm, the center is a circle with the diameter of 40mm and alternating black and white, the center adopts a precise cross wire, the line width of the cross wire at the center of the GCP artificial mark is 0.5mm as shown in figure 2(c), not only is the precise observation convenient, but also the figure is beneficial to the extraction of the center of the GCP artificial mark of the digital photo. The GCP artificial mark is fixed on the metal rod by using the super glue, and the relative position of the GCP artificial mark is kept stable for a long time.
Establishing a free coordinate system with the indium steel ruler O in combination with the graph 21The point is taken as the origin O of the free coordinate systemWA right-handed coordinate system is used, as in fig. 2 (a). The plane coordinates of 2 measuring stations A, B are determined by a rear intersection method, and 4 points are uniformly selected on an indium steel ruler. The two Nikon CX-102 total stations are respectively erected on A, B two measurement stations, the horizontal angle and the vertical angle of three points on the indium steel ruler are respectively measured, the other point is used as a redundant observation value to be subjected to adjustment, 8 returns are measured, and the three-dimensional coordinates of the measurement stations are obtained through calculation based on a back intersection calculation formula and a triangle elevation calculation formula. And uniformly selecting a plurality of GCPs and Check Points (CP) on three layers of steel frames of the calibration field to carry out fisheye camera calibration experiments. The horizontal angle and the vertical angle of the GCP are measured at A, B two measuring stations respectively by using the same instrument, and the measured number is 4. And calculating three-dimensional coordinates of the GCP and the CP by using a forward intersection and triangulation elevation measurement formula.
And 2, deducing a mathematical relationship between the fisheye image point and the spatial three-dimensional coordinate point by using the relation between the equidistant projection function of the fisheye camera and the perspective projection function of the traditional optical camera, solving a projection transformation matrix, and calculating an initial value of the basic matrix by using the epipolar geometry theory. The specific implementation mode is as follows:
perspective projection model
r'＝ftanθ (1)
Equidistant projection model
r＝fθ (2)
θ is the angle of incidence of the incident ray and f is the focal length.
The relation between the fisheye image point and the perspective projection image point obtained by the fisheye camera equidistant projection model and the perspective projection model will be described with reference to fig. 4(a) and (b)
Figure BDA0003144507500000051
Obtaining a mathematical relation between the fisheye image point and the spatial three-dimensional coordinate point according to Direct Linear Transformation (DLT):
Figure BDA0003144507500000052
where H is a projective transformation matrix. Order to
Figure BDA0003144507500000053
Then, a Levenberg-Marquardt algorithm is used for minimizing the reprojection error, and the optimized projection transformation matrix is solved based on least square adjustment.
min(∑id(xi，HZi)2)
Suppose HLAnd HRThe first fisheye image and the second fisheye image are respectively projection transformation matrixes. H can be obtained by performing the same procedure as above using two fisheye imagesLAnd HR。
The basic matrix is obtained by calculation according to a polar line geometric constraint theoretical formula:
Figure BDA0003144507500000061
wherein
Figure BDA0003144507500000062
The initial values of the interior orientation elements required to solve equation (4) are described with reference to fig. 5, 6, and 7. The specific steps of solving the initial value of the internal orientation element are as follows:
because the fisheye camera cannot be manufactured without errors, the imaging of the fisheye image shows an ellipse which is approximate to a circle, according to the characteristic of the fisheye camera, curve fitting denoising is carried out by using the two images, and the centers and the radiuses of the two images are obtained according to a curve equation. Wherein fig. 5(a) and 5(b) are calibration field fisheye images taken in two orientations, fig. 6(a) and 6(b) are images which are first denoised by median filtering and then segmented according to the threshold segmentation principle, and fig. 7(a) and 7(b) are the results of the final fitting. Some curve edge points are selected in fig. 7(a) and 7(b) and then calculated according to the general equation of the ellipse and the optical principle of the focal length to obtain the initial value of the internal orientation element.
And 3, adding the distortion model into the polar line geometric constraint formula, and solving the distortion parameter of the fisheye camera by using a Newton iteration method. The specific embodiment is explained with reference to fig. 3:
distortion models for fisheye cameras are generally considered only two, the radial distortion model and the tangential distortion model are shown in fig. 3. Radial distortion model
Figure BDA0003144507500000063
Tangential distortion model
Figure BDA0003144507500000064
The coordinates of the undistorted fish-eye image point A can be obtained from the distortion model, see FIG. 3
Figure BDA0003144507500000065
In the above formula, k1，k2Is the radial distortion parameter, p1，p2Is the tangential distortion parameter.
Δ(u-u0)rAnd Δ (v-v)0)rRadial distortion in the u and v directions, respectively;
Δ(u-u0)tand Δ (v-v)0)tTangential distortions in the u and v directions, respectively;
(u(id)，v(id)) Is the coordinates of the undistorted fisheye image points.
The mathematical relationship between the fisheye image point and the perspective projection image point can obtain the perspective projection image coordinate without distortion:
Figure BDA0003144507500000071
of the above formula (u'(id)，v'(id)) Are the undistorted perspective projection image coordinates.
The distortion parameter calibration model of the fisheye camera can be obtained by adding the undistorted perspective projection image coordinates into the polar line geometric constraint formula
Figure BDA0003144507500000072
Of the above formula (u'L(id),v'L(id)) Is the virtual perspective projection image coordinate (u ') corresponding to the first fish-eye image'R(id),v'R(id)) Is the virtual perspective projection image coordinate corresponding to the second fisheye image.
And (5) unfolding and simplifying the calibration model (10) and solving by using a Newton method.
After the expansion, the equation is as follows,
Figure BDA0003144507500000073
the formula, A, B, C, D, E, G, I, J, L, M, N, S, U, Y and T are (U)0，v0，fu,fv) Constant of (u)0,v0,fu,fv) The value of (2) is obtained by the expression (4).
The solution process is as follows.
The expanded equation (11) is denoted as f (x), and the newton's iterative equation for solving the nonlinear system of equations is:
X(k+1)＝X(k)-f'(X(k))-1f(X(k)),k＝0,1,... (12)
namely, it is
Figure BDA0003144507500000074
In the formula, f' (X)(k)) Is the Jacobian matrix of f (X), with the initial value X ═ 0,0,0,0]T. Solving to obtain fish-eye camera distortion parameter X ═ k1,k2,p1,p2]T。
And 4, correcting the fisheye image points by using distortion, and solving a projection transformation matrix and a basic matrix. And decomposing the projection transformation matrix to obtain the internal and external orientation elements. The specific embodiment is as follows.
Correcting coordinates of fish eye image points by distortion parameters and solving projection transformation matrix H(id)(ideal projective transformation matrix):
[u'(id),v'(id),1]T＝H(id)Z (13)
order to
Figure BDA0003144507500000081
The inside orientation elements are:
Figure BDA0003144507500000082
Figure BDA0003144507500000083
and calculating to obtain the inner orientation element and the distortion parameter.
The above embodiments are merely illustrative, and not restrictive, and those skilled in the relevant art can make various changes and modifications without departing from the spirit and scope of the invention. Therefore, all equivalent technical solutions also belong to the scope of the present invention, and the protection scope of the present invention should be defined by the claims.
The technical contents not described in detail in the present invention are all known techniques.

Claims (5)

1.极线几何约束的鱼眼相机标定方法，其特征在于包括以下步骤：1. The fisheye camera calibration method of epipolar geometry constraint is characterized in that comprising the following steps: 步骤1，建立室内三维标定场，对标定场中的人工标志点进行坐标测量，然后对标定场进行多个角度的摄影，测量影像上的人工标志点的鱼眼图像点坐标；Step 1, establish an indoor three-dimensional calibration field, measure the coordinates of the artificial marker points in the calibration field, and then photograph the calibration field from multiple angles to measure the coordinates of the fisheye image points of the artificial marker points on the image; 步骤2，利用鱼眼相机等距投影函数以及传统光学相机的透视投影函数的关系，然后推导出鱼眼图像点与空间三维坐标点之间的数学关系，用LM算法最小化误差，得到最优化的投影变换矩阵，利用极线几何的理论计算出基本矩阵的初始值；Step 2: Using the relationship between the equidistant projection function of the fisheye camera and the perspective projection function of the traditional optical camera, then deduce the mathematical relationship between the fisheye image point and the three-dimensional coordinate point in space, and use the LM algorithm to minimize the error and obtain the optimal The projection transformation matrix of , and the initial value of the fundamental matrix is calculated by using the theory of epipolar geometry; 步骤3，将畸变模型加入极线几何约束式，用牛顿迭代法求出鱼眼相机畸变参数；Step 3: Add the distortion model to the epipolar geometric constraint, and use the Newton iteration method to obtain the distortion parameters of the fisheye camera; 步骤4，用畸变改正鱼眼图像点，求解投影变换矩阵以及基本矩阵和投影变换矩阵，分解投影变换矩阵得到内外方位元素。Step 4: Correct the fisheye image point with distortion, solve the projection transformation matrix, the fundamental matrix and the projection transformation matrix, and decompose the projection transformation matrix to obtain the inner and outer orientation elements. 2.根据权利要求1所述的极线几何约束的鱼眼相机标定方法，其特征在于：所述步骤1建立有深度的高精度的室内三维标定场，并且需要长期维护，用全站仪对三维标定场进行高精度测量；因为鱼眼图像的大畸变特点，对鱼眼像点的采集要精细。2. The fisheye camera calibration method of epipolar geometry constraint according to claim 1, is characterized in that: described step 1 establishes the indoor three-dimensional calibration field with high precision with depth, and needs long-term maintenance, and uses total station to calibrate. The three-dimensional calibration field is used for high-precision measurement; because of the large distortion characteristics of the fish-eye image, the collection of fish-eye image points should be fine. 3.根据权利要求1所述的极线几何约束的鱼眼相机标定方法，其特征在于：所述步骤2是以透视投影为媒介，通过透视投影图像坐标与空间三维坐标之间的关系推导出鱼眼图像点坐标与空间三维坐标之间的关系，在这个过程中首先要确定内方位元素的初始值。3. the fisheye camera calibration method of epipolar geometry constraint according to claim 1, is characterized in that: described step 2 is to take perspective projection as a medium, deduce by the relation between perspective projection image coordinates and space three-dimensional coordinates The relationship between the coordinates of the fisheye image point and the three-dimensional coordinates of the space, in this process, the initial value of the inner orientation element must be determined first. 4.根据权利要求1所述的极线几何约束的鱼眼相机标定方法，其特征在于：所述步骤3将畸变模型加入极线几何约束式中，式中的基本矩阵是已知数，此时的极线几何约束式的未知数只有相机的畸变参数，用若干个高精度同名像点可以解算出畸变参数。4. the fisheye camera calibration method of epipolar geometry constraint according to claim 1, is characterized in that: described step 3 adds distortion model in epipolar geometry constraint formula, the basic matrix in formula is a known number, this The unknowns of the epipolar geometric constraint formula are only the distortion parameters of the camera, and the distortion parameters can be calculated by using several high-precision image points with the same name. 5.根据权利要求1所述的极线几何约束的鱼眼相机标定方法，其特征在于：所述步骤4是进一步提高内外方位元素的求解精度，因为第一次求解投影变换矩阵的过程中，鱼眼图像点没有进行畸变的改正，为了提高精度，用求得的畸变改正鱼眼图像点进行计算。5. the fisheye camera calibration method of epipolar geometry constraint according to claim 1, is characterized in that: described step 4 is to further improve the solution precision of inner and outer orientation elements, because in the process of solving projection transformation matrix for the first time, There is no distortion correction for the fisheye image points. In order to improve the accuracy, the calculated distortion is used to correct the fisheye image points for calculation.
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