CN115147498A - Method for calibrating camera by using asymptote property of main shaft known quadratic curve - Google Patents
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Abstract
A method for calibrating a camera by using the asymptote property of a main shaft known quadratic curve. The image of the principal axis with respect to the pole of the conic and the image of the asymptote are solved using the property that the absolute point with respect to the epipolar line of the conic, i.e. the tangent line passing through the absolute point, is an asymptote. The asymptote and the absolute point satisfy the polar line relation of the pole with respect to the quadratic curve, and the image of the absolute point can be obtained by combining with the projective invariance, so that the vanishing line is recovered. The center of the quadratic curve and the relation of the infinite straight line and the quadratic curve meet the polar line of the pole, and the image of the center of the quadratic curve is obtained by combining the projective invariance. The image of the principal axis intersects the vanishing line at the vanishing point, and forms a set of orthogonal vanishing points with the image of the principal axis about the pole of the quadratic curve. Then 5 sets of orthogonal vanishing points can be obtained from 5 images, so that the parameters in the pinhole camera can be solved by using the image constraints of the orthogonal vanishing points and the absolute quadratic curve.
Description
Technical Field
The invention belongs to the field of computer vision, and relates to a method for solving internal parameters of a pinhole camera by using the property of a quadratic curve asymptote with a known main shaft in space.
Background
The central task of computer vision is to understand images, and its ultimate goal is to make the computer have the ability to recognize three-dimensional environmental information through two-dimensional images. This capability will not only enable machines to perceive geometric information of objects in a three-dimensional environment, including shape, pose, motion, etc., but also to describe, store, recognize and understand them. Camera calibration, which is the step necessary for many computer vision applications, is determining the mapping from a three-dimensional spatial point to its two-dimensional image point. In order to determine the mapping process, a geometric imaging model of the camera needs to be established, parameters of the geometric model are called as camera parameters, and the camera parameters can be divided into an internal parameter and an external parameter. The intrinsic parameters describe the imaging geometry of the imaging system and the extrinsic parameters describe the orientation and position of the imaging system with respect to the world coordinate system. Camera calibration can be divided into traditional calibration, self-calibration and calibration based on geometric entities. No matter which calibration method is used, the aim is to establish a constraint relation, particularly a linear constraint relation, between a two-dimensional image and parameters in a camera, which is a target pursued by the current camera calibration and is one of hot spots of research in the field of computer vision at present.
The pinhole camera imaging model is simple, is convenient to manufacture, does not need special mirror surfaces, and is one of the hotspots of the research in the panoramic vision field. Objects with special geometric properties are visible everywhere in a real scene, and the objects serving as templates become an important research direction for calibrating the camera. Such as a point, line, circle, sphere, etc. The conic curve with the known main shaft, the circle with the known diameter in special cases and the confocal conic curve also have good geometrical characteristics and are easy to obtain in real scenes. The documents "The common self-triangle of polar circuits and its application to The camera calibration", (H.F.Huang, H.Zhang and Y.M.Cheng.Procedents of The IEEE International Conference on Computer Vision and Pattern registration, 2015 (1): 4065-4072) use The properties of common autostereospheres of concentric circles to obtain vanishing lines by generalized eigenvalue decomposition, each image being able to obtain two constraints like absolute conic. In the presence of noise, the concentric circle images also need to be rectified. The literature "geometrical and geometrical constraints of projected systematic circuits and the third applications to camera calibration", (J.S.Kim, and P.Gurdjos and I.S.Kweon.IEEE Transactions on Pattern Analysis and Machine Analysis, 2005,27 (4): 637-642) using concentric circles as calibration templates, yields a line quadratic curve bundle based on the image of the envelope of two circles, the image of the envelope of a circle point being the case of degradation of the line quadratic curve bundle. The image of the envelope of the circle points can be obtained by generalized eigenvalue decomposition, a homography matrix is obtained. Each image can get two constraints of the image of the absolute quadratic curve. In the presence of noise, the method is applicable at all. The documents "Occidean structure from confocal structures: theory and application to camera calibration", (J.S.Kim, P.Gurdjos, I.S.KWeon.Procedents of the IEEE International Conference on Computer Vision and Pattern Recognition,2010,11 (4): 803-812) use confocal quadratic curves as calibration templates, contain the special case of confocal quadratic curves, i.e. concentric circles, and give a bundle of linear quadratic curves based on the image of the envelope of the confocal quadratic curves, the image of the envelope of the points of the circles being the case of a degenerated linear quadratic curve. The image of the envelope of the circular point can be obtained through generalized eigenvalue decomposition, and a homography matrix is obtained. A quadratic curve with a known focus, two confocal quadratic curves with known semi-axial product ratios and two confocal quadratic curves with unknown focus are discussed separately. However, the image of the envelope of the circular point is calculated after the minimum characteristic value is judged, the running time is long, and the robustness is not strong enough under the condition of large noise. In the literature, "Camera calibration using vertical-axes aligned central control", (x.h.ying and h.b.zha. Proceedings of the alignment of the two principal axes on Computer Vision,2007 (1): 138-148), the quadratic curves aligned with the two principal axes are used as a template, which is more general than the confocal quadratic curves, and the eigenvalue decomposition is performed on the quadratic curves aligned with the two principal axes, so that a set of images of orthogonal vanishing points and centers can be obtained, and finally, the Camera internal parameters are linearly solved. The literature "Central control with a Common Axis of Symmetry: properties and Applications to Camera Calibration", (Z.J.Zhao.Proc.Twenty-Second int. Joint Conf.Artificial Intelligence,2011 (1): 2079-2084) takes the secondary curves of two Common Symmetry axes as Calibration templates, carries out generalized eigenvalue decomposition on the images of the two secondary curves to obtain a vanishing line, and finally linearly solves the parameters in the Camera. The calibration template researched by the method is more general, but after the generalized characteristic value decomposition is carried out on the image of the quadratic curve, the characteristic value needs to be judged to determine the vanishing line, so that the robustness is not strong enough under the condition of large noise.
Disclosure of Invention
The invention provides a target for solving the internal parameters of the pinhole camera, which is simple to manufacture, wide in application and good in stability, and the target is formed by a quadratic curve with a known main shaft. In the process of solving the internal parameters of the pinhole camera, 5 images of a target need to be shot by the pinhole camera, and then 5 internal parameters of the pinhole camera can be linearly solved.
The invention adopts the following technical scheme:
5 images containing a conic with a known principal axis were taken from different positions with a pinhole camera. The invention relates to a method for solving parameters in a pinhole camera by using a quadratic curve with a known main shaft in a plane as a target. Firstly, respectively extracting edge points of a target image from the image, and obtaining a quadratic curve image equation and an equation of a principal axis image by using least square fitting. The image of the principal axis with respect to the pole of the conic and the image of the asymptote are solved using the property that the absolute point with respect to the epipolar line of the conic, i.e. the tangent line passing through the absolute point, is an asymptote. The asymptote and the absolute point satisfy the polar line relation of the pole with respect to the quadratic curve, and the image of the absolute point can be obtained by combining with the projective invariance, so that the vanishing line is recovered. The center of the quadratic curve and the relation of the infinite straight line and the quadratic curve meet the polar line of the pole, and the image of the center of the quadratic curve is obtained by combining the projective invariance. The image of the principal axis intersects the vanishing line at the vanishing point, and forms a set of orthogonal vanishing points with the image of the principal axis about the pole of the quadratic curve. Then 5 sets of orthogonal vanishing points can be obtained from 5 images, so that the parameters in the pinhole camera can be solved by using the image constraints of the orthogonal vanishing points and the absolute quadratic curve. The method comprises the following specific steps: and fitting a target projection equation, solving the images of the principal axis about the poles of the quadratic curve and the asymptote to obtain the images of the absolute points, recovering the images of the vanishing line and the center of the quadratic curve to obtain orthogonal vanishing points, determining the images of the absolute quadratic curve, and solving the internal parameters of the pinhole camera.
And judging and obtaining the image of the circular ring point, determining the image of the absolute quadratic curve, and solving the internal parameters of the pinhole camera.
1. Fitting target projection equation
And extracting the pixel coordinates of the Edge points of the target image by using an Edge function in a Matlab program, and fitting by using a least square method to obtain a target projection equation.
2. Solving for the image of the principal axis with respect to the poles of the quadratic curve and the image of the asymptote
Let C be the projection of a known conic C on any principal axis L in space. Establishing a world coordinate system O by taking the center O of the quadratic curve C as an origin w -x w y w z w Wherein the support plane pi on which the quadratic curve C lies is limited to O w x w y w And (4) a plane. Then the optical center O of the camera c Establishing a camera coordinate system O as an origin c -x c y c z c Wherein the image planes pi and O c x c y c The planes are parallel. Then any point X on the quadratic curve C and the image point X on the image C of the quadratic curve satisfy X = K R | t]X, where R and t are each the camera coordinate system O c -x c y c z c And the world coordinate system O w -x w y w z w A rotation matrix and a translation vector in between. With O c The intrinsic parameter matrix of the camera with the optical center is
Wherein [ u ] 0 v 0 1] T Being the homogeneous coordinate of the principal point of the camera, s being the tilt factor, f u And f v Is a scale factor. Wherein f is u ，f v ，s，u 0 ，v 0 5 intrinsic parameters of the camera. Here by c n (n =1,2,3,4,5) represents a coefficient matrix of an image of a quadratic curve in the nth image, l n (n =1,2,3,4,5) represents a coefficient matrix of an image of the principal axis of the quadratic curve in the nth image. For simplicity of description herein, the curve and its coefficient matrix are represented by the same letter.
Due to c n Is the image of a quadratic curve l n Is the image of the principal axis. According to the property of the asymptote of the quadratic curve, the image of the pole of the quadratic curve and the image of the asymptote l can be determined IA ，l JA 。
3. Get the image of the absolute point
Because of the fact thatThe asymptote and the absolute point satisfy the polar line relation of the pole with respect to the quadratic curve, and the image m of the absolute point can be obtained by combining the projective invariance IA ，m JA 。
4. Restoring the image of the center of the vanishing line and the quadratic curve
The asymptote is two special diameters of the quadratic curve, intersects the center of the quadratic curve, and combines with the projective invariance to obtain an image o of the center of the quadratic curve. The center of the quadratic curve and the infinite straight line satisfy the polar line relation of the pole about the quadratic curve, so the vanishing line l can be recovered by combining with the projective invariance ∞ 。
5. Obtaining orthogonal vanishing points
The image of the principal axis intersects the vanishing line at a vanishing point v' ∞ Pole v of a quadratic curve relating to the principal axis ∞ Form a set of orthogonal vanishing points.
6. Determining images of absolute quadratic curves
5 images provide 5 groups of vanishing points, and an image omega of an absolute quadratic curve is obtained according to the constraint relation between the vanishing points and the image of the absolute quadratic curve.
7. Solving pinhole camera intrinsic parameters
According to ω = K -T K -1 And carrying out Cholesky decomposition on the image omega of the absolute quadratic curve and then carrying out inversion to obtain an internal parameter matrix K, namely obtaining 5 internal parameters of the camera.
The invention has the advantages that:
(1) The target is simple to manufacture, and only any one secondary curve with a known main shaft is needed.
(2) The image boundary points of the target can be almost extracted, and the accuracy of curve fitting is improved, so that the calibration accuracy is improved.
Drawings
FIG. 1 is a projection model of a target under a pinhole camera.
Detailed Description
The invention provides a target for solving parameters in a pinhole camera, which is formed by a quadratic curve with a known main axis in space, as shown in figure 1. The method for solving the internal parameters of the pinhole camera by using the novel target comprises the following specific steps:
1. fitting target projection equation
And extracting the pixel coordinates of the Edge points of the target image by using an Edge function in a Matlab program, and fitting by using a least square method to obtain a target projection equation.
2. Solving for the image of the principal axis with respect to the poles of the quadratic curve and the image of the asymptote
Let C be the projection of a known conic C on any principal axis L in space. Establishing a world coordinate system O by taking the center O of the quadratic curve C as an origin w -x w y w z w . Limiting II of plane where quadratic curve C is located at O w x w y w And (4) a plane. With camera optical center O c Establishing a camera coordinate system O as an origin c -x c y c z c . Then any point X on the quadratic curve C and an image point X on the image C of the quadratic curve satisfy X = K R | t]X, where R and t are the rotation matrix and translation vector between the camera coordinate system and the world coordinate system, respectively. If the order is O c The intrinsic parameter matrix of the camera with the optical center is
Wherein [ u ] 0 v 0 1] T Being the homogeneous coordinate of the principal point of the camera, s being the tilt factor, f u And f v Is a scale factor. Wherein f is u ，f v ，s，u 0 ，v 0 5 intrinsic parameters of the camera. Here by c n (n =1,2,3,4,5) represents the coefficient matrix of the image of the quadratic curve in the nth image, l n (n =1,2,3,4,5) represents a coefficient matrix of an image of the principal axis of the quadratic curve in the nth image. For simplicity of description herein, the curve and its coefficient matrix are represented by the same letter.
Taking the 1 st image as an example, as shown in FIG. 1, the quadratic curve C and the infinite straight line L on the support plane II ∞ Cross over to absolute point I A ，J A The following equation set is satisfied:
absolute point I A ，J A Polar lines with respect to the quadratic curve C, i.e. over I A ，J A Are respectively marked as L IA ，L JA 。L IA ，L JA Is a curve passing through the second order point of infinity I on C A ，J A Tangent line of (1), then L IA ，L JA Are the two asymptotes of the quadratic curve C. From the pole-polar relationship:
quadratic curve C and absolute point I A ，J A Envelope C of quadratic curve C according to dual principle and satisfying formula (1) * And the asymptote L IA ，L JA Satisfies the following conditions:
if the matrix C is a reversible matrix, then C * And C -1 By a scaling factor λ, so equation (3) is equivalent to
Absolute point I A ，J A At infinity line L ∞ And (3) satisfying the following conditions:
L ∞ ＝I A ×J A ， (5)
where is the x outer product. The principal axis L of the quadratic curve C must pass through the center O. The polar line of the center O with respect to the quadratic curve C is an infinite straight line L ∞ . The extreme point of the principal axis L with respect to the quadratic curve C is an infinite point V ∞ ：
V ∞ ＝C -1 L。 (6)
Point of infinity V ∞ At infinity line L ∞ In the above, there are:
V ∞ T L ∞ ＝0。 (7)
according to the formula (2), (5) and (7) are:
under projective transformation, the image m of the absolute point IA1 ，m JA1 Is a pair of images l of conjugate complex points, asymptotes IA1 ，l JA1 Is a set of conjugate complex lines. Asymptotic image l IA1 ，l JA1 Has a homogeneous coordinate of l IA1 ＝[k 1 -k 2 i k 3 -k 4 i 1] T ，l JA1 ＝[k 1 -k 2 i k 3 -k 4 i 1] T . According to projective invariance, the method is obtained by (4):
image c of a quadratic curve 1 Image of the principal axis l 1 It is known that the vanishing point v depends on projective invariance ∞1 Can be obtained by the formula (6):
v ∞1 ＝c 1 -1 l 1 。 (10)
satisfies projective invariance according to the relationship between associativity and polar line, and has the following formula (8)
l IA1 ，l JA1 Is a set of complex conjugate lines providing the same real and imaginary constraints, so that only l is considered here IA1 The real and imaginary parts of (a). Obtained by the formulae (9) and (11):
where Re represents the real part and Im represents the imaginary part.
Image of quadratic curvec 1 Image of the principal axis l 1 It is known that the vanishing point v can be obtained from the formula (10) ∞1 . By solving equation (12), the asymptotic image l can be obtained IA1 ，l JA1 。
3. Get the image of the absolute point
Asymptote L IA ，L JA Is an infinite straight line L ∞ Upper absolute point I A ，J A Epipolar line with respect to the quadratic curve C. Image m of absolute point according to projective invariance IA1 ，m JA1 And the image of asymptote l IA1 ，l JA1 The polar line relation is satisfied:
4. restoring images of vanishing lines and centers of quadratic curves
Asymptote L IA ，L JA Are two particular diameters, intersecting at the center O of the quadratic curve. Asymptotic image l based on projective invariance IA1 ，l JA1 Image o intersecting the center of the quadratic curve 1 ：
o 1 ＝l IA1 ×l JA1 。 (15)
Image o of the center of the quadratic curve 1 And a vanishing line l ∞1 The pole polar relationship is satisfied:
l ∞1 ＝c 1 o 1 。 (16)
5. obtaining orthogonal vanishing points
Image C of quadratic curve C 1 Image of the principal axis l 1 It is known that the vanishing point v can be obtained from the formula (10) ∞1 Asymptotic image l IA1 ，l JA1 Image m of absolute point IA1 ，m JA1 Image of the center of a quadratic curve o 1 And a vanishing line l ∞1 . Image of principal axis l 1 And a vanishing line l ∞1 Is at vanishing point v' ∞1 ：
v′ ∞1 ＝l 1 ×l ∞1 。 (17)
Image of principal axis l 1 And vanishing point v ∞1 Satisfies the polar-polar relationship of poles, thus v ∞1 ，v′ ∞1 Is a set of orthogonal vanishing points.
6. Determining an image of an absolute quadratic curve
5 groups of orthogonal vanishing points can be obtained from 5 images according to the orthogonal vanishing points v ∞i ，v′ ∞i A constraint relation of one to the image ω of the absolute quadratic curve can be found:
7. solving pinhole camera intrinsic parameters
According to ω = K -T K -1 And carrying out Cholesky decomposition on the image omega of the absolute quadratic curve and then carrying out inversion to obtain an internal parameter matrix K, namely obtaining 5 internal parameters of the camera.
Examples
The invention provides a method for linearly determining internal parameters of a pinhole camera by using a quadratic curve with a known main shaft in space as a target. The schematic diagram of the template structure adopted by the invention is shown in figure 1. The embodiments of the present invention are described in more detail below by way of an example.
1. Fitting image boundary and target curve equation
5 images of the target are shot by a pinhole camera, the images are read in, pixel coordinates of Edge points of the target image are extracted by utilizing an Edge function in Matlab, and an equation of an image of a quadratic curve is obtained by fitting with a least square method. Here by c n (n =1,2, …, 5) represents a coefficient matrix of an image of a quadratic curve in the nth image, l n (n =1,2, …, 5) represents a coefficient matrix of an image of the principal axis of the quadratic curve in the nth image, with the following results:
2. solving for the image of the principal axis with respect to the poles of the quadratic curve and the image of the asymptote
The image v of the principal axis with respect to the pole of the quadratic curve can be obtained from the equation (10) ∞1 Coefficient matrix of (a):
similarly, the images of the envelopes of the absolute points of the other 4 images can be obtained, i.e. their coefficient matrix representation:
the asymptotic image l can be obtained from the formula (12) IA1 ，l JA1 Coefficient matrix of (a):
similarly, the images of the asymptotes of the other 4 images can be obtained, i.e. their coefficient matrix representation:
3. get the image of the absolute point
The absolute point image m can be obtained from the formula (14) I1 ，m J1 Coefficient matrix of (2):
similarly, the images of the absolute points of the other 4 images can be obtained, i.e. their coefficient matrix representation:
4. restoring the image of the center of the vanishing line and the quadratic curve
The image o of the center of the conic curve can be obtained from the equation (15) 1 Coefficient matrix of (2):
similarly, the image of the center of the quadratic curve of the other 4 images can be obtained, i.e. their coefficient matrix representation:
the vanishing line l is obtained from the formula (16) ∞1 Coefficient matrix of (a):
similarly, the vanishing lines of the other 4 images can be obtained, i.e. their coefficient matrix representation:
5. obtaining orthogonal vanishing points
V is obtained from the formula (17) ∞1 Of orthogonal vanishing point v' ∞1 Coefficient matrix of (2):
similarly, one of the orthogonal vanishing points for the other 4 images can be obtained, i.e. their coefficient matrix representation:
6. determining an image of an absolute quadratic curve
Substituting expressions (29), (30), (31) (62), (33), (64), (65), (66), (67), and (68) into expression (18) simultaneously results in a linear system of equations for ω, and decomposing the system of equations SVD results in a matrix of coefficients for ω, with the following results:
7. solving pinhole camera intrinsic parameters
According to ω = K -T K -1 K is obtained by performing Cholesky decomposition on ω in formula (69) and then inverting, and the results are as follows:
therefore, the 5 internal parameters of the pinhole camera are respectively: f. of u ＝849.934190814953，f v ＝900.017406208901，s＝0.192886751110487，u 0 ＝399.978635628115，v 0 ＝299.991507097548。
Claims (1)
1. A method for calibrating a camera by utilizing the asymptote property of a main shaft known quadratic curve is characterized in that a main shaft known quadratic curve is used as a target; the method comprises the following specific steps: firstly, respectively extracting edge points of a target image from the image, and obtaining a quadratic curve image equation and an equation of a main axis image by using least square fitting; solving the image of the principal axis relative to the pole of the quadratic curve and the image of the asymptote by utilizing the property that the absolute point relative to the polar line of the quadratic curve, namely the tangent line passing through the absolute point is the asymptote; the asymptote and the absolute point satisfy the polar line relation of the pole with respect to the quadratic curve, and the projective invariance is combined to obtain the image of the absolute point, so as to recover the vanishing line; the center of the quadratic curve and the relation of the infinite straight line and the quadratic curve meet the polar line of the pole, and the image of the center of the quadratic curve is obtained by combining the projective invariance; the image of the principal axis intersects with the vanishing line at the vanishing point, and forms a group of orthogonal vanishing points with the image of the principal axis about the pole of the quadratic curve; then 5 images are obtained to obtain 5 groups of orthogonal vanishing points, so that the parameters in the pinhole camera are solved by utilizing the image constraints of the orthogonal vanishing points and the absolute quadratic curve;
(1) Solving for the image of the principal axis with respect to the poles of the quadratic curve and the image of the asymptote
Setting the projection of a quadric curve C with a known main axis L in space as C; establishing a world coordinate system O by taking the center O of the quadratic curve C as an origin w -x w y w z w Wherein the support plane pi on which the quadratic curve C lies is limited to O w x w y w A plane; then the optical center O of the camera c Establishing a camera coordinate system O as an origin c -x c y c z c Wherein the image planes pi and O c x c y c The planes are parallel; then any point X on the quadratic curve C and the image point X on the image C of the quadratic curve satisfy X = K R | t]X, where R and t are each the camera coordinate system O c -x c y c z c And the world coordinate system O w -x w y w z w A rotation matrix and a translation vector therebetween; with O c The intrinsic parameter matrix of the camera is
Wherein [ u ] 0 v 0 1] T Being the homogeneous coordinate of the principal point of the camera, s being the tilt factor, f u And f v Is a scale factor; wherein f is u ，f v ，s，u 0 ，v 0 5 intrinsic parameters for the camera; here by c n Where n =1,2,3,4,5, coefficient matrix representing the image of the quadratic curve in the nth image, l n Where n =1,2,3,4,5 represents a coefficient matrix of the image of the principal axis of the quadratic curve in the nth image; for the sake of simplicity of description herein, the curve and its coefficient matrix are represented by the same letter;
due to c n Is the image of a quadratic curve,/ n Is the image of the principal axis; determining the image of the pole of the quadratic curve and the image of the asymptote l according to the property of the asymptote of the quadratic curve IA ，l JA ；
(2) Get the image of the absolute point
Because the asymptote and the absolute point satisfy the polar line relation of the pole with respect to the quadratic curve, the projective invariance is combined to obtain the image m of the absolute point IA ，m JA ；
(3) Restoring the image of the center of the vanishing line and the quadratic curve
The asymptote is two special diameters of a quadratic curve, intersects the center of the quadratic curve, and combines with projective invariance to obtain an image o of the center of the quadratic curve; the center of the quadratic curve and the infinite straight line satisfy the polar line relation of the pole about the quadratic curve, so the vanishing line l is recovered by combining the projective invariance ∞ ；
(4) Obtaining orthogonal vanishing points
The image of the main shaft intersects the vanishing line at a vanishing point v' ∞ Pole v of the quadratic curve with the principal axis ∞ Form a set of orthogonal vanishing points.
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