CN108090934B - Method for calibrating parabolic catadioptric camera according to geometric characteristics of spherical section - Google Patents
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Abstract
The invention relates to a method for calibrating a parabolic catadioptric camera according to the geometric characteristics of a spherical section. The projection model of the space sphere on the unit view sphere is a small circle formed on the unit view sphere, and the intersection line of an oblique cone formed by the optical center and the spherical image and the unit view sphere is the small circle. There is a large circle with its center coinciding with the center of the unit sphere parallel to this small circle, i.e. the large circle can be obtained by translating the small circle on the unit sphere surface. The geometric invariance of the perpendicular and perspective projection of the connecting line of one point on the circle outside the diameter and two end points of the diameter can be obtained, and a spherical image can provide a pair of orthogonal vanishing points. On the basis of obtaining the principal point, the three spherical images can obtain the internal parameters of the camera.
Description
Technical Field
The invention belongs to the field of computer vision, and relates to a method for solving intrinsic parameters of a parabolic catadioptric camera by using geometric characteristics of a sphere and a spherical section of the sphere in space.
Background
Computer vision is a simulation of biological vision using a computer and associated equipment. The main task of the method is to process the acquired pictures to obtain the three-dimensional information of the corresponding scene. The origin of computer vision is internationally recognized as the three-dimensional real object analysis research initiated by r.roberts in the united states in the mid sixties, and the three-dimensional real object analysis method utilizes a model of a known polyhedron to analyze the real position of a corresponding object in a depicted two-dimensional picture in a three-dimensional space, and populates the past two-dimensional image analysis to three-dimensional scene analysis. Computer vision has begun to develop since the early seventies.
With the large number of applications of the computational vision technology in various fields, the small visual range of the traditional camera cannot meet the requirements of the computational vision technology. In order to meet the requirements of the computational vision technology, the center refraction and reflection camera is inoculated, so that the camera calibration also becomes the key point of the research of the computational vision technology. The camera calibration is to obtain the camera internal parameters, and the calibration process is to find the geometric projection relation between the real three-dimensional actual object and the image plane two-dimensional object image. According to the two-dimensional points of the image, the matched space three-dimensional point coordinates can be obtained through related algebraic calculation. The document Auto-calibration and absolute calibration (Triggs B., proceedings of Computer Vision and Pattern Recognition,1997:604-614.) firstly introduces the relevant knowledge in high-level geometry into the camera self-calibration, greatly simplifies the process of calibrating the absolute quadratic curve, and the calibration accuracy is still high, thus having great significance in the process of calibrating the camera. The literature "geometrical properties of central catadioptric line images and the third application in calibration", (Barreto J.P., Araujo H., IEEE Transactions on Pattern Analysis and Machine understanding, 27(8),2005, 1327-. Documents "Calibration of central catadioptric camera using a DLT-like approach", (Puig l., basic y., Sturm p., et al. international Journal of Computer Vision,93(1),2011, 101-.
Spheres, straight lines are several geometric shapes common in three-dimensional space. Although both straight lines and spheres form a quadratic curve in the image plane through the unit sphere model, the most important advantage of spheres for straight lines is that there is no self-occlusion, the closed contour of a sphere in space viewed from any direction is always a circle, and its projected contour can be extracted in its entirety. Camera calibration using a ball has become a hot spot in recent years due to the rich visual geometry of the ball. The use of a ball to calibrate a central Catadioptric camera was first proposed in the literature "catadiotactic calibration using geometric innovations" (Ying x., Hu z., IEEE Transactions on Pattern Analysis and Machine Analysis, 26(10),2004, 1260-. They demonstrated that the image of a sphere under the unit sphere projection model of a central catadioptric camera is elliptical and that the projected conic of one sphere provides two invariants in the non-degenerate case. In order to reduce the complexity of the solution, they propose a step-by-step calibration method which requires at least 4 spherical projections to complete the calibration of the camera. However, the calibration method proposed in this document is nonlinear, the computational complexity is high, and the calibration method can only calibrate partial intrinsic parameters of the parabolic catadioptric camera. The literature "Camera Calibration from Images of Spheres" (Zhang h., Wong k.y.k., Zhang g., IEEE Transactions on Pattern Analysis and Machine Intelligence,29(3), 2007, 499-. The document "identification objective geometrical Properties of Central catalytic Line Images and Sphere Images with Applications to Calibration" (Ying x., Zha h., International Journal of computer vision,78(1),2008:89-105.) specifically states that the camera is calibrated under a Catadioptric camera using the "double contact" theory, it is proposed that under a Catadioptric camera each Line image and Sphere image is calibrated with a modified absolute quadratic curve like "double contact", and then the Catadioptric camera is calibrated by the double contact theory using three straight lines or three spheres, a straight Line being a special case of a Sphere. The document "a calibration method for a paracatadioptric camera from space images" (Duan h., Wu y., Pattern Recognition Letters,33(6), 2012, 677-.
Disclosure of Invention
The method for solving the internal parameters of the parabolic catadioptric camera by utilizing the space ball is simple to manufacture, wide in application and good in robustness. In the process of solving the intrinsic parameters of the parabolic catadioptric camera, the five intrinsic parameters of the parabolic catadioptric camera are solved linearly by using three images of the space sphere shot by the parabolic catadioptric camera.
The invention adopts the following technical scheme:
three spherical images were taken from different positions with parabolic catadioptric cameras. The invention considers the projection model of the space sphere on the unit view sphere, the intersection line of the oblique cone formed by the optical center and the spherical image and the unit view sphere is a small circle. There is a large circle with its center coinciding with the center of the unit sphere parallel to this small circle, i.e. the large circle can be obtained by translating the small circle on the unit sphere surface. A spherical image can provide a pair of orthogonal vanishing points according to the geometric properties (a point on a circle outside the diameter is perpendicular to a connecting line of two end points of the diameter) and the geometric invariance of perspective projection. On the basis of obtaining the principal point, the three spherical images can obtain the internal parameters of the camera. The method comprises the following specific steps: fitting a mirror surface contour projection equation and a spherical image equation, solving an equation of an image of a large circle with the circle center coincident with the sphere center of the unit view sphere and parallel to a small circle, determining a principal point of the camera, determining an orthogonal vanishing point, and solving an internal parameter matrix of the parabolic catadioptric camera.
1. Fitting mirror surface contour projection equation and spherical image equation
And extracting the pixel coordinates of the edge points of the mirror profile projection and the edge points of the shot spherical image by using a function in a Matlab program, and fitting by using a least square method to obtain a mirror profile projection equation and an equation of the spherical image.
2. Equation for solving image of large circle with center coincident with unit visual sphere center and parallel to small circle
The small circle s on the unit sphere is an oblique cone HSAnd the intersection of the unit viewing sphere. First, the lower oblique cone H of the parabolic catadioptric camera is obtainedSThe solution of the equation is divided into three steps, namely, the first step, obtaining a spherical image C through fittingSAnd obtaining a homogeneous coordinate expression of the coordinate expression under a world coordinate system. CSIs a quadratic curve in the image plane, in the world coordinate system, zwThe value of the axis being z1F +1, where f is the effective focal length, i.e., the homogeneous equation for the tilted conic directrix. Second, get the ball image CSA point m on, a connection m and an optical center OCThe obtained straight line is an oblique cone generatrix, namely an equation of the oblique cone generatrix. Thirdly, simultaneous generatrix equation and directrix equation are used to obtain the oblique cone H under the world coordinate systemSA homogeneous equation of (c). Then simultaneous unit sphere of view homogeneous equation and oblique cone H in world coordinate systemSThe homogeneous equation of (2) can be used for solving the equation of the projection of the small circle s of the space sphere on the unit sphere. The normal vector [ n ] of the small circle s and the plane of the small circlex ny nz]TThe vector product of (a) is 0, and the two points are also located on the unit sphere, so that any point on the small circle s satisfies:
wherein is d0The distance from the spherical center O of the unit visual sphere to the plane (base plane) of the small circle s can be used to obtain the normal vector nx ny nz]TAnd d0。
Under a parabolic catadioptric camera, if the unit normal vector [ n ] of the plane where the small circle s is located is knownx ny nz]TAn image of a large circle S having a center coincident with the center of the unit view sphere and parallel to the small circle S on the image plane is C'SThen, then
Wherein
For a virtual camera OCThe initial value of the internal parameter is set,
is the initial value of the aspect ratio,
is the initial value of the effective focal length,
is the initial value of the tilt factor and,
is the initial homogeneous coordinate matrix form of the principal point of the catadioptric camera and is recorded as
H′SIs an equation coefficient matrix of a large circle S with the center coinciding with the center of a unit sphere and parallel to a small circle S.
3. Determining camera principal points
In the parabolic catadioptric system, the center of the image is the projection of the principal point, i.e. the unit sphere center O, on the image plane. Three space spheres are selected, and the projection of the three space spheres on the unit sphere is three small circles s1，s2，s3Three large circles S with the centers coincident with the unit visual sphere and parallel to the small circle can be obtained by translation1，S2，S3. Let S1，S2，S3Projection on the image plane is CL1,CL2,CL3The intersection points are respectively H12,K12，H13，K13，H23，K23The corresponding intersection connecting lines form three line segments H12K12，H13K13，H23K23The three lines must intersect at a point p, which is the center of the image, i.e., the principal point
p＝H12K12∧H13K13∧H23K23，
I.e. the principal point coordinates p ═ (u) can be determined0,v0)。
4. Determining orthogonal vanishing points
By shifting the origin of the image plane coordinate system to the principal point, i.e. by a matrix Tp：
All points on the image are then translated, and the image of the absolute quadratic curve can be represented as ω 'at the new coordinates'＝K'-TK'-1Wherein K' is TpK. The projections of the three great circles intersect at 6 intersection points, and according to the geometric knowledge of the circles, a point on the circle outside the diameter is perpendicular to a connecting line of two end points of the diameter, so that a group of orthogonal directions can be determined. Three sets of orthogonal vanishing points can be determined on the image plane according to the orthogonal invariance of the projection geometry.
5. Solving internal parameters of parabolic catadioptric camera
Three great circles on the unit sphere can determine three groups of orthogonal direction vanishing points on the image plane, and three constraint conditions for absolute quadratic curves can be provided. Knowing the principal point coordinates, the equation of the absolute quadratic curve can be determined, and then Cholesky decomposition and inversion are carried out on the equation to obtain the internal parameter matrix
Where r is the aspect ratio, f is the effective focal length, s is the tilt factor, [ u [ u ] ]0 v0 1]TThe homogeneous coordinate matrix form of the camera principal point p is obtained, namely the parabolic catadioptric camera internal parameters are obtained.
The invention has the advantages that:
(1) the methods herein require only spheres as targets.
(2) The method has generality, and can be applied to various central catadioptric cameras under the condition that the mirror parameters are known.
Drawings
Fig. 1 is a projection model of a space sphere on a unit view sphere.
Fig. 2 is a set of orthogonal vanishing points for three large circles.
Detailed Description
The invention provides a method for solving intrinsic parameters of a parabolic catadioptric camera by utilizing a space sphere. The space sphere is projected to the unit view sphere to form a small circle, a large circle with the center coinciding with the center of the unit view sphere is parallel to the small circle, and the image of the large circle projected to the image plane under the camera can be obtained through the imaging relation of the small circle and the large circle on the image plane. A spherical image can provide a pair of orthogonal vanishing points according to the geometric properties (a point on a circle outside the diameter is perpendicular to a connecting line of two end points of the diameter) and the geometric invariance of perspective projection. On the basis of obtaining the principal point, the three spherical images can obtain the internal parameters of the camera. The method comprises the following specific steps: fitting a mirror surface contour projection equation and a spherical image equation, solving an equation of an image of a large circle with the circle center coincident with the sphere center of the unit view sphere and parallel to a small circle, determining a principal point of the camera, determining an orthogonal vanishing point, and solving internal parameters of the parabolic catadioptric camera. The method comprises the following specific steps:
1. fitting mirror surface contour projection equation and spherical image equation
And extracting the pixel coordinates of the edge points of the mirror profile projection and the edge points of the shot spherical image by using a function in a Matlab program, and fitting by using a least square method to obtain a mirror profile projection equation and an equation of the spherical image. Here by C0Coefficient matrices representing the projected curve of the mirror profile, CiCoefficient matrix representing spherical image in ith image, pass C0Obtaining an initial value of the parameter matrix K in the camera
Specifically, formula (1):
here, C0(p, q) (p 1, 2; q 1,2,3) represents a matrix C0The p-th row of (a) and the q-th column of (b),
phi is half of the angle of view of the camera, and rho is a parabolic mirror profile projection ellipse C0Long semi-axis length of (2).
2. Equation for solving image of large circle with center coincident with unit visual sphere center and parallel to small circle
As shown in FIG. 1, O is the center of the unit view sphere, OcA point on the spherical surface of the unit view sphere, the image plane and the axis OOcPerpendicular and with point of intersection p, small circle s on unit sphere being oblique cone HSIntersecting line with unit viewing sphere, zwAxis and zcShaft and OOcCoincidence, xwAxis, xcAxis and ywAxis, ycThe axes are parallel to the u-axis and the v-axis of the image plane, respectively. First, the lower oblique cone H of the parabolic catadioptric camera is obtainedSThe solution of the equation is divided into three steps:
first, obtaining a spherical image C by fittingSThen it is in the world coordinate system O-xwywzwThe following homogeneous coordinate expression is:
wherein [ x ]1 y1 z1 1]TIs CSA homogeneous coordinate matrix of a point of (a), whereinij＝βji(ij＝ji),CSA quadratic curve in the image plane, z being in the world coordinate systemwThe value of the axis being z1F +1, f is the effective focal length, i.e. the homogeneous equation of the oblique cone directrix is as follows:
is denoted as F (x)1,y1,z11) is 0, whereinij＝βij(ij) and f is the effective focal length.
Second, get the ball image CSOne point of (m ═ x)1 y1 z1 1]TConnecting m and optical center OCThe obtained straight line is an oblique cone generatrix, namely the parameter equation of the oblique cone generatrix is as follows:
where λ is the parameter, (x, y, z) is in the camera coordinate system Oc-xcyczcThe following points, fig. 1.
Thirdly, simultaneous generatrix equation and directrix equation are used to obtain the oblique cone H under the world coordinate systemSA homogeneous equation of (c).
Is marked as Fs' (x, y, z,1) ═ 0 where [ x y z 1-]TIs HSHomogeneous coordinate matrix of points, betaij＝βji(ij＝ji)。
Unit sphere of view at O-xwywzwThe following homogeneous equation can be expressed as:
is denoted as F' (x, y, z,1) ═ 0, where [ x y z 1 ═ 1]TIs a homogeneous coordinate matrix point on the unit view sphere. Then the unit sphere and the oblique cone H are combinedSThe homogeneous equation of (2) can be obtained, namely the equation of the projection small circle s of the space sphere on the unit sphere is obtained:
the normal vector [ n ] of the small circle s and the plane of the small circlex ny nz]TIs 0, and these two points are also located on the unit sphere, so that any point [ x ] on the small circle ss ys zs 1]TThe homogeneous coordinate matrix satisfies:
wherein is d0The distance from the spherical center O of the unit visual sphere to the plane (base plane) of the small circle s can be used to obtain the normal vector nx ny nz]TAnd d0。
Under a parabolic catadioptric camera, if the unit normal vector [ n ] of the plane where the small circle s is located is knownx ny nz]TCenter of circleAn image of a large circle S superposed on the center of the unit view sphere and parallel to the small circle S is C 'on the image plane'SThen, then
Wherein
For a virtual camera OCThe initial value of the internal parameter is set,
H′Sis an equation coefficient matrix of a large circle S with the center coinciding with the center of a unit sphere and parallel to a small circle S.
3. Determining camera principal points
In the parabolic catadioptric system, the center of the image is the projection of the principal point, i.e. the unit sphere center O, on the image plane. Three space spheres are selected, and the projection of the three space spheres on the unit sphere is three small circles s1，s2，s3. By means of the translation it is possible to obtain: three large circles S with the circle centers coincident with the unit visual sphere and parallel to the small circle1，S2，S3. As shown in FIG. 2, let S1，S2，S3Projection on the image plane is CL1,CL2,CL3The intersection points are respectively H12,K12，H13，K13，H23，K23The corresponding intersection connecting lines form three line segments H12K12，H13K13，H23K23The three lines must intersect at a point p, which is the center of the image, i.e., the principal point
p＝H12K12∧H13K13∧H23K23， (10)
Where Λ represents the intersection, i.e. the principal point coordinate p ═ can be determined (u)0,v0)。
4. Determining orthogonal vanishing points
The projections of the three great circles intersect at 6 intersection points, and according to the geometric knowledge of the circles, a point on the circle outside the diameter is perpendicular to a connecting line of two end points of the diameter, so that a group of orthogonal directions can be determined. Three sets of orthogonal vanishing points can be determined on the image plane according to the orthogonal invariance of the projection geometry.
By shifting the origin of the image plane coordinate system to the principal point, i.e. by a matrix Tp：
All points on the image are translated, the diameter endpoint Hij,KijIs translated to obtain H'ij,K'ijThen, then
In the new coordinate system, the image of the absolute quadratic curve may be represented as ω '═ K'-TK'-1Wherein K' is TpK. Thus K' depends only on the three parameters r, f, s.
Let three great circles be S1，S2，S3The diameters coplanar with each great circle are only two, depending on the geometry. In FIG. 2, the projection of the space sphere is CL1,CL2,CL3In which H is12,K12Is CL1,CL2True intersection of (A), H13,K13Is CL1,CL3True intersection point, H23,K23Is CL2,CL3Real intersection point, in-plane diameter projection of H12K12And H13K13. In the new coordinate system, the translation is H'12K'12And H'13K'13. So as to obtain vanishing point in orthogonal direction under new coordinate
f'23＝H'12H'13∧K'12K'13， (13)
u'23＝H'12K'13∧H'13K'12。 (14)
Solving to obtain a group of dead points f 'in orthogonal directions'23And u'23I.e. f as shown in FIG. 223And u23A set of orthogonal directions of vanishing points. Can obtain f 'in the same way'24And u'24，f'34And u'34Are the other two orthogonal direction vanishing points.
5. Solving internal parameters of parabolic catadioptric camera
Three great circles on the unit sphere can determine three groups of orthogonal direction vanishing points on the image plane, and three constraint conditions for absolute quadratic curves can be provided. As in equation (15):
knowing the principal point coordinates, an equation of an absolute quadratic curve can be determined, and then Cholesky decomposition and inversion are carried out on the equation to obtain an internal parameter matrix K, namely the internal parameters of the parabolic catadioptric camera.
Examples
The invention provides a method for calibrating internal parameters of a parabolic catadioptric camera by using the geometric characteristics of a spherical section of a space. The schematic structure of the experimental template adopted by the invention is shown in figure 1. In real experiments, the experimental device we used was a parabolic catadioptric camera, the viewing angle of which was 180 °. The method comprises the following specific steps:
1. fitting mirror surface contour projection equation and spherical image equation
The size of the image used in the present invention is 2510 × 2400. A space ball is used as a calibration object, and three images are shot by the parabolic catadioptric camera according to different positions of the ball. The three images taken were processed with Canny edge operators and the mirror profile projection equation and the spherical equation were obtained by least squares fitting. The coefficient matrix of the projection equation of the mirror surface contour of the first image is C0The coefficient matrixes of the three spherical image equations are respectively Cn(n ═ 1,2,3), the results were as follows:
2. equation for solving image of large circle with center coincident with unit visual sphere center and parallel to small circle
The matrix obtained by substituting (16) into (1)
The results are as follows:
the compound represented by the formula (9):
H′Sis the equation coefficient matrix of the big circle S with the center coinciding with the sphere center of the unit view sphere and parallel to the small circle S, the coefficient matrix of the equation of the image of the big circle with the center coinciding with the sphere center of the unit view sphere and parallel to the small circle in the 3 images can be obtained as C'1、C′2And C'3The concrete results are as follows:
3. determining camera principal points
Equations of the images of the three great circles are respectively linked up to obtain a homogeneous coordinate matrix of the images of the six intersection points, and the specific result is as follows:
H12＝[2330.181283374150 -554.940572757582 1]T， (24)
K12＝[1141.363894479753 -135.334194009578 1]T； (25)
H13＝[1870.30583280823 -644.155170765446 1]T， (26)
K13＝[1123.074004583982 -40.481938995892 1]T； (27)
H23＝[1058.749027771881 -243.90564493188 1]T， (28)
K23＝[1331.355884369049 -78.720280120007 1]T. (29) substituting the expressions (24), (25), (26), (27), (28) and (29) into the expression (10), and solving the expression (10) by using an SVD method to obtain a homogeneous coordinate matrix of the principal point p of the camera:
p＝[1253.316286535539 1200.324159267578 1]T。 (30)
4. determining orthogonal vanishing points
Substituting equation (30) into equation (11) yields a translation matrix T that shifts the origin of the planar coordinate system to the principal pointpThe results are as follows:
substituting the equations (24), (25), (26), (27), (28) and (29) into the equation (12) respectively to obtain a homogeneous coordinate matrix of six intersection images under a new coordinate system, and the specific results are as follows:
H′12＝[1076.864996838616 359.2046990935430 1]T， (32)
K′12＝[-1119.523920557881 254.3448142546260 1]T； (33)
H′13＝[6169.922967452878 -4409.847472574096 1]T， (34)
K′13＝[-1302.422819515584 1123.823818546789 1]T； (35)
H′23＝[-1945.672587636570 -738.0052485117630 1]T， (36)
K′23＝[78039.597833512289 76413.902490904604 1]T。 (37)
substituting the expressions (32) to (37) into the expressions (13) and (14) to obtain a homogeneous coordinate matrix of vanishing points in the orthogonal direction in the new coordinate system, and the specific results are as follows:
f′23＝[8.908102047018100 -549.1215310471364 1]T， (38)
u′23＝[-432.2127108993893 230.3547904569334 1]T； (39)
f′24＝[-350.9545753799571 -38.6955534583341 1]T， (40)
u′24＝[-30.20792156985200 123.6239885324170 1]T； (41)
f′34＝[-899.8051516840750 245.2792340804432 1]T， (42)
u′34＝[-87.69793925413430 235.5231862491332 1]T。 (43)
5. and solving the intrinsic parameters of the parabolic catadioptric camera.
Substituting expressions (38) to (43) into expression (15) yields a linear system of equations for the elements in ω, and solving the linear system of equations using SVD decomposition yields a coefficient matrix for ω, with the following results:
and finally, performing Cholesky decomposition on omega in the formula (44) and then inverting to obtain an internal parameter matrix K, namely obtaining the internal parameter matrix of the parabolic catadioptric camera, wherein the result is as follows:
therefore, the 5 intrinsic parameters of the parabolic catadioptric camera are respectively: r-2.796857599, f-221.3601894290854,
s＝-2.899145753753384，u0＝1253.316286535539，v0＝1200.3241592675780。
Claims (1)
1. a method for calibrating a parabolic catadioptric camera according to the geometric characteristics of the cross section of a sphere is characterized in that a sphere in the space is used as a calibration object; the method comprises the following specific steps: shooting three spherical images from different positions by using a parabolic catadioptric camera; according to a projection model of the space sphere on the unit view sphere, a small circle is formed on the unit view sphere, and an intersection line of an oblique cone formed by the optical center and the spherical image and the unit view sphere is the small circle; a large circle with the center coinciding with the spherical center of the unit visual sphere is parallel to the small circle, namely the large circle is obtained by translating the small circle on the spherical surface of the unit visual sphere; according to the geometric invariance of the perpendicular and perspective projection of the connecting line of one point on a circle outside the diameter and two end points of the diameter, a pair of orthogonal vanishing points are provided for a spherical image; on the basis of obtaining the principal point, three spherical images are used for obtaining the internal parameters of the camera;
(1) equation for solving image of large circle with center coincident with unit visual sphere center and parallel to small circle
The small circle s on the unit sphere is an oblique cone HSThe intersection line of the unit sight ball; first, the lower oblique cone H of the parabolic catadioptric camera is obtainedSThe solution of the equation is divided into three steps, namely, the first step, obtaining a spherical image C through fittingSObtaining a homogeneous coordinate expression of the coordinate system in the world; cSIs a quadratic curve in the image plane, in the world coordinate system, zwThe value of the axis being z1F +1, wherein f is the effective focal length, and a homogeneous equation of the oblique cone collimation line is obtained; second, get the ball image CSA point m on, a connection m and an optical center OCThe obtained straight line is an oblique cone generatrix to obtain an equation of the oblique cone generatrix; third step, simultaneous bus equations andthe equation of the standard line is that the oblique cone H is obtained under the world coordinate systemSA homogeneous equation of (a); then simultaneous unit sphere of view homogeneous equation and oblique cone H in world coordinate systemSThe homogeneous equation of (2) is an equation for solving the projection of a small circle s of a space sphere on a unit sphere; the normal vector [ n ] of the small circle s and the plane of the small circlex ny nz]THas a vector product of 0, so that any point [ x ] on the small circle ss ys zs 1]TSatisfies the following conditions:
wherein is d0The distance from the sphere center O of the unit visual sphere to the plane where the small circle s is located, and the normal vector [ n ] is obtained from the equation setx ny nz]TAnd d0(ii) a Under a parabolic catadioptric camera, if the unit normal vector [ n ] of the plane where the small circle s is located is knownx ny nz]TAn image of a large circle S having a center coincident with the center of the unit view sphere and parallel to the small circle S on the image plane is C'SThen, then
Wherein
For virtual cameras at the optical center OCIs set to the initial value of the internal parameter of (1),
H′Sis an equation coefficient matrix of a large circle S with the circle center coincident with the sphere center of the unit visual sphere and parallel to a small circle S;
(2) determining camera principal points
In the parabolic catadioptric system, the center of the image is the principal point, namely the projection of the spherical center O of the unit visual sphere on the image plane; three space spheres are selected, and the projection of the three space spheres on the unit sphere is three small circles s1，s2，s3Obtaining three large circles S with the centers coincident with the unit visual sphere and parallel to the small circle by translation1，S2，S3(ii) a Let S1，S2，S3Projection on the image plane is CL1,CL2,CL3The intersection points are respectively H12,K12，H13，K13，H23，K23The corresponding intersection connecting lines form three line segments H12K12，H13K13，H23K23The three lines must intersect at a point p, which is the center of the image, i.e. the principal point p ═ H12K12∧H13K13∧H23K23Where Λ represents the intersection, i.e. the principal point coordinate p is determined (u ═ u)0,v0)；
(3) Determining orthogonal vanishing points
By shifting the origin of the image plane coordinate system to the principal point, i.e. by a matrix Tp：
All points on the image are then translated, and the image of the absolute quadratic curve is denoted ω ' ═ K ' at the new coordinates '-TK'-1Wherein K' is TpK, K is an internal parameter matrix; the projections of the three great circles intersect at 6 intersection points, and according to the geometric knowledge of the circles, one point on the circle outside the diameter is vertical to a connecting line of two end points of the diameter, so that a group of orthogonal directions are determined; determining three groups of vanishing points in the orthogonal direction on an image plane according to the orthogonal invariance of the projection geometry;
(4) solving internal parameters of parabolic catadioptric camera
Three great circles on the unit sphere can determine three groups of vanishing points in the orthogonal direction on the image plane, and three constraint conditions for an absolute quadratic curve are provided; knowing the principal point coordinates, determining an equation of an absolute quadratic curve, performing Cholesky decomposition on the equation, and then performing inversion to obtain an internal parameter matrix K, namely obtaining the internal parameter matrix of the parabolic catadioptric camera.
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