1. A tooth-shaped structure assembling and measuring method based on monocular vision is characterized by comprising the following steps:

s1: extracting tooth vertex points of an output shaft and an input shaft of a tooth profile structure based on an SUSAN algorithm of a self-adaptive core and a self-adaptive threshold;

s2: acquiring coordinates of the missing teeth of the output shaft and the middle point of the spline of the input shaft in the image according to a path search method and distance constraint;

s3: adopting RANSAC idea clustering to eliminate candidate points with larger errors, thereby facilitating the subsequent ellipse fitting of the tooth top circle and the auxiliary circle of the output shaft and the input shaft and obtaining ellipse parameters;

s4: and the spatial six-degree-of-freedom pose is calculated through addendum circle projection ellipse, auxiliary circle projection ellipse, missing tooth and spline midpoint information under the image coordinate system.

2. The tooth-shaped structure assembling and measuring method based on the monocular vision as recited in claim 1, wherein: the step S1 includes the following:

a 1: based on the gray level histogram, searching a binarization threshold value by using an approximate one-dimensional Means method, and calculating the SUSAN template radius:

wherein R1 is the radius of the output shaft template, R2 is the radius of the input shaft template, n (IA1) and n (IA2) are the number of pixels occupied by the foreground of the respective annular ROI (region of interest), n (IB1) and n (IB2) are the number of pixels occupied by the background in the respective annular ROI, K is_1,2The number of pixels occupied by each annular ROI is Q, and the total resolution of the image is Q;

a 2: calculating the gray difference between each pixel and the central pixel in the neighborhood of the template, and recording the initial iteration value L₁Wherein R is the template radius:

setting the gray scale difference less than or equal to L₀The gray values of the gray values are classified into a set A, the rest are classified into a set B, the next iteration is carried out, and the L is recorded₀Wherein q is the gray difference value A (q) between a certain pixel and the central pixel in the template field, B (q) is a set A respectively, and the number of pixel points equal to the q value in B;

by analogy with the above formula, continuously and iteratively calculating L₃…, Ln, and quitting iteration until the absolute value of the difference value between Ln and Ln-1 is zero, and assigning the last iteration value to the gray level difference threshold value t in the similar comparison function.

3. The tooth-shaped structure assembling and measuring method based on the monocular vision as recited in claim 1, wherein: the step S2 includes the following:

b 1: calculating the distance between adjacent tooth crests along an elliptical path, wherein two points corresponding to the maximum distance are two angular points of the missing tooth, pi and pj are two adjacent points along the elliptical path in the tooth crest point set C, and finally, the obtained two-point coordinate marks p1 and p2 are as follows:

max(p_i,p_j)＝||p_i-p_j||₂(i≠j,i,j∈C)；

b 2: an ellipse fitted by the addendum points is outwards made into an annular ROI area containing a spline part, the positions of corner points of the spline part are detected based on the SUSAN method, two end points of the spline are obtained and are marked as Q1 and Q2, and a spline midpoint coordinate P and a tooth-missing midpoint coordinate Q are as follows:

4. the tooth-shaped structure assembling and measuring method based on the monocular vision as recited in claim 1, wherein: the step S3 includes the following:

c 1: after respective ROI areas are established, Si is assumed to be a collected tooth vertex point set or an arc segment point set through RANSAC clustering, S is a point set after clustering is stored, the point set is initially empty, values in the Si are marked as pi (i is 1,2 … N), the abscissa of the point set is pix, the ordinate of the point set is piy, and aj is (pix, piy,1) T, the first 5 values are taken out, and the following formula is substituted, wherein [ ] represents a determinant, and lambda is a given threshold;

-λ＜[a₁,a₂,a₅][a₃,a₄,a₅][a₁,a₃,a_i][a₂,a₄,a_i]-[a₁,a₃,a₅][a₂,a₄,a₅][a₁,a₂,a_i][a₃,a₄,a_i]＜λ；

c 2: for aj meeting the formula, corresponding pi is an inner point under the condition of taking the 5 values, the process is repeated by taking 5 different values for Si, and the set with the most inner points is compared as a clustered set S to obtain an output shaft tooth top point set S₁Auxiliary arc section point set S of output shaft₂Input shaft tooth top point set S₃Auxiliary arc segment point set S of input shaft₄。

5. The tooth-shaped structure assembling and measuring method based on the monocular vision as recited in claim 1, wherein: the step S4 includes the following:

d 1: when the projection of the space circle on the image plane is not a perfect circle, the ambiguity problem exists, the auxiliary circles of the output shaft and the input shaft are used as a group of coaxial parallel circles, and the center O of the tooth top circle of the output shaft or the input shaft is obtained through calculation_A1、O_A2Normal vector n_A1、n_A2And auxiliary circle center O_B1、O_B2Normal vector n_B1、n_B2Since the directions of the vectors of the circle normal are approximately equal, n is calculated respectively_A1·n_B1、n_A1·n_B2、n_A2·n_B1、n_A2·n_B2Obtaining two groups with larger numerical values, comparing the actual depth distance of the addendum circle and the auxiliary circle center with the distance calculated by the corresponding circle center coordinates, and solving when the difference value is smaller;

d 2: because the addendum circle plane is determined, the middle points of the missing teeth and the splines are coplanar with the addendum circles in space, and a circular plane normal equation and a space point perspective projection equation are combined:

wherein (u, v) is the coordinate of the center point of the missing tooth or the spline under a pixel coordinate system;is coke ratio, f^IIs the focal length of the lens, and is,is the width of the pixel in the direction U, V under its coordinate system; p ═ x_C,y_C,z_C)^TThe coordinates are the coordinates under the coordinate system of the corresponding midpoint camera; n ═ n (n)₁,n₂,n₃)^T、Q＝(x₁,y₂,z₃)^TThe center coordinates and normal vectors of the addendum circle in the camera coordinate system are respectively.

Background

The measurement auxiliary assembly is widely applied to the fields of manufacturing engineering such as aviation, aerospace and the like. In the past, measurement auxiliary assembly mainly depends on large-size measurement systems such as a laser tracker, a laser radar, an iGPS and the like, and the devices are high in price and time-consuming and labor-consuming to install and disassemble. In the assembling process of key parts of a tooth-shaped structure, because the output shaft and the input shaft have a shielding relation, if a laser tracker is used for assembling and guiding, the pose is difficult to measure, and the mode of leading out target balls and the like belongs to indirect measurement under the condition of complex field conditions, so that collision is easy to generate, and the requirement on measuring precision is difficult to guarantee.

Therefore, there is a need to improve the prior art to overcome the deficiencies of the prior art.

Disclosure of Invention

The invention aims to provide a tooth profile structure assembling and measuring method based on monocular vision so as to solve the problems in the background technology.

In order to achieve the purpose, the invention provides the following technical scheme: a tooth-shaped structure assembling and measuring method based on monocular vision comprises the following steps:

s1: extracting tooth vertex points of an output shaft and an input shaft of a tooth profile structure based on an SUSAN algorithm of a self-adaptive core and a self-adaptive threshold;

s2: acquiring coordinates of the missing teeth of the output shaft and the middle point of the spline of the input shaft in the image according to a path search method and distance constraint;

s3: adopting RANSAC idea clustering to eliminate candidate points with larger errors, thereby facilitating the subsequent ellipse fitting of the tooth top circle and the auxiliary circle of the output shaft and the input shaft and obtaining ellipse parameters;

s4: and the spatial six-degree-of-freedom pose is calculated through addendum circle projection ellipse, auxiliary circle projection ellipse, missing tooth and spline midpoint information under the image coordinate system.

As a preferred embodiment of the present invention, the step S1 includes the following steps:

a 1: based on the gray level histogram, searching a binarization threshold value by using an approximate one-dimensional Means method, and calculating the SUSAN template radius:

wherein R1 is the radius of the output shaft template, R2 is the radius of the input shaft template, n (IA1) and n (IA2) are the number of pixels occupied by the foreground of the respective annular ROI (region of interest), n (IB1) and n (IB2) are the number of pixels occupied by the background in the respective annular ROI, K is_1,2The number of pixels occupied by each annular ROI is Q, and the total resolution of the image is Q;

a 2: calculating the gray difference between each pixel and the central pixel in the neighborhood of the template, and recording the initial iteration value L₁Wherein R is the template radius:

setting the gray scale difference less than or equal to L₀The gray values of the gray values are classified into a set A, the rest are classified into a set B, the next iteration is carried out, and the L is recorded₀Wherein q is the gray difference value A (q) between a certain pixel and the central pixel in the template field, B (q) is a set A respectively, and the number of pixel points equal to the q value in B;

by analogy with the above formula, continuously and iteratively calculating L₃…, Ln, and quitting iteration until the absolute value of the difference value between Ln and Ln-1 is zero, and assigning the last iteration value to the gray level difference threshold value t in the similar comparison function.

As a preferred embodiment of the present invention, the step S2 includes the following steps:

b 1: calculating the distance between adjacent tooth crests along an elliptical path, wherein two points corresponding to the maximum distance are two angular points of the missing tooth, pi and pj are two adjacent points along the elliptical path in the tooth crest point set C, and finally, the obtained two-point coordinate marks p1 and p2 are as follows:

max(p_i,p_j)＝||p_i-p_j||₂(i≠j,i,j∈C)；

b 2: an ellipse fitted by the addendum points is outwards made into an annular ROI area containing a spline part, the positions of corner points of the spline part are detected based on the SUSAN method, two end points of the spline are obtained and are marked as Q1 and Q2, and a spline midpoint coordinate P and a tooth-missing midpoint coordinate Q are as follows:

as a preferred embodiment of the present invention, the step S3 includes the following steps:

c 1: after respective ROI areas are established, Si is assumed to be a collected tooth vertex point set or an arc segment point set through RANSAC clustering, S is a point set after clustering is stored, the point set is initially empty, values in the Si are marked as pi (i is 1,2 … N), the abscissa of the point set is pix, the ordinate of the point set is piy, and aj is (pix, piy,1) T, the first 5 values are taken out, and the following formula is substituted, wherein [ ] represents a determinant, and lambda is a given threshold;

-λ＜[a₁,a₂,a₅][a₃,a₄,a₅][a₁,a₃,a_i][a₂,a₄,a_i]-[a₁,a₃,a₅][a₂,a₄,a₅][a₁,a₂,a_i][a₃,a₄,a_i]＜λ；

c 2: for aj meeting the formula, corresponding pi is an inner point under the condition of taking the 5 values, the process is repeated by taking 5 different values for Si, and the set with the most inner points is compared as a clustered set S to obtain an output shaft tooth top point set S₁Auxiliary arc section point set S of output shaft₂Input shaft tooth top point set S₃Auxiliary arc segment point set S of input shaft₄。

As a preferred embodiment of the present invention, the step S4 includes the following steps:

d 1: when the projection of the space circle on the image plane is not a perfect circle, the ambiguity problem exists, the auxiliary circles of the output shaft and the input shaft are used as a group of coaxial parallel circles, and the center O of the tooth top circle of the output shaft or the input shaft is obtained through calculation_A1、O_A2Normal vector n_A1、n_A2And auxiliary circle center O_B1、O_B2Normal vector n_B1、n_B2Since the directions of the vectors of the circle normal are approximately equal, n is calculated respectively_A1·n_B1、n_A1·n_B2、n_A2·n_B1、n_A2·n_B2Obtaining two groups with larger numerical values, comparing the actual depth distance of the addendum circle and the auxiliary circle center with the distance calculated by the corresponding circle center coordinates, and solving when the difference value is smaller;

d 2: because the addendum circle plane is determined, the middle points of the missing teeth and the splines are coplanar with the addendum circles in space, and a circular plane normal equation and a space point perspective projection equation are combined:

wherein (u, v) is the coordinate of the center point of the missing tooth or the spline under a pixel coordinate system;is coke ratio, f^IIs the focal length of the lens, and is,is the width of the pixel in the direction U, V under its coordinate system; p ═ x_C,y_C,z_C)^TThe coordinates are the coordinates under the coordinate system of the corresponding midpoint camera; n ═ n (n)₁,n₂,n₃)^T、Q＝(x₁,y₂,z₃)^TThe center coordinates and normal vectors of the addendum circle in the camera coordinate system are respectively.

Compared with the prior art, the invention has the beneficial effects that:

1. the mode that the traditional mode that the output shaft and the input shaft are assembled in the involution state through human eye observation is replaced, and the state evaluation is more accurate.

2. Compared with the assembly mode of the tooth-shaped structure guided by large-size measurement such as a laser tracker, the method has the advantages that the assembly cost is greatly reduced, the site adaptability is remarkably enhanced, the influence of the environment such as humidity and temperature is small, and the disassembly, the assembly and the maintenance are convenient.

3. The image processing and pose calculation related algorithm is quick and effective, the assembly time consumption is greatly shortened, and the efficiency is improved.

Drawings

FIG. 1 is a flow chart of spatial circle monocular vision pose measurement;

FIG. 2 is a schematic diagram of a principle of measuring a spatial circle monocular vision pose;

FIG. 3 is a SUSAN circular template with a neighborhood of 56 pixels;

FIG. 4 is a schematic diagram of the extraction results of the tooth vertex points of the output shaft and the input shaft;

FIG. 5 is a schematic view of the output shaft spline, the input shaft missing tooth midpoint;

FIG. 6 is a schematic diagram of the fitting of the addendum ellipse and the auxiliary ellipse of the output shaft and the input shaft.

Detailed Description

The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.

Referring to fig. 1-6, the present invention provides a tooth profile structure assembling and measuring method based on monocular vision, which comprises the following steps: the method comprises the following steps:

s1: extracting the top points of the gear rings of the output shaft and the input shaft;

s2: auxiliary round edge extraction of the output shaft and the input shaft;

s3: extracting the middle points of the output shaft spline and the input shaft missing tooth;

s4: fitting a key characteristic ellipse;

s5: and (5) resolving the pose.

The first embodiment is as follows:

s1 extracting the vertexes of the output shaft and the input shaft gear ring:

setting the radius of the circular template as R (R belongs to N)₊) Containing 2R²+6R+1(R≥3,R∈N₊) The size of each pixel and the template are key for influencing the angular point extraction effect, annular (approximate tooth root to tooth top part) ROI division is respectively carried out on an output shaft and an output shaft gear ring, respective gray level histograms are obtained, a binarization threshold value is searched by using an approximate one-dimensional Means method, a foreground and a background are divided, and an R value is obtained as follows:

wherein R1 is the output shaft template radius, R2 is the input shaft template radius, n (I)_A1)，n(I_A2) Number of pixels occupied by the respective annular ROI foreground, n (I)_B1)，n(I_B2) For the number of pixels occupied by the background in the respective annular ROI, K_1,2Q is the total resolution of the image for the number of pixels occupied by the respective annular ROI, and if R is 5, the template is shown in FIG. 2.

In order to extract the corner, the pixel gray levels in the neighborhood of the template need to be compared with the central gray level, and the similarity comparison function is as follows:

in the formula: r and r0 represent coordinates of the template center and the template field pixels, I (r)₀) The gray value of the pixel under the corresponding coordinate is taken, t is a gray difference threshold value, 20 is taken conventionally, and the false detection rate is the lowest when the index is taken to be 6 by combining theory and practice.

And counting the similarity degree as follows:

and then, by using a corner response function of the following formula, if the USAN value of a certain pixel point is smaller than a geometric threshold g, the pixel point is considered as an initial corner, namely, the smaller the USAN value is, the larger the corner response is.

Since the tooth crest angle of the output shaft and the input shaft is about 120 degrees in the image plane, g is 0.67n_max。

The difference in gray levels between each pixel in the neighborhood of the template and the center pixel is calculated as follows, noting the initial iteration value L1, where R is the template radius.

Setting the gray scale difference less than or equal to L₀The gray values of the gray values are classified into a set A, the rest are classified into a set B, the next iteration is carried out, and the L is recorded₂Wherein q is the gray level difference between a certain pixel and the central pixel in the template field, and A (q) and B (q) are the pixel points with the same value as q in the set A and B respectively.

And repeating iteration in the same way, stopping iteration until the absolute value of the interpolation of the previous iteration value and the next iteration value is zero, and assigning the last iteration value to the gray difference threshold value t.

Through the calculation of the threshold t, the angular points acquired through the angular point response function have tooth root angular points, if the screening of the tooth apical angular points needs to be completed, alternative angular point neighborhood gray level difference sets A and B need to be involved, if the number of elements in the set A is greater than that in the set B, the point can be considered as an apical angular point, and if the number of the elements in the set A is not greater than that in the set B, the point is removed, and the angular point extraction effect is as shown in FIG. 3.

Example two:

s2 auxiliary circle edge extraction of output shaft and input shaft:

in order to acquire the end face arc segment information, the edge detection needs to be carried out on the image, a fixed value needs to be manually set in advance for a dual threshold in the traditional Canny algorithm, the edge is discontinuous due to overlarge threshold, and a false edge can be extracted due to undersize threshold, so that an OTSU threshold is adopted for substitution.

Dividing the collected image into foreground and background, recording the total pixel of the image as N, recording the threshold value as T, recording the ratio of the number of pixels belonging to the foreground to the total pixel as alpha, and recording the average gray level as g₁(ii) a The ratio of the number of pixels belonging to the background to the total pixels is recorded as beta, and the average gray scale is recorded as g₂. The average value of the total gray scale of the image is recorded as g, and the variance between the classes is recorded as sigma². Then there are:

N₁+N₂＝N,α+β＝1

g＝α×g₁+β×g₂

σ²＝α×(g-g₁)²+β×(g-g₂)²

the best segmentation threshold value, i.e. the gray value at which the inter-class variance is maximal. And (5) optimizing the g one by one to obtain a maximum threshold value T, wherein the T is a high threshold value in Canny, and the low threshold value is 0.5T. After the single-pixel edge is detected, a discrete arc section is further required to be obtained and used as an input set of ellipse fitting, and then a three-way area is removed by adopting an edge skeleton pruning algorithm and the curvature mutation position is cut off by Shi-Thomas corner detection. And finally, establishing corresponding ROI area division, wherein the ROI area division belongs to the discrete arc segment parts of the auxiliary circle of the output shaft and the input shaft.

Example three:

s3 extracting the middle points of the output shaft spline and the input shaft missing tooth:

for the missing tooth of the input shaft, because all the tooth crest point coordinates of the missing tooth are obtained and the ellipse is fitted, the distance between the adjacent tooth crests can be calculated along the path of the ellipse, and when the distance obtains the maximum value, the corresponding two points are the two angular points of the missing tooth. Let pi, pj be two adjacent points along the elliptical path in the addendum point set C, and the resulting two-point seating mark p1, p2 is given by:

max(p_i,p_j)＝||p_i-p_j||₂(i≠j,i,j∈C)

for the output shaft spline, if the relative teeth are more protruded, an ellipse fitted by the tooth top points is outwards used as an annular ROI (region of interest), the region contains spline features, and the positions of the angular points of the spline are detected by the SUSAN (surface Acoustic wave analysis) method to obtain two end points of the spline, which are marked as q₁，q₂. The spline midpoint coordinate P and the tooth-missing midpoint coordinate Q are as follows:

the extraction effect of the output shaft spline and the input shaft missing tooth is shown in figure 4.

Example four:

s4 key feature ellipse fitting:

after the addendum point and the edge arc section are obtained, ellipse fitting is carried out, an output shaft tooth top ellipse and an input shaft tooth top ellipse under an image coordinate system and respective end face auxiliary circles are respectively fitted, and an ellipse equation expansion formula and a homogeneous coordinate conversion thereof are as follows: ax²+By²+ Cxy + Dx + Ey + F ═ 0, with Q representing the matrix:

after respective ROI areas are established, clustering is carried out through RANSAC to avoid interference and improve robustness, and then an ellipse fitting method based on geometric distance is adopted to improve fitting accuracy. Ransac (random Sample consensus) is a method for removing cluster data to obtain an effective set based on a repeated minimum Sample set. Si is assumed to be a collected tooth top point set or arc segment point set, and S is a point set after cluster storage and is empty initially. The values in Si are labeled as pi (i ═ 1,2 … N), with the abscissa of pix and the ordinate of piy, and the aj ═ T (pix, piy,1) takes the first 5 values, which is substituted by the following formula, where [ ] represents the determinant and λ is the given threshold:

||[a₁,a₂,a₅][a₃,a₄,a₅][a₁,a₃,a_i][a₂,a₄,a_i]-[a₁,a₃,a₅][a₂,a₄,a₅][a₁,a₂,a_i][a₃,a₄,a_i]||＜λ

the left end of the inequality is a 6-point representation of the ellipse. For aj satisfying the above formula, corresponding pi is the interior point under the condition of taking the 5 values, the process is repeated by taking different 5 values for Si, and the set with the most interior points is compared as the clustered set S. The algorithm can respectively search an output shaft tooth vertex set, an output shaft auxiliary arc section point set, an input shaft tooth vertex set and an input shaft auxiliary arc section point set which are prepared for ellipse fitting.

And after the outliers are removed, ellipse fitting can be directly carried out, each point is substituted into an ellipse equation to obtain a coefficient matrix of the ellipse equation, and an initial fitting result can be obtained after SVD decomposition. The fitting is then based on the geometric distance as follows:

wherein Q is a matrix of quadratic form,is a conjugate matrix of Q and is,establishing a nonlinear optimization objective function:

after the corresponding Q is obtained, optimization is carried out by a Levenber-Marquardt algorithm, and the ellipse fitting condition is shown in figure 5.

Example five:

s5 resolving the pose with six degrees of freedom:

the space circle is imaged as an ellipse on the image plane after perspective transformation, and the ellipse can be understood as being intercepted by an elliptic viewing cone formed by the image plane, the space circle and the optical center, as shown in figure 1, O_w-X_wY_wZ_wAs a world coordinate system, O_c-X_cY_cZ_cAs camera coordinate system, O_I-X_IY_IIs an image plane coordinate system. After the pose of the space circle is solved in the camera coordinate system, the pose of the circle in the world coordinate system can be solved through an external reference matrix obtained by calibrating the camera.

Space circular coordinate P under camera coordinate system_i ^C＝(x_i ^C,y_i ^C,z_i ^C)^TThe homogeneous coordinate P of the point in the image coordinate system_i ^I＝(x_i ^I,y_i ^I,1)^TAnd homogeneous coordinate P in pixel coordinate system_i ^I”＝(u_i ^I,v_i ^I,1)^TThe conversion relationship is as follows:

in the formula (u)₀ ^I,v₀ ^I) Is the camera principal point, in units of pixels;is coke ratio, f^IIs the focal length of the lens, and is,the width of a pixel in the direction U, V under its coordinate system is given in mm.

The view cone equation can be obtained from the imaging model:

Ax²+By²+Cxy+Dxz+Eyz+Fz²＝0

converting the above equation into a matrix form, wherein Q is a quadratic matrix:

the real symmetric matrix Q is diagonalized such that:

P^-1QP＝P^TQP＝diag(λ₁,λ₂,λ₃)

the pose of the spatial circle with known radius is solved, namely the intersection problem of the conical plane is solved, after a viewing cone under a camera coordinate system is converted into a standard coordinate system, the center coordinate and the normal vector of the spatial circle can be obtained by calculating and converting the viewing cone under the camera coordinate system into the standard coordinate system, and the center coordinate and the normal vector of the spatial circle are respectively as follows:

wherein P is^RIs a transformation matrix of the viewing cone from the camera coordinate system to the standard coordinate system.

According to the formula, if the projection of the space circle on the image plane is not a perfect circle, the center coordinates and normal vectors of the space circle have two groups of solutions, namely the ambiguity problem exists.

Obtaining the center O of the addendum circle of the output shaft by calculation_A1、O_A2Normal vector n_A1、n_A2And auxiliary circle center O_B1、 O_B2Normal vector n_B1、n_B2. Since the vector directions of the circle normal are approximately equal, n is respectively calculated_A1·n_B1、n_A1·n_B2、n_A2·n_B1、 n_A2·n_B2And obtaining two groups with larger numerical values, comparing the actual depth distance of the addendum circle and the auxiliary circle center with the distance calculated by the corresponding circle center coordinates, and solving the solution when the difference is smaller.

All information of the output shaft and the input shaft except the yaw angle can be obtained so far, and the last dimension is determined by the output shaft spline and the input shaft missing tooth respectively. When the camera internal reference of monocular vision is known, a three-dimensional space point cannot be reversely solved only by the internal reference matrix due to the lack of depth information of one space three-dimensional point, but because the addendum circle plane is determined, and the tooth-missing and spline midpoint is coplanar with the respective addendum circles in the space, only a circular plane normal equation and a space point perspective projection equation need to be simultaneously established, as follows:

wherein (u, v) is the coordinate of the center point of the missing tooth or the spline under a pixel coordinate system;is coke ratio, f^IIs the focal length of the lens, and is,is the width of the pixel in the direction U, V under its coordinate system; p ═ x_C,y_C,z_C)^TThe coordinates are the coordinates under the coordinate system of the corresponding midpoint camera; n ═ n (n)₁,n₂,n₃)^T、Q＝(x₁,y₂,z₃)^TThe center coordinates and normal vectors of the addendum circle in the camera coordinate system are respectively.

In the description of the present invention, it is to be understood that the indicated orientations or positional relationships are based on the orientations or positional relationships shown in the drawings and are only for convenience in describing the present invention and simplifying the description, but are not intended to indicate or imply that the indicated devices or elements must have a particular orientation, be constructed and operated in a particular orientation, and are not to be construed as limiting the present invention.

In the present invention, unless otherwise explicitly specified or limited, for example, it may be fixedly attached, detachably attached, or integrated; can be mechanically or electrically connected; the terms may be directly connected or indirectly connected through an intermediate, and may be communication between two elements or interaction relationship between two elements, unless otherwise specifically limited, and the specific meaning of the terms in the present invention will be understood by those skilled in the art according to specific situations.

Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that changes, modifications, substitutions and alterations can be made in these embodiments without departing from the principles and spirit of the invention, the scope of which is defined in the appended claims and their equivalents.