A method and system for determining the central axis of a body of revolution based on iterative projection optimization

Abstract

The application discloses a method and system for determining the central axis of a rotary body based on iterative projection optimization, comprising: obtaining three-dimensional point cloud data of a part of the surface of the rotary body in a three-dimensional space; based on the three-dimensional point cloud data, using multi-initial point iterative optimization to obtain an optimal projection direction vector of the whole, the direction being the direction of the central axis of the rotary body, and based on this, calculating other geometric parameters of the rotary body; the iterative optimization process aims to find a projection direction which can make the projection point set of the three-dimensional point cloud data on the orthogonal projection plane have maximum circular symmetry. The application can still robustly and accurately determine the central axis of the rotary body in the case of incomplete data.

Description

Technical Field

[0001] This invention relates to the fields of 3D computer vision and point cloud processing, and particularly to a method and system for determining the central axis of a solid of revolution based on iterative projection optimization. This method and system robustly and accurately extracts the geometric parameters, especially the central axis of revolution, of a solid of revolution (e.g., a cone, cylinder, frustum, etc.) from incomplete or noisy 3D point cloud data. This invention can be widely applied in reverse engineering, industrial automation inspection, robot vision navigation, and digital preservation of cultural relics. Background Technology

[0002] In modern industrial manufacturing, quality control, and 3D reconstruction, identifying and fitting geometric primitives from 3D point cloud data acquired by devices such as laser scanners and structured light cameras is a fundamental and crucial task. Especially for solids of revolution formed by rotating curved surfaces around a straight line, accurately determining their central axis of rotation and contour parameters is of great significance for evaluating part machining accuracy and realizing object grasping and assembly.

[0003] Existing geometric fitting methods, such as algorithms implemented in mainstream point cloud processing libraries (e.g., Point Cloud Library, hereinafter referred to as PCL), typically employ a strategy of directly fitting model parameters in 3D space. These methods, such as those based on Random Sample Consensus (RANSAC), assume a 3D model with multiple degrees of freedom (e.g., the vertices, axes, and half-vertex angles of a cone; or the axis and radius of a cylinder) by randomly sampling a set of points, and then evaluate the degree of fit between this model and the entire point cloud.

[0004] However, these direct 3D fitting methods have significant limitations when dealing with incomplete data commonly found in the real world. In many applications, due to occlusion, scanning angle limitations, or the characteristics of the object itself, the collected point cloud data often only covers a local area of ​​the surface of the body of revolution. For such partial point cloud data, directly fitting a high-degree-of-freedom 3D model becomes very difficult. This is because sparse or unevenly distributed point sets may not provide sufficient geometric constraints for all model parameters, leading to unstable fitting processes, low accuracy, or even complete failure. Furthermore, the noise and outliers prevalent in point cloud data further interfere with the fitting process, reducing the robustness of the algorithm.

[0005] Therefore, how to robustly and accurately extract the central axis of rotation from an incomplete, noisy partial point cloud of a body of revolution is a pressing problem that needs to be solved in current technology. Summary of the Invention

[0006] To address the technical problems of poor robustness, low accuracy, and easy failure when fitting a solid of revolution from partial or noisy 3D point cloud data in existing technologies, this invention proposes a method and system for determining the central axis of a solid of revolution based on iterative projection optimization. This method can still robustly and accurately determine the central axis of a solid of revolution even when the data is incomplete.

[0007] The specific technical solution is as follows:

[0008] A method for determining the central axis of a body of revolution based on iterative projection optimization includes the following steps:

[0009] S1: Obtain the three-dimensional point cloud data of the surface of the rotating part in three-dimensional space;

[0010] S2: Based on the three-dimensional point cloud data, a multi-initial-point strategy is adopted to generate a set of widely distributed initial projection direction vectors on the unit sphere, with each initial point corresponding to an initial projection direction vector;

[0011] S3: Perform iterative optimization on each initial point to obtain the optimal projection direction vector of that initial point; the iterative optimization process aims to find a projection direction that maximizes the circular symmetry of the projection point set of the 3D point cloud data on its orthogonal projection plane.

[0012] S4: Compare the optimal projection direction vectors of all initial points to obtain the overall optimal projection direction vector, which is the direction of the central axis of the body of revolution, and calculate other geometric parameters of the body of revolution based on this.

[0013] Furthermore, in S1, the acquired three-dimensional point cloud data is preprocessed, including: removing noise points by using a statistical outlier removal filter.

[0014] Alternatively, a voxel grid filter can be used to downsample overly dense point clouds.

[0015] Furthermore, in step S3, iterative optimization is performed on each initial point, including the following sub-steps:

[0016] (S3.1) For each initial projection direction vector, parameterize it using the spherical coordinate system;

[0017] (S3.2) Project the three-dimensional point cloud data orthogonally onto a two-dimensional plane with the current projection direction vector as the normal, to generate a two-dimensional projection point set P′;

[0018] (S3.3) Perform geometric shape fitting on the two-dimensional projection point set P′ to obtain a circular or elliptical contour containing the most interior points, and integrate the interior points that conform to the circular or elliptical contour to obtain the interior point set I′.

[0019] (S3.4) Calculate the objective function used to measure the circular symmetry of the fitted circular or elliptical contour;

[0020] (S3.5) Based on the value of the objective function, update the projection direction vector using a derivative-free optimization algorithm so that the objective function value calculated in subsequent iterations tends to be minimized;

[0021] (S3.6) Repeat S3.2-S3.5 until the convergence condition is met, and obtain the optimal projection direction vector of the initial point.

[0022] Furthermore, the objective function is obtained by weighted summation of each cost item, which includes an interior point ratio penalty term and a geometric roundness error term;

[0023] The interior point ratio penalty term is used to penalize points in the two-dimensional projection point set P′ that do not belong to the interior point set I′. Its value is 1 minus the ratio of the total number of interior points in the interior point set I′ to the total number of points in the two-dimensional projection point set P′.

[0024] The geometric roundness error term is used to penalize the non-roundness of the geometric shape formed by the interior points, and its value is the eccentricity of the fitted circular or elliptical profile.

[0025] Furthermore, the cost item also includes a concentricity error item, used to measure concentricity, the value of which is the square root of the average of the sum of squares of the deviations of the center coordinates of all concentric circles from the overall average center coordinates.

[0026] Furthermore, the cost item also includes a multi-layer circle center parallelism error item, which is used to measure the parallelism. Its value is the square of the cross product of the direction vector of the multi-layer circle center fitting line and the current projection direction vector.

[0027] Furthermore, in step (S3.5), the derivative-free optimization algorithm employs the Nelder-Mead simplex method or the Powell conjugate direction method.

[0028] Furthermore, in step (S3.3), a random sampling consensus algorithm is used to fit the geometric shape on the two-dimensional projection point set P′.

[0029] Furthermore, in the word iteration of the random sampling consensus algorithm, the fitting method used to calculate the geometric parameters from the sample points adopts the Taubin algebra fitting method.

[0030] A system for determining the central axis of a body of revolution based on iterative projection optimization includes a memory and one or more processors. The memory stores executable code, and the one or more processors are coupled to the memory. When the executable code is executed, it is used to implement the method for determining the central axis of a body of revolution based on iterative projection optimization.

[0031] The beneficial effects of this invention are:

[0032] (1) Extremely high robustness and universality: This invention transforms the three-dimensional fitting problem into the problem of finding the best two-dimensional projection, which greatly reduces the requirements for data integrity and can uniformly handle various solids of revolution, including cones, cylinders, and frustums. Even if the input point cloud is only a small segment of the surface of the solid of revolution, the method of this invention can still work stably and effectively resist the fitting failure problem caused by data sparsity and partial occlusion.

[0033] (2) Global optimal solution: This invention combines the application of a multi-initial-point optimization strategy. By searching from multiple different initial projection directions, it effectively avoids the algorithm from falling into the trap of local optima, significantly improves the probability of finding the global optimal solution, and thus ensures the correctness of the final result. Attached Figure Description

[0034] Figure 1 This is a flowchart of the method for determining the central axis of a rotating body based on iterative projection optimization in an embodiment of the present invention.

[0035] Figure 2 This is a flowchart illustrating the iterative optimization process for each initial point in this embodiment of the invention.

[0036] Figure 3 This is a schematic diagram showing the changes in the projected shape of a cylinder under different projection directions in an embodiment of the present invention. Among them, (a) is a schematic diagram of the point cloud of the rotating body to be processed, (b) is the result of projecting the point cloud of the rotating body onto the vector (1,1,0) direction, (c) is the result of projecting the point cloud of the rotating body onto the vector (1,1,1) direction, (d) is the result of projecting the point cloud of the rotating body onto the vector (0.3,0,1) direction, (e) is the result of projecting the point cloud of the rotating body onto the vector (0.2,0,1) direction, and (f) is the result of projecting the point cloud of the rotating body onto the vector (0,0,1) direction.

[0037] Figure 4 The diagram shows the results of each step of the RANSAC algorithm in this embodiment of the invention. (a) is a diagram of the original projection point data, (b) is a diagram of the random sampling of the minimum point set in the algorithm, (c) is a diagram of the algorithm identifying interior points and outliers, and (d) is a diagram of the final fitted circle result.

[0038] Figure 5This is a schematic diagram of the Nelder-Mead method iteration results in an embodiment of the present invention, wherein (a) is a general schematic diagram of the 3D surface and the optimization path, (b) is the true global optimal optimization path, (c) is a comparison of the final convergence of each different iteration initial point (i.e., starting point), (d) is a comparison of the iterative optimization paths among multiple starting points, (e) is a schematic diagram of the simplex evolution of the optimal path, and (f) is a schematic diagram of the gradient field optimization trajectory.

[0039] Figure 6 This is a system structure block diagram of determining the central axis of a rotating body based on iterative projection optimization in an embodiment of the present invention.

[0040] The diagram shows a data interface module 501, a data storage module 502, a data processing module 503, and a data output module 504. Detailed Implementation

[0041] The present invention will be described in detail below with reference to the accompanying drawings and preferred embodiments. The objectives and effects of the present invention will become clearer as a result. The present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.

[0042] This implementation proposes a method for determining the central axis of a solid of revolution based on iterative projection optimization. The core idea is to transform the complex problem of fitting a three-dimensional geometric solid into an optimization problem of finding the optimal two-dimensional projection direction. This method is based on a key geometric principle: a solid of revolution exhibits perfect circular symmetry if and only if its orthogonal projection direction on a plane is parallel to the central axis of revolution. For example, the projections of a cone and cylinder are perfect circles, while the projection of a frustum is two perfect concentric circles. Any deviation from the projection direction will disrupt this circular symmetry, resulting in an ellipse or an off-center shape.

[0043] like Figure 1 As shown, the method for determining the central axis of a body of revolution based on iterative projection optimization includes the following steps:

[0044] S1: Receive point cloud data. Acquire a 3D point cloud dataset P={p1,p2,…,p...} of the surface of the object under test using a 3D scanning device. N}, where N is the total number of sampling points; the object to be tested is a rotating body with a central axis, and the three-dimensional point cloud data in the dataset P is sampled from a portion of the surface of this rotating body.

[0045] Optionally, to improve the efficiency and robustness of subsequent processing, the original 3D point cloud data can be preprocessed. For example, a Statistical Outlier Removal (SOR) filter can be used to remove obvious noise points in order to extract the main point cloud of the solid of revolution to be processed and reduce noise. When computing power is insufficient, a Voxel Grid filter can be used to downsample the overly dense point cloud to improve efficiency.

[0046] S2: Initialize the multi-initial-point optimization process.

[0047] To avoid getting trapped in local optima, a multi-initial-point strategy is adopted. M widely distributed (uniformly distributed in this embodiment) initial points are selected on the unit sphere as starting points. Each initial point corresponds to an initial projection direction vector, generating a set of widely distributed initial projection direction vectors {n} on the unit sphere. start,1 ,n start,2 ,…,n start,M}

[0048] S3: Perform iterative optimization for each initial point to obtain the optimal projection direction vector for that initial point, i.e., the local optimum. For example... Figure 2 As shown, S3 is implemented through the following sub-steps:

[0049] (S3.1) For each initial projection direction vector n start,j Parameterize the vector, j = 1, 2, 3, ..., M. Specifically, the three-dimensional unit vector (i.e., the projection direction vector n) is... start,j The optimization problem is parameterized into an optimization problem for two angular variables, namely the zenith angle φ∈[0,π] and the azimuth angle θ∈[0,2π), by using a spherical coordinate system.

[0050] (S3.2) Projected Point Cloud: The 3D point cloud data is orthogonally projected onto a 2D plane with the current projection direction vector as the normal, generating a 2D projected point set P′. When the projection direction is not parallel to the central axis, the projection loses its circular symmetry, and the graphic exhibits non-circular characteristics. With optimization iterations, it transforms into an ellipse or an eccentric graphic. Taking a cylinder as an example, the changes in the 2D projected point set during this process are as follows: Figure 3 As shown; when parallel, they appear as one or more concentric circles.

[0051] (S3.3) Plane fitting: Perform a robust geometry fitting on the two-dimensional projection point set P′ to identify the set of interior points I′ that conform to the potential circular profile and estimate the geometric parameters of the circular profile.

[0052] like Figure 4 As shown, this step preferably uses the Random Sampling Consensus (RANSAC) algorithm, which includes the following fitting sub-steps:

[0053] ① Read the original projected point data containing noise and outliers, such as Figure 4 As shown in (a).

[0054] ②For example Figure 4 As shown in (b), multiple minimum point sets are randomly sampled to obtain multiple initial circle models.

[0055] ③ Traverse all points in the two-dimensional point set P′, and identify interior points and outliers, such as... Figure 4 As shown in (c). Preferably, the Taubin algebraic fitting method is used for fitting the geometric contour because it has excellent stability and computational efficiency for incomplete circular arc data; the Taubin fitting degree is used to analyze the discrimination of interior points and outliers.

[0056] ④ Take the initial circle model with the most interior points as the final fitting result, such as... Figure 4 As shown in (d).

[0057] Furthermore, in order to more accurately quantify the degree of deviation from circular symmetry, the target for fitting can be an ellipse.

[0058] (S3.4) Evaluate projection quality: Calculate the value of the objective function (or cost function) used to measure the circular symmetry of the two-dimensional projection point set. This objective function is designed to be minimized when the projection has perfect circular symmetry and to increase when it deviates from it.

[0059] Furthermore, the objective function consists of at least two weighted terms (applicable when the solid of revolution is a cylinder), namely, an interior point proportional penalty term and a geometric roundness error term, as detailed below:

[0060] Interior point ratio penalty: This term penalizes points in the 2D projected point set P′ that cannot be interpreted by a uniform circular or elliptical contour (i.e., points that do not belong to the interior point set I′). The value of the interior point ratio penalty is calculated based on the ratio of the number of interior points identified during geometric fitting of the 2D projected point set P′ to the total number of points (hereinafter referred to as the interior point ratio), and the expression is as follows:

[0061]

[0062] In the formula, For interior point proportional penalty, The number of interior points in the set of interior points. This represents the total number of points in the two-dimensional projection point set.

[0063] When the projection direction is correct, more points will converge on the circular outline, resulting in a high proportion of inner points and low cost.

[0064] Geometric roundness error term: This term penalizes the non-roundness of the geometry formed by the interior points (i.e., the planar fitted shape obtained in S3.3). Preferably, the eccentricity e of the ellipse fitted to the set of interior points (especially the outermost contour) is used as a measure of geometric roundness error, i.e., the geometric roundness error term Cost. circularity =e. A perfect circle has an eccentricity of 0, and its projection is flatter; the larger the eccentricity, the higher the cost of the geometric roundness error term.

[0065] Furthermore, for solids of revolution with multiple concentric circle projections (such as frustums of cones), the objective function may also include a concentricity error term to measure concentricity, as expressed below:

[0066]

[0067] In the formula, Cost concentricity N represents the concentricity error term. concentric The number of concentric circles in the plane. The coordinates of the center of the corresponding concentric circles are... The coordinates of the average center of all concentric circles are given.

[0068] Furthermore, for conical solids of revolution (such as cones), the objective function also includes multiple layers of circle center parallelism error terms. A single projection layer of a conical solid of revolution contains an infinite number of concentric circles; therefore, along the current projection direction vector... Divide the point cloud into K layers, and then process each layer layer by layer. Perform fitting as described in (S3.3), retaining only the largest circle obtained from the fitting for each layer. , by K layer By fitting a straight line in three-dimensional space, the direction vector of the line can be obtained. Preferably, the cross product value is used as a measure of parallelism error, and the expression is as follows:

[0069]

[0070] In the formula, Cost parallelism Indicates the parallelism error of multiple circular layers, n cadidate Represents the current projection direction vector, (x j ,y j ,z j (x) represents the three-dimensional parameter of the current projection direction vector in the Cartesian coordinate system. c ,y c ,z c ) represents the three-dimensional parameter in the rectangular coordinate system of the direction vector of the line fitted by the largest circle in each layer.

[0071] The weighted sum of the cost items yields the objective function Cost(n), expressed as follows:

[0072]

[0073] In the formula, w1 is the weight of the interior point proportional penalty term, w2 is the weight of the geometric roundness error term, w3 is the weight of the concentricity error term, and w4 is the weight of the multi-layer circle center parallelism error term.

[0074] (S3.5) Update the projection direction: A derivative-free optimization algorithm is used to update the projection direction vector based on the objective function value calculated in the previous step, so that the new projection direction vector can lead to a smaller objective function value.

[0075] Due to the introduction of RANSAC, the objective function Cost(n) is non-convex, non-smooth, and non-differentiable. In this embodiment, the derivative-free optimization algorithm employs the Nelder-Mead simplex method or the Powell conjugate direction method. The Nelder-Mead simplex method iteratively moves a simplex (such as a triangle) to find the minimum point on the complex cost function terrain, obtaining the minimum objective function value. The corresponding candidate projection direction is the updated projection direction vector, serving as the local optimum for the current initial point. The iterative results are as follows: Figure 5 As shown, in the Nelder-Mead algorithm calculation process, taking 5 initial points (i.e., starting points) as an example, refer to... Figure 5 In (d), the five initial points represent five initial projection direction vectors. X1 and X2 are two parameters of the projection direction vectors (i.e., zenith angle and azimuth angle). As the iteration progresses, the parameters move towards a local optimum on the complex objective function terrain. The final convergence of different initial points is compared to... Figure 5 As shown in (c), the evolution process is as follows Figure 5 As shown in (e).

[0076] (S3.6) Repeat S3.2-S3.5 until the convergence condition is met (in this embodiment, the convergence condition is: the objective function no longer decreases for five consecutive iterations, i.e., it fluctuates within a small value range; or the preset maximum number of iterations is reached). At this point, the objective function reaches its minimum, and the corresponding projection direction vector is taken as the optimal projection direction vector n for the initial point. local_opt,j .

[0077] S4: Determine the optimal axis direction.

[0078] After the optimization processes starting from all initial points have converged, the objective function values ​​corresponding to their respective local optima are compared, and the minimum value is selected. This minimum objective function value is Cost. min The corresponding projection direction vector n global_opt This is the optimal estimated direction of the central axis of the body of revolution (hereinafter referred to as the optimal axis direction n). axisIn this embodiment, the global optimal solution is obtained based on five local optimal solutions, and the corresponding optimization trajectory is as follows. Figure 5 As shown in (f).

[0079] S5: Extract complete geometric parameters.

[0080] Based on the determined optimal axis direction n axis Further calculations are performed on other geometric parameters of the body of revolution, including determining the specific location of the central axis in space and the profile parameters that describe its radial dimensions.

[0081] The specific method for calculating the position of the central axis is as follows: [The following text appears to be incomplete and requires further context: "in the optimal projection direction vector n..."] global_opt On the plane, perform a final two-dimensional circle fitting on the interior points of the three-dimensional point cloud to obtain the optimal two-dimensional circle center C. 2D By projecting the center of this two-dimensional circle back into three-dimensional space, we can obtain a point P on the axis of rotation. axis At this point, the central axis has been uniquely determined.

[0082] The contour parameters are calculated as follows: For all three-dimensional interior points, calculate the perpendicular distance (i.e., radius r) from each point to the determined central axis. i ) and the projection position of that point on the axis (i.e., height h) i ). By analyzing (h) i ,r i The data pairs are analyzed or fitted to determine the profile function r(h) of the solid of revolution. If r(h) is a constant, it is a cylinder; if it is a linear function, it is a cone or a frustum.

[0083] To implement the above-mentioned method for determining the central axis of a solid of revolution based on iterative projection optimization, this embodiment also proposes a system for determining the central axis of a solid of revolution based on iterative projection optimization, such as... Figure 6 As shown, the system includes: a data interface module 501, a data storage module 502, a data processing module 503, and a data output module 504.

[0084] The data interface module 501 is used to receive point cloud data, and is electrically connected to the data storage module 502; the data processing module 503 is the core module, used to execute the above-mentioned method for determining the central axis of the body of revolution based on iterative projection optimization, and the final output of the body of revolution parameters is output by the data output module 504; the data storage module 502 is used to store point cloud data, intermediate calculation results of the data processing module 503, and the final body of revolution parameters.

[0085] This invention exhibits strong noise resistance: by employing the RANSAC framework in the two-dimensional fitting stage and combining it with an objective function based on interior point scaling penalties and geometric roundness errors, this invention can effectively identify and isolate noise points and "structural outliers" generated by imperfect projections, thereby obtaining more reliable fitting results. On the other hand, this invention also boasts high accuracy: by optimizing an objective function that directly reflects the geometric quality of the projection (circular symmetry), it can converge to a very accurate estimate of the axis direction; the use of a derivative-free optimization algorithm ensures that high-quality solutions can be found even on complex, non-convex cost function landscapes.

[0086] It will be understood by those skilled in the art that the above descriptions are merely preferred examples of the invention and are not intended to limit the invention. Although the invention has been described in detail with reference to the foregoing examples, those skilled in the art can still modify the technical solutions described in the foregoing examples or make equivalent substitutions for some of the technical features. All modifications and equivalent substitutions made within the spirit and principles of the invention should be included within the scope of protection of the invention.

Claims

1. A method for determining the central axis of a body of revolution based on iterative projection optimization, characterized in that, Includes the following steps: S1: Obtain the three-dimensional point cloud data of the surface of the rotating part in three-dimensional space; S2: Based on the three-dimensional point cloud data, a multi-initial-point strategy is adopted to generate a set of widely distributed initial projection direction vectors on the unit sphere, with each initial point corresponding to an initial projection direction vector; S3: Perform iterative optimization on each initial point to obtain the optimal projection direction vector of that initial point; the iterative optimization process aims to find a projection direction that maximizes the circular symmetry of the projection point set of the 3D point cloud data on its orthogonal projection plane. In step S3, iterative optimization is performed on each initial point, including the following sub-steps: S3.1: For each initial projection direction vector, parameterize it using the spherical coordinate system; S3.2: Project the three-dimensional point cloud data orthogonally onto a two-dimensional plane with the current projection direction vector as the normal, to generate a two-dimensional projection point set P′; S3.3: Perform geometric shape fitting on the two-dimensional projection point set P′ to obtain a circular or elliptical contour containing the most interior points, and integrate the interior points that conform to the circular or elliptical contour to obtain the interior point set I′. S3.4: Calculate the objective function used to measure the circular symmetry of the fitted circular or elliptical contour; S3.5: Based on the value of the objective function, update the projection direction vector using a derivative-free optimization algorithm so that the objective function value calculated in subsequent iterations tends to be minimized; S3.6: Repeat S3.2-S3.5 until the convergence condition is met, and obtain the optimal projection direction vector of the initial point; S4: Compare the optimal projection direction vectors of all initial points to obtain the overall optimal projection direction vector, which is the direction of the central axis of the body of revolution, and calculate other geometric parameters of the body of revolution based on this.

2. The method for determining the central axis of a body of revolution based on iterative projection optimization according to claim 1, characterized in that, In step S1, the acquired three-dimensional point cloud data is preprocessed, including: removing noise points by using a statistical outlier removal filter. Alternatively, a voxel grid filter can be used to downsample overly dense point clouds.

3. The method for determining the central axis of a body of revolution based on iterative projection optimization according to claim 1, characterized in that, The objective function is obtained by weighted summation of each cost item, which includes an interior point ratio penalty term and a geometric roundness error term. The interior point ratio penalty term is used to penalize points in the two-dimensional projection point set P′ that do not belong to the interior point set I′. Its value is 1 minus the ratio of the total number of interior points in the interior point set I′ to the total number of points in the two-dimensional projection point set P′. The geometric roundness error term is used to penalize the non-roundness of the geometric shape formed by the interior points, and its value is the eccentricity of the fitted circular or elliptical profile.

4. The method for determining the central axis of a body of revolution based on iterative projection optimization according to claim 3, characterized in that, The cost item also includes a concentricity error item, which is used to measure concentricity. Its value is the square root of the average of the sum of squares of the deviations of the center coordinates of all concentric circles from the overall average center coordinates.

5. The method for determining the central axis of a body of revolution based on iterative projection optimization according to claim 3, characterized in that, The cost item also includes a multi-layer circle center parallelism error item, which is used to measure the parallelism. Its value is the square of the cross product of the direction vector of the multi-layer circle center fitted line and the current projection direction vector.

6. The method for determining the central axis of a body of revolution based on iterative projection optimization according to claim 1, characterized in that, In S3.5, the derivative-free optimization algorithm adopts the Nelder-Mead simplex method or the Powell conjugate direction method.

7. The method for determining the central axis of a body of revolution based on iterative projection optimization according to claim 1, characterized in that, In step S3.3, a random sampling consensus algorithm is used to fit the geometric shape on the two-dimensional projection point set P′.

8. The method for determining the central axis of a body of revolution based on iterative projection optimization according to claim 7, characterized in that, In a single iteration of the random sampling consensus algorithm, the fitting method used to calculate the geometric parameters from the sample points is the Taubin algebra fitting method.

9. A system for determining the central axis of a body of revolution based on iterative projection optimization, characterized in that, The device includes a memory and one or more processors, wherein the memory stores executable code, and the one or more processors are coupled to the memory. When the executable code is executed, it is used to implement the method for determining the central axis of a body of revolution based on iterative projection optimization as described in any one of claims 1-8.

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