基于平面路径的扫掠面高精度重建

doi:10.11996/JG.j.2095-302X.2025020425

图学学报 ›› 2025, Vol. 46 ›› Issue (2): 425-436.DOI: 10.11996/JG.j.2095-302X.2025020425

• 计算机图形学与虚拟现实 • 上一篇下一篇

基于平面路径的扫掠面高精度重建

刘圣军¹(), 陶珊珊¹, 王海波¹, 李钦松², 刘新儒¹()

1.中南大学数学与统计学院，湖南长沙 410083
2.中南大学大数据研究院，湖南长沙 410083

收稿日期:2024-08-22 接受日期:2024-10-25 出版日期:2025-04-30 发布日期:2025-04-24
通讯作者:刘新儒(1982-)，男，副教授，博士。主要研究方向为几何造型、数值建模、数据分析与智能算法。E-mail：liuxinru@csu.edu.cn
第一作者:刘圣军(1979-)，男，教授，博士。主要研究方向为几何计算与分析、数字图像处理、智能算法及应用。E-mail：shjliu.cg@csu.edu.cn
基金资助:
国家自然科学基金(62172447);国家自然科学基金(62302530);湖南省自然科学基金(2023JJ40769)

High-precision reconstruction of swept surfaces with a planar path

LIU Shengjun¹(), TAO Shanshan¹, WANG Haibo¹, LI Qinsong², LIU Xinru¹()

1. School of Mathematics and Statistics, Central South University, Changsha Hunan 410083, China
2. Big Data Institute, Central South University, Changsha Hunan 410083, China

Received:2024-08-22 Accepted:2024-10-25 Published:2025-04-30 Online:2025-04-24
First author：LIU Shengjun (1979-), professor, Ph.D. His main research interests cover geometric calculation and analysis, digital image processing, intelligent algorithms, and applications. E-mail：shjliu.cg@csu.edu.cn
Supported by:
National Natural Science Foundation of China(62172447);National Natural Science Foundation of China(62302530);Hunan Natural Science Foundation(2023JJ40769)

摘要/Abstract

摘要：

从三角网格数据重建CAD建模过程是逆向工程中的研究重点之一，高效、高精的曲面重建具有重要的工程价值。针对由直线段和圆弧段组成的平面路径扫掠生成的三角网格表示曲面，提出了基于轮廓和路径曲线自动提取的扫掠过程重建，实现扫掠曲面的高精度重建。首先，基于统一三角网格模型曲率矢量场自动获得初始路径，再利用高斯映射迭代和配准拟合的方法生成了扫掠的轮廓曲线；然后，逆向计算扫掠路径的离散有序点集，通过引入切空间表示方法来识别路径中的直线段和圆弧段，并基于相切几何约束条件建立了拟合的优化模型，对初始路径进一步优化；最后，由计算得到的轮廓曲线和路径曲线执行扫掠操作，以获得重建的扫掠曲面。实验结果表明，该方法实现了自动提取轮廓曲线和路径曲线，进而重建扫掠模型的建模过程，减少了繁琐的人工交互，提取的轮廓和路径有效地避免了离散误差累积，使得最终重建的扫掠曲面精度更高，且适用于有噪声的数据和存在缺失数据的扫掠曲面。

关键词: 逆向工程, 曲面重建, 扫掠曲面, 高斯映射, 几何约束

Abstract:

The reconstruction of CAD modeling process from triangular mesh is a key focus in reverse engineering, and efficient, high-precision swept surface reconstruction is of great engineering value. Targeting the mesh representation surface generated by planar path sweeping composed of line and arc segments, sweeping reconstruction based on profile and path automatic extraction was proposed to achieve high-precision reconstruction of swept surfaces. Firstly, the initial path was automatically obtained based on the unified curvature vector field of the triangular mesh, and the profile was generated using Gaussian mapping iteration, registration, and fitting. Then, the scattered point set of the path was computed inversely. The straight line and arc segments in the path were identified using a tangent space representation, and the fitting optimization model was established based on the tangent geometric constraints to optimize the initial path. Finally, the swept surface was reconstructed by performing a sweeping operation with the calculated profile and path. Experimental results demonstrated that the proposed method achieved automatic extraction of profile and path curves, thereby reconstructing the modeling process of the sweeping model. This approach reduced tedious manual interactions, and the extracted profiles and paths effectively avoided the accumulation of discrete errors, resulting in a higher precision in the final reconstructed sweeping surface. The method was also applicable to models with noisy data and those with missing data.

Key words: reverse engineering, surface reconstruction, swept surface, Gaussian mapping, geometric constraint

中图分类号:

TP391

刘圣军, 陶珊珊, 王海波, 李钦松, 刘新儒. 基于平面路径的扫掠面高精度重建[J]. 图学学报, 2025, 46(2): 425-436.

LIU Shengjun, TAO Shanshan, WANG Haibo, LI Qinsong, LIU Xinru. High-precision reconstruction of swept surfaces with a planar path[J]. Journal of Graphics, 2025, 46(2): 425-436.

图/表 22

图1 算法流程图

Fig. 1 Algorithm pipeline

图2 统一曲率场((a)原始最小曲率场；(b)区分不同类型曲率点；(c)强曲率点对齐；(d)平滑弱曲率区域；(e)初始路径跟踪结果)

Fig. 2 Unify curvature field ((a) Original minimum curvature field; (b) Distinguish different types of curvature points; (c) Align strong curvature points; (d) Smooth the area with weak curvature points; (e) Initial tracking path result)

图3 初始路径跟踪规则((a) pi+1落在顶点或边缘；(b) pi+1落在三角片内部)

Fig. 3 Rules for initial path tracking ((a) pi+1 falls at the apex or edge; (b) pi+1 falls inside the triangle)

图4 高斯映射迭代过程

Fig. 4 Gaussian mapping iterative process

图5 一组横截面轮廓示例图

Fig. 5 A set of cross-sectional profile examples

图6 路径跟踪的离散误差累计示意图

Fig. 6 Accumulation of discrete error in path tracking

图7 准均匀非有理三次B样条逼近结果((a)~(c)控制点数分别为6，16和64的准均匀非有理三次B样条逼近结果；(d)准均匀非有理三次B样条逼近的曲率梳(64个控制点)；(e)直线-圆弧拟合的曲率梳)

Fig. 7 Quasi uniform non rational cubic B-spline approximation results ((a)~(c) Quasi uniform non rational cubic B-Spline approximation results with 6, 16, and 64 control points, respectively; (d) Curvature comb for quasi uniform non rational cubic B-Spline approximation (64 control points); (e) Curvature comb for line-arc fitting)

图8 有序点集的切空间表示

Fig. 8 Tangent space representation of ordered point sets

图9 切空间表示方法检测直线-圆弧段示例图((a)切空间表示下的中点折线段；(b)路径分段结果)

Fig. 9 Detection of line arc segments using tangent space representation method ((a) The midpoint line segment in the tangent space representation; (b) Path segmentation results)

图10 轮廓曲线对比图((a)全局视图；(b)侧视图)

Fig. 10 Comparison chart of profile ((a) Global view; (b) Side view))

表1 轮廓曲线误差对比

Table 1 Comparison of profile curve errors

方法	最大误差	中间误差	平均误差
文献[10]	0.054 5	0.009 4	0.013 2
本文方法	0.014 7	0.004 0	0.006 9

图11 不同方法生成的路径与真实路径的比较以及对应的扫掠结果对比((a)~(c)不同方法拟合路径的结果；(d)~(f)真实轮廓沿不同拟合路径扫掠的曲面)

Fig. 11 Comparison of paths generated by different methods with the real path and corresponding sweeping results ((a)~(c) The results of fitting paths using different methods; (d)~(f) The swept surfaces of the true contour along different fitting path)

表2 路径曲线误差对比

Table 2 Comparison of path curve errors

方法	最大误差	中间误差	平均误差
文献[10]	0.099 9	0.038 6	0.041 6
B样条拟合	0.035 0	0.017 7	0.015 7
直线-圆弧段拟合	0.031 0	0.006 8	0.009 0

表3 曲面重建误差对比

Table 3 Comparison of surface reconstruction errors

模型	方法	误差
模型	方法	MAE	RMSE
①	文献[10]	0.001 84	0.002 99
①	本文方法	0.000 70	0.000 88
②	文献[10]	0.000 37	0.000 60
②	本文方法	0.000 11	0.000 15
③	文献[10]	0.000 20	0.0002 96
③	本文方法	0.000 16	0.000 18
④	文献[10]	0.000 88	0.001 19
④	本文方法	0.000 32	0.000 39

图12 模型①轮廓、路径提取结果及曲面重建误差((a)原始三角网格；(b)轮廓侧视图；(c)路径俯视图；(d)文献[10]；(e)本文方法)

Fig. 12 Profile, path extraction results, and surface reconstruction errors of model ① ((a) Original triangular mesh; (b) Side view of the profile; (c) Top view of the path; (d) Reference[10]; (e) Ours)

图13 模型②轮廓、路径提取结果及曲面重建误差((a)原始三角网格；(b)轮廓侧视图；(c)路径俯视图；(d)文献[10]；(e)本文方法)

Fig. 13 Profile, path extraction results, and surface reconstruction errors of model ② ((a) Original triangular mesh; (b) Side view of the profile; (c) Top view of the path; (d) Reference [10]; (e) Ours)

图14 模型③轮廓、路径提取结果及曲面重建误差((a)原始三角网格；(b)轮廓侧视图；(c)路径俯视图；(d)文献[10]；(e)本文方法)

Fig. 14 Profile, path extraction results, and surface reconstruction errors of model ③ ((a) Original triangular mesh; (b) Side view of the profile; (c) Top view of the path; (d) Reference [10]; (e) Ours)

图15 模型④轮廓、路径提取结果及曲面重建误差((a)文献[10]；(b)本文方法)

Fig. 15 Profile, path extraction results, and surface reconstruction errors of model ④ ((a) Reference [10]; (b) Ours)

图16 高斯噪声模型重建结果

Fig. 16 Reconstruction results of noisy models with Gaussian noise ((a) Std=1.00; (b) Std=0.10; (c) Std=0.01)

表4 噪声模型重建误差

Table 4 Reconstruction error of noisy models

噪声标准差	误差
噪声标准差	MAE	RMSE
Std=1.00	-	-
Std=0.10	0.000 523	0.000 62
Std=0.01	0.000 356	0.000 43

图17 数据缺失模型的重建结果((a)部分缺角；(b)沿路径缺失)

Fig. 17 Reconstruction results of data missing models ((a) Partial missing corners; (b) Missing along the path)

表5 数据缺失模型重建误差

Table 5 Reconstruction error of data missing models

模型	对比	误差
模型	对比	MAE	RMSE
部分缺角	缺失数据	0.000 335	0.000 414
部分缺角	模型④	0.000 344	0.000 420
沿路径缺失	缺失数据	0.000 403	0.000 490
沿路径缺失	模型④	0.000 402	0.000 489

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基于平面路径的扫掠面高精度重建

High-precision reconstruction of swept surfaces with a planar path

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