基于调和映射理论进行曲面数控雕刻的研究
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摘要
随着数控雕刻技术的发展，雕刻产品的精细度越来越高。但是进行曲面雕刻时存在一个棘手的难题，就是图案的扭曲变形。如何生成曲面雕刻的刀具运动轨迹来解决这个难题是打破曲面数控雕刻过程中的瓶颈的关键。
为了避免在复杂曲面的数控雕刻过程中出现图案的扭曲变形，本文提出了新的生成曲面数控雕刻刀具运动轨迹的方法。我们把应用平面数控雕刻技术生成的刀具运动轨迹，按照调和映射的对应关系映射到曲面上，生成曲面雕刻的刀具运动轨迹。由于调和映射具有等角变换的性质，所以本文提出的方法可以保证刻在曲面上图案基本不变形。
本文介绍了调和映射的基本理论，重点研究平面到曲面的调和映射。文中的曲面或用参数方程表达或用符号距离函数表达。对于这两种表达方式，分别应用变分法推导出平面到曲面的调和映射所必须满足的偏微分方程。这两个偏微分方程都是非线性调和方程。
直接求解非线性调和方程是很困难的，我们把非线性调和方程转化为非线性扩散方程，通过求解非线性扩散终值问题来得到调和映射。本文给出了求解非线性扩散终值问题的数值算法，并讨论了算法的稳定性，同时给出了编程实现算法的主要思路和重要程序的框图。
为了演示本文提出的生成曲面数控雕刻刀具运动轨迹的方法的有效性，我们进行了几个数值实验。文中分别依照初始映射、中间计算结果和调和映射的对应关系将图像贴到曲面上，并把贴图后的曲面透视投影到观察面上，从最开始的初始映射到最后的调和映射，透视效果逐渐改善，证明本文提出的方法是有效的。
With the development of numerical control engraving technology, the precision of sculptures is better and better. But there is a thorny problem on sculpting on a complex surface. It is the pattern's distortion. And it is the key to break the bottleneck to the process of numerical control sculpting on surfaces, that how to generate the tool-path of engraving on surfaces to solve the tough problem.
To avoid the pattern being distorted in the process of numerical control sculpting on a complex surface, a new method is put forward to generate the tool-path of engraving on the surface. It maps the tool-path, which is generated by using the technology of sculpting on the plan, to the surface based on the harmonic mapping. Because the harmonic mapping has a conformal property, this method can preserve the pattern sculpted on surface to be unchanged.
The basic theory of harmonic mapping is introduced, and the harmonic mapping from a plan to a surface is emphasized. In this paper, the surface is presented with a parameterized equation or a signed distance function. And to the both expression of the surface, two partial differential equations are achieved, which the harmonic mapping must satisfy, by using the variation approach. These partial differential equations are all nonlinear harmonic equations.
It is difficult to solve these nonlinear harmonic equations directly, so these nonlinear harmonic equations are changed to nonlinear diffusion equations, and the harmonic mapping is obtained by solving the nonlinear diffusion equations in the stable state. The numerical algorithm for getting the solution of the nonlinear diffusion equations in the stable state is introduced, meanwhile the stability of the algorithm is discussed. The main idea for carrying out the algorithm is given, and the emphases program schemata are also presented.
In order to show the effectiveness of the method, which is used to generate the tool-path of engraving on the surface, some numerical examples are given. The image is mapped to the surface with the initial mapping, the mappings of some computing steps and the harmonic mapping respectively, and then the surface is projected to the vision plan. These project effect are better and better from the initial mapping to the harmonic mapping, so the method is proved to be effective.
为了避免在复杂曲面的数控雕刻过程中出现图案的扭曲变形，本文提出了新的生成曲面数控雕刻刀具运动轨迹的方法。我们把应用平面数控雕刻技术生成的刀具运动轨迹，按照调和映射的对应关系映射到曲面上，生成曲面雕刻的刀具运动轨迹。由于调和映射具有等角变换的性质，所以本文提出的方法可以保证刻在曲面上图案基本不变形。
本文介绍了调和映射的基本理论，重点研究平面到曲面的调和映射。文中的曲面或用参数方程表达或用符号距离函数表达。对于这两种表达方式，分别应用变分法推导出平面到曲面的调和映射所必须满足的偏微分方程。这两个偏微分方程都是非线性调和方程。
直接求解非线性调和方程是很困难的，我们把非线性调和方程转化为非线性扩散方程，通过求解非线性扩散终值问题来得到调和映射。本文给出了求解非线性扩散终值问题的数值算法，并讨论了算法的稳定性，同时给出了编程实现算法的主要思路和重要程序的框图。
为了演示本文提出的生成曲面数控雕刻刀具运动轨迹的方法的有效性，我们进行了几个数值实验。文中分别依照初始映射、中间计算结果和调和映射的对应关系将图像贴到曲面上，并把贴图后的曲面透视投影到观察面上，从最开始的初始映射到最后的调和映射，透视效果逐渐改善，证明本文提出的方法是有效的。
With the development of numerical control engraving technology, the precision of sculptures is better and better. But there is a thorny problem on sculpting on a complex surface. It is the pattern's distortion. And it is the key to break the bottleneck to the process of numerical control sculpting on surfaces, that how to generate the tool-path of engraving on surfaces to solve the tough problem.
To avoid the pattern being distorted in the process of numerical control sculpting on a complex surface, a new method is put forward to generate the tool-path of engraving on the surface. It maps the tool-path, which is generated by using the technology of sculpting on the plan, to the surface based on the harmonic mapping. Because the harmonic mapping has a conformal property, this method can preserve the pattern sculpted on surface to be unchanged.
The basic theory of harmonic mapping is introduced, and the harmonic mapping from a plan to a surface is emphasized. In this paper, the surface is presented with a parameterized equation or a signed distance function. And to the both expression of the surface, two partial differential equations are achieved, which the harmonic mapping must satisfy, by using the variation approach. These partial differential equations are all nonlinear harmonic equations.
It is difficult to solve these nonlinear harmonic equations directly, so these nonlinear harmonic equations are changed to nonlinear diffusion equations, and the harmonic mapping is obtained by solving the nonlinear diffusion equations in the stable state. The numerical algorithm for getting the solution of the nonlinear diffusion equations in the stable state is introduced, meanwhile the stability of the algorithm is discussed. The main idea for carrying out the algorithm is given, and the emphases program schemata are also presented.
In order to show the effectiveness of the method, which is used to generate the tool-path of engraving on the surface, some numerical examples are given. The image is mapped to the surface with the initial mapping, the mappings of some computing steps and the harmonic mapping respectively, and then the surface is projected to the vision plan. These project effect are better and better from the initial mapping to the harmonic mapping, so the method is proved to be effective.
引文
[1] J. Eells and J. H. Sampson, "Harmonic mapping of Riemannian manifolds", Amer. J. Math. (86), 109-160, 1964.
[2] J. Eells and L. Lemaire, "A report on harmonic maps", Bull. London Math. Soc. 10: 1, pp. 1-68, 1978.
[3] J. Eells and L. Lemaire, "Another report on harmonic maps", Bull. London Math. Soc. 20:5, pp. 385-524, 1988.
[4] B. Fuglede, "Harmonic Morphisms between Riemannian manifolds", Ann. Inst. Fourier(Grenoble)(28), 107-144, 1978.
[5] P. Baird, Harmonic Maps with Symmetry, Harmonic Morphisms and Deformation of Metrics, Research Notes in Math. (87), Pitman, Boston, London, Melbourne, 1983.
[6] P. Baird and J. Eells, "A conservation law for harmonic maps", In E. Looijenga, D. Siersma and F. Takens, Geometry Symposium. Utrecht 1980, 1-25, Lecture Notes in Math. (894), Springer-Verlag, Berlin, Heidelberg, New York, 1981.
[7] P. Baird and J. C. Wood, "Harmonic morphisms and conformal foliations by geodesics of three-dimensional space forms", Journal Austral. Math. Soc. Ser. A(51), 118-153, 1991.
[8] S. Gudmundsson, "Harmonic morphisms from complex projective spaces", Geometry Dedicata (53), 155-161, 1994.
[9] S. Gudmundsson, "On the existence of harmonic morphisms from symmetric spaces of rank one", Preprint, Lund University, 1996.
[10] S. Gudmundsson and R. Sigurdsson, "A note on the classification of holomorphic harmonic morphisms", Potential Analysis(2), 295-298, 1993.
[11] S. Gudmundsson, S. Montaldo and T. Mustafa, "The atlas of harmonic morphisms", http://www.maths.1th.se/matematiklu/personal/sigma/harmonic/atlas.html.
[12] S. Gudmundsson, "The bibliography of harmonic morphisms", http://www.maths.1th.se/matematiklu/personal/sigma/harmonic/bibliography.html.
[13] F. E. Burstall, L. Lemaire and J. Rawnsley, "Harmonic maps bibliography", http://www.bath.ac.uk/masfeb/harmonic, html.
[14] G. Gigantes, "A note on harmonic morphisms", Preprint, Univ. Camerino, 1983.
[15] Y. L. Ou, "Quadratic harmonic morphisms and O-systems", Preprint, University of leeds, 1995.
[16] Y. L. Ou and J. C. Wood, "On the classification of quadratic harmonic morphisms between Euclidean spaces", Algebras, Groups and Geometry (13), 41-53, 1996.
[17] J. C. Wood, "Harmonic morphisms between Riemannian manifolds", In T. Kotake, S. Nishikawa and R. Schoen, Geometry and Global Anaiysis, 413-422, Tohoku Univ., Sendai, 1993.
[18] J. C. Wood, "Harmonic maps and morphisms in 4 dimensions", Proceedings of the first Brazilian-USA Workshop on Geometry, Topology and Physics.
[19] T. Ishihara, "A mapping of Riemannian manifolds which preserves harmonic functions", Journal Mathematics Kyoto University (19), 215-229, 1979.
[20] 陈维恒，“微分几何初步”，北京，北京大学出版社，1990．
[21] 伍鸿熙，“黎曼几何初步”，北京，北京大学出版社，1989．
[22] M. P. Do Carmo, "Differential Geometry of Curves and Surface", Pentice hall, 1976.
[23] N. J. Hicks, "Notes on Differential Geometry", D. Van Nostrand, Princeton, 1965.
[24] R. Alexander and S. Alexander, "Geodesics in Riemannian manifolds with boundary", Indiana University Mathematics Journal 30: 4, 1981.
[25] S. Alexander, I. D. Berg and R. L. Bishop, "The Riemannian obstacle problem", Illinois Journal of Mathematics 31: 1, Spring 1987.
[26] J. M. Corcuera and W. S. Kendall, "Riemannian barycenters and geodesic convexity", Preprint, University of Warwick, 1983.
[27] V. Caselles, R. Kimmel, G. Sapiro and C. Sbert, "Minimal surfaces based object segmentation", IEEE-PAMI, 19: 4, pp. 394-398, April 1997.
[28] H. B. Lawson, "Lectures on minimal submanifolds", Volume Ⅰ, Mathematics Lecture Series(9), Publish or Perish, 1980.
[29] J. J. Koenderink, "The structure of images", Biological Cybernetics 50, pp. 363-370, 1984.
[30] A. K. Jain, "Partial differential equations and finite-difference methods in image processing, part 1: Image representation", Journal of Optimization Theory and Applications 23, pp. 65-91, 1977.
[31] R. A. Hummel, "Representations based on zero-crossing in scale-space", Proceeding of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 204-299, IEEE New York, 1986.
[32] L. Alvarez, P. L. Lions and J. M. Morel, "Image selective smoothing and edge detection by nonlinear diffusion", SIAM J. Numer. Anal. 29, pp. 845-866, 1992.
[33] B. Romeny, "Geometry Driven Diffusion in Computer Vision", Kluwer, The Netherlands, 1994.
[34] S. J. Osher and L. I. Rudin, "Feature-oriented image enhancement using shock filters", SIAM J. Numer. Anal. 27, pp. 919-940, 1990.
[35] Marcelo Bertalmio, "Processing of flat and non-flag image information on arbitrary manifolds using partial differential equations", Ph. D thesis, University of Minnesota, 2001.
[36] R. Kimmel and N. Sochen, "Orientation diffusion", Journal of Visual Communication and Image Representation, to appear.
[37] T. Chan and J. Shen, "Variationsl restoration of non-flat image feature: Models and algorithms", UCLA CAM-TR 99-20, June 1999.
[38] B. Tang, G Sapiro and V. Caselles, "Diffusion of general data on non-flat manifolds via harmonic maps theory: The direction diffusion case", Int. Journal Computer Vision 36: 2, pp. 149-161, February 2000.
[39] Dongmei Zhang and Martial Hebert, "Harmonic maps and their applications in surface matching", In IEEE Proceedings of Computer Vision Pattern Recognition, 1999.
[40] S. F. Frisken, R. N. Perry, A. Rockwood and T. Jones, "Adaptively sampled fields: A general representation of shape for computer graphics" , Computer Graphics (SIGGRAPH), New Orleans, July 2000.
[41] 胡健伟、汤怀民，“微分方程数值方法”，北京，科学出版社，1999．
[42] R. Courant, K. O. Friedrichs and H. Lewy, "On the partial difference equations of mathematical physics", IBM Journal of Research and Development 11 (1967), 215-234.
[43] A. R. Mitchell and D. F. Griffiths, "The finite difference methods in partial differential equations", John Wiley & Sons, 1980.
[44] R. D. Richtmyer and K. W. Morton, "Difference methods for initial value problem", John Wiley & Sons, 1967.
[45] J. H. Bramble, "Multigrid method", Longman, Harlow, 1993.
[46] 李荣华、冯果忱，“微分方程数值解法”，高等教育出版社，1996．
[47] F. Alouges, "An energy decreasing algorithm for harmonic maps", in J. M. Coron et al., Editors, Nematics, Nato ASI Series, Kluwer Academic Publishers, Netherlands, pp. 1-13, 1991.
[48] L. Rudin, S. Osher and E. Fatemi, "Nonlinear total variation based noise removal algorithms", Physics D, 60, pp. 259-268, 1992.
[49] L. Ambrosio and N. Dancer, Calculus of Variations and Partial Differential Equations, Topics on Geometrical Evolution Problem and Degree Theory, Spring-Verlag, New York, 2000.
[50] M. Bertalmio, L. T. Cheng, S. Osher and G. Sapiro, "Variational problems and partial differential equations on implicit surfaces: The framework and example in image processing and pattern formation", IMA University of Minnesota Preprint, June 2000.
[51] 周济、周艳红，“数控加工技术”，北京，国防工业出版社，2002．
[52] 王春海、樊锐、赵先仲，“数字化加工技术”，北京，化学工业出版社，2003．
[53] 韩鸿鸾、荣维芝，“数控机床加工程序的编制”，北京，机械工业出版社，2002．
[54] 雷宇、梁剑新，“平面数控雕刻数控自动编程系统的开发”，机械制造(008)：25-26，2001．
[55] 李俊山、陆鹏，“遥感图像边缘检测人工神经网络”，模式识别与人工智能(11)：479-485，1998．
[56] 林金明、朱国力，“基于图象的刀具路径生成技术”，制造业自动化(011)：49-51，2001．
[57] 王青松、傅建中，“面向对象的计算机数控雕刻控制软件”，机电一体化(006)：40-43，2000．
[58] 邓中亮，“数控雕刻加工刀具轨迹自动生成方法”，制造技术与机床(007)：27-30，1997．
[59] 吴石林，“任意回转体圆周表面数控雕刻技术”，制造技术与机床(009)：33-35，2002．
[2] J. Eells and L. Lemaire, "A report on harmonic maps", Bull. London Math. Soc. 10: 1, pp. 1-68, 1978.
[3] J. Eells and L. Lemaire, "Another report on harmonic maps", Bull. London Math. Soc. 20:5, pp. 385-524, 1988.
[4] B. Fuglede, "Harmonic Morphisms between Riemannian manifolds", Ann. Inst. Fourier(Grenoble)(28), 107-144, 1978.
[5] P. Baird, Harmonic Maps with Symmetry, Harmonic Morphisms and Deformation of Metrics, Research Notes in Math. (87), Pitman, Boston, London, Melbourne, 1983.
[6] P. Baird and J. Eells, "A conservation law for harmonic maps", In E. Looijenga, D. Siersma and F. Takens, Geometry Symposium. Utrecht 1980, 1-25, Lecture Notes in Math. (894), Springer-Verlag, Berlin, Heidelberg, New York, 1981.
[7] P. Baird and J. C. Wood, "Harmonic morphisms and conformal foliations by geodesics of three-dimensional space forms", Journal Austral. Math. Soc. Ser. A(51), 118-153, 1991.
[8] S. Gudmundsson, "Harmonic morphisms from complex projective spaces", Geometry Dedicata (53), 155-161, 1994.
[9] S. Gudmundsson, "On the existence of harmonic morphisms from symmetric spaces of rank one", Preprint, Lund University, 1996.
[10] S. Gudmundsson and R. Sigurdsson, "A note on the classification of holomorphic harmonic morphisms", Potential Analysis(2), 295-298, 1993.
[11] S. Gudmundsson, S. Montaldo and T. Mustafa, "The atlas of harmonic morphisms", http://www.maths.1th.se/matematiklu/personal/sigma/harmonic/atlas.html.
[12] S. Gudmundsson, "The bibliography of harmonic morphisms", http://www.maths.1th.se/matematiklu/personal/sigma/harmonic/bibliography.html.
[13] F. E. Burstall, L. Lemaire and J. Rawnsley, "Harmonic maps bibliography", http://www.bath.ac.uk/masfeb/harmonic, html.
[14] G. Gigantes, "A note on harmonic morphisms", Preprint, Univ. Camerino, 1983.
[15] Y. L. Ou, "Quadratic harmonic morphisms and O-systems", Preprint, University of leeds, 1995.
[16] Y. L. Ou and J. C. Wood, "On the classification of quadratic harmonic morphisms between Euclidean spaces", Algebras, Groups and Geometry (13), 41-53, 1996.
[17] J. C. Wood, "Harmonic morphisms between Riemannian manifolds", In T. Kotake, S. Nishikawa and R. Schoen, Geometry and Global Anaiysis, 413-422, Tohoku Univ., Sendai, 1993.
[18] J. C. Wood, "Harmonic maps and morphisms in 4 dimensions", Proceedings of the first Brazilian-USA Workshop on Geometry, Topology and Physics.
[19] T. Ishihara, "A mapping of Riemannian manifolds which preserves harmonic functions", Journal Mathematics Kyoto University (19), 215-229, 1979.
[20] 陈维恒，“微分几何初步”，北京，北京大学出版社，1990．
[21] 伍鸿熙，“黎曼几何初步”，北京，北京大学出版社，1989．
[22] M. P. Do Carmo, "Differential Geometry of Curves and Surface", Pentice hall, 1976.
[23] N. J. Hicks, "Notes on Differential Geometry", D. Van Nostrand, Princeton, 1965.
[24] R. Alexander and S. Alexander, "Geodesics in Riemannian manifolds with boundary", Indiana University Mathematics Journal 30: 4, 1981.
[25] S. Alexander, I. D. Berg and R. L. Bishop, "The Riemannian obstacle problem", Illinois Journal of Mathematics 31: 1, Spring 1987.
[26] J. M. Corcuera and W. S. Kendall, "Riemannian barycenters and geodesic convexity", Preprint, University of Warwick, 1983.
[27] V. Caselles, R. Kimmel, G. Sapiro and C. Sbert, "Minimal surfaces based object segmentation", IEEE-PAMI, 19: 4, pp. 394-398, April 1997.
[28] H. B. Lawson, "Lectures on minimal submanifolds", Volume Ⅰ, Mathematics Lecture Series(9), Publish or Perish, 1980.
[29] J. J. Koenderink, "The structure of images", Biological Cybernetics 50, pp. 363-370, 1984.
[30] A. K. Jain, "Partial differential equations and finite-difference methods in image processing, part 1: Image representation", Journal of Optimization Theory and Applications 23, pp. 65-91, 1977.
[31] R. A. Hummel, "Representations based on zero-crossing in scale-space", Proceeding of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 204-299, IEEE New York, 1986.
[32] L. Alvarez, P. L. Lions and J. M. Morel, "Image selective smoothing and edge detection by nonlinear diffusion", SIAM J. Numer. Anal. 29, pp. 845-866, 1992.
[33] B. Romeny, "Geometry Driven Diffusion in Computer Vision", Kluwer, The Netherlands, 1994.
[34] S. J. Osher and L. I. Rudin, "Feature-oriented image enhancement using shock filters", SIAM J. Numer. Anal. 27, pp. 919-940, 1990.
[35] Marcelo Bertalmio, "Processing of flat and non-flag image information on arbitrary manifolds using partial differential equations", Ph. D thesis, University of Minnesota, 2001.
[36] R. Kimmel and N. Sochen, "Orientation diffusion", Journal of Visual Communication and Image Representation, to appear.
[37] T. Chan and J. Shen, "Variationsl restoration of non-flat image feature: Models and algorithms", UCLA CAM-TR 99-20, June 1999.
[38] B. Tang, G Sapiro and V. Caselles, "Diffusion of general data on non-flat manifolds via harmonic maps theory: The direction diffusion case", Int. Journal Computer Vision 36: 2, pp. 149-161, February 2000.
[39] Dongmei Zhang and Martial Hebert, "Harmonic maps and their applications in surface matching", In IEEE Proceedings of Computer Vision Pattern Recognition, 1999.
[40] S. F. Frisken, R. N. Perry, A. Rockwood and T. Jones, "Adaptively sampled fields: A general representation of shape for computer graphics" , Computer Graphics (SIGGRAPH), New Orleans, July 2000.
[41] 胡健伟、汤怀民，“微分方程数值方法”，北京，科学出版社，1999．
[42] R. Courant, K. O. Friedrichs and H. Lewy, "On the partial difference equations of mathematical physics", IBM Journal of Research and Development 11 (1967), 215-234.
[43] A. R. Mitchell and D. F. Griffiths, "The finite difference methods in partial differential equations", John Wiley & Sons, 1980.
[44] R. D. Richtmyer and K. W. Morton, "Difference methods for initial value problem", John Wiley & Sons, 1967.
[45] J. H. Bramble, "Multigrid method", Longman, Harlow, 1993.
[46] 李荣华、冯果忱，“微分方程数值解法”，高等教育出版社，1996．
[47] F. Alouges, "An energy decreasing algorithm for harmonic maps", in J. M. Coron et al., Editors, Nematics, Nato ASI Series, Kluwer Academic Publishers, Netherlands, pp. 1-13, 1991.
[48] L. Rudin, S. Osher and E. Fatemi, "Nonlinear total variation based noise removal algorithms", Physics D, 60, pp. 259-268, 1992.
[49] L. Ambrosio and N. Dancer, Calculus of Variations and Partial Differential Equations, Topics on Geometrical Evolution Problem and Degree Theory, Spring-Verlag, New York, 2000.
[50] M. Bertalmio, L. T. Cheng, S. Osher and G. Sapiro, "Variational problems and partial differential equations on implicit surfaces: The framework and example in image processing and pattern formation", IMA University of Minnesota Preprint, June 2000.
[51] 周济、周艳红，“数控加工技术”，北京，国防工业出版社，2002．
[52] 王春海、樊锐、赵先仲，“数字化加工技术”，北京，化学工业出版社，2003．
[53] 韩鸿鸾、荣维芝，“数控机床加工程序的编制”，北京，机械工业出版社，2002．
[54] 雷宇、梁剑新，“平面数控雕刻数控自动编程系统的开发”，机械制造(008)：25-26，2001．
[55] 李俊山、陆鹏，“遥感图像边缘检测人工神经网络”，模式识别与人工智能(11)：479-485，1998．
[56] 林金明、朱国力，“基于图象的刀具路径生成技术”，制造业自动化(011)：49-51，2001．
[57] 王青松、傅建中，“面向对象的计算机数控雕刻控制软件”，机电一体化(006)：40-43，2000．
[58] 邓中亮，“数控雕刻加工刀具轨迹自动生成方法”，制造技术与机床(007)：27-30，1997．
[59] 吴石林，“任意回转体圆周表面数控雕刻技术”，制造技术与机床(009)：33-35，2002．