平面隐式曲线的Hermite插值逼近
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摘要
隐式曲线在医学图像处理、地理信息系统、数值场可视化等领域中有着重要应用。在分析点采样和曲线逼近理论的基础上,提出一种运用Hermite插值方法逼近平面隐式曲线的算法。首先将曲线绘制区域网格化,在网格单元各边中通过线性插值计算曲线采样点;其次通过计算采样点精简前后构成的曲线段之间产生的误差优化采样点;最后通过Hermite插值法逼近隐函数曲线。实验表明,通过该算法绘制出的曲线在采样点数量较少的情况下,其光滑度和准确度仍较高。
Implicit curves play an important role in medical image processing, geographic information system, and numerical field visualization. On the basis of sampling point analysis and curve approximation method, we introduce an algorithm for approximating planar implicit curves by means of Hermite interpolation. The sampling points were firstly obtained by linearly interpolating each edge of the grid cells distributed uniformly in the grid region. Then, we calculated the error between curve segments before and after optimizing. Once the error meets the optimizing requirements, the sampling points are consequently optimized. Finally, the algorithm approximated the implicit curves by the Hermite interpolation method. Experiments have shown that even when the number of sampling points is small, the curves drawn by the algorithm still have relatively higher smoothness and accuracy.
Implicit curves play an important role in medical image processing, geographic information system, and numerical field visualization. On the basis of sampling point analysis and curve approximation method, we introduce an algorithm for approximating planar implicit curves by means of Hermite interpolation. The sampling points were firstly obtained by linearly interpolating each edge of the grid cells distributed uniformly in the grid region. Then, we calculated the error between curve segments before and after optimizing. Once the error meets the optimizing requirements, the sampling points are consequently optimized. Finally, the algorithm approximated the implicit curves by the Hermite interpolation method. Experiments have shown that even when the number of sampling points is small, the curves drawn by the algorithm still have relatively higher smoothness and accuracy.
引文
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[12]吴坚,张接信,蔡宗琰.用短线截交法绘制隐式曲线[J].机械科学与技术,2006,25(8):965-966.
[13]SUFFERN K G.Quadtree algorithms for contouring functions of two variables[J].Computer Journal,1990,33(5):402-407.
[14]DUFF T.Interval arithmetic recursive subdivision for implicit functions and constructive solid geometry[J].ACM Computer Graphics,1992,26(2):131-138.
[15]寿华好,何苹,缪永伟.自动微分在隐式曲线绘制中的应用[J].计算机工程与应用,2009,46(1):150-153.
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[17]李洪发.分段三次Hermite插值的同时逼近[J].天津师范大学学报:自然版,2012,32(2):38-40.
[18]FUHRMANN S,KAZHDAN M,GOESELE M.Accurate isosurface interpolation with hermite data[C]//2015 International Conference on 3D Vision.New York:IEEE Press,2015:256-263.
[2]刘颖.复杂的隐式函数曲线绘制算法的研究[D].长春:长春大学,2006.
[3]张三元,孙守迁,蒋方炎,等.数字化仿真人体模型的设计方法[J].系统仿真学报,2000,12(1):49-50.
[4]温维亮,郭新宇,陆声链,等.隐式曲面在三维植物建模中的应用研究综述[J].图学学报,2010,31(3):1-10.
[5]赵伟,赵卓宁,李五生.一种有效的离散数据场等值线生成方法[J].成都信息工程学院学报,2007,22(1):116-121.
[6]SHELBERG M C,MOELLERING H,LAM N.Measuring the fractal dimensions of empirical cartographic curves[C]//Auto-Carto 5.Virginia:American Academy of Photometry and American Congress on Surveying and Mapping,1982:481-490.
[7]SUNDARAMOORTHI G,YEZZI A,MENNUCCI A.Coarse-to-fine segmentation and tracking using Sobolev active contours[J].IEEE Transactions on Pattern Analysis&Machine Intelligence,2008,30(5):851-864.
[8]TAUBIN G.Distance approximations for rasterizing implicit curves[J].ACM Transactions on Graphics,1994,13(1):3-42.
[9]童若锋,汪国昭,金通光.轮廓跟踪的TN方法[C]//第一届全国几何设计与计算学术会议论文集.东营:中国石油大学出版社,2002:579-582.
[10]余正生.隐式曲面造型与绘制算法研究[D].杭州:浙江大学,1999.
[11]蔡耀志.正负法数控绘图[J].工程图学学报,1984,5(Z1):3-9.
[12]吴坚,张接信,蔡宗琰.用短线截交法绘制隐式曲线[J].机械科学与技术,2006,25(8):965-966.
[13]SUFFERN K G.Quadtree algorithms for contouring functions of two variables[J].Computer Journal,1990,33(5):402-407.
[14]DUFF T.Interval arithmetic recursive subdivision for implicit functions and constructive solid geometry[J].ACM Computer Graphics,1992,26(2):131-138.
[15]寿华好,何苹,缪永伟.自动微分在隐式曲线绘制中的应用[J].计算机工程与应用,2009,46(1):150-153.
[16]LORENSEN W E,CLINE H E.Marching cubes:a high resolution 3D surface construction algorithm[J].ACM Computer Graphics,1987,21(4):163-169.
[17]李洪发.分段三次Hermite插值的同时逼近[J].天津师范大学学报:自然版,2012,32(2):38-40.
[18]FUHRMANN S,KAZHDAN M,GOESELE M.Accurate isosurface interpolation with hermite data[C]//2015 International Conference on 3D Vision.New York:IEEE Press,2015:256-263.