隐参数曲线曲面及基于PHT样条的间断图像缩放
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摘要
几何造型是计算机辅助几何设计和计算机图形学中的核心研究内容,主要解决计算机图像系统环境中曲线、曲面的表示、设计和分析等问题。随着时代的发展和科技的日新月异,在工业、影视及娱乐业等应用领域出现了越来越多的曲线、曲面处理方面的研究课题,推动着几何造型技术不断发展。本论文的主要工作是围绕着曲线曲面的表示而展开。
本文首先介绍了CAD/CAM造型系统中曲线、曲面的参数表示方法和隐式表示方法,分别讨论它们的优点和缺陷,并概述两种表示方法之间的相互转化方式和这两种表示方法在数据重构方面的研究进展。
为了能更好地利用这两种表示方法的优点,且希望给出一种便于两种表示方法之间进行相互转化的方式,我们在第二章中提出了一种新的表示形式一一隐参数曲线、曲面。结合参数曲线、曲面的微分几何性质研究了新表示形式的微分几何性质,并给出了隐参数表示与参数表示、隐式表示之间的相互转换方式。基于张量积B样条函数表示,我们演示了隐参数曲线、曲面易于表示复杂拓扑的曲线、曲面和可以通过改变控制网而修改曲线、曲面形状的特性。
隐参数表示在参数表示和隐式表示之间的相互转化的过程中起到桥梁的作用。而其中隐式表示到隐参数表示的转化是一个难点,这也是论文中所要解决的主要问题。在第三章中我们给出了对于给定隐式曲线构造其对应的近似隐参数曲线的数学模型,其中所得映射是拓扑同胚于对应标准型的连续映射。并把该模型推广到三维空间中,构造隐式曲面的隐参数曲面表示。第四章中,我们对隐式曲线的隐参数曲线构造模型进行推广,用来构造给定离散数据点到多边形框架的映射,并将该模型应用于数据图像处理中。
论文的第五章是利用T网格上样条理论研究带有间断特征的数字图像缩放问题,即利用具有间断效果的PHT样条进行数据重构和数据重采样。根据PHT样条插值基点几何信息的特性,在间断的基点处使用不同的几何信息,使数据重构结果具有间断效果。由于这种方法仅能处理间断特征位于水平和竖直位置上的数据,所以对于一般的间断问题需先将间断特征线映射到水平和竖直位置。文中结合最优化方法提供了将间断特征线映射到水平和竖直位置上的参数化模型,并利用所得参数化结果进行数据拟合。该方法解决了一般的样条函数本身由于整体连续性所造成的信息平滑问题,保留了图像中某些位置上的高频信息,且图像的缩放结果不会产生模糊现象,实验结果说明该方法的有效性。最后,在第六章中我们总结了本文所做的工作,并对将来的工作做了展望。
Geometric modeling is one of the key technologies in the fields of computer aided geometric design and computer graphics. Its main research topics include represen-tation, design, display, and analysis of curves and surfaces. With the progress of science and technology in recent years, research relevant to the processing of curves and surfaces in industry, film and entertainment is required, which promotes con-tinuous development of geometric modeling techniques. This thesis addresses the representation of curves and surfaces and its applications.
Firstly, the two major representations of curves and surfaces in CAD/CAM systems, which are parametric and implicit presentation, are briefly recalled. The inherent advantages and disadvantages of both representations are detailed. Re-search work of conversion between them and data reconstructions with them is summarized.
In order to better take advantage of these two representations and give a con-venient way for the conversion between them, we present a new representation of curves and surfaces in chapter2, named parametric curves with implicit domains and parametric surfaces with implicit domains. According to the differential geom-etry properties of parametric curves and surfaces, we give the relevant differential geometry concepts of the new representation forms. And the conversion between the new representation and any of the other two representations which are parametric and implicit is provided. Finally examples are provided to show its application in representation of curves or surfaces with complex topology and the local control of curves or surfaces for the parametric curves with implicit domains or parametric surfaces with implicit domains with the tensor-product B-splines forms.
PCID provides a bridge between parametric curve and implicit curve because the conversion of parametric form or implicit form to PCID is convenient and effi-cient. Since the conversion from implicit form to PCID form is difficult and impor-tant, this is the main problem to be solved in the paper. In chapter3, we propose a framework model for constructing the approximate PCID of given implicit curve. It can map given points to the implicit domain homeomorphically. The resulting map is smooth and overlapping-free. The model is also extended to three-dimensional space, constructing approximate parametric surfaces with implicit domains of im-plicit surfaces. In chapter4, we extend the model to more general cases, which constructs the mapping between two groups of given sample points. And we also successfully apply the proposed models to several different fields of digital image processing.
In chapter5, image zooming with intermittent features is studied based on the theory of PHT-spline, namely data reconstruction and data resampling using PHT-spline with mixed orders of continuity. According to the characteristics of in-terpolation Hermitian information at basis vertices for PHT-spline, the image data reconstruction can achieve the intermittent effect by using different Hermitian in-formation at the intermittent basis vertices. Since this method can just handle the data whose intermittent characteristic line is in the horizontal and vertical position, a parametric model combined optimization method is provided for general inter-mittent data, which can map the intermittent curves to horizontal or vertical lines. Based on the parameterization result, reconstruction image is got. This method solves the image smoothing problem of general spline functions due to the overall continuity. Reconstruction image can not only maintain a higher resolution in some locations, but also have good smoothness in other locations. The image scaling will not produce blurring in the feature lines, and experimental results show the effectiveness of this method.
In the end, we conclude our thesis with future research problems.
本文首先介绍了CAD/CAM造型系统中曲线、曲面的参数表示方法和隐式表示方法,分别讨论它们的优点和缺陷,并概述两种表示方法之间的相互转化方式和这两种表示方法在数据重构方面的研究进展。
为了能更好地利用这两种表示方法的优点,且希望给出一种便于两种表示方法之间进行相互转化的方式,我们在第二章中提出了一种新的表示形式一一隐参数曲线、曲面。结合参数曲线、曲面的微分几何性质研究了新表示形式的微分几何性质,并给出了隐参数表示与参数表示、隐式表示之间的相互转换方式。基于张量积B样条函数表示,我们演示了隐参数曲线、曲面易于表示复杂拓扑的曲线、曲面和可以通过改变控制网而修改曲线、曲面形状的特性。
隐参数表示在参数表示和隐式表示之间的相互转化的过程中起到桥梁的作用。而其中隐式表示到隐参数表示的转化是一个难点,这也是论文中所要解决的主要问题。在第三章中我们给出了对于给定隐式曲线构造其对应的近似隐参数曲线的数学模型,其中所得映射是拓扑同胚于对应标准型的连续映射。并把该模型推广到三维空间中,构造隐式曲面的隐参数曲面表示。第四章中,我们对隐式曲线的隐参数曲线构造模型进行推广,用来构造给定离散数据点到多边形框架的映射,并将该模型应用于数据图像处理中。
论文的第五章是利用T网格上样条理论研究带有间断特征的数字图像缩放问题,即利用具有间断效果的PHT样条进行数据重构和数据重采样。根据PHT样条插值基点几何信息的特性,在间断的基点处使用不同的几何信息,使数据重构结果具有间断效果。由于这种方法仅能处理间断特征位于水平和竖直位置上的数据,所以对于一般的间断问题需先将间断特征线映射到水平和竖直位置。文中结合最优化方法提供了将间断特征线映射到水平和竖直位置上的参数化模型,并利用所得参数化结果进行数据拟合。该方法解决了一般的样条函数本身由于整体连续性所造成的信息平滑问题,保留了图像中某些位置上的高频信息,且图像的缩放结果不会产生模糊现象,实验结果说明该方法的有效性。最后,在第六章中我们总结了本文所做的工作,并对将来的工作做了展望。
Geometric modeling is one of the key technologies in the fields of computer aided geometric design and computer graphics. Its main research topics include represen-tation, design, display, and analysis of curves and surfaces. With the progress of science and technology in recent years, research relevant to the processing of curves and surfaces in industry, film and entertainment is required, which promotes con-tinuous development of geometric modeling techniques. This thesis addresses the representation of curves and surfaces and its applications.
Firstly, the two major representations of curves and surfaces in CAD/CAM systems, which are parametric and implicit presentation, are briefly recalled. The inherent advantages and disadvantages of both representations are detailed. Re-search work of conversion between them and data reconstructions with them is summarized.
In order to better take advantage of these two representations and give a con-venient way for the conversion between them, we present a new representation of curves and surfaces in chapter2, named parametric curves with implicit domains and parametric surfaces with implicit domains. According to the differential geom-etry properties of parametric curves and surfaces, we give the relevant differential geometry concepts of the new representation forms. And the conversion between the new representation and any of the other two representations which are parametric and implicit is provided. Finally examples are provided to show its application in representation of curves or surfaces with complex topology and the local control of curves or surfaces for the parametric curves with implicit domains or parametric surfaces with implicit domains with the tensor-product B-splines forms.
PCID provides a bridge between parametric curve and implicit curve because the conversion of parametric form or implicit form to PCID is convenient and effi-cient. Since the conversion from implicit form to PCID form is difficult and impor-tant, this is the main problem to be solved in the paper. In chapter3, we propose a framework model for constructing the approximate PCID of given implicit curve. It can map given points to the implicit domain homeomorphically. The resulting map is smooth and overlapping-free. The model is also extended to three-dimensional space, constructing approximate parametric surfaces with implicit domains of im-plicit surfaces. In chapter4, we extend the model to more general cases, which constructs the mapping between two groups of given sample points. And we also successfully apply the proposed models to several different fields of digital image processing.
In chapter5, image zooming with intermittent features is studied based on the theory of PHT-spline, namely data reconstruction and data resampling using PHT-spline with mixed orders of continuity. According to the characteristics of in-terpolation Hermitian information at basis vertices for PHT-spline, the image data reconstruction can achieve the intermittent effect by using different Hermitian in-formation at the intermittent basis vertices. Since this method can just handle the data whose intermittent characteristic line is in the horizontal and vertical position, a parametric model combined optimization method is provided for general inter-mittent data, which can map the intermittent curves to horizontal or vertical lines. Based on the parameterization result, reconstruction image is got. This method solves the image smoothing problem of general spline functions due to the overall continuity. Reconstruction image can not only maintain a higher resolution in some locations, but also have good smoothness in other locations. The image scaling will not produce blurring in the feature lines, and experimental results show the effectiveness of this method.
In the end, we conclude our thesis with future research problems.
引文
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