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Laplacian坐标在三角网格变形中的应用研究

The Research of Laplacian Coordinates for Mesh Deformation

作者: 专业:软件工程 导师:刘秀平 年度:2009 学位:硕士  院校: 大连理工大学

Keywords

Laplacian Coordinates, Unit Weight, Cotangent Weight, The Mean Curvature Vector

        本文对网格编辑中的Laplacian坐标进行了详细的介绍。Laplacian坐标是用来存储网格的几何信息和一些相关的参数信息的,本文分别从Laplacian坐标的不同的权重系数的角度,对不同形式的权的坐标讨论了它们的几何意义,并给出了实验结果。近几年,用微分信息来存储局部内在特征已经被广泛的应用于网格编辑中。给定约束条件,通过使微分信息变化最小来实现变形网格的重建。因为微分信息在全局坐标系中体现,所以不具有旋转不变性。我们看到要是变形网格保持某种视觉效果,就要保持网格的局部参数化和几何信息,为了表示这两个信息,Laplacian算子的特殊性质起到了重要作用。特别的,可以把Laplacian算子的系数当作参数信息,把Laplacian坐标的大小看作几何信息。这两组信息集合都是没有方向且非线性的,只与点的位置有关。Unit权Laplacian坐标的意义是,顶点i的所有临域点围成的几何图形的中心点到这个顶点的向量;Cotangent权Laplacian坐标的意义是,网格在这个顶点处的离散曲率法向量,它可以很好的存储网格的特征信息;Tangent权是在Cotangent权的基础上出现的,是为了解决一定范围内的大角度变形问题中系数的非负性。本文还介绍了近几年解Laplacian编辑系统的常用方法,主要是对旋转问题产生的扭曲的处理方法,还有在某些机器加工领域中,对变形问题的保持特定区域不变的要求的特殊处理方法。并且使用了其中一种编辑系统,对Unit权Laplacian坐标和Cotangent权Laplacian坐标,以实验结果的形式进一步说明了它们的几何意义和适用范围,与理论上得到的结果完全一致。
    In this thesis, we introduced the Laplacian Coordinates for mesh editing. We considered the different weights of Laplacian Coordinates and discussed their geometric meaning. The meaning of Laplacian Coordinates with unit weights is the vector from the vertex to the center of its adjacent vertices.In recent years, differential information has been greatly used for mesh editing. Given certain constraints, a deformed mesh is reconstructed by minimizing the changes in the differential information. Since the differential information is encoded in the global coordinate system, it must be transformed to fit the orientation of details in the deformed surface. Otherwise distortion will appear. We observe that visually desired deformed meshes should preserve both local parameterization and geometry details.To find suitable representations for the two information, the special property of Laplacian coordinates is useful.Specially, we can consider the coefficients of Laplacian operator as the parametrization information and the magnitudes of the Laplacian coordinates as the geometry information. Both sets of informations are non-directional and non-linearly dependent on the vertex positions. The meaning of Laplacian Coordinates with cotangent weights is the mean value curvature of the vertex. The cotangent weights may be negative and problematic to define for very large angles due to the properties of cotangent nearπ;convex weights that mimic the cotangent weights are called mean-value coordinates.We also introduced the method of Laplacian Editing in recent years.The main problem is that reconstructing the mesh by minimizing deviation from the original Laplacian Coordinates would lead to undesired distortion, specifically, shearing and stretching distortion. There are many people to study this problem. We chose one of their method in our experiment and got the comparison between unit weights and cotangent weights. From the experiment results, we can see there are same With theory.
        

Laplacian坐标在三角网格变形中的应用研究

摘要4-5
Abstract5
1 绪论8-17
    1.1 非自由变形8-9
    1.2 自由变形9-10
    1.3 多分辨率编辑10-11
    1.4 微分方法11-15
        1.4.1 网格的表示方法11-12
        1.4.2 网格降噪12-13
        1.4.3 网格编辑13-14
        1.4.4 网格重新计算14-15
    1.5 本文主要工作15-17
2 基于Laplacian坐标的网格变形17-35
    2.1 Laplacian坐标17-23
        2.1.1 Laplacian坐标的性质17-18
        2.1.2 Unit权Laplacian坐标18-20
        2.1.3 Cotangent权Laplacian坐标20-21
        2.1.4 Tangent权Laplacian坐标21-23
    Laplacian去噪算法23-27
        2.2.1 Taubin算法24-25
        2.2.2 HC算法25-26
        2.2.3 Desbrun算法26-27
    2.3 Laplacian编辑系统27-35
        2.3.1 Lipman对旋转问题的解决方法28-30
        2.3.2 Sorkine对旋转问题的解决方法30-32
        2.3.3 Oscar对旋转问题的解决方法32-35
3 保特征变形35-39
    3.1 位置约束35-36
    3.2 旋转约束36-37
    3.3 保持特定区域不变37-39
        3.3.1 保持特定区域的形状不变表示法37-38
        3.3.2 特定区域位置移动限制表示法38-39
4 实验部分39-51
    4.1 实验步骤39-40
    4.2 实验流程图40-41
    4.3 核心代码41-44
    4.4 实验效果图及分析44-51
结论51-52
参考文献52-55
攻读硕士学位期间发表学术论文情况55-56
致谢56-58
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