数字几何处理中球面参数化和重新网格化研究
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摘要
随着现代科学技术的不断发展,三维几何模型成为继声音、图像和视频之后的第4代多媒体数据类型在工业界得到了广泛的应用。因此,一门新的处理三维几何数据的学科——数字几何处理也应运而生,并逐渐成为数字信号处理的研究热点。但三维几何模型的特点显著区别于其它传统媒体,它所表达的信号的拓扑结构是任意复杂的,即可能有任意的亏格数以及任意条边界。这导致一些传统的信号处理工具不能直接用于处理三维几何模型,而需要研究更加复杂的数字几何处理算法。本文以三角网格数据为基础,在球面参数化和重新网格化两个方面对数字几何处理进行了研究,主要工作包括:
1.针对封闭的亏格为0的三角网格,提出了一种基于质心坐标和M(o|¨)bius变换的均匀准保角球面参数化方法(UQCSP方法)。算法首先通过立体投影和现有的固定圆边界的平面参数化方法得到一个初始的球面映射;然后引入原始网格顶点在切平面上的质心坐标对初始的球面参数化进行优化来减少变形;最后用将球面网格顶点的质心变换到球心的M(o|¨)bius变换均匀化最终的球面网格。该方法能够在球面参数化角度变形较小的情况下减少面积变形,而且也能够保证一般保角球面参数化方法所不具有的有效性。此外,本文也将该方法推广到单边界亏格为0的网格上,所得参数化在复杂网格的纹理映射中的均匀化效果较现有的保角、保面积变换有明显的改善。
2.针对封闭的亏格为0的三角网格,提出了基于对称性分析的球面参数化思想。该思想利用物体的几何对称性质进行分割,将复杂的球面参数化问题转化为更为简单的平面参数化问题。通过该思想可以得到两种均衡角度和面积变形的球面参数化方法:基于拉伸度量的球面参数化方法(BSMSP方法)和基于拟调和映射的球面参数化方法(BQHSP方法)。它们都能够解决保角球面映射对于复杂结构的物体面积变形较大的问题。对于形状接近于球面的物体它们都能够获得几乎等距的球面参数化,而且只需要求解几次线性方程组,非常高效。其中BSMSP方法能够严格保证球面参数化的有效性。尽管BQHSP方法在理论上无法保证,但是相对于BSMSP方法来说,它能够减少复杂结构物体的球面参数化的角度变形。此外,我们将BQHSP方法所得的球面参数化应用到细分连接性的重新网格化中。通过实验结果可以看出,这种方法能够获得高质量的细分连接性网格。
3.针对任意亏格、任意边界的三角网格,提出了一种基于距离场的自适应各向同性的重新网格化方法。各向同性的重新网格化中研究得比较多的一类方法是通过对原始网格进行全局参数化来实现的。但是对于高亏格带边界的复杂的三角网格模型,全局参数化本身是一个非常复杂的问题。因此,本研究直接对三角网格进行自适应操作,不采用全局参数化。算法首先通过对网格的边进行折叠或者分裂得到需要的顶点数目;然后通过优化网格顶点的位置和连接性来改进网格中三角形的形状和顶点位置分布。为了减少网格优化过程中误差的累积,通过引入网格曲面的距离场将新生成的顶点保留到原始网格模型上。实验证明,这是一种简单、快速而有效的方法。
With the development of modern science techniques. 3D geometrical models are recently becoming a new type of medium after sound, images and video, which are applied in industry widely. Consequently, a new discipline processing 3D geometrical models is born, which is called digital geometry processing. It has been a focus of research activity in digital signal processing. Unfortunately, 3D geometrical models are significantly different from other traditional mediums. The topology of 3D geometrical models are very complicated, that is to say, they have any genus and boundary. So traditional signal processing techniques can not be extended to process 3D geometrical models. We must develop some more intricate digital geometry processing algorithms. In this dissertation. we take the triangular mesh models as our research object, and do some research on spherical parameterization and remeshing in digital geometry processing. The main work can be summarized as follows:
1. We present a uniform quasi-conformal spherical parameterization method (UQCSP method) based on barycentric coordinates and M(o|¨)bius transformation for closed genus-zero triangular meshes. An initial spherical parameterization is generated by lifting fixed circle boundary planar parameterization methods to the sphere through the stereographic projection. Then we develop a new approach to improve the initial spherical parameterization by barycentric coordinates. At last, a uniform spherical parameterization is generated by a M(o|¨)bius transformation. This method can reduce the area distortion of the spherical parameterization whose angle distortion is low. Furthermore, it can avoid fold-overs which may exist for other angle preserving spherical parameterization approaches. We also extend this method to the singular boundary genus-zero meshes. Experiments reveal that it has obvious improvement in uniform texture mapping for complicated meshes than the available discrete conformal mapping and the discrete authalic mapping.
2. We present a new idea based on symmetry analysis for the construction of spherical parameterization for closed genus-zero triangular meshes. It reduces the intricate spherical parameterization problem to a planar one by splitting a closed genus-zero mesh into two pieces with the help of its geometry symmetry traits. According to this simplification we can get two spherical parameterization methods balancing angle and area distortion: the spherical parameterization method based on the stretch metric (BSMSP method) and the spherical parameterization method based on the quasi-harmonic maps (BQHSP method). Both of them can solve the problem that spherical conformal mapping is often having no control in area distortion. They can produce almost isometric parameterizations for the objects close to the sphere. Furthermore, their computation time is dominated by solving only linear systems. The BSMSP method can guarantee the validity of the spherical parameterization. Although the BQHSP method can not guarantee it in theory, it can reduce the angle distortion of the spherical parameterizations for the complicated objects. Besides, we also apply the spherical parameterization generated by the BQHSP method to the subdivision connectivity remeshing. Experiments reveal that this method can obtain high quality subdivision connectivity mesh.
3. We present an adaptive isotropic remeshing method based on distance field for arbitrary genus and boundary 3D meshes. Many approaches generate the isotropic remeshing by global parameterization. However, as for complicated meshes with high genus and arbitrary boundary, the construction of the global parameterization is a very complicated problem. Consequently, we process the mesh directly and do not use global parameterization. Firstly, a mesh with the required number of vertices can be obtained according to edge-collapse or edge-split. Then the vertex sampling and the triangle quality can be improved by geometry optimization and connectivity optimization. In order to reduce the accumulation of the error, we introduce a novel method based on mesh distance field to remain the optimized vertices on the original mesh. Experiments and comparisons are taken with some nontrivial 3D models, which reveals that our approach is effective, fast and robust.
1.针对封闭的亏格为0的三角网格,提出了一种基于质心坐标和M(o|¨)bius变换的均匀准保角球面参数化方法(UQCSP方法)。算法首先通过立体投影和现有的固定圆边界的平面参数化方法得到一个初始的球面映射;然后引入原始网格顶点在切平面上的质心坐标对初始的球面参数化进行优化来减少变形;最后用将球面网格顶点的质心变换到球心的M(o|¨)bius变换均匀化最终的球面网格。该方法能够在球面参数化角度变形较小的情况下减少面积变形,而且也能够保证一般保角球面参数化方法所不具有的有效性。此外,本文也将该方法推广到单边界亏格为0的网格上,所得参数化在复杂网格的纹理映射中的均匀化效果较现有的保角、保面积变换有明显的改善。
2.针对封闭的亏格为0的三角网格,提出了基于对称性分析的球面参数化思想。该思想利用物体的几何对称性质进行分割,将复杂的球面参数化问题转化为更为简单的平面参数化问题。通过该思想可以得到两种均衡角度和面积变形的球面参数化方法:基于拉伸度量的球面参数化方法(BSMSP方法)和基于拟调和映射的球面参数化方法(BQHSP方法)。它们都能够解决保角球面映射对于复杂结构的物体面积变形较大的问题。对于形状接近于球面的物体它们都能够获得几乎等距的球面参数化,而且只需要求解几次线性方程组,非常高效。其中BSMSP方法能够严格保证球面参数化的有效性。尽管BQHSP方法在理论上无法保证,但是相对于BSMSP方法来说,它能够减少复杂结构物体的球面参数化的角度变形。此外,我们将BQHSP方法所得的球面参数化应用到细分连接性的重新网格化中。通过实验结果可以看出,这种方法能够获得高质量的细分连接性网格。
3.针对任意亏格、任意边界的三角网格,提出了一种基于距离场的自适应各向同性的重新网格化方法。各向同性的重新网格化中研究得比较多的一类方法是通过对原始网格进行全局参数化来实现的。但是对于高亏格带边界的复杂的三角网格模型,全局参数化本身是一个非常复杂的问题。因此,本研究直接对三角网格进行自适应操作,不采用全局参数化。算法首先通过对网格的边进行折叠或者分裂得到需要的顶点数目;然后通过优化网格顶点的位置和连接性来改进网格中三角形的形状和顶点位置分布。为了减少网格优化过程中误差的累积,通过引入网格曲面的距离场将新生成的顶点保留到原始网格模型上。实验证明,这是一种简单、快速而有效的方法。
With the development of modern science techniques. 3D geometrical models are recently becoming a new type of medium after sound, images and video, which are applied in industry widely. Consequently, a new discipline processing 3D geometrical models is born, which is called digital geometry processing. It has been a focus of research activity in digital signal processing. Unfortunately, 3D geometrical models are significantly different from other traditional mediums. The topology of 3D geometrical models are very complicated, that is to say, they have any genus and boundary. So traditional signal processing techniques can not be extended to process 3D geometrical models. We must develop some more intricate digital geometry processing algorithms. In this dissertation. we take the triangular mesh models as our research object, and do some research on spherical parameterization and remeshing in digital geometry processing. The main work can be summarized as follows:
1. We present a uniform quasi-conformal spherical parameterization method (UQCSP method) based on barycentric coordinates and M(o|¨)bius transformation for closed genus-zero triangular meshes. An initial spherical parameterization is generated by lifting fixed circle boundary planar parameterization methods to the sphere through the stereographic projection. Then we develop a new approach to improve the initial spherical parameterization by barycentric coordinates. At last, a uniform spherical parameterization is generated by a M(o|¨)bius transformation. This method can reduce the area distortion of the spherical parameterization whose angle distortion is low. Furthermore, it can avoid fold-overs which may exist for other angle preserving spherical parameterization approaches. We also extend this method to the singular boundary genus-zero meshes. Experiments reveal that it has obvious improvement in uniform texture mapping for complicated meshes than the available discrete conformal mapping and the discrete authalic mapping.
2. We present a new idea based on symmetry analysis for the construction of spherical parameterization for closed genus-zero triangular meshes. It reduces the intricate spherical parameterization problem to a planar one by splitting a closed genus-zero mesh into two pieces with the help of its geometry symmetry traits. According to this simplification we can get two spherical parameterization methods balancing angle and area distortion: the spherical parameterization method based on the stretch metric (BSMSP method) and the spherical parameterization method based on the quasi-harmonic maps (BQHSP method). Both of them can solve the problem that spherical conformal mapping is often having no control in area distortion. They can produce almost isometric parameterizations for the objects close to the sphere. Furthermore, their computation time is dominated by solving only linear systems. The BSMSP method can guarantee the validity of the spherical parameterization. Although the BQHSP method can not guarantee it in theory, it can reduce the angle distortion of the spherical parameterizations for the complicated objects. Besides, we also apply the spherical parameterization generated by the BQHSP method to the subdivision connectivity remeshing. Experiments reveal that this method can obtain high quality subdivision connectivity mesh.
3. We present an adaptive isotropic remeshing method based on distance field for arbitrary genus and boundary 3D meshes. Many approaches generate the isotropic remeshing by global parameterization. However, as for complicated meshes with high genus and arbitrary boundary, the construction of the global parameterization is a very complicated problem. Consequently, we process the mesh directly and do not use global parameterization. Firstly, a mesh with the required number of vertices can be obtained according to edge-collapse or edge-split. Then the vertex sampling and the triangle quality can be improved by geometry optimization and connectivity optimization. In order to reduce the accumulation of the error, we introduce a novel method based on mesh distance field to remain the optimized vertices on the original mesh. Experiments and comparisons are taken with some nontrivial 3D models, which reveals that our approach is effective, fast and robust.
引文
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