三角形网格上曲面重构研究
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摘要
随着计算机应用技术和计算机图形学技术的不断发展,使用细节丰富的多边形网格来描述数字几何模型越来越普遍。和传统算法向比,利用多边形网格描述与处理数字模型的算法灵活多样,具有计算效率高,便于绘制等优点,使其在计算机辅助几何设计(CAGD),计算机游戏、动画、电影、数字模拟,以及计算机科学可视化等领域得到广泛地应用。但是利用多边形模型简洁,高效地描述数字几何模型还面临一系列问题。首先,由于直接产生的多边形模型往往存在洞、自交、重叠、复边、法向不一致、非流体等问题,阻碍了该技术除绘制以外领域的广泛应用；其次,多边形模型往往含有过多细节,大量退化元素等问题,不仅影响占用的存储空间,而且影响算法稳定性、复杂性,以及计算精确度。与多边形模型相比,隐式曲面模型不存在前面提到的问题,所以如何利用隐式曲面来处理多边形模型越来越受到人们的广泛关注,成为数字几何处理研究的热点和重点问题之一。近年来,使用隐式曲面模型逼近或者插值多边形模型的研究取得了一系列重要成果,但依然存在大量需要解决的问题,例如,如何保证隐式曲面模型和被逼近的多边形模型之间的误差在给定的范围内；如何使构造的隐式曲面模型表示形式简洁,并且存储高效；如何在满足误差的前提下,快速地将多边形模型转变成满足要求的隐式曲面模型；以及如何快速绘制隐式曲面模型等。
基于前面提到的问题,本文在,1)高效鲁棒的多边形——隐式曲面构造方法；2)隐式曲面的快速采样方法；3)基于误差构造简化曲面模型,三方面做了针对性的研究工作,具体内容和成果如下：
1、高效鲁棒的多边形——隐式曲面构造方法
基于加权多边形——隐式曲面误差项,提出了一种鲁棒的多层次单位隐式曲面划分算法。新算法能够使构造的隐式曲面在满足误差的条件下,高效稳定地逼近原多边形网格。利用提出的能够正确区分多边形内部顶点和网格顶点的加权多边形——隐式曲面误差项,新算法构造的局部隐式曲面(一般隐式二次曲面)不仅具有局部多边形模型所建议的形状,而且多边形顶点和局部隐式曲面之间的误差更小。另外,将多边形网格模型的对偶模型和加权多边形——隐式曲面误差项应用到传统的多层次单位隐式曲面划分算法中,提出一个稳定,高效的隐式曲面重构算法。同传统的多层次单位隐式曲面划分算法向比,在同样的逼近误差下,新算法更稳定,并且产生的隐式曲面模型含有更少的核函数。
2、隐式曲面的快速采样方法
提出了一种新的隐式曲面快速采样方法。首先提出了一种新的采样点互斥能量目标函数,基于该目标函数,通过一种混合优化方法来求解采样点的分布。第1步为采样点的局部优化,通过对采样点移动速度的控制参数调整,避免了大量Hessian矩阵的求逆操作,使得采样点能够根据互斥半径快速覆盖整个隐式曲面,得到初始采样点集；第2步为采样点的全局优化,采用L-BFGS方法对所有采样点进行优化,得到最终的高质量采样结果。通过实验表明,新方法的采样速度大大提高,并能够获得较好的隐式曲面采样点分布。并且基于该方法的均匀分布或者依据曲率的自适应分布采样结果,能够生成简洁并且高质量的多边形模型。
3、基于误差构造简化曲面模型
提出了一种新的隐式曲面构造方法,构造的隐式曲面由每个三角形上构造的曲面片组成,且能够插值或者逼近原始的多边形网格。每个三角形上的曲面片由顶点处构造的隐式二次曲面加权产生,并且为了使构造的曲面模型含有的曲面片数尽可能少,我们把三角形网格简化算法引入到曲面重构过程中。算法输入一个三角形网格和一个拟合误差,依据顶点到构造的曲面模型之间的距离是不是满足给定的误差迭代地删除网格顶点,直至算法收敛。既然该算法是基于曲面重构和顶点删除的,那么我们可以精确地控制删除的顶点和构造的曲面模型之间的误差,并且大量的试验结果显示,该算法产生的曲面模型能够较好地逼近原始模型。
With the development of computer applications technology and computer graphics tech-nology, using polygonal meshes with a convincing level of realism to represent3D dig-ital geometry models is now generally acceptable. Due to polygonal meshes' highly desirable properties for rendering and calculating, processing and representing3D dig-ital models with them are widely applied in the fields of Computer Aided Geometric Design (CAGD), Computer Games, Animation, Movies, Simulation and Computer Sci-entific Visualization. However, using polygonal meshes to efficiently represent digi-tal models faces a series of problems. First, most polygonal meshes generated from other kind of data set often contain holes, gaps, T-junctions, self-intersections, and non-manifold structure which make them unsuitable for many purposes other than rendering; Next, most polygonal meshes often contain a lot of unnecessary details and degenera-tive polygons which affect not only their storage space, but also the stability, com-plexity, and precision of the algorithms worked on the polygonal meshes. Compared with the method which employed polygonal meshes to represent digital models, such problems discussed above have never arisen within the approaches that make use of im-plicit surface models to represent digital models. Therefore, how to effectively process polygonal meshes with implicit surface models is a hotspot and an essential problem of Digital Geometry Processing (DGP). In recent years, the research of using implicit surface model to interpolate or approximate polygonal mesh has obtained fast develop-ment, and achieved a series of significant progress, but many problems are pressing for solution. For example, how to control the approximation precision between the con-structed implicit surface model and the polygonal mesh; how to express the implicit surface model and store it effectively; how to generate an implicit surface model from its corresponding polygonal mesh under a given approximation error; how to fast render the implicit surface.
According to the problems mentioned above, the research of this thesis includes:1) Robust and effective polygonal meshes-implicit surface modeling approach;2) Rapid sampling on implicit surfaces via hybrid optimization;3) Surface construction with fewer patches within precision, and following is the specific content:
1. Robust and effective polygonal meshes-implicit surface modeling approach
we present a new robust multi-level partition of unity (MPU) method, which con-structs an implicit surface from a triangular mesh via the new error metric between the mesh and the implicit surface. The new error metric employs a weighted function of inner points and vertices of a triangle to fit an implicit surface, which can control the approximation error between the surface and vertices of the triangle. Furthermore, it is applied to the MPU method by utilizing the dual graph of a triangular mesh, and the general quadric implicit surface is used for surface representation. Compared with the MPU method, the new method generates fewer subdivision cells with the same approx-imation error and performs more steadily especially when given triangular mesh with fewer vertices.
2. Rapid sampling on implicit surfaces via hybrid optimization
A new method for implicit surfaces sampling is presented. At first, a new objec-tive function of repulsive energy is given for constraining sampling points distributed on the surface uniformly. Based on this new objective function, the distribution of sam-pling points can be solved by a hybrid optimization. The first step is a local optimiza-tion of sampling points through a parameter to control the velocity of a sampling point for avoiding inverse matrices computation of Hessian, so a set of sampling points can cover given implicit surface rapidly. The second step is global optimization of sampling points. L-BFGS method is used in this step. We can have ideal sampling distribution via these two steps. Experimental results show that the new method can sample fast on implicit surfaces, and the points' distribution is also satisfactory.
3. Surface construction with fewer patches within precision
We present an algorithm to generate an interpolation or approximation model con-sisting of many patches from a triangle mesh, and each patch is a weighted combination of the three surfaces associated with the vertices of a triangle. Moreover, to make the whole surface include fewer patches, mesh simplification is introduced into the process of surface construction. The algorithm takes a triangle mesh and a given error as in- put, and iteratively deletes vertex whose distance to the surface model constructed from the simplified mesh is less than or equal to the given error until convergence. Since the method is based on surface approximation and vertex deletion, it allows us to control the error between the generated model and the original mesh precisely. Furthermore, many experimental results show that the generated models approximate the original models good.
基于前面提到的问题,本文在,1)高效鲁棒的多边形——隐式曲面构造方法；2)隐式曲面的快速采样方法；3)基于误差构造简化曲面模型,三方面做了针对性的研究工作,具体内容和成果如下：
1、高效鲁棒的多边形——隐式曲面构造方法
基于加权多边形——隐式曲面误差项,提出了一种鲁棒的多层次单位隐式曲面划分算法。新算法能够使构造的隐式曲面在满足误差的条件下,高效稳定地逼近原多边形网格。利用提出的能够正确区分多边形内部顶点和网格顶点的加权多边形——隐式曲面误差项,新算法构造的局部隐式曲面(一般隐式二次曲面)不仅具有局部多边形模型所建议的形状,而且多边形顶点和局部隐式曲面之间的误差更小。另外,将多边形网格模型的对偶模型和加权多边形——隐式曲面误差项应用到传统的多层次单位隐式曲面划分算法中,提出一个稳定,高效的隐式曲面重构算法。同传统的多层次单位隐式曲面划分算法向比,在同样的逼近误差下,新算法更稳定,并且产生的隐式曲面模型含有更少的核函数。
2、隐式曲面的快速采样方法
提出了一种新的隐式曲面快速采样方法。首先提出了一种新的采样点互斥能量目标函数,基于该目标函数,通过一种混合优化方法来求解采样点的分布。第1步为采样点的局部优化,通过对采样点移动速度的控制参数调整,避免了大量Hessian矩阵的求逆操作,使得采样点能够根据互斥半径快速覆盖整个隐式曲面,得到初始采样点集；第2步为采样点的全局优化,采用L-BFGS方法对所有采样点进行优化,得到最终的高质量采样结果。通过实验表明,新方法的采样速度大大提高,并能够获得较好的隐式曲面采样点分布。并且基于该方法的均匀分布或者依据曲率的自适应分布采样结果,能够生成简洁并且高质量的多边形模型。
3、基于误差构造简化曲面模型
提出了一种新的隐式曲面构造方法,构造的隐式曲面由每个三角形上构造的曲面片组成,且能够插值或者逼近原始的多边形网格。每个三角形上的曲面片由顶点处构造的隐式二次曲面加权产生,并且为了使构造的曲面模型含有的曲面片数尽可能少,我们把三角形网格简化算法引入到曲面重构过程中。算法输入一个三角形网格和一个拟合误差,依据顶点到构造的曲面模型之间的距离是不是满足给定的误差迭代地删除网格顶点,直至算法收敛。既然该算法是基于曲面重构和顶点删除的,那么我们可以精确地控制删除的顶点和构造的曲面模型之间的误差,并且大量的试验结果显示,该算法产生的曲面模型能够较好地逼近原始模型。
With the development of computer applications technology and computer graphics tech-nology, using polygonal meshes with a convincing level of realism to represent3D dig-ital geometry models is now generally acceptable. Due to polygonal meshes' highly desirable properties for rendering and calculating, processing and representing3D dig-ital models with them are widely applied in the fields of Computer Aided Geometric Design (CAGD), Computer Games, Animation, Movies, Simulation and Computer Sci-entific Visualization. However, using polygonal meshes to efficiently represent digi-tal models faces a series of problems. First, most polygonal meshes generated from other kind of data set often contain holes, gaps, T-junctions, self-intersections, and non-manifold structure which make them unsuitable for many purposes other than rendering; Next, most polygonal meshes often contain a lot of unnecessary details and degenera-tive polygons which affect not only their storage space, but also the stability, com-plexity, and precision of the algorithms worked on the polygonal meshes. Compared with the method which employed polygonal meshes to represent digital models, such problems discussed above have never arisen within the approaches that make use of im-plicit surface models to represent digital models. Therefore, how to effectively process polygonal meshes with implicit surface models is a hotspot and an essential problem of Digital Geometry Processing (DGP). In recent years, the research of using implicit surface model to interpolate or approximate polygonal mesh has obtained fast develop-ment, and achieved a series of significant progress, but many problems are pressing for solution. For example, how to control the approximation precision between the con-structed implicit surface model and the polygonal mesh; how to express the implicit surface model and store it effectively; how to generate an implicit surface model from its corresponding polygonal mesh under a given approximation error; how to fast render the implicit surface.
According to the problems mentioned above, the research of this thesis includes:1) Robust and effective polygonal meshes-implicit surface modeling approach;2) Rapid sampling on implicit surfaces via hybrid optimization;3) Surface construction with fewer patches within precision, and following is the specific content:
1. Robust and effective polygonal meshes-implicit surface modeling approach
we present a new robust multi-level partition of unity (MPU) method, which con-structs an implicit surface from a triangular mesh via the new error metric between the mesh and the implicit surface. The new error metric employs a weighted function of inner points and vertices of a triangle to fit an implicit surface, which can control the approximation error between the surface and vertices of the triangle. Furthermore, it is applied to the MPU method by utilizing the dual graph of a triangular mesh, and the general quadric implicit surface is used for surface representation. Compared with the MPU method, the new method generates fewer subdivision cells with the same approx-imation error and performs more steadily especially when given triangular mesh with fewer vertices.
2. Rapid sampling on implicit surfaces via hybrid optimization
A new method for implicit surfaces sampling is presented. At first, a new objec-tive function of repulsive energy is given for constraining sampling points distributed on the surface uniformly. Based on this new objective function, the distribution of sam-pling points can be solved by a hybrid optimization. The first step is a local optimiza-tion of sampling points through a parameter to control the velocity of a sampling point for avoiding inverse matrices computation of Hessian, so a set of sampling points can cover given implicit surface rapidly. The second step is global optimization of sampling points. L-BFGS method is used in this step. We can have ideal sampling distribution via these two steps. Experimental results show that the new method can sample fast on implicit surfaces, and the points' distribution is also satisfactory.
3. Surface construction with fewer patches within precision
We present an algorithm to generate an interpolation or approximation model con-sisting of many patches from a triangle mesh, and each patch is a weighted combination of the three surfaces associated with the vertices of a triangle. Moreover, to make the whole surface include fewer patches, mesh simplification is introduced into the process of surface construction. The algorithm takes a triangle mesh and a given error as in- put, and iteratively deletes vertex whose distance to the surface model constructed from the simplified mesh is less than or equal to the given error until convergence. Since the method is based on surface approximation and vertex deletion, it allows us to control the error between the generated model and the original mesh precisely. Furthermore, many experimental results show that the generated models approximate the original models good.
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