低秩和紧致表示在几何建模中的应用

发布时间：2018-04-04 16:53

本文选题：隐式曲面　切入点：曲面重建　出处：《中国科学技术大学》2017年博士论文

【摘要】：由离散的点云构造光滑的曲面表示是几何建模与计算机图形学中的基本问题。在过去的二十多年中,隐式曲面重建已成为研究的焦点之一,这是由于隐式表示具有许多优点,例如其易于实现交、差、并等集合操作,能够表示几何和拓扑比较复杂的物体。但是对于复杂的模型,现有的隐式曲面重建方法所得到的隐式表示的存储量往往是巨大的。本文第三章、第四章正是结合这一背景,就如何降低隐式表示的存储量进行了相关研究。在第三章,我们基于一种新的隐式表示形式——多层有理代数样条曲面提出了一种自适应的曲面重建算法。为了得到具有紧致表示的隐式曲面,我们结合张量的低秩逼近技术建立了局部拟合模型,然后将此模型转化为凸优化问题并用ADMM算法和CP分解算法求解。实验表明,与现有的方法相比,我们的方法可以大大降低隐式表示的存储量,同时能够得到高质量的重建结果。此外,我们的方法还具有较强的自适应能力。然而,上述的多层有理代数样条曲面只有零阶连续。为了能够重建出具有更高光滑性的隐式曲面,同时进一步降低隐式表示的存储量,本文第四章提出了一种基于相场引导的隐式曲面重建方法。我们采用层次B样条作为曲面表示形式,以近似的相场函数作为引导,通过求解曲面拟合模型重构得隐式函数,该函数在隐式曲面附近的值由-1平滑地变化到1,而在其他区域的值为常数(1或-1)。和现有的方法相比,我们的方法重构得到的隐式曲面的存储量更小,并且此曲面具有更高阶连续性。此外,绝大部分曲面重建方法需要输入点云的法向信息。然而,许多实际扫描的数据的部分法向是缺失的,或者法向信息不可靠,这给曲面重建带来了挑战和困难。而我们的方法无需输入点云的法向信息,这是本文方法的一大优势。在等几何分析中,区域的参数化和矩阵的装配是两个关键的步骤。我们研究发现,如果参数域和计算域间的映射的秩比较低,那么求解方程过程中矩阵装配所需的时间将大大减少。本文第五章在这一背景下,就如何降低等几何分析中参数表示的秩进行了相关研究。我们使用张量的低秩逼近技术构造了一种低秩的拟共形映射。为了计算该映射,我们给出了一种有效的算法,其主要思想是交替地求解两个子问题。实验表明,我们的方法可以得到计算域的低秩参数表示。此外,相比于已有的方法,我们的参数化结果的扭曲更低。
[Abstract]:It is a basic problem in geometric modeling and computer graphics to construct smooth surface representation from discrete point clouds.However, for complex models, the storage capacity of the implicit representation obtained by the existing implicit surface reconstruction methods is often enormous.The third chapter and the fourth chapter focus on how to reduce the storage capacity of implicit representation.In chapter 3, we propose an adaptive surface reconstruction algorithm based on a new implicit representation, multilayer rational algebraic spline surface.In order to obtain an implicit surface with compact representation, a local fitting model is established by using Zhang Liang's low rank approximation technique. Then the model is transformed into a convex optimization problem and solved by ADMM algorithm and CP decomposition algorithm.Experimental results show that compared with the existing methods, our method can greatly reduce the storage capacity of implicit representation and obtain high quality reconstruction results.In addition, our method also has a strong adaptive ability.However, the above multilayer rational algebraic spline surfaces have only zero order continuity.In order to reconstruct implicit surfaces with higher smoothness and further reduce the storage capacity of implicit representations, a phase field-guided implicit surface reconstruction method is proposed in chapter 4.We use the hierarchical B-spline as the surface representation and the approximate phase field function as the guide, and reconstruct the implicit function by solving the surface fitting model.The value of the function changes smoothly from 1 to 1 near the implicit surface, while the value in other regions is constant 1 or -1.Compared with the existing methods, the implicit surface reconstructed by our method has less storage and higher order continuity.In addition, most surface reconstruction methods need to input normal information of point cloud.However, part of the normal direction of the scanned data is missing, or the normal information is unreliable, which brings challenges and difficulties to surface reconstruction.Our method does not need to input the normal information of the point cloud, which is one of the advantages of this method.In isometric analysis, the parameterization of region and the assembly of matrix are two key steps.It is found that if the rank of the mapping between the parameter domain and the computational domain is low, the time required for matrix assembly in the process of solving the equation will be greatly reduced.In the fifth chapter, we study how to reduce the rank of parameter representation in isometric analysis.We use Zhang Liang's low rank approximation technique to construct a low rank quasiconformal mapping.In order to calculate the mapping, we present an efficient algorithm, whose main idea is to solve two subproblems alternately.Experiments show that our method can obtain the low rank parameter representation in the computational domain.In addition, the distortion of our parameterized results is lower than that of the existing methods.
【学位授予单位】：中国科学技术大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O186.11

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