低扭曲几何映射的研究

发布时间：2018-05-30 05:47

本文选题：数字几何数据 + 低扭曲几何映射　；参考：《中国科学技术大学》2016年博士论文

【摘要】：随着三维数据采集技术的发展,数字几何数据成为继声音、图像和视频之后的新媒体形式,并在曲面造型、计算机动画与视觉、地理信息系统、物理仿真、虚拟现实、科学计算的可视化等领域有着广泛的应用。本文的研究是基于三角网格曲面这一基本的几何数据。在计算机图形学、计算机视觉和医学图像处理等领域,寻找曲面间的低扭曲几何映射一直是一个重要的基本性问题。其中,曲面参数化和曲面注册就是其中的两个重要技术。随着数字几何处理的发展及曲面微分几何在计算机科学中的广泛应用,越来越多基于低扭曲几何映射的工程技术难题被解决。本文以三角网格数据为基础,围绕低扭曲几何映射问题,对低扭曲的球面参数化及平面网格上低扭曲映射进行研究。对于亏格为0的封闭三角网格曲面,我们提出了一种尽可能保刚性的球面参数化方法(ARAP方法)。该方法是平面域上尽可能保刚性参数化在球面域的推广。我们的目标是寻找一个具有最优半径的球面网格,使得每个三角形都能尽量保刚性地映射到该球面上。通过分析二维及三维空间连续和离散的ARAP能量,给出基于ARAP能量的球面参数化的优化模型。该模型的求解涉及到一个带有非线性约束的二次规划优化问题。针对该问题,我们提出了有效的两步迭代算法,包括局部／全局算法更新球面顶点坐标和迭代更新半径。该方法克服了以往方法仅仅优化角度扭曲或者面积扭曲的缺点,直接优化刚性(长度)扭曲。实验结果表明,与现有方法比较,该方法具有最好的保持三角形刚性的能力。对于亏格为0的封闭三角网格曲面,我们提出了一种低度量扭曲的有效球面参数化方法(BLD方法),包括保角度和保长度两类。以往的方法要么不能控制最大度量扭曲,要么不能保证球面参数化的有效性。针对这些不足,我们提出了一种基于AMIPS能量优化的球面参数化模型。给定一个初始的有效参数化,即使初始的扭曲比较大,我们通过求解一个带有非线性约束的非线性优化问题来逐步改进它,惩罚参数化过程中最大扭曲的同时,保证优化过程中三角形不出现翻转。为了有效地求解该优化问题,我们设计了一种基于参数动态调整的不精确Block Coordinate Descent方法的迭代优化算法,得到了最优半径球面上的低扭曲的有效球面参数化。实验结果表明,无论是在几何简单还是复杂的模型上,该方法得到的球面参数化结果均能在保证双射的基础上,具有最低的最大扭曲和平均扭曲。此外,数值实验表明,该方法快速、有效,不依赖于初值,且对参数的选择不敏感。对于图像特征点匹配问题,我们通过构造平面网格上低扭曲映射的方法来寻找两张图像之间具有几何一致性的特征点对应。输入是由SIFT算法得到的含有噪音的若干图像特征点对应,我们希望能从含有大量噪音的特征点对应中寻找出尽可能多的具有几何信息的对应点。针对该问题,我们提出了基于拟共形映射函数空间的过滤方法,将该问题转化成关于Beltrami系数和拟共形映射函数的带约束的优化问题,并提出了一种基于变量分离方法和迭代最小二乘方法的有效迭代算法。在每一步的迭代中,我们要求解两个子问题：关于拟共形映射函数的稀疏线性系统求解问题和关于Beltrami系数的带有线性约束的凸二次规划问题。为了衡量算法的准确率,我们定义了刻画特征点匹配准确性的统计量F-measure,并分别在合成数据和真实图像上做了测试。实验结果表明,我们的方法能够筛选出更多的具有几何一致性的特征点对应,并且对参数的选择和噪音均不敏感。
[Abstract]:With the development of 3D data acquisition technology, digital geometric data has become a new media form following sound, image and video. It has a wide application in surface modeling, computer animation and vision, geographic information system, physical simulation, virtual reality, scientific computing visualization and other fields. This paper is based on triangular mesh. This basic geometric data. In the fields of computer graphics, computer vision and medical image processing, it has always been an important basic problem to find low distorted geometric mapping between surfaces. Among them, surface parameterization and surface registration are two important technologies. With the development of digital geometric processing and surface differential With the wide application of geometry in computer science, more and more engineering and technical problems based on low distorted geometric mapping are solved. Based on triangular mesh data, this paper studies low twisted geometric mapping problem, low twisted spherical parameterization and low distortion mapping on plane grids. For a closed triangular mesh with a deficiency of 0 We propose a spherical parameterization method (ARAP method) to preserve the rigidity as much as possible. This method is the extension of the rigid parameterization in the spherical domain as far as possible. Our goal is to find a spherical mesh with the optimal radius, so that every triangle can be mapped to the sphere as rigid as possible. The optimization model of spherical Parameterization Based on ARAP energy is given in the continuous and discrete ARAP energy of two-dimensional and three-dimensional space. The solution of this model involves a two time programming optimization problem with nonlinear constraints. In view of this problem, we propose an effective two step iterative algorithm, including the local / global algorithm to update the spherical vertex coordinates. The method overcomes the disadvantage that the previous method only optimizes the angle distortion or area distortion, and optimizes the rigid (length) distortion directly. The experimental results show that the method has the best ability to keep the triangle rigidity compared with the existing method. For the closed triangular mesh surface with the deficiency of 0, we put forward a new method. The effective spherical parameterization method (BLD method) for low degree distortion, including two classes of preserving angle and length of length. The previous method either can not control the maximum metric distortion or can not guarantee the validity of the spherical parameterization. In view of these shortcomings, we propose a spherical parameterization model based on the AMIPS energy optimization. It is effective parameterized, even if the initial distortion is large, we gradually improve it by solving a nonlinear optimization problem with nonlinear constraints, punish the maximum distortion in the parameterization process and ensure that the triangle does not turn over in the optimization process. In order to effectively solve the optimization problem, we design a kind of parameter based on the parameter. The iterative optimization algorithm of the dynamically adjusted inexact Block Coordinate Descent method obtains the low distortion effective spherical parameterization on the optimal radius spherical surface. The experimental results show that the spherical parameterized results obtained by this method can be the lowest on the basis of guaranteeing the double fire on both the geometric simple and the complex models. In addition, the numerical experiments show that the method is fast, effective, not dependent on the initial value and is insensitive to the selection of parameters. For the image feature point matching problem, we find the corresponding feature points with geometric consistency between the two images by constructing the low distortion mapping method on the plane grid. The input is SI. A number of image feature points with noise are obtained by the FT algorithm. We hope to find out as many geometric information as possible from the corresponding feature point correspondence with a large number of noises. In this case, we propose a filtering method based on the quasi conformal mapping function space, which translates the problem into the Beltrami coefficient and the quasi conformal mapping function. A constrained optimization problem of conformal mapping functions, and an efficient iterative algorithm based on variable separation method and iterative least square method is proposed. In each iteration, we require two sub problems: solving the problem of sparse linear systems with quasi conformal mapping functions and linear constraints on Beltrami coefficients In order to measure the accuracy of the algorithm, to measure the accuracy of the algorithm, we define a statistic F-measure that characterizes the accuracy of feature points matching, and tests both the synthetic data and the real image. The experimental results show that our method can select more geometric consistency of the corresponding feature points, and the selection of the parameters. Both selection and noise are not sensitive.
【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】：2016
【分类号】：TP391.41

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