径向基函数隐式曲面的研究及应用
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摘要
在计算机图形学中,隐式曲面是几何形体的一种重要表示方法,因为它是单一的解析函数,适合于碰撞检测、变形、融合、扭曲、布尔运算等许多方面。其中,径向基函数(简称RBF)隐式曲面能准确、稳定地解决散乱点数据的三维重建问题,是近年来最重要的隐式曲面算法之一。本论文创新地将RBF隐式曲面引入到基于图像的建模、极小曲面问题等领域,并研究了RBF隐式曲面的几何变换问题。相关的研究成果可应用于虚拟现实、场景建模、计算机视觉、图形学、可视化等领域,关于极小曲面的研究成果可应用于整形手术和牙科手术、商品包装设计、分子工程和材料科学、艺术、现代膜工程等许多领域。因此,本文的研究具有重要的学术意义和应用价值。
本文以RBF隐式曲面为研究主线,在基于图像的建模、RBF隐式曲面的几何变换、极小曲面问题三个方面进行了深入研究并提出一些新的算法,主要内容包括：
1)在基于图像的建模领域,给出了一个基于网格的特征检测算法,可以在图片序列中得到更多的匹配点,并引入RBF隐式曲面重建算法来重建出目标的三维模型。其主要思想是在首帧中手动指定相对密集的网格,在网格点附近确定最容易跟踪的特征点,利用迭代方法得到子像素精度的角点坐标,然后用稀疏特征集的多尺度光流跟踪算法跟踪这些角点。之后使用自标定算法,就可以重建出相对均匀和稠密的三维点云。最后,引入RBF隐式曲面重建算法,生成目标的三维表面模型。多个图像序列的重建结果表明,本文算法对纹理丰富的场景能够获得较为满意的曲面模型。
2)推导了RBF隐式曲面在几何变换前后的系数变化公式。当使用RBF隐式曲面时,可能需要对它进行几何变换。传统的RBF隐式曲面几何变换算法是：在给定点使用逆变换,然后用逆变换后的点,在原函数中计算函数值。传统算法需要保持初始的RBF中心。有些时候,则需要几何变换后的RBF中心。在这种情况下,如果仍然使用传统算法,就需要同时保持初始的和几何变换后的RBF中心。显然,这是一个浪费存储空间的问题。本文推导的RBF系数变化公式则解决了上一问题。本文算法只需要保留变换后的RBF中心,因此在需要变换后的RBF中心时会节省很多存储空间。利用这个公式,可以很快计算出新的RBF曲面,并且对紧支撑和全局支撑的RBF都有效。在运算时间方面,本文也详细比较了本文算法和传统算法。理论分析和实验结果表明,本文算法在很多情况下比传统算法更快。将本文算法,应用到了基于RBF隐式曲面的碰撞检测和布尔运算等。并提出一个优化算法,可以加快隐式曲面进行布尔运算的速度。同时,对紧支撑RBF隐式曲面布尔运算中出现的鼓包问题,本文也提出了一个解决方案。
3)极小曲面问题是数学领域的经典难题,特别是它的多解性问题尚未解决。本文提出一种新的算法可以从一个初始曲面迭代得到多个不同的极小曲面(即得到多个不同的解),包括不稳定的极小曲面。给定空间曲线,寻找以此曲线作为边界的面积最小或极小的曲面,称为Plateau问题,或极小曲面问题。对给定的边界曲线,本文算法从一个以此为边界的任意曲面开始迭代,最后得到一个极小曲面。对同一个初始曲面,当选择不同的参数时,本文算法就可能得到不同的极小曲面。目前检索到的其他同类算法都只能收敛到一个极小曲面。本文算法通过模拟一个非线性弹簧模型在空气阻力下的动态行为,来得到最终的极小曲面。对于如何生成该初始曲面的问题,目前很少有文献报道相关的算法。本文提出一个算法,使用RBF隐式曲面,可以半自动地生成指定亏格的初始曲面,只需要少量的用户操作,对任意多个边界曲线都有效。另外,对于规则的单根边界曲线,提出一个算法可以全自动地生成初始曲面。
In computer graphics field, implicit surface is an important.geometric representation, because it is a single analytic function, and it is suitable for collision detection, deformation, blending, warping, Boolean operations and many other aspects. Among them, the radial basis function (short for RBF) implicit surface can accurately and stably solve the 3D reconstruction issue of scattered points, and it is the most important implicit surface algorithms in recent years. This paper innovatively introduces the RBF implicit surface to the image-based modeling, minimal surface problem, and study on the geometric transformation of RBF implicit surface. Relevant research results can be applied to virtual reality, scene modeling, computer vision, graphics, visualization and other fields, research on minimal surfaces can be applied to plastic surgery and dental surgery, packaging design, molecular engineering and materials science, art, modern membranes engineering and many other fields. Therefore, this study has important significance on science and worthiness in practical application.
Focused on RBF implicit surface, the image-based modeling, the geometric transformation of RBF implicit surface and minimal surface problem are studied and a number of novel algorithms are proposed in this dissertation. The main contributions include:
1) In image-base modeling field, we present a grid-based feature detection algorithm, which can match more points among a sequence of images, and then we introduce the RBF implicit surface reconstruction to obtain the 3D model of the object. We make relatively dense grids and look for the best features near the grid points. Then, the sub-pixel coordinates of the corners are found by iteration. Then we calculate optical flow for a sparse feature set using iterative Lucas-Kanade method in pyramids. By this technique, we can reconstruct comparatively even and dense 3D point-cloud with self-calibration algorithm. Implicit surface reconstruction was introduced to generate the surface model of the object. By experimenting with a number of image sequences, the reconstruction results show that the algorithm can obtain satisfied surfaces for a rich texture scene.
2) We derived the relationship between the initial and the transformed RBF coefficients. When we use a Radial Basis Function (RBF) implicit surface, we may need to transform it. The conventional algorithm to transform an RBF implicit surface is to apply inverse transformation to the given point and to evaluate the original function at the inversely transformed point. The algorithm keeps the initial RBF centers. Sometimes, we need the transformed RBF centers. In these cases, if we still use the conventional algorithm, we need to keep both the initial and the transformed RBF centers. Obviously, this is a problem that wastes the memory. We have derived the relationship between the initial and the transformed RBF coefficients, which can solve the previous problem. Our method only needs to keep the transformed RBF centers, and save much memory. By this method, we can get the new RBF surface quickly and it works for both globally and compactly supported RBF. We also compare our algorithm with the conventional algorithm about the time efficiency in details. The theoretical analysis and experiment results show that our algorithm is faster than the conventional algorithm in many cases. We also applied our method on RBF-based collision detection and Boolean operations。We also propose a method to speed up the Boolean operations of implicit surface. Moreover, we present a solution to relieve the bumps in CSRBF Boolean operations.
3) The minimal surface problem is a classic problem in mathematics, especially its multiple-solutions problem has not yet resolved. This paper proposed a new algorithm, which may iteratively obtain a number of different minimal surfaces from an initial surface (i.e. get a number of different solutions), including unstable minimal surface. Giving a space curve as the boundary, finding out the surfaces which have minimal area is called "Plateau's problem" or "minimal suface problem". For a given boundary, our algorithm will take an initial arbitrary surface and then derive a minimal surface. For the same initial surface, when we select different parameters, the algorithm may derive different minimal surfaces, which other existing methods cannot achieve. Our algorithm simulates a non-linear spring model in the impact of air drag to derive the results. For the problem of how to generate the initial surface, there are a few literatures that report the related algorithms. We propose an algorithm using RBF implicit surface, which can generate specified genus initial surface semi-automatically, only a small amount of user interaction is needed. Our method is valid for any number of boundary curves. Moreover, for a single regular boundary curve, we propose an algorithm to generate the initial surface automatically.
本文以RBF隐式曲面为研究主线,在基于图像的建模、RBF隐式曲面的几何变换、极小曲面问题三个方面进行了深入研究并提出一些新的算法,主要内容包括：
1)在基于图像的建模领域,给出了一个基于网格的特征检测算法,可以在图片序列中得到更多的匹配点,并引入RBF隐式曲面重建算法来重建出目标的三维模型。其主要思想是在首帧中手动指定相对密集的网格,在网格点附近确定最容易跟踪的特征点,利用迭代方法得到子像素精度的角点坐标,然后用稀疏特征集的多尺度光流跟踪算法跟踪这些角点。之后使用自标定算法,就可以重建出相对均匀和稠密的三维点云。最后,引入RBF隐式曲面重建算法,生成目标的三维表面模型。多个图像序列的重建结果表明,本文算法对纹理丰富的场景能够获得较为满意的曲面模型。
2)推导了RBF隐式曲面在几何变换前后的系数变化公式。当使用RBF隐式曲面时,可能需要对它进行几何变换。传统的RBF隐式曲面几何变换算法是：在给定点使用逆变换,然后用逆变换后的点,在原函数中计算函数值。传统算法需要保持初始的RBF中心。有些时候,则需要几何变换后的RBF中心。在这种情况下,如果仍然使用传统算法,就需要同时保持初始的和几何变换后的RBF中心。显然,这是一个浪费存储空间的问题。本文推导的RBF系数变化公式则解决了上一问题。本文算法只需要保留变换后的RBF中心,因此在需要变换后的RBF中心时会节省很多存储空间。利用这个公式,可以很快计算出新的RBF曲面,并且对紧支撑和全局支撑的RBF都有效。在运算时间方面,本文也详细比较了本文算法和传统算法。理论分析和实验结果表明,本文算法在很多情况下比传统算法更快。将本文算法,应用到了基于RBF隐式曲面的碰撞检测和布尔运算等。并提出一个优化算法,可以加快隐式曲面进行布尔运算的速度。同时,对紧支撑RBF隐式曲面布尔运算中出现的鼓包问题,本文也提出了一个解决方案。
3)极小曲面问题是数学领域的经典难题,特别是它的多解性问题尚未解决。本文提出一种新的算法可以从一个初始曲面迭代得到多个不同的极小曲面(即得到多个不同的解),包括不稳定的极小曲面。给定空间曲线,寻找以此曲线作为边界的面积最小或极小的曲面,称为Plateau问题,或极小曲面问题。对给定的边界曲线,本文算法从一个以此为边界的任意曲面开始迭代,最后得到一个极小曲面。对同一个初始曲面,当选择不同的参数时,本文算法就可能得到不同的极小曲面。目前检索到的其他同类算法都只能收敛到一个极小曲面。本文算法通过模拟一个非线性弹簧模型在空气阻力下的动态行为,来得到最终的极小曲面。对于如何生成该初始曲面的问题,目前很少有文献报道相关的算法。本文提出一个算法,使用RBF隐式曲面,可以半自动地生成指定亏格的初始曲面,只需要少量的用户操作,对任意多个边界曲线都有效。另外,对于规则的单根边界曲线,提出一个算法可以全自动地生成初始曲面。
In computer graphics field, implicit surface is an important.geometric representation, because it is a single analytic function, and it is suitable for collision detection, deformation, blending, warping, Boolean operations and many other aspects. Among them, the radial basis function (short for RBF) implicit surface can accurately and stably solve the 3D reconstruction issue of scattered points, and it is the most important implicit surface algorithms in recent years. This paper innovatively introduces the RBF implicit surface to the image-based modeling, minimal surface problem, and study on the geometric transformation of RBF implicit surface. Relevant research results can be applied to virtual reality, scene modeling, computer vision, graphics, visualization and other fields, research on minimal surfaces can be applied to plastic surgery and dental surgery, packaging design, molecular engineering and materials science, art, modern membranes engineering and many other fields. Therefore, this study has important significance on science and worthiness in practical application.
Focused on RBF implicit surface, the image-based modeling, the geometric transformation of RBF implicit surface and minimal surface problem are studied and a number of novel algorithms are proposed in this dissertation. The main contributions include:
1) In image-base modeling field, we present a grid-based feature detection algorithm, which can match more points among a sequence of images, and then we introduce the RBF implicit surface reconstruction to obtain the 3D model of the object. We make relatively dense grids and look for the best features near the grid points. Then, the sub-pixel coordinates of the corners are found by iteration. Then we calculate optical flow for a sparse feature set using iterative Lucas-Kanade method in pyramids. By this technique, we can reconstruct comparatively even and dense 3D point-cloud with self-calibration algorithm. Implicit surface reconstruction was introduced to generate the surface model of the object. By experimenting with a number of image sequences, the reconstruction results show that the algorithm can obtain satisfied surfaces for a rich texture scene.
2) We derived the relationship between the initial and the transformed RBF coefficients. When we use a Radial Basis Function (RBF) implicit surface, we may need to transform it. The conventional algorithm to transform an RBF implicit surface is to apply inverse transformation to the given point and to evaluate the original function at the inversely transformed point. The algorithm keeps the initial RBF centers. Sometimes, we need the transformed RBF centers. In these cases, if we still use the conventional algorithm, we need to keep both the initial and the transformed RBF centers. Obviously, this is a problem that wastes the memory. We have derived the relationship between the initial and the transformed RBF coefficients, which can solve the previous problem. Our method only needs to keep the transformed RBF centers, and save much memory. By this method, we can get the new RBF surface quickly and it works for both globally and compactly supported RBF. We also compare our algorithm with the conventional algorithm about the time efficiency in details. The theoretical analysis and experiment results show that our algorithm is faster than the conventional algorithm in many cases. We also applied our method on RBF-based collision detection and Boolean operations。We also propose a method to speed up the Boolean operations of implicit surface. Moreover, we present a solution to relieve the bumps in CSRBF Boolean operations.
3) The minimal surface problem is a classic problem in mathematics, especially its multiple-solutions problem has not yet resolved. This paper proposed a new algorithm, which may iteratively obtain a number of different minimal surfaces from an initial surface (i.e. get a number of different solutions), including unstable minimal surface. Giving a space curve as the boundary, finding out the surfaces which have minimal area is called "Plateau's problem" or "minimal suface problem". For a given boundary, our algorithm will take an initial arbitrary surface and then derive a minimal surface. For the same initial surface, when we select different parameters, the algorithm may derive different minimal surfaces, which other existing methods cannot achieve. Our algorithm simulates a non-linear spring model in the impact of air drag to derive the results. For the problem of how to generate the initial surface, there are a few literatures that report the related algorithms. We propose an algorithm using RBF implicit surface, which can generate specified genus initial surface semi-automatically, only a small amount of user interaction is needed. Our method is valid for any number of boundary curves. Moreover, for a single regular boundary curve, we propose an algorithm to generate the initial surface automatically.
引文
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