三维几何模型的形状编辑技术研究
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摘要
随着三维扫描设备的发展,点云以及网格模型的获取越来越简单,针对点云以及网格模型进行形状编辑,在计算机动画、游戏以及影视特效的制作中扮演着越来越重要的角色,逐渐成为计算机图形学中的一项重要研究课题。前人虽然对网格的形状编辑技术研究很多,如微分域上特征保持的网格变形技术,多分辨率网格变形技术,但这类方法为满足用户需求,通常需要建模者具有一定的三维建模经验以及艺术修养。另一方面,点云曲面模型,相对于网格模型,具有数据结构简单,无需维护全局一致的拓扑关系的优点,尤其是点云曲面固有的任意复杂形态的表达能力,使得点云曲面成为与三角网格相辅相成的又一曲面表达形式。而目前针对点云曲面的变形编辑的研究工作相对而言比较少,尤其是在变形编辑的易用性以及重用性方面。
本文针对上述存在的主要问题,对于网格模型的变形编辑,给出了一种二维图像驱动的三维网格风格化变形方法,该方法可以利用二维手绘或是二维照片来产生三维网格变形,使得变形后的网格具有与二维图像相似的风格。由于二维手绘更适合于初级用户的交互习惯,降低了三维变形编辑对于用户的建模经验要求,另一方面,二维照片的使用则降低了变形编辑对于用户艺术修养的要求,初级用户也可编辑出栩栩如生的变形效果。对于点云曲面的变形编辑,则主要围绕关键帧动画中变形编辑的需求,在特征保持的点云曲面的自由变形,关键帧之间的形状插值以及点云曲面的变形传输方面进行了探讨,给出了一系列的算法与实现,并通过算法实例证明了算法的可行性与有效性。
本文的主要研究内容和创新点在于:
●提出了一种特征保持的大规模点云曲面的自由变形方法。该方法结合了传统的自由变形比较快速的优点以及微分网格变形技术中特征保持的优点,从而在提高变形效率的同时,能够较好的保持点云曲面的局部几何细节。首先基于OBBTree的思想以及有向包围盒的相交测试,构建点云曲面的代理模型,而后通过代理模型驱动原始模型的变形。由于约束了代理模型的每个顶点的仿射变换尽量为刚性变换,点云曲面的局部几何细节能够得到较好的保持。针对传统的自由变形中不太直观的缺点,即用户的交互在代理模型上进行,本文方法允许用户直接在点云模型上操作,按照最近邻原则将所操作的顶点射到代理模型上。
●给出了点云曲面上顶点变形梯度的定义及其闭合解。不同于传统的定义于三角面片上的变形梯度,点云曲面的顶点变形梯度直接定义于点云曲面的每个顶点,从而无需类似网格的拓扑结构。另外,文章从最小二乘的角度出发,得到了该定义下顶点变形梯度的一个闭合解。
●将上述定义的顶点变形梯度应用于点云曲面的形状插值以及变形传输中,很好的解决了其中顶点对应关系的构建以及变形点云曲面的重建问题。形状插值问题中,通过将顶点变形梯度进行矩阵极分解,而后对分解后的旋转分量以及缩放分量分别进行插值,避免了线性插值过程中出现的萎缩现象。变形传输问题中目标点云曲面每个顶点的变形梯度的合成则由矩阵的指数映射(Exponential Mapping)完成,避免了球面线性插值中与插值顺序的相关性,即不同的插值顺序,会得到不同的插值结果。
●提出了一种利用二维图像驱动三维网格进行风格化变形的方法。该方法首先将二维图像表示成一个平面网格,并建立三维网格与该平面网格的对应关系。二维图像的风格则利用物体的轮廓线,特征线,物体的局部几何细节以及用户标定的特征四种不同层次上的特征加以描述,每种风格的迁移可表示成一个二次能量的形式,最后三维网格的风格化变形则通过一个整体的二次能量优化实现。用户可以通过调节不同风格能量在整体优化方程中的权重系数来改变最后风格化变形的结果,从而为用户在风格化变形中提供更多的控制。
With the development of the 3D scanning technology, the acquisition of point set surface and mesh models becomes easier and easier. Shape editing with these 3D models are now intensively involved in many applications, such as computer animation, computer game, film industry, and so on, which also makes shape editing as an important research topic in the field of computer graphics. Although many researches on mesh editing have been existed, e.g. detail-preserving gradient domain mesh deformation, multi-resolution mesh editing, in order to create some vivid models, the user has to learn 3D modeling skills and aesthetic knowledge. On the other hand, point set surface becomes another important representation of 3D models, due to its simple data structure and intrinsic power of expressing arbitrary complex shapes. However, the editing methods for point set surface are rare, especially for the case of easy for use and reusability.
This thesis focused on the aforementioned problems. As to the mesh editing, a method of image driven shape deformation with styles was proposed. It used a 2D drawing or 2D picture to produce the 3D mesh deformation. The deformed 3D mesh has the similar styles with the original 2D image. Since novels used to use 2D sketch as interaction interface, and 2D picture can lead to vivid deformed results, this method can reduce the requirement of the 3D modeling experience and aesthetic knowledge. As to the point set surface editing, discussions on detail-preserving shape deformation for large scale point set surface, shape interpolation between key frames and deform transfer for point set surface were intensively involved, all of which are the most frequently used technologies in key-framed animation system. This thesis also presented some algorithms and their implementation for these aspects. The examples shown in this thesis proved the feasibility and effectiveness of these methods.
The main research topics and contributions of this thesis include:
·Proposed a method of detail-preserving deformation for large scale point set surface. This method inherits the advantages of free form deformation and gradient domain deformation. It can well preserve the local details of the point set surface and keep a high efficiency during deformation. Firstly, the proxy model of the point set surface was constructed based on the idea of OBBTree and the intersection test of the oriented bounding box. The deformation of the original point set surface was driven by the deformation of the proxy model and the local details preservation was implemented via constraining the affine transformation of each node of the proxy model to be rigid transformation. Since the interaction for the traditional free form deformation is performed on the proxy model, which was not so intuitive for the users, this method allows direct actions on the point set surface through a nearest-neighbor mapping technology.
·Proposed the definition of vertex deformation gradient of point set surface. The closed form solution of which was also presented. Compared with the traditional deformation gradient which is defined on each triangle face of the mesh, the vertex deformation gradient is defined on each point of the point set surface directly and can be got from its k nearest points. From the view of quadratic energy optimization, we also presented the closed form solution of the vertex deformation gradient.
·By applying the vertex deformation gradient into the problem of shape interpolation and deformation transfer for the point set surface, correspondence between the source point set surface and the target point set surface was well established. The reconstruction for the deformed point set surface can be also well solved. For the case of shape interpolation, the shrinkage problem was avoided by decomposing the vertex deformation gradient into rational part and scale part with polar decomposition and each part was interpolated individually. The vertex deformation gradient of the target point set surface during deformation transfer was interpolated with matrix exponential mapping, which avoids the non-commutative property of the spherical linear interpolation, namely, different order of interpolation will lead to different result.
·Proposed a method of image driven shape deformation with styles. This method represents the shape styles with four kinds of shape descriptions from different levels in the context of triangular mesh. Namely, the object contour, the context curves, local geometric details, perception-driven features. The transfer of each of these styles can be formulated as a quadratic energy optimization. The whole deformation of the 3D mesh was implemented as energy optimization via combining all these energy items together. The coefficient of each energy item in the total energy equation can be adjusted with users' intents, so that user can get different resulting styles and get more controls for the stylization of the 3D mesh.
本文针对上述存在的主要问题,对于网格模型的变形编辑,给出了一种二维图像驱动的三维网格风格化变形方法,该方法可以利用二维手绘或是二维照片来产生三维网格变形,使得变形后的网格具有与二维图像相似的风格。由于二维手绘更适合于初级用户的交互习惯,降低了三维变形编辑对于用户的建模经验要求,另一方面,二维照片的使用则降低了变形编辑对于用户艺术修养的要求,初级用户也可编辑出栩栩如生的变形效果。对于点云曲面的变形编辑,则主要围绕关键帧动画中变形编辑的需求,在特征保持的点云曲面的自由变形,关键帧之间的形状插值以及点云曲面的变形传输方面进行了探讨,给出了一系列的算法与实现,并通过算法实例证明了算法的可行性与有效性。
本文的主要研究内容和创新点在于:
●提出了一种特征保持的大规模点云曲面的自由变形方法。该方法结合了传统的自由变形比较快速的优点以及微分网格变形技术中特征保持的优点,从而在提高变形效率的同时,能够较好的保持点云曲面的局部几何细节。首先基于OBBTree的思想以及有向包围盒的相交测试,构建点云曲面的代理模型,而后通过代理模型驱动原始模型的变形。由于约束了代理模型的每个顶点的仿射变换尽量为刚性变换,点云曲面的局部几何细节能够得到较好的保持。针对传统的自由变形中不太直观的缺点,即用户的交互在代理模型上进行,本文方法允许用户直接在点云模型上操作,按照最近邻原则将所操作的顶点射到代理模型上。
●给出了点云曲面上顶点变形梯度的定义及其闭合解。不同于传统的定义于三角面片上的变形梯度,点云曲面的顶点变形梯度直接定义于点云曲面的每个顶点,从而无需类似网格的拓扑结构。另外,文章从最小二乘的角度出发,得到了该定义下顶点变形梯度的一个闭合解。
●将上述定义的顶点变形梯度应用于点云曲面的形状插值以及变形传输中,很好的解决了其中顶点对应关系的构建以及变形点云曲面的重建问题。形状插值问题中,通过将顶点变形梯度进行矩阵极分解,而后对分解后的旋转分量以及缩放分量分别进行插值,避免了线性插值过程中出现的萎缩现象。变形传输问题中目标点云曲面每个顶点的变形梯度的合成则由矩阵的指数映射(Exponential Mapping)完成,避免了球面线性插值中与插值顺序的相关性,即不同的插值顺序,会得到不同的插值结果。
●提出了一种利用二维图像驱动三维网格进行风格化变形的方法。该方法首先将二维图像表示成一个平面网格,并建立三维网格与该平面网格的对应关系。二维图像的风格则利用物体的轮廓线,特征线,物体的局部几何细节以及用户标定的特征四种不同层次上的特征加以描述,每种风格的迁移可表示成一个二次能量的形式,最后三维网格的风格化变形则通过一个整体的二次能量优化实现。用户可以通过调节不同风格能量在整体优化方程中的权重系数来改变最后风格化变形的结果,从而为用户在风格化变形中提供更多的控制。
With the development of the 3D scanning technology, the acquisition of point set surface and mesh models becomes easier and easier. Shape editing with these 3D models are now intensively involved in many applications, such as computer animation, computer game, film industry, and so on, which also makes shape editing as an important research topic in the field of computer graphics. Although many researches on mesh editing have been existed, e.g. detail-preserving gradient domain mesh deformation, multi-resolution mesh editing, in order to create some vivid models, the user has to learn 3D modeling skills and aesthetic knowledge. On the other hand, point set surface becomes another important representation of 3D models, due to its simple data structure and intrinsic power of expressing arbitrary complex shapes. However, the editing methods for point set surface are rare, especially for the case of easy for use and reusability.
This thesis focused on the aforementioned problems. As to the mesh editing, a method of image driven shape deformation with styles was proposed. It used a 2D drawing or 2D picture to produce the 3D mesh deformation. The deformed 3D mesh has the similar styles with the original 2D image. Since novels used to use 2D sketch as interaction interface, and 2D picture can lead to vivid deformed results, this method can reduce the requirement of the 3D modeling experience and aesthetic knowledge. As to the point set surface editing, discussions on detail-preserving shape deformation for large scale point set surface, shape interpolation between key frames and deform transfer for point set surface were intensively involved, all of which are the most frequently used technologies in key-framed animation system. This thesis also presented some algorithms and their implementation for these aspects. The examples shown in this thesis proved the feasibility and effectiveness of these methods.
The main research topics and contributions of this thesis include:
·Proposed a method of detail-preserving deformation for large scale point set surface. This method inherits the advantages of free form deformation and gradient domain deformation. It can well preserve the local details of the point set surface and keep a high efficiency during deformation. Firstly, the proxy model of the point set surface was constructed based on the idea of OBBTree and the intersection test of the oriented bounding box. The deformation of the original point set surface was driven by the deformation of the proxy model and the local details preservation was implemented via constraining the affine transformation of each node of the proxy model to be rigid transformation. Since the interaction for the traditional free form deformation is performed on the proxy model, which was not so intuitive for the users, this method allows direct actions on the point set surface through a nearest-neighbor mapping technology.
·Proposed the definition of vertex deformation gradient of point set surface. The closed form solution of which was also presented. Compared with the traditional deformation gradient which is defined on each triangle face of the mesh, the vertex deformation gradient is defined on each point of the point set surface directly and can be got from its k nearest points. From the view of quadratic energy optimization, we also presented the closed form solution of the vertex deformation gradient.
·By applying the vertex deformation gradient into the problem of shape interpolation and deformation transfer for the point set surface, correspondence between the source point set surface and the target point set surface was well established. The reconstruction for the deformed point set surface can be also well solved. For the case of shape interpolation, the shrinkage problem was avoided by decomposing the vertex deformation gradient into rational part and scale part with polar decomposition and each part was interpolated individually. The vertex deformation gradient of the target point set surface during deformation transfer was interpolated with matrix exponential mapping, which avoids the non-commutative property of the spherical linear interpolation, namely, different order of interpolation will lead to different result.
·Proposed a method of image driven shape deformation with styles. This method represents the shape styles with four kinds of shape descriptions from different levels in the context of triangular mesh. Namely, the object contour, the context curves, local geometric details, perception-driven features. The transfer of each of these styles can be formulated as a quadratic energy optimization. The whole deformation of the 3D mesh was implemented as energy optimization via combining all these energy items together. The coefficient of each energy item in the total energy equation can be adjusted with users' intents, so that user can get different resulting styles and get more controls for the stylization of the 3D mesh.
引文
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