基于显式表达的高效网格形变技术
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摘要
随着扫描获取技术的发展,多边形网格表示方法凭借其普适的形状描述能力,得到了人们的高度关注和广泛的应用。今天,人们可以很容易地从真实世界获取高精度的网格模型。如何对这些已有模型进行有效地编辑和复用,以满足造型和动画等领域的应用需求,成为了计算机图形学中非常重要的问题。高效的网格形变技术作为解决此问题的关键,是近年来研究的热点。
根据形变函数的定义和计算方式,目前的形变方法主要可以分为两类。一类基于顶点坐标的显示表达式进行计算,以蒙皮、自由形变等空间插值方法为代表。此类方法算法简单、运行速度快、独立于模型表达,但是对结果缺乏有效控制,难以保证形变质量。另一类以微分域方法为代表,通过建立保持几何细节的目标能量形式,使用优化过程对顶点坐标进行隐式求解。此类方法能够得到高质量的形变结果,并支持多种约束控制,缺点是依赖于特定的模型表达,且优化过程需要求解大规模系统方程,在处理复杂模型时算法复杂度很高。
本文围绕复杂网格模型的形变问题展开研究,针对性地提出了多个基于显式表达的高效网格形变方法。其核心思想是通过引入新的显式表达式,以提高形变函数的表达能力和形变的自由度,并结合微分属性的优化进一步对形变效果进行控制。所提出的方法在保证算法效率的基础上,有效提高了形变结果的质量。本文的主要贡献有：
·提出了一种基于四面体网格的光顺形变插值方法。针对传统重心坐标插值在单元边界处的一阶不连续现象,我们通过在四面体网格顶点处引入一个额外的仿射变换,为边界处形变梯度的调整提供了新的自由度,并借助此自由度,实现了各单元内形变梯度的全局优化,进而得到了光顺的插值形变效果。该插值方法具有显式计算、局部支撑、适合GPU并行处理等优点,具有很高的计算效率。该方法还可以拓展用于二维图像形变以及四面体网格的交互形变。
·针对不同模型表达之间的形变迁移问题,提出了一种基于包围网格插值的形变迁移方法。我们将目标网格嵌入一个包围网格中,利用从源形变中提取的形变梯度信息驱动包围网格进行形变,再通过显式插值技术变形目标网格。该方法分为两个阶段：包围网格形变阶段基于格林坐标函数构成的形变子空间,根据内部控制点处的形变梯度约束,利用优化过程反向求取包围网格的对应形变,其算法复杂度与目标网格无关；目标网格形变阶段的格林坐标插值利用GPU并行计算,可以达到很高的计算效率。该形变迁移方法突破了传统方法只能处理流形网格的限制,可以支持离散多边形集合、多部分模型等多种目标网格类型,而源形变可以是动作捕捉、网格序列等多种数据格式。
·针对形状插值中的路径问题,提出了一种基于预计算扭曲轨迹的高效形状插值方法。该方法的核心是一种新的显式计算的扭曲轨迹表达式,通过引入一个非线性分量来描述顶点处的局部朝向变化,解决了传统的线性插值轨迹在处理大旋转形变时出现的塌缩问题。我们将插值过程分为两个阶段,预计算阶段使用对应于插值效果尽可能刚性的目标能量,优化求解每个顶点处的最佳轨迹参数；实时插值阶段使用GPU高效地并行计算各顶点的轨迹坐标。与微分域的隐式路径插值方法相比,我们的方法在插值阶段不需要求解方程,性能得到了大幅提升。此外,该方法还可以拓展到多目标插值和四面体模型插值的情况。
As an important research topic of computer graphics which has been intensively studied, mesh deformation is widely used in many applications such as entertainment and virtual reality. Recently, as the increasing of mesh complexity, current methods are difficult to meet both the performance and quality requirements of applications.
According to the formulation of the deformation functions, current mesh deformation meth-ods can be divided into two categories:explicit form based methods and implicit form based meth-ods. Explicit form based methods, including Skeletal-Subspace Deformation (SSD) and Free Form Deformation (FFD), share the following advantages:simple, fast and independent of model repre-sentation. But the drawback of these methods is the lack of efficient way to control the deformation effects, and it is hard to achieve high quality results under large deformation for these methods. Implicit form based methods, represented by gradient domain methods, are able to achieve high quality detail-preserving deformation results and support various constraints. But they mostly de-pend on manifold mesh and require solving large linear or non-linear systems, which lead to high memory and computation costs for complex models.
This thesis focuses on the development of the explicit form based deformation techniques. Base on new explicit form deformation functions with high expression ability and optimization of gradient domain properties, several efficient deformation techniques are presented, which are able to achieve high quality deformation results in high performance, even for complex models. The main contributions include:
●Propose a smooth interpolation deformation method based on tetrahedron control mesh. The traditional barycentric coordinates generate apparent first-order discontinuity artifacts across the boundary. To avoid such artifacts, we add a local transformation at each control vertex for interpolation, so that we can minimize the first-order discontinuity by optimizing the local transformations. Based on the explicit and local support formulation, the interpolation can be efficiently evaluated using modern GPUs and achieves high performance. Our method can also be applied to 2D image objects and deforming the control mesh.
●Propose a cage based deformation transfer method. Previous methods can only transfer the deformation between manifold mesh. To avoid such limitation, the target model is first em-bedded into a cage by explicit interpolation function. The deformation gradient sequences of some user-selected points on the source are then extracted and used as the gradient con-straints in the cage to guide the target deformation. By employing such method, we can transfer deformation from various sources to geometric models in variant representations. The key part of this method is a green coordinates based subspace method, whose opti-mization is independent of the target model. It ensures that the deformation transfer can be independent of model representation and the nonlinear optimization can be efficiently solved. Besides, position, orientation and length constraints can be applied for supporting more controls to the deformation transfer results.
●Propose a shape interpolation method based on precomputed trajectory warping. We present a new explicit form of trajectory, which regulates the linear interpolation trajectory with additional angular velocity component for tracing the local rotations. This nonlinear trajec-tory can efficiently eliminate the shrinking artifacts introduced by linear trajectory. We find trajectory parameters for each vertex by optimization with the consideration of as-rigid-as-possible deformation in the pre-computing stage. During run-time, the vertices coordinates on the intermediate shape can be computed in parallel according to the explicit formulation. Comparing with the implicit methods which reconstruct the intermediate shape by optimiza-tion, our method does not need to solve large linear system and is able to achieve extremely high performance. This technique can also be extended easily to multi-pose interpolation and tetrahedron models.
根据形变函数的定义和计算方式,目前的形变方法主要可以分为两类。一类基于顶点坐标的显示表达式进行计算,以蒙皮、自由形变等空间插值方法为代表。此类方法算法简单、运行速度快、独立于模型表达,但是对结果缺乏有效控制,难以保证形变质量。另一类以微分域方法为代表,通过建立保持几何细节的目标能量形式,使用优化过程对顶点坐标进行隐式求解。此类方法能够得到高质量的形变结果,并支持多种约束控制,缺点是依赖于特定的模型表达,且优化过程需要求解大规模系统方程,在处理复杂模型时算法复杂度很高。
本文围绕复杂网格模型的形变问题展开研究,针对性地提出了多个基于显式表达的高效网格形变方法。其核心思想是通过引入新的显式表达式,以提高形变函数的表达能力和形变的自由度,并结合微分属性的优化进一步对形变效果进行控制。所提出的方法在保证算法效率的基础上,有效提高了形变结果的质量。本文的主要贡献有：
·提出了一种基于四面体网格的光顺形变插值方法。针对传统重心坐标插值在单元边界处的一阶不连续现象,我们通过在四面体网格顶点处引入一个额外的仿射变换,为边界处形变梯度的调整提供了新的自由度,并借助此自由度,实现了各单元内形变梯度的全局优化,进而得到了光顺的插值形变效果。该插值方法具有显式计算、局部支撑、适合GPU并行处理等优点,具有很高的计算效率。该方法还可以拓展用于二维图像形变以及四面体网格的交互形变。
·针对不同模型表达之间的形变迁移问题,提出了一种基于包围网格插值的形变迁移方法。我们将目标网格嵌入一个包围网格中,利用从源形变中提取的形变梯度信息驱动包围网格进行形变,再通过显式插值技术变形目标网格。该方法分为两个阶段：包围网格形变阶段基于格林坐标函数构成的形变子空间,根据内部控制点处的形变梯度约束,利用优化过程反向求取包围网格的对应形变,其算法复杂度与目标网格无关；目标网格形变阶段的格林坐标插值利用GPU并行计算,可以达到很高的计算效率。该形变迁移方法突破了传统方法只能处理流形网格的限制,可以支持离散多边形集合、多部分模型等多种目标网格类型,而源形变可以是动作捕捉、网格序列等多种数据格式。
·针对形状插值中的路径问题,提出了一种基于预计算扭曲轨迹的高效形状插值方法。该方法的核心是一种新的显式计算的扭曲轨迹表达式,通过引入一个非线性分量来描述顶点处的局部朝向变化,解决了传统的线性插值轨迹在处理大旋转形变时出现的塌缩问题。我们将插值过程分为两个阶段,预计算阶段使用对应于插值效果尽可能刚性的目标能量,优化求解每个顶点处的最佳轨迹参数；实时插值阶段使用GPU高效地并行计算各顶点的轨迹坐标。与微分域的隐式路径插值方法相比,我们的方法在插值阶段不需要求解方程,性能得到了大幅提升。此外,该方法还可以拓展到多目标插值和四面体模型插值的情况。
As an important research topic of computer graphics which has been intensively studied, mesh deformation is widely used in many applications such as entertainment and virtual reality. Recently, as the increasing of mesh complexity, current methods are difficult to meet both the performance and quality requirements of applications.
According to the formulation of the deformation functions, current mesh deformation meth-ods can be divided into two categories:explicit form based methods and implicit form based meth-ods. Explicit form based methods, including Skeletal-Subspace Deformation (SSD) and Free Form Deformation (FFD), share the following advantages:simple, fast and independent of model repre-sentation. But the drawback of these methods is the lack of efficient way to control the deformation effects, and it is hard to achieve high quality results under large deformation for these methods. Implicit form based methods, represented by gradient domain methods, are able to achieve high quality detail-preserving deformation results and support various constraints. But they mostly de-pend on manifold mesh and require solving large linear or non-linear systems, which lead to high memory and computation costs for complex models.
This thesis focuses on the development of the explicit form based deformation techniques. Base on new explicit form deformation functions with high expression ability and optimization of gradient domain properties, several efficient deformation techniques are presented, which are able to achieve high quality deformation results in high performance, even for complex models. The main contributions include:
●Propose a smooth interpolation deformation method based on tetrahedron control mesh. The traditional barycentric coordinates generate apparent first-order discontinuity artifacts across the boundary. To avoid such artifacts, we add a local transformation at each control vertex for interpolation, so that we can minimize the first-order discontinuity by optimizing the local transformations. Based on the explicit and local support formulation, the interpolation can be efficiently evaluated using modern GPUs and achieves high performance. Our method can also be applied to 2D image objects and deforming the control mesh.
●Propose a cage based deformation transfer method. Previous methods can only transfer the deformation between manifold mesh. To avoid such limitation, the target model is first em-bedded into a cage by explicit interpolation function. The deformation gradient sequences of some user-selected points on the source are then extracted and used as the gradient con-straints in the cage to guide the target deformation. By employing such method, we can transfer deformation from various sources to geometric models in variant representations. The key part of this method is a green coordinates based subspace method, whose opti-mization is independent of the target model. It ensures that the deformation transfer can be independent of model representation and the nonlinear optimization can be efficiently solved. Besides, position, orientation and length constraints can be applied for supporting more controls to the deformation transfer results.
●Propose a shape interpolation method based on precomputed trajectory warping. We present a new explicit form of trajectory, which regulates the linear interpolation trajectory with additional angular velocity component for tracing the local rotations. This nonlinear trajec-tory can efficiently eliminate the shrinking artifacts introduced by linear trajectory. We find trajectory parameters for each vertex by optimization with the consideration of as-rigid-as-possible deformation in the pre-computing stage. During run-time, the vertices coordinates on the intermediate shape can be computed in parallel according to the explicit formulation. Comparing with the implicit methods which reconstruct the intermediate shape by optimiza-tion, our method does not need to solve large linear system and is able to achieve extremely high performance. This technique can also be extended easily to multi-pose interpolation and tetrahedron models.
引文
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