基于对偶网格的变形技术
|
摘要
网格变形是数字几何处理中最基本的问题之一,随着数字几何模型逐渐走入人们的日常生活,网格变形在实际应用中也变得越来越有用,大型建模软件都集成了变形这个工具。它的应用包括模型重用和重建,虚拟场景的物体模拟,计算机动画中的帧序列,甚至三维特效的生成。
鉴于网格变形的应用前景,本文提出了一种新的网格变形的框架,步骤分为:(1)输入三角网格(2)求解对偶网格(3)对对偶网格变形(4)从变形后的对偶网格恢复三角网格得到变形后的结果。
本文回顾了三角网格的对偶网格的定义及其邻域特征。分析了对偶网格与原三角网格的关系,考虑了如何从变形后的对偶网格恢复到原三角网格得到变形结果的问题。本文的解决办法是利用细节编码将原三角网格的顶点编码为此顶点对应的对偶网格面上的顶点的局部金字塔坐标,随后将变形后的对偶网格加上编了码的细节,就能得到无收缩的三角网格上的变形结果,实现了由对偶网格驱动的三角网格变形框架。
本文利用三角网格的对偶网格的1-邻域的简单规则结构、基面的共面性、以及法向计算的稳定性来给出新的算法或者改进原有算法。
第一部分:
改进金字塔坐标变形算法,使得变形结果(1)迭代更加稳定(2)不失迭代速度(3)保持相同的变形效果。也就是说金字塔坐标算法的稳定性得到了非常大的改进,迭代速度也没有削弱,变形结果非常相似。
第二部分:
相对于其它变形算法(如ARAP(保刚性形状变形),线性旋转不变内在量等),引进了一种新的表示形状保持的内在量,即伸缩不变内在量。该内在量具有旋转、平移、伸缩不变性。首先在三角网格上面进行定义,发现矩阵的行列数难以估计,然后给出对偶网格上伸缩不变内在量的定义,并且提出了基于对偶网格上保持伸缩不变内在量的变形方法。最后给出了一个基于Laplace方法的一个改进。
Mesh deformation is one of the basic problems in digital geometry processing. While the digital geometry model is going in people's lives, deformation becomes much useful than any time before. Large-scale modeling software integrate the deformation as their necessary tools. The application of mesh deformation consists of re-using models, reconstruction of models, making computer animation frames, and making effects in movies.
This paper proposes a new framework for triangle mesh deformation.. The main steps follows(1) input a triangle mesh (2)compute the dual mesh(3)deform the dual mesh (4)get triangle mesh from deformed dual mesh.
This paper recalls the definition of the dual mesh of a given triangle mesh and the 1 -ring neighbor characteristics and analyzes the relationship between dual and primal triangle mesh. The transfer from deformed dual mesh to triangle mesh is taken into consideration. To get non-shrinking deformed triangle mesh from deformed dual mesh, this paper uses pyramid detail encoding to record each vertex of triangle mesh according to the corresponding dual mesh face's vertices and during step (4) in last section uses this encoding to recover the triangle mesh vertices. So far, the whole four steps in last section can lead to an triangle mesh deformation framework based on dual mesh.
By taking advantage of the simple, regular, coplanar and stabile structure of the dual mesh, this paper uses the dual mesh framework to improve some algorithm or propose new algorithm.
The first part:
Improve pyramid coordinate deformation method and achieves three results: (1) more stable iteration (2) cost almost the same iteration time to get pleasing result (3) get almost the same results. So the pyramid coordinates method is improved in stability but without lose iteration time and deformation effect.
The second part:
Related to other deformation algorithm, e.g. as-rigid-as-possible which keeps rigidity of local shape, This paper proposes new intrinsic variables on triangle meshes to keep shapes, so called scale invariant intrinsic variables which are invariant under translation, rotation, and scaling. Firstly give definition on triangle mesh found that the matrix's row num has nothing to do with column num, then this paper proposes the definition on dual mesh. Based on the dual mesh definition of scale invariant intrinsic variables, several deformation methods based on optimize objective functions with preserving intrinsic invariant quantities are proposed while some pleasing results are achieved. And at last this paper give a new algorithm based on Naive Laplace deformation method.
鉴于网格变形的应用前景,本文提出了一种新的网格变形的框架,步骤分为:(1)输入三角网格(2)求解对偶网格(3)对对偶网格变形(4)从变形后的对偶网格恢复三角网格得到变形后的结果。
本文回顾了三角网格的对偶网格的定义及其邻域特征。分析了对偶网格与原三角网格的关系,考虑了如何从变形后的对偶网格恢复到原三角网格得到变形结果的问题。本文的解决办法是利用细节编码将原三角网格的顶点编码为此顶点对应的对偶网格面上的顶点的局部金字塔坐标,随后将变形后的对偶网格加上编了码的细节,就能得到无收缩的三角网格上的变形结果,实现了由对偶网格驱动的三角网格变形框架。
本文利用三角网格的对偶网格的1-邻域的简单规则结构、基面的共面性、以及法向计算的稳定性来给出新的算法或者改进原有算法。
第一部分:
改进金字塔坐标变形算法,使得变形结果(1)迭代更加稳定(2)不失迭代速度(3)保持相同的变形效果。也就是说金字塔坐标算法的稳定性得到了非常大的改进,迭代速度也没有削弱,变形结果非常相似。
第二部分:
相对于其它变形算法(如ARAP(保刚性形状变形),线性旋转不变内在量等),引进了一种新的表示形状保持的内在量,即伸缩不变内在量。该内在量具有旋转、平移、伸缩不变性。首先在三角网格上面进行定义,发现矩阵的行列数难以估计,然后给出对偶网格上伸缩不变内在量的定义,并且提出了基于对偶网格上保持伸缩不变内在量的变形方法。最后给出了一个基于Laplace方法的一个改进。
Mesh deformation is one of the basic problems in digital geometry processing. While the digital geometry model is going in people's lives, deformation becomes much useful than any time before. Large-scale modeling software integrate the deformation as their necessary tools. The application of mesh deformation consists of re-using models, reconstruction of models, making computer animation frames, and making effects in movies.
This paper proposes a new framework for triangle mesh deformation.. The main steps follows(1) input a triangle mesh (2)compute the dual mesh(3)deform the dual mesh (4)get triangle mesh from deformed dual mesh.
This paper recalls the definition of the dual mesh of a given triangle mesh and the 1 -ring neighbor characteristics and analyzes the relationship between dual and primal triangle mesh. The transfer from deformed dual mesh to triangle mesh is taken into consideration. To get non-shrinking deformed triangle mesh from deformed dual mesh, this paper uses pyramid detail encoding to record each vertex of triangle mesh according to the corresponding dual mesh face's vertices and during step (4) in last section uses this encoding to recover the triangle mesh vertices. So far, the whole four steps in last section can lead to an triangle mesh deformation framework based on dual mesh.
By taking advantage of the simple, regular, coplanar and stabile structure of the dual mesh, this paper uses the dual mesh framework to improve some algorithm or propose new algorithm.
The first part:
Improve pyramid coordinate deformation method and achieves three results: (1) more stable iteration (2) cost almost the same iteration time to get pleasing result (3) get almost the same results. So the pyramid coordinates method is improved in stability but without lose iteration time and deformation effect.
The second part:
Related to other deformation algorithm, e.g. as-rigid-as-possible which keeps rigidity of local shape, This paper proposes new intrinsic variables on triangle meshes to keep shapes, so called scale invariant intrinsic variables which are invariant under translation, rotation, and scaling. Firstly give definition on triangle mesh found that the matrix's row num has nothing to do with column num, then this paper proposes the definition on dual mesh. Based on the dual mesh definition of scale invariant intrinsic variables, several deformation methods based on optimize objective functions with preserving intrinsic invariant quantities are proposed while some pleasing results are achieved. And at last this paper give a new algorithm based on Naive Laplace deformation method.
引文
[1]A.Sheffer and V.Krayevoy.Pyramid coordinates for morphing and deformation.In Proceedings of International Symposium on 3D Data Processing.Visualization,and Transmission,Thessaloniki:68-75,2004.
[2]A.Nealen,O.Sorkine,M.Alexa,and D.Cohen-Or.A Sketch-Based Interface for Detail-Preserving Mesh Editing.ACM Trans.Graphics,vol.24,no.3,2005.
[3]M.Botsch and L.Kobbelt.Multi-resolution surface representation based on displacement volumes.Comput.Graph.Forum(Eurographics 2003)22(3),483-491,2003.
[4]D.Zorin,P.Schr(o|¨)der,and W.Sweldens.Interactive multi-resolution mesh editing.In Proceedings of ACM SIGGRAPH 1997,pages 259-268.ACM Press.1997.
[5]K.Hormann and M.S.Floater.Mean value coordinates for arbitrary planar polygons.ACM Transactions on Graphics,Mar.Accepted,2006.
[6]M.S.Floater.Mean value coordinates.Computer Aided Geometric Design,20(1):19-27,2003.
[7]M.S.Floater,K.Geza,and M.Reimers.Mean value coordinates in 3d.Computer Aided Geometric Design 22,7,623-631,2005.
[8]M.Botsch and L.Kobbelt.An Intuitive Framework for Real-Time Freeform Modeling,ACM Trans.Graphics,vol.23,no.3,pp.630-634,2004.
[9]M.Alexa.Differential coordinates for local mesh morphing and deformation.The Visual Computer 19,2,105-114,2003.
[10]任绍忠,刘利刚,王国瑾.保细节平面形状编辑.计算机辅助设计与图形学学报,Vol.19,No.1,Jan.2007.
[11]O.Au,C.Tai,H.Fu,and L.Liu.Mesh Editing with Curvature Flow Laplacian.Eurographics Symposium on Geometry Processing,2005.
[12]O.Sorkine.State-of-the-art report:Laplacian mesh processing.In Proceedings of the Eurographics 2005.Eurographics Association.2005.
[13]O.Sorkine,Y.Lipman,D.Cohen-Or,M.Alexa,C.R(o|¨)ssl,and H.-P.Seidel.Laplacian Surface Editing,Proc.Symp.Geometry Processing,pp.179-188,2004.
[14]O.Chung Au,C.Tai,L.Liu,and H.Fu.Dual Laplacian editing for meshes..IEEE Transactions on Visualization and Computer Graphics,12(3):386-395,2006.
[15]Olga Sorkine and Marc Alexa.As-rigid-as-possible surface modeling.In Proceedings of Eurographics/ACM SIGGRAPH Symposium on Geometry Processing,July 2007.
[16] 许栋. Differential Mesh Processing. 浙江大学博士论文, 2006.
[17]T. W. Sederberg and S. R. Parry. Free-form deformation of solid geometric models. 1986.In SIGGRAPH '86: Proceedings of the 13th annual conference on Computer graphics and interactive techniques, pages 151-160, New York, NY, USA. ACM Press.
[18] V. Krayevoy and A. Sheffer. Mean-value geometry encoding. IJSM 12, 1, 29-46, 2006.
[19] W. Sweldens and P. Schroder. Digital geometric signal processing, course notes 50. In Proceedings of ACM SIGGRAPH 2001. ACM Press.
[20] Y. Lipman, D. Levin, and D. Cohen-Or. Green Coordinates. ACM SIGGRAPH2008.
[21] Y. Lipman, O. Sorkine, D. Levin, and D. Cohen-Or. Linear Rotation-Invariant Coordinates for Meshes. ACM Trans. Graphics, vol. 24. 2005.
[22]Y. Lipman, O.Sorkine, D.Cohen-Or, and D.Levin. Differential coordinates for interactive mesh editing. In Proceedings of Shape Modeling International, IEEE Computer Society Press. To appear. 2004.
[23] Y. Yu, K. Zhou, D. Xu, X.Shi, H. Bao, B. Guo, and H. Shum. Mesh editing with poisson-based gradient field manipulation. ACM Trans. Graph.23(3):644-651. 2004.
[2]A.Nealen,O.Sorkine,M.Alexa,and D.Cohen-Or.A Sketch-Based Interface for Detail-Preserving Mesh Editing.ACM Trans.Graphics,vol.24,no.3,2005.
[3]M.Botsch and L.Kobbelt.Multi-resolution surface representation based on displacement volumes.Comput.Graph.Forum(Eurographics 2003)22(3),483-491,2003.
[4]D.Zorin,P.Schr(o|¨)der,and W.Sweldens.Interactive multi-resolution mesh editing.In Proceedings of ACM SIGGRAPH 1997,pages 259-268.ACM Press.1997.
[5]K.Hormann and M.S.Floater.Mean value coordinates for arbitrary planar polygons.ACM Transactions on Graphics,Mar.Accepted,2006.
[6]M.S.Floater.Mean value coordinates.Computer Aided Geometric Design,20(1):19-27,2003.
[7]M.S.Floater,K.Geza,and M.Reimers.Mean value coordinates in 3d.Computer Aided Geometric Design 22,7,623-631,2005.
[8]M.Botsch and L.Kobbelt.An Intuitive Framework for Real-Time Freeform Modeling,ACM Trans.Graphics,vol.23,no.3,pp.630-634,2004.
[9]M.Alexa.Differential coordinates for local mesh morphing and deformation.The Visual Computer 19,2,105-114,2003.
[10]任绍忠,刘利刚,王国瑾.保细节平面形状编辑.计算机辅助设计与图形学学报,Vol.19,No.1,Jan.2007.
[11]O.Au,C.Tai,H.Fu,and L.Liu.Mesh Editing with Curvature Flow Laplacian.Eurographics Symposium on Geometry Processing,2005.
[12]O.Sorkine.State-of-the-art report:Laplacian mesh processing.In Proceedings of the Eurographics 2005.Eurographics Association.2005.
[13]O.Sorkine,Y.Lipman,D.Cohen-Or,M.Alexa,C.R(o|¨)ssl,and H.-P.Seidel.Laplacian Surface Editing,Proc.Symp.Geometry Processing,pp.179-188,2004.
[14]O.Chung Au,C.Tai,L.Liu,and H.Fu.Dual Laplacian editing for meshes..IEEE Transactions on Visualization and Computer Graphics,12(3):386-395,2006.
[15]Olga Sorkine and Marc Alexa.As-rigid-as-possible surface modeling.In Proceedings of Eurographics/ACM SIGGRAPH Symposium on Geometry Processing,July 2007.
[16] 许栋. Differential Mesh Processing. 浙江大学博士论文, 2006.
[17]T. W. Sederberg and S. R. Parry. Free-form deformation of solid geometric models. 1986.In SIGGRAPH '86: Proceedings of the 13th annual conference on Computer graphics and interactive techniques, pages 151-160, New York, NY, USA. ACM Press.
[18] V. Krayevoy and A. Sheffer. Mean-value geometry encoding. IJSM 12, 1, 29-46, 2006.
[19] W. Sweldens and P. Schroder. Digital geometric signal processing, course notes 50. In Proceedings of ACM SIGGRAPH 2001. ACM Press.
[20] Y. Lipman, D. Levin, and D. Cohen-Or. Green Coordinates. ACM SIGGRAPH2008.
[21] Y. Lipman, O. Sorkine, D. Levin, and D. Cohen-Or. Linear Rotation-Invariant Coordinates for Meshes. ACM Trans. Graphics, vol. 24. 2005.
[22]Y. Lipman, O.Sorkine, D.Cohen-Or, and D.Levin. Differential coordinates for interactive mesh editing. In Proceedings of Shape Modeling International, IEEE Computer Society Press. To appear. 2004.
[23] Y. Yu, K. Zhou, D. Xu, X.Shi, H. Bao, B. Guo, and H. Shum. Mesh editing with poisson-based gradient field manipulation. ACM Trans. Graph.23(3):644-651. 2004.