Real-time. Meshless Deformation. Xiaohu Guo, Hong Qin Center for Visual Computing Department of Computer Science Stony Brook

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1 Real-time Meshless Deformation Xiaohu Guo, Hong Qin Center for Visual Computing Department of Computer Science Stony Brook

2 Outline Introduction & previous work Meshless methods Computational techniques Hierarchical discretization Modal analysis Experimental results Conclusion & future work

3 Introduction Increasing data acquisition power has made large-scale sampled surfaces prevalent in digital geometry processing pipeline. Laser scanner, structured light, holoimage, Developing new techniques for point- centered digital processing has become a common and long-term mission. This paper focuses on real-time (or interactive) physical animation and manipulation of sampled geometry. Holoimage scanning system in Stony Brook (Courtesy of Gu et al.)

4 Contribution Physical simulation no mesh Point-sampled geometry has no differential properties MLS method Node distribution control and numerical quadrature octree structure Real-time simulation modal warping

5 Computational Framework

6 Previous Work Physics-based animation D. Terzopoulos, J. Platt, A. Barr, and K. Fleischer, Elastically Deformable Models, SIGGRAPH 1987 D. L. James and D. K. Pai, ArtDefo:: Accurate Real-time Deformable Objects, SIGGRAPH 1999 G. Debunne,, M. Desbrun,, M. P. Cani,, and A. H. Barr, Dynamic Real-time Deformations using Space & Time Adaptive Sampling, SIGGRAPH 2001 Modal analysis A. Pentland and J. Williams, Good Vibrations: Model Dynamics for Graphics and Animation, SIGGRAPH 1989 D. L. James and D. K. Pai, Dyrt:: Dynamic Response Textures for Real Time Deformation Simulation with Graphics Hardware, SIGGRAPH 2002 K. Hauser, C. Shen, and J. O Brien, O Interactive Deformation using Modal Analysis with Constraints, Graphics Interface 2003 M. G. Choi and H. S. Ko, Modal Warping: Real-time Simulation of Large Rotational Deformation and Manipulations, IEEE TVCG 2005

7 Previous Work Meshless methods T. Belytschko,, Y. Y. Lu, and L. Gu, Element-free Galerkin Methods, Int. J. Num. Meth.. Eng T. Belytschko,, Y. Y. Lu, and L. Gu, Fracture and Crack Growth by Element-free Galerkin Methods, Model. Simul.. Mater. Sci.. Eng M. Desbrun and M. P. Cani, Animating Soft Substances with Implicit Surfaces, SIGGRAPH 1995 M. Desbrun and M. P. Cani, Smoothed Particles: A New Paradigm for Animating Highly Deformable Bodies, EG workshop on Animation and Simulation 1996 M. Muller, R. Keiser, A. Nealen,, M. Pauly,, M. Gross, and M. Alexa, Point Based Animation of Elastic, Plastic and Melting Objects, SCA 2004 M. Muller, B. Heidelberger, M. Teschner,, and M. Gross, Meshless Deformations Based on Shape Matching, SIGGRAPH 2005 M. Pauly,, R. Keiser, B. Adams, P. Dutre,, M. Gross, and L. J. Guibas, Meshless Animation of Fracturing Solids, SIGGRAPH 2005

8 Elastic Deformation We use the Euler-Lagrange equations for the elastic deformation: d dt T u& ( u& ) V ( u) + μu& + u where the kinetic energy: M is the mass matrix: and the elastic potential energy: ε is the strain tensor; = F T ext M = IJ 1 2 Ω Ω ρ( x ) u& u& dω V = 1 2 I, J = ρ( φ ( φ ( dω I J M IJ u& I u& ν 2 ij kl = G tr ( ε ) + δ δ ε ikε jl dω Ω 2 1 ν J φi is the MLS shape function value of node I.

9 Meshless Methods Developed in Mechanical Engineering to solve PDEs/ODEs numerically based on scattered nodes without the need of additional mesh structure. Advantages of Meshless Methods: No mesh for the scattered nodes Spatial adaptivity (node insertion/elimination) Shape function polynomial order adaptivity Minimized data management overhead We use the Moving Least Squares (MLS) shape function [Lancaster and Salkauskas,, 1981], employed in the Element-free Galerkin (EFG) method [Belytchko[ et al. 1994]

x, x ) > 0} = I I The approximation of the field function f at a parametric position x is only

10 MLS Shape Functions Each node I is associated with a positive weight function w I of compact support. The support of the weight function defines the influence domain of the node: Ω I { x R 2 : w ( = w( x, x ) > 0} = I I The approximation of the field function f at a parametric position x is only affected by those nodes whose weights are non-zero at x. Influence Domain Analysis Domain Node Object Boundary

11 MLS Shape Functions If a function f ( defined on the domain is sufficiently smooth, s we can l define a local approximation around a fixed point x Ω : l f ( x, L f ( = p ( a ( = p x m i= 1 ( a( where p i ( are polynomial basis functions, a i ( are their r coefficients. We can derive ( by minimizing the weighted L norm: J T 2 i.e. T x ) p ( x ) a( ) J = ( Pa f ) W( ( Pa f ) we obtain a( by setting: = A( a( B( f = 0 where the m m matrix A( is called moment matrix: i a 2 [ ] = w I ( x f I I I J a f ( x, i Ω T T T A ( = P W( P B( = P W( So we obtain: 1 a ( = A ( B( f

12 MLS Shape Functions So the approximate field function can be written as: l T 1 f ( x ) f ( x, = p ( A ( B( f = Φ( f where Φ( is the vector of the MLS shape functions: Φ( = T 1 [ φ (, φ (,... ( ] = p ( A ( B( ) 1 2 φ n x The moment matrix A( will be ill-conditioned when: 1. The basis functions p( are (almost) linearly dependent; 2. There are not enough nodal supports overlapping at the given point; Note that the necessary condition for the moment matrix to be invertible is: x Ω card{ I : x Ω. I } > m (patch covering condition) 3. The nodes whose supports overlap at the point are arranged in a special pattern, such as a plane for a complete linear basis p(, or a conic section for a quadratic polynomial basis.

13 MLS Shape Functions The continuity of the shape function is directly related to the continuity of both weight functions and the polynomial basis. The FEM equivalents can be reached if the weight functions are piecewise- constant over each influence domain.

14 Computational Techniques We need to generate overlapping patches Ω I comprising a cover { Ω I } of the domain Ω. Unstructured distribution of nodes will cause many algorithmic difficulties Determination of the patches that contribute to a certain integration point needs an expensive global search. The moment matrix may not be invertible if the patch covering conditions are not satisfied. The interaction of scattered nodes with the geometric boundary becomes b difficult to handle. Hierarchical discretization for meshless dynamics Octree-based distance field for surface geometry Octree-based volumetric node placement and patch generation Octree-based Gaussian integration for matrix assembly

15 Hierarchical Discretization We utilize the Multi-level level Partition of Unity (MPU) implicit surface construction method of [ [Ohtake et al. 2003]

16 Hierarchical Discretization Volumetric node placement and patch generation Gaussian integration for matrix assembly Object boundary α size( OI ) Integration point Integration cell node I Patch Ω I Octant O I

17 Modal Analysis for Meshless Dynamics Basics of Modal Analysis: M u& + Cu& + Ku = F (Rayleigh damping) C = α M + βk Let Ψ and Λ be the solution matrices of the generalized eigenvalue problem KΨ = MΨΛ, such that MΨ I and., = ( α I + βλ, and K z = Λ are all diagonal. M z = I ) C z Ψ T = Ψ T KΨ = Λ is called the modal displacement matrix, and we can let u ( t) = Ψz( t). The ith column of Ψ represents the ith mode shape and z(t) contains the corresponding modal amplitudes. We can take only the dominant l columns of Ψ by examining the eigenvalues. Ψ M && z z + Czz& + K zz = Ψ T F

18 Modal Warping for Rotational Deformation Modal Warping [ [Choi and Ko,, 2005] = 1 T 1 u ( u + u ) + ( u u T ) = ε +ω 2 2 strain tensor ε = 1 ( u + u T ) 2 1 rotation tensor ω = 1 ( u u T ) = ( u) = w rotation vector w ( = ( u) = ( ) Φ( Ψz 2 2 Φ( is the vector of MLS shape functions

Modal Warping for Rotational Deformation Embed a local

with a rotation matrix R computed from its rotation vector w.

coordinate frame as a linear Euler-Lagrangian equation : Mu&& L

decomposition : i && z L + C z z& L + K Manipulation constraints

19 Modal Warping for Rotational Deformation Embed a local coordinate frame at each simulation node, which is associated with a rotation matrix R computed from its rotation vector w. The non-linear dynamic system can be approximated at each local coordinate frame as a linear Euler-Lagrangian equation : Mu&& L + Cu& L + Ku L = R T F L L u ( t) = Φz ( t) By modal decomposition : i && z L + C z z& L + K Manipulation constraints (position/rotation) can be integrated using u Lagrange multipliers. i z z L = Ψ T ( T R F)

20 Video

21 Experimental Results Implemented on Windows XP PC with dual Intel Xeon 2.8GHz CPUs, 2.0GB RAM, and GeForce Fx 5900 Ultra GPU. For all the data sets, the MLS pre-computation for the system matrices takes < 10 minutes, and modal decomposition takes < 1 minute. model points nodes modes sec/frame bar 5,634 1, Igea 134, balljoint 137, rabbit 67,038 1, Santa 75,781 1,

22 Conclusion A A real-time meshless animation and simulation paradigm for point-sampled volumetric objects. Both interior and surface representation only consist of point samples. Exploit the Modal Warping technique in the meshless framework to achieve real-time manipulation and deformation.

23 Future Work Meshless physical simulation can be slow, especially when changing topology (such as cracks, or surgical cutting process). Are there any geometry-driven semi-physical simulation possible?... Volumetric mapping for material modeling, human-computer interaction

24 Acknowledgements NSF grant ACI ITR grant IIS Alfred P. Sloan Fellowship The Igea,, rabbit, balljoint,, and santa models are courtesy of Cyberware Inc. Colleagues at Center for Visual Computing, SUNY Stony Brook.

Real-time Meshless Deformation

Real-time Meshless Deformation Xiaohu Guo Hong Qin State University of New York at Stony Brook email: {xguo, qin}@cs.sunysb.edu http://www.cs.sunysb.edu/{ xguo, qin} Abstract In this paper, we articulate