Meshless Modeling, Animating, and Simulating Point-Based Geometry

Transcription

1 Meshless Modeling, Animating, and Simulating Point-Based Geometry Xiaohu Guo Stony Brook xguo@cs.sunysb.edu

2 Graphics Primitives - Points The emergence of points as the underlying graphics primitive: Increasing data acquisition power A dramatic increase in the polygonal complexity The average size of a rendered polygon is less than the size of a screen pixel Processing of many small triangles leads to bandwidth bottlenecks and excessive rasterization requirements Overhead of managing, processing and manipulating mesh connectivity information, especially: topological change in shape manipulation, or fracture in dynamic simulation, etc. Polygonal Mesh Points (Courtesy of Levoy et al.)

3 Research Agenda and Objectives Dynamic Points : : point-based digital clay which can be directly manipulated, edited, deformed through human-computer interaction for various applications (without the need of converting point- sampled geometry to polygonal meshes and/or higher-order polynomial representations) In a very general sense, Dynamic Points are governed by partial differential equations in the variational framework (e.g., Lagrangian mechanics, level-set formulation, meshless finite element techniques, etc.)

4 Research Issues Geometric level Point Cloud Volumetric Surface Material modeling Implicit Explicit Local parameterization Global parameterization

5 Research Issues Physics level Solid Model Simple mass-spring spring FEM with continuum mechanics Mesh-based FEM Meshless Method

6 Presentation Overview A geometric processing paradigm for point-based geometry: Dynamic Points the integration of geometric representation and physical simulation for point surfaces and volumes Global conformal parameterization of point- sampled surfaces (at the geometric level) Meshless,, thin-shell finite element formulation for point geometry (at the physics level) Applications: interactive simulation and animation, shape deformation and editing, crack generation and propagation, shape morphing, etc.

7 Presentation Overview Point Cloud Geometry Implicit Surface Parameterization Local Global Surface Mapping Volume Level-set surface editing Surface completion shape & texture Material, HCI Physics Mass-Spring Meshless FEM Dynamic surface editing Volumetric deformation Thin-shell simulation Physical morphing

8 Level-set set-based Point Surface Editing Local and global surface editing Level-set editing techniques Grid-based collision detection and topology change

9 Level-set Based Point Surface Editing Xiaohu Guo, Jing Hua, Hong Qin, Point Set Surface Editing Techniques based on Level- Sets, in Computer Graphics International, Xiaohu Guo, Jing Hua, and Hong Qin, Scalar-Function-Driven Editing on Point Set Surfaces, in IEEE Computer Graphics and Applications, 2004.

10 Presentation Overview Point Cloud Geometry Implicit Surface Parameterization Local Global Surface Mapping Volume Level-set surface editing Surface completion shape & texture Material, HCI Physics Mass-Spring Meshless FEM Dynamic surface editing Volumetric deformation Thin-shell simulation Physical morphing

11 Dynamic Point Surface Editing Dynamic surface editing Mass-spring spring dynamic system Physically-based haptic user interface force force

12 Dynamic Point Surface Editing Xiaohu Guo, Hong Qin, Dynamic Dynamic Sculpting and Deformation of Point Set Surfaces, in Pacific Graphics, Xiaohu Guo, Jing Hua, and Hong Qin, Touch-Based Based Haptics for Interactive Editing on Point Set Surfaces, in IEEE Computer Graphics and Applications, 2004.

13 Presentation Overview Point Cloud Geometry Implicit Surface Parameterization Local Global Surface Mapping Volume Level-set surface editing Surface completion geometry & texture Material, HCI Physics Mass-Spring Meshless FEM Dynamic surface editing Volumetric deformation Thin-shell simulation Physical morphing

14 Surface Completion for Shape & Texture Repair noisy, defective, incomplete point set surfaces Hole-filling capability for both shape and texture by local parameterization Active contour method for locating holes Curvature-centered and Texture- centered digital signature for comparing and selecting similar patches Shape and texture Poisson warping Parameterized hole region Hole filling Parameterization Parameterized warped region

15 Surface Completion for Shape & Texture Seyoun Park, Xiaohu Guo, Hayong Shin, and Hong Qin, Shape and Appearance Repair for Incomplete Point Surfaces, in IEEE ICCV 2005.

16 Presentation Overview Point Cloud Geometry Implicit Surface Parameterization Local Global Surface Mapping Volume Level-set surface editing Surface completion shape & texture Material, HCI Physics Mass-Spring Meshless FEM Dynamic surface editing Volumetric deformation Thin-shell simulation Physical morphing

17 Point-based Parameterization geometric intuition Seeking two vector fields directly defined over point samples. The global parameterization (u, v) are derived from the two vector fields, respectively, by integration. Compute the canonical homology basis of a genus g object a, b, a, b,..., a g, b } { g Cut the surface open along each topological handle, and map each patch to [0, u max ] X [0, v max ].

18 Global Conformal Parameterization theoretic background Finding a global conformal parameterization = computing a pair of smooth vector fields ( ω 1, ω2 on the surface: Both ω1 and ω2 have zero curl. Both ω1 and ω2 have zero divergence. ω ω = ω 2 and are conjugate to each other, 2 *ω 1 1. Vector fields ω1, ω2 with zero curl and zero divergence are called harmonic 1-forms1 forms.. The pair of conjugate harmonic 1-forms 1 ( ω 1, ω2) is called holomorphic 1-form. )

19 Global Conformal Parameterization theoretic background After we get the holomorphic 1-form ( ω 1, ω2), we map the surface to the ( u, v) plane by integration: Fix a base vertex v0, for any vertex vk, we select a curve γ on the surface from v to v 0 k, then we define the parameter value of equals: ( u( vk ), v( v k )) = ( ω 1, ω2) γ v k therefore, locally, ω = u, ω 1 2 = v. The parameter does not depend on the choice of γ, but depends on the homotopy class of γ.

20 Global Conformal Parameterization algorithmic overview + If we cut the surface along a k to get two boundaries a k and a k. We can define a harmonic function f : S R, such that f = 0 a k and f = 1, and f minimize the harmonic energy a + E( f ) k then the gradient is a harmonic 1-form on. = f S f S 2 We can construct 2g harmonic 1-forms: ω, ω,..., ω } corresponding to each homology basis. { 1 2 2g

21 Global Conformal Parameterization algorithmic overview If there are some boundary loops (they may be manually selected to improve uniformity of conformal factor), add them to the loop set Σ and remove one boundary loop from Σ (because the dimension of cohomology group is 2g + ( b 1) ). For each loop τ from Σ, compute the harmonic function f : S R, such that: f f =1 τ = 0 γ γ Σ γ τ Δf = 0 Together with the g harmonic 1-forms, now we have harmonic 1-forms f. 2 2g + ( b 1)

22 Global Conformal Parameterization algorithmic overview At each point on S, rotate f about the normal a right angle to obtain another vector field * f ; the pair vector fields ( f,* f) is a holomorphic 1-form corresponding to each homology basis. These 2g + ( b 1) holomorphic 1-forms compose a basis for all the holomorphic 1-forms on the surface. Once we get the holomorphic 1-forms, we can find the map from the surface to the plane by integration.

23 Video Global Conformal Parameterization some results

24 Meshless Method for Point Surface Physical Simulation Upon global parameterization of point samples, it is the next, natural step to directly build physical model on top of point geometry without converting point samples to meshes Finite element principle is ideal for this goal, however, popular, frequently-used used finite element formulations are typically mesh-based We shall apply meshless techniques over point geometry for dynamic simulation We propose to simulate meshless thin-shell elastic deformation and crack propagation directly over point- sampled geometry enabled by the global conformal parameterization.

25 Meshless Thin-shell Simulation 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 ( x) = w( x, x ) > 0} = I I Influence Domain Node 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. The approximate field function can be written as: f T [ P W( x) P] 1 l T T ( x) f ( x, x) = p ( x) P W( x) f = Analysis Domain Φ( x) f Object Boundary

26 Meshless Thin-shell Simulation some basics about thin-shell For any point-sampled surfaces, if we assume that one dimension (thickness) of the surface body is significantly smaller than the other two dimensions, we can consider it as a thin-shell. We can describe the positions r and r of any material point in the reference and deformed configurations by: r ( θ, θ, θ ) = x( θ, θ ) + θ x r ( θ, θ, θ ) = x( θ, θ ) + θ x,3,3 1 ( θ, θ 2 ) 1 2 ( θ, θ ) where θ and θ are parameters of the point-surface, and θ is in the thickness direction: h 3 h θ. 2 2

27 Meshless Thin-shell Simulation some basics about thin-shell The first fundamental form: g 1 θ 2 ij dθ d measures the length of an arc element on the surface, and its coefficients g ij are components of the metric tensor: g ij ( x) = x, i x, j The second fundamental form: b 1 θ 2 ij dθ d measures the curvature e of the surface and its coefficients b ij are components of the curvature tensor: b ij ( x) = x, ij n = x, i n, j The Green-Lagrange strain tensor can be derived from the first and second fundamental forms of the middle surface. Membrane strain tensor: Bending strain tensor: αij = ( x 2 βij = ( x , i x, j x, i x, j ) 1 0 0, i x,3 j x, i x,3 j )

28 Meshless Thin-shell Simulation some basics about thin-shell We use the Euler-Lagrange equations for the thin-shell elastic deformation: d dt T u& ( u& ) V ( u) + μu& + u where the kinetic energy: = F T = ext 1 2 Ω hρ( x ) u& u& dω = 1 2 I, J M IJ u& I u& J M is the mass matrix: M IJ = hρ( x) φ ( x) φ ( x) dω Ω I J and the elastic potential energy: V 3 Eh T ~ Eh T ~ = dω Ω α Hα + β Hβ ν 12(1 ν ) φi is the MLS shape function value of node. I

29 Meshless Thin-shell Simulation sampling on the domain We utilize quad-tree structure on the parametric domain. We place the sampling nodes at the center of each quad-tree cell. The subdivision depth of the quad-tree is dependent on the conformal factor λ, so that initially the sampling nodes are uniformly distributed on the t manifold surface. The quad-tree cells can be also utilized as integration cells to perform numerical quadrature for the mass and stiffness matrices.

30 Meshless Thin-shell Simulation modeling cracks The physical model undergoing crack evolution: We use the simplified condition of maximal principle stress to decide d both where and how the material cracks. If the maximum eigenvalue of the stress exceeds a threshold, a crack line (with cracking speed proportional to the maximum eigenvalue of the stress) should be generated. Secondary fractures can be given higher thresholds to help reduce spurious branching in practice. The representation of the evolving geometry: For thin-shell crack simulation, the evolving geometry can be simply represented as line segments on the 2-D 2 D parametric domain.

31 Meshless Thin-shell Results Video Xiaohu Guo, Xin Li, Yunfan Bao, Xianfeng Gu,, and Hong Qin, Meshless Thin-shell Simulation Based on Global Conformal Parameterization, accpted by IEEE TVCG, 2005.

32 Presentation Overview Point Cloud Geometry Implicit Surface Parameterization Local Global Surface Mapping Volume Level-set surface editing Surface completion shape & texture Material, HCI Physics Mass-Spring Meshless FEM Dynamic surface editing Volumetric deformation Thin-shell simulation Physical morphing

33 Meshless Point Surface Morphing Surface morphing by interpolating thin- shell membrane and bending strain energy, making the morphing physically plausible. Incremental update of the stiffness matrix K to correct the linearization artifacts (of membrane and bending strain). We use Meshless method to perform numerical simulation. The method is based on local parameterization of the underlying point-set surface and is computationally efficient.

34 Meshless Point Surface Morphing Yunfan Bao,, Xiaohu Guo, and Hong Qin, Physically Based Morphing of Point-sampled Surfaces, in Computer Animation and Virtual Worlds, 2005.

35 Presentation Overview Point Cloud Geometry Implicit Surface Parameterization Local Global Surface Mapping Volume Level-set surface editing Surface completion shape & texture Material, HCI Physics Mass-Spring Meshless FEM Dynamic surface editing Volumetric deformation Thin-shell simulation Physical morphing

36 Meshless Volumetric Deformation A real-time meshless animation and simulation paradigm for point-sampled volumetric objects. Both interior and surface representation only compose point samples. Exploit the Modal Warping technique to the meshless framework to achieve real-time manipulation and deformation.

37 Meshless Volumetric Deformation Video Xiaohu Guo, and Hong Qin, Real-time Meshless Deformation, in Computer Animation and Virtual Worlds, 2005.

38 Current Projects Point Cloud Geometry Implicit Surface Parameterization Local Global Surface Mapping Volume Level-set surface editing Surface completion shape & texture Material, HCI Physics Mass-Spring Meshless FEM Dynamic surface editing Volumetric deformation Thin-shell simulation Physical morphing

39 Current Projects Point-based surface mapping Point-based volumetric mapping Animation behavior reuse

40 Conclusion Meshless modeling, animation, and simulation of point- based geometry. Local and global point set surface editing based on dynamic implicit icit functions and level-sets. Point-surface surface completion (geometry & texture) based on local parameterization. Point-based global conformal parameterization founded upon Riemann surface theory and Hodge theory. The conformal structure of surfaces can be derived from point samples based on their vicinity information. A meshless thin-shell simulation framework based on global conformal parameterization of the point-sampled surfaces. Extend the meshless thin-shell shell model to physics-based morphing of point-sampled surfaces. Real-time meshless volumetric deformation based on modal analysis.

41 Acknowledgements National Science Foundation Sloan Fellowship My advisor: Hong Qin Faculty member: Xianfeng Gu My colleagues: Jing Hua, Xin Li, Yunfan Bao, Seyoun Park Special thanks to IBM research for this symposium!