Dynamic Points: When Geometry Meets Physics. Xiaohu Guo Stony Brook

Transcription

1 Dynamic Points: When Geometry Meets Physics Xiaohu Guo Stony Brook xguo@cs.sunysb.edu

2 Point Based Graphics Pipeline Acquisition Modeling Rendering laser scanning structured light holography etc. Point cloud from physical objects processing modeling animation etc. Manipulation of point samples (images courtesy of Zwicker et al.) surface splatting hardware acc. etc. Uses points as rendering primitive

3 Research Agenda and Objectives Our research objectives are to develop deformable models for various applications in computer graphics, computational vision, reverse engineering, geometric design, user interaction, shape editing, haptic interface, etc. Dynamic Points : : point-based digital clay which can be directly manipulated, edited, deformed through human- computer interaction for various, aforementioned 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 Research Overview my previous and current work 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

7 Presentation Overview A geometric processing paradigm for point set surfaces: 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.

8 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.)

9 Point Acquisition Point samples are acquired from scanning devices (laser scanning, structured light, holography,...) Point sampled geometry is ubiquitous for visual computing and other applications How to maximize the utility of point samples? The previous solutions are to convert point-sampled sampled geometry to continuous representations via implicit surfaces and volumetric shape techniques Holoimage scanning system in Stony Brook (curtesy of David Gu et al.)

10 Implicit Surface Modeling Idea: Represent n-manifold n as zero- set of a scalar function in (n+1) space 3 On the manifold: { p R f ( p) = 0} 3 Inside: { p R f ( p) > 0} 3 Outside: { p R f ( p) < 0} Surface reconstruction using implicits: Find a volumetric implicit function f,, which implicitly defines a surface interpolating those scattered point clouds Implicit function representing an X (Courtesy of G. Turk et al.)

11 Reconstruction from Point Clouds Local and global surface distance field Fitting a volumetric implicit function to the local distance field of each point (Scalar trivariate B-spline function) Global Shepard s s blending

12 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

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

14 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, pp , Xiaohu Guo, Jing Hua, and Hong Qin, Scalar-Function-Driven Editing on Point Set Surfaces, in IEEE Computer Graphics and Applications, Vol. 24, No. 4, pp , Video

15 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

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

17 Dynamic Point Surface Editing Xiaohu Guo, Hong Qin, Dynamic Dynamic Sculpting and Deformation of Point Set Surfaces, in Pacific Graphics, pp , Xiaohu Guo, Jing Hua, and Hong Qin, Touch-Based Based Haptics for Interactive Editing on Point Set Surfaces, in IEEE Computer Graphics and Applications, Vol. 24, No. 6, pp , 2004.

18 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

19 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

20 Surface Completion for Shape & Texture Seyoun Park, Xiaohu Guo, Hayong Shin, and Hong Qin, Shape and Appearance Repair for Incomplete Point Surfaces, in The Tenth IEEE International Conference on Computer Vision (ICCV) 2005.

21 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

22 Point Surface Global Parameterization We develop a global conformal parameterization method for point-sampled surfaces of arbitrary topology The global parameterization for point samples will facilitate a large variety of graphics applications, such as reverse engineering, surface physical simulation, shape registration, morphing, texture synthesis, scientific computation, etc. Our new computational techniques have a great potential to maximize the utility of point-sampled surfaces, while simultaneously retaining their structural simplicity

23 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 ].

24 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. )

25 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.

26 Global Conformal Parameterization open surface for improving uniformity In order to improve the quality of global conformal parameterizations, we need to introduce some boundaries to the surface. See the proof in [M. Jin et al. VIS 2004]. In general, we need to introduce boundaries to the tips of long tubes of the surfaces. Suppose the surface is of genus g and has b boundaries, then it has the cohomology group of 2g + ( b 1) dimension.

27 Global Conformal Parameterization algorithmic overview The canonical homology basis of a genus g object, are closed curves: a, b, a, b,..., a g, b } { g 2g The two-hole torus is cut into two patches by handle separators, which connect the two zero points (yellow dot).

28 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

29 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)

30 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.

31 Global Conformal Parameterization detailed implementation Approximating the differential operators over point-set surfaces using only local neighborhood information. Approximating the gradient: by minimizing the energy E( f( p)) = f ( q) f ( p) f ( p) ( q p) Approximating the Laplacian: q 1 Δ f ( p) = ( f( q) f( p)) N Locating extremal points: find the cluster of points with smaller gradient lengths and compute the center of gravity of each cluster. q 2

32 Global Conformal Parameterization detailed implementation Tracing integral curves: add a new tracing point in the direction of f ( p) on the tangent plane of p, then project to the original surface by MLS projection. Compute homology basis using fair Morse function: we define a function on the surface, and make the function as smooth as possible to reduce the number of extremal points. The integral curve along the gradient of the function gives the homology basis. Approximating integration: uv( q) = ( ω1, ω2) r The integration process is path independent. We fix a base point p, for any point q, we choose a path r arbitrarily from p to q, the path r = { p0, p1, p2,..., pn}, then n uv( q) = ( ω ( p ) ( p p ), ω ( p ) ( p p )) 1 i 1 i i 1 2 i 1 i i 1 i= 1

33 Video Global Conformal Parameterization some results

34 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

35 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

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

37 Meshless Thin-shell Simulation MLS shape functions So the approximate field function can be written as: l T 1 f ( x ) f ( x, x) = p ( x) A ( x) B( x) f = Φ( x) f where Φ(x) is the vector of the MLS shape functions: Φ( x) = T 1 [ φ ( x), φ ( x),... ( x) ] = p ( x) A ( x) B( ) 1 2 φ n x The moment matrix A(x) will be ill-conditioned when: 1. The basis functions p(x) 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(x), or a conic section for a quadratic polynomial basis.

38 Meshless Thin-shell Simulation MLS shape functions To obtain the consistency of any desirable order of approximation, it is necessary to have a complete basis. These are two examples of complete bases in 2-D 2 D for first and second order consistency: Linear: p { 1, x y} T ( m = 3) =, w I (x) Quadratic: { 1, x, y, x 2, xy y 2 } T ( m = 6 ) =, The weight functions play important roles in constructing the shape functions. They should be positive to guarantee a unique solution n for a(x) ; they should decrease in magnitude as the distance to the node increases to enforce local neighbor influence; they should have compact support, which ensure sparsity of the global matrices. They can differ in both the shape of the influence domain (e.g. rectangle centered at the node for tensor nsor-product weights, or sphere), and in functional form (e.g. polynomials of varying degrees, or non-polynomials such as the truncated Gaussian weight). The continuity of the shape function is directly related to the continuity of weight functions and polynomial basis. The FEM equivalents can be reached if the weight functions are piecewise-constant constant over each influence domain. p

39 Motivation why Meshless 1) Simulation of fracture modeling the propagation of cracks along arbitrary and complex paths: Notoriously difficult task for mesh-based computational techniques remeshing in each time step of integration, degradation of both accuracy and complexity of implementation. Moving discontinuities (such as cracks) can be naturally facilitated, since no new mesh needs to be constructed, and the computational cost of remeshing at each time step can be avoided. (images courtesy of Pauly et al.)

40 Motivation why Meshless 2) Simulation of large deformation: Mesh distortion can either end the computation altogether or result in drastic deterioration of accuracy; Adaptive simulation is not naturally facilitated by mesh structure. The connectivity of meshless simulation nodes can be generated as part of the computation and can change over time; Accuracy can be controlled more easily, since nodes can be added flexibly in areas where more refinement is needed. (images courtesy of Muller et al.)

41 Motivation surface vs. volume Conventional methods for modeling point-based geometry are based on volumetric approaches. No connectivity information, hard to establish the simulation domain for surfaces. Raising the analysis domain to 3-D 3 D space is not natural for pure surface simulation (like thin-shells)

42 Motivation local vs. global param. Local Parameterization: Enforcing consistency among neighboring local parameterizations would be computationally intractable, which make it impossible to perform quadrature in the Galerkin weak form (such as the mass and stiffness matrices) to ensure numerical accuracy and stability. For complex branching cracks, simply cutting connections between simulation nodes is too coarse an approach, which can not guarantee continuity and consistency requirements. Global Conformal Parameterization: The quadrature can be performed on the global parametric domain. Crack branching becomes 2-D 2 D line- intersection problem. Easy to enforce continuity requirements by utilizing transparency criterion and dynamic up-sampling for stable and accurate simulation of cracks.

43 Meshless Thin-shell Simulation We propose to simulate meshless thin-shell elastic deformation and crack propagation directly over point- sampled geometry enabled by the global conformal parameterization. Compared with other local parameterization approaches, the global parameterization makes the physical simulation more accurate and stable.

44 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

45 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 )

46 Meshless Thin-shell Simulation some basics about thin-shell Assuming linearized kinematics, the displacement field is introduced as: 1 2 u( θ, θ ) = 1 2 x( θ, θ ) x ( θ, θ ) The linearized membrane and bending strain can be written as: α ij = 1 ( x 2 u + x u 0 0, i, j, j, i 0,1 0,2 ) [ u ( x x ) + u ( x x )] βij = u, ij x,3 +,1, ij,2,2,1 x x We can arrange the strain tensors into vectors:, ij α = α11 α 22 2α 12 β = β11 β22 2β 12

47 Meshless Thin-shell Simulation some basics about thin-shell If we only consider the isotropic elasticity, the membrane and bending b stress can be written as: n~ 11 α11 m~ 11 β 3 11 n ~ = ~ Eh ~ n22 = 2 α 22 ~ 1 ν H ~ = ~ Eh ~ E : Young s s modulus m m22 = H β 2 22 n12 2α ~ 12(1 ν ) ν : Poisson s s ratio 12 m12 2β12 where H ~ is the constitutive matrix: ~ H = ( ) ( )( ) a ν a a + 1 ν a ( ) 22 2 a sym a a a a ( ) ( )( ) ν a a + 1+ ν a ij a are the first fundamental form in the dual basis: a ij g jk i = δ k = 0 1 i i = k k

48 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

49 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. Since λ is equivalent in u and v directions, we can choose e simple symmetric supporting region (such as squares) for each node. For a node i reside in cell Qi, its supporting region is a square of size η size( Qi ) centered around node i.

50 Meshless Thin-shell Simulation sampling on the domain We restrict the quad-tree to be one-level adjusted, where the level difference of all terminal cells and their neighbors is no more than one. This T can facilitate automatic satisfaction of the patch covering condition to make the t moment matrix invertible. The quad-tree cells can be also utilized as integration cells to perform numerical quadrature for the mass and stiffness matrices. For multiple parametric planes, the support of each simulation node n can be expanded to its neighboring parametric plane when it is close to the boundary of its parametric plane. So the behaviors of the simulation nodes s across the boundaries of parametric planes are consistent with non-boundary nodes.

51 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.

52 Meshless Thin-shell Simulation modeling cracks When a crack is generated, the support of the nodes affected by the discontinuities needs to be modified accordingly to incorporate the proper behavior of the shape functions and its derivatives. The simplest way to introduce discontinuities into meshless approximations is to use the visibility criterion. However, it would w cause undesirable discontinuities of the shape functions inside the domain. So we choose the transparency criterion to allow partial interaction of nodes in the vicinity of the crack front. These operations o can be easily performed on the 2-D 2 D parametric domain. Dynamic up-sampling (node insertion) needs to be performed during crack simulation in the vicinity of crack lines. It can be also easily implemented utilizing the quad-tree structure by subdividing the quad- tree cells.

53 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, submitted to IEEE Trans. Vis. Comput.. Graph

54 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

55 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.

56 Meshless Point Surface Morphing Yunfan Bao,, Xiaohu Guo, and Hong Qin, Physically Based Morphing of Point-sampled Surfaces, in Computer Animation and Virtual Worlds, Vol. 16, No. 3-4, 2005.

57 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

58 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.

59 Meshless Volumetric Deformation Video Xiaohu Guo, and Hong Qin, Real-time Meshless Deformation, in Computer Animation and Virtual Worlds, Vol. 16, No. 3-4, 2005.

60 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

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

62 Possible Future Work The global conformal parameterization is expected to pave the way for some possible future projects in point-based based graphics, such as registration/analysis, segmentation, matching, attribute transfer, texture synthesis, spline surface fitting,... We only assume that the point surface is sufficiently and regularly sampled. The sampling issue is far from trivial. How to compute topology directly from raw point cloud?... 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?... What is deformation? How to parameterize the space of deformation?...

63 Acknowledgements National Science Foundation Sloan Fellowship My advisor: Hong Qin Faculty member: Xianfeng Gu My colleagues: Jing Hua, Xin Li, Yunfan Bao, Seyoun Park