Deformable Body Simulation

1. Deformable Body Simulation Ipek OGUZ COMP259 – 04.12.2005

2. Outline • M-rep based deformations • Introduction to m-reps • Deformable m-reps for image segmentation • Skeleton-driven deformations • Character animation

3. Deformable body simulationPossible approaches • Finite differences method, using Lagrangian motion equations • Hierarchical models, octrees, etc. • Stability problem: Implicit solvers • Quasi-static solutions: Compute equilibrium state, than animate • BEM assuming constant material properties inside the object • Anatomical modelling

4. Free-form deformation • Embed the object into a domain that is more easily parametrized than the object. • Advantages: • You can deform arbitrary objects • Independent of object representation

5. Basics of m-reps • Atom(‘hub’) position, x • Spoke length, r • Spoke bisector, b • Object angle Ө • b, b┴ and n form local coordinate frame • Left, internal atom • Right, end atom

6. Object representation • Mesh of medial atoms • Here, the middle one is internal, the rest are end atoms

7. Discrete vs Continuous

8. Properties of m-reps • The medial locus of an object is represented explicitly. • A fuzzy approximate representation of the object's boundary is implied by the medial locus representation. • An accurate description of the boundary is given by a smooth fine-scale deformation of the fuzzy implied boundary.

9. M-rep based deformation • Mostly used for image segmentation • Can be used for PBS as well Initial model (ex: from an atlas) Target image Deformation M-rep FEM model FEM-based deformation

10. Segmentation using m-reps • Bayesian approach • P(w|f): a posteriori probability • P(f|w): likelihood • P(w): a priori probability • p(f): scaling factor • Optimize P(w|f) over all possible deformations • maximum a posteriori (MAP)

11. Algorithm • Start with an initial model m • Optimize F(m|Itarget) • F is a sum of two terms, log prior, and log likelihood • Can be applied at different scales • The initial model can come from • Geometrical analysis of a set of hand-segmented training images • A single hand-segmented training image

12. Algorithm details • Manually place the model in the 3D image, • Find and apply the similarity transform which optimizes F(m|Itarget) • Until convergence, do • For each medial atom in m • {Transform the atom to optimize F(m|Itarget)} • For each boundary tile implied by m • {Shift the position of the tile along the tile’s normal to optimize F(m|Itarget)}

13. Results • Video

14. Advantages of m-rep based deformations • Capability for deformation of the interior • Provides a way for appropriate locality based on medical relevance • Multiple scale levels (multi-object, object, object section, boundary) • Correspondences are preserved

15. Interactive Skeleton-Driven Dynamic Deformations Steve Capell, Seth Green, Brial Curless, Tom Duchamp, Zoran Popovic

16. What is this paper all about? • Character animation • We want to tell how the character should act • But we don’t want to tell how the character should move! • The answer: elastically deformable characters modeled with a simple skeleton

17. Application Areas • Movies: You want to let the animator construct the model easily • Previous work included muscle, skin, etc models – painful process • Games, VR: You want to have interactive rates • Previous work included purely kinematic deformations – not realistic

18. Challenges • We need: • A lot of physical principles • A lot of geometric modelling • A lot of computational tools • Interactive simulation rate • Ease of use • What else could you possibly need?

19. Skeleton and control lattice • Each object Ω has: • A skeleton, S (in red) • A control lattice, K (in black) • The control lattice can have hierarchical scale • K0 (in black) vs K1 (in green)

20. General Ideas • Coarse volumetric control lattice provides the elements for FEM • Motion control: Put line constraints along “bones” of the skeleton • Form regions around bones, and simulate linearly in regions • Hierarchical control lattice: Level-of-detail simulation

21. One simulation step • For each regiondo • Extract regional variables from global system • Compute displacement from “rest state” transformed according to the transformation of the “bone” • Build the linear system for solving local equations of motion • Solve the linear system using conjugate gradients • Merge solution from each region, weighted (user-assigned weights) • Update the global system state

22. Mesh Requirements • The mesh can be coarse, but it should encompass the geometric model, to ensure complete integration over interior. • Not necessarily regular grid: could even contain mix of tetrahedra and hexahedra • Could have hierarchical basis, for adaptive level of detail simulation

23. Ideas to achieve our goals • Motion control via skeleton: add line (“bone”) constraints to the finite element model • Make computation simpler: Have an edge in the mesh for each bone • Interactive rate: Linearly solve motion equations around each bone, blending deformation at overlapping regions • Similar approach to free form deformations, but here principles of continuum elasticity are used

24. Components • The object (or the character): Domain Ω • The skeleton: a graph S, is a subset of Ω • Joints: Vertices of S • Bones: Edges of S • Motion: p(x, t) • Restriction Map: a piecewise linear function on S, ps(x, t)

25. Goal • Solve for the dynamic motion of the object given the motion of the skeleton • Basically a PDE system with constraint: p(x, t) = ps(x, t) for all xєS • Separate p(x, t) into a rest state r(x) and a displacement d(x, t)

26. The Hierarchical Basis • Control lattice: “Lazy wavelets” • For all i, j, i≠j, the intersection Ci,j= Ci UCjis either empty or a face, edge, or vertex of both Ciand Cj. • The edges of S are edges of cells of K. • The domain is contained in the interior of K. • For all i, each vertex of Cihas valence 3 (within Ci).

27. Equations of Motion • Kinetic energy T ( similar to ½mv2) • Elastic potential energy V • Both T and V depend on q, q’ • Euler-Lagrange equations

28. Computing V • Strain tensor: degree of metric distortion • Green’s strain tensor • Stress tensor: forces acting on the interior of a continuum • Shear modulus, G • Poisson’s ratio,

29. “Body Forces” • Act on the whole body, rather than a specific cell in the mesh • Ex: gravity • Similar to the familiar mg

30. Numerical integration • Simple trick for speed up: precompute the integrals • Subdivide K • Compute basis functions at each vertex • Tetrahedralize • Compute integrals over each tetrahedron using piecewise linear approximations • Then use nonlinear Newton-Raphson to solve the system

31. Skeletal Simulation • Did you actually believe they did all those expensive computations at interactive rate? • Of course not! • First animate the skeleton (real quick) • Then approximate nonlinear dynamics

32. Regions

33. Solving the nonlinear system • Approximate by linearizing motion equation at each step • Make sure you use small time steps • h: timestep • μ: Damping coefficient • S: stiffness matrix • Solve using Conjugate Gradients solver

34. Conjugent Gradient Method • Initialize at P[0]; • g[0] = h[0] := F(P[0]); • for i = 0 to n-1 • P[i+1] := minimum of F along the line h[i] through P[i],i.e., choose λ[i] to minimize F(P[i+1])=F(P[i]+ λ[i] h[i]); • g [i+1] := F(P[i+1]); • γ[i+1] := (g[i+1]- g [i])  g [i+1] / g [i] g [i]; • h [i+1]:= g[i+1]+ γ[i] h[i];

35. Bone Constraints • Skeleton is directly controlled by keyframe data • Requiring the bones to be on edges of S makes things very simple • A control point on an edge  a component of ∆v that is known a priori • ∆vk – known components • ∆vu – unknown components

36. Linear Subspace Constraints • What if we have more than one object? • Position constraints • Extension of bone constraints • RHS is constant a, for each timestep • C =

37. Blended Local Linearization • Problem: computation of the stiffness matrix at each time step • Soln: Linearize the strain tensor • Problem: Severe distortions when deformation is large • Better soln: Locally linearize • Idea: no large deformation from nearby bones • User-specified regions • Blend at places where regions overlap • Define a binary Q matrix, where Q[a][b]=1 means that the basis function a is nonzero for the region b

38. One Simulation Step - revisited For each region i do • Extract regional variables from global system [ri, qi, q’i] = [Qi r,Qi q, Q’i q] • qi corresponds to displacement from rest state transformed according to the transformation of the bone For each a do qia = qia – Ti (ria) + ria • Build the linear system for solving local equations of motion Construct Ai and bi from bone constraint • Solve the linear system using conjugate gradients Solve Ai ∆vi = bi • Merge solution from each region, weighted ∆v =∑i Wi QiT ∆vi • Update the global system state q’ = q’ + ∆v q = q + hq’

39. Twist Constraint • We don’t twist much around our bones • Soln: soft constraint to penalize all displacement near bones • Quadratic  its Hessian is constant • Add to the stiffness matrix

40. Adaptation • More details at places where large deformations occur • Little deformation  go up one level • Large deformation  go down one level • Precompute and store related info

41. Contributions • Crafting the function space to handle constraints • Blended local linearization of non-linear equations • Method of solving constraints using linear subspace projection • New constraint for allowing 1D bones to behave like 3D

42. ...contributions • None of these are terribly novel, in fact • But this is the first time so many techniques have been put together • Result: interactive animation of arbitrary shaped characters with user control over skeleton

43. Results • Show video

44. References • Interactive skeleton-driven dynamic deformations Steve Capell, Seth Green, Brian Curless, Tom Duchamp, Zoran Popović July 2002  ACM Transactions on Graphics (TOG) , Proceedings of the 29th annual conference on Computer graphics and interactive techniques, • Collisions and deformations: A multiresolution framework for dynamic deformations Steve Capell, Seth Green, Brian Curless, Tom Duchamp, Zoran Popović July 2002  Proceedings of the 2002 ACM SIGGRAPH/Eurographics symposium on Computer animation • S.M. Pizer, T. Fletcher, Y. Fridman, D.S. Fritsch, A.G. Gash, J.M. Glotzer, S. Joshi, A. Thall, G Tracton, P. Yushkevich, and E.L. Chaney, "Deformable M-Reps for 3D Medical Image Segmentation," International Journal of Computer Vision - Special UNC-MIDAG issue, (O Faugeras, K Ikeuchi, and J Ponce, eds.), vol. 55, no. 2, pp. 85-106, Kluwer Academic, November-December 2003. • PT Fletcher, SM Pizer, G Gash, and S Joshi, "Deformable M-rep segmentation of object complexes," in IEEE International Symposium on Biomedical Imaging (ISBI), pp. 26-29, 2002. • Pizer S, S Joshi, PT Fletcher, M Styner, G Tracton, and Z Chen, "Segmentation of Single-Figure Objects by Deformable M-reps," in Medical Image Computing and Computer-Assisted Intervention (MICCAI), (WJ Niessen and MA Viergever, eds.), (New York), pp. 862-871, Oct. 2001. • Yushkevich, Paul (2003). Statistical Shape Characterization Using the Medial Representation. PhD dissertation. Advisor: Prof. Stephen Pizer