1. Deformable Body Simulation
Ipek OGUZ
COMP259 –

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

3. Deformable body simulation Possible 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)
Deformation
Target image
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: What else could you possibly 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 region do
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 U Cj is either empty or a face, edge, or vertex of both Ci and 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 Ci has 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 , 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 , 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 , Oct
Yushkevich, Paul (2003). Statistical Shape Characterization Using the Medial Representation. PhD dissertation. Advisor: Prof. Stephen Pizer