Deformable Body Simulation

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Deformable Body Simulation

• Ipek OGUZ
• COMP259 04.12.2005

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Outline

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

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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

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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

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Basics of m-reps

• Atom(hub) position, x
• Spoke length, r
• Spoke bisector, b
• Object angle ?
• b, b- and n form local coordinate frame
• Left, internal atom
• Right, end atom

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Object representation

• Mesh of medial atoms
• Here, the middle one is internal, the rest are
  end atoms

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Discrete vs Continuous
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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.

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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
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Segmentation using m-reps

• Bayesian approach
• P(wf) a posteriori probability
• P(fw) likelihood
• P(w) a priori probability
• p(f) scaling factor
• Optimize P(wf) over all possible deformations
• maximum a posteriori (MAP)

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Algorithm

• Start with an initial model m
• Optimize F(mItarget)
• 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

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Algorithm details

• Manually place the model in the 3D image,
• Find and apply the similarity transform which
  optimizes F(mItarget)
• Until convergence, do
• For each medial atom in m
• Transform the atom to optimize F(mItarget)
• For each boundary tile implied by m
• Shift the position of the tile along the tiles
  normal to optimize F(mItarget)

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Results

• Video

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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

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Interactive Skeleton-Driven Dynamic Deformations

• Steve Capell, Seth Green, Brial Curless, Tom
  Duchamp, Zoran Popovic

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What is this paper all about?

• Character animation
• We want to tell how the character should act
• But we dont want to tell how the character
  should move!
• The answer elastically deformable characters
  modeled with a simple skeleton

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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

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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?

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Skeleton and control lattice

• Each object O 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)

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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

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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

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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

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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

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Components

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

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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 x?S
• Separate p(x, t) into a rest state r(x) and a
  displacement d(x, t)

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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).

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Equations of Motion

• Kinetic energy T ( similar to ½mv2)
• Elastic potential energy V
• Both T and V depend on q, q
• Euler-Lagrange equations

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Computing V

• Strain tensor degree of metric distortion
• Greens strain tensor
• Stress tensor forces acting on the interior of a
  continuum
• Shear modulus, G
• Poissons ratio,

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Body Forces

• Act on the whole body, rather than a specific
  cell in the mesh
• Ex gravity
• Similar to the familiar mg

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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

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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

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Regions
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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

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Conjugent Gradient Method

• Initialize at P0
• g0 h0 ?F(P0)
• for i 0 to n-1
• Pi1 minimum of F along the line hi
  through Pi, i.e., choose ?i to minimize
• F(Pi1)F(Pi ?i hi)
• g i1 ?F(Pi1)
• ? i1 (gi1- g i) ? g i1 / g i ? g
  i
• h i1 gi1 ? i hi

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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

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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

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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 Qab1 means
  that the basis function a is nonzero for the
  region b

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One Simulation Step - revisited

• For each region i do
• Extract regional variables from global system
• ri, qi, qi Qi r,Qi q, Qi 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

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Twist Constraint

• We dont twist much around our bones
• Soln soft constraint to penalize all
  displacement near bones
• Quadratic ? its Hessian is constant
• Add to the stiffness matrix

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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

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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

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...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

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Results

• Show video

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References

• Interactive skeleton-driven dynamic deformations
  Steve Capell, Seth Green, Brian Curless, Tom
  Duchamp, Zoran Popovic 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
  Popovic 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