Introduction to Modeling Of NonRigid Objects

1
Introduction to Modeling Of Non-Rigid Objects

• COMP 259 Physically Based Modeling
• Russell Gayle

2
Lecture Outline

• Why Use Deformable Objects?
• Very Quick FFD Review
• Spring-Mass Systems
• Continuum Methods
• FEM - Mathematical Background
• FEM For Deformations
• Skeleton Driven Deformations
• Global Deformations
• Quick Mention of other methods

3
Why Deformable Objects

• Many things arent rigid!
• In any cases, it is not sufficient to have rigid
  objects
• Cloth, skin, etc
• Comes into play even in image analysis for
  segmentation

4
Free-Form Deformation

• For details, refer to Jeremys talk
• Basic idea
• Deform the space in which the object lies
• Can fit the object into lattices of varying
  resolution and shape
• Can use it to deform only specific areas or the
  entire object

5
FFD Pros and Cons

• Advantages
• Quick and simple to use
• Disadvantages
• Difficult for precise control
• Or, you must have a LOT of patience
• A geometric method, so not intrinsically
  physically accurate

6
Physical Methods

• For realistic simulation, FFD can be too much
  work
• It requires tedious moving of control points
• We would like methods that will do all this for
  us
• Thats where physical methods come in!

7
Spring-Mass Systems

• Similar to the springs we modeled in homework 1
• But, now we model an object as a collections of
  point masses connected by springs

8
Spring-Mass Math

• Just like before, we can solve an ODE to update
  the system
• Fortunately, this generalizes nicely into a
  matrix form
• - Where M, C, K and 3Nx3N matrices of mass,
  damping, and stiffness, and x and v are position
  and velocity. Note that in general, each of these
  are very sparse.

9
Applications

• Everywhere!
• Skin and facial animation
• Terzopolous and Waters, 90
• State Transitions
• Terzopolous et al, 89
• Artificial Fish
• Tu and Terzopolous, 94
• Many, many, many more!

10
Advantages

• Very simple and well understood from a dynamics
  standpoint
• Easy to implement and construct
• Can easily be rendered at interactive rates
• Easy parallelizable

11
Disadvantages

• Discrete model
• Difficult to get good spring constants
• Certain constraints are hard to express
• Incompressible volumes, thin surfaces, complex
  objects
• May be able to get around some of this with extra
  springs, but that increases computational cost
• Angular springs can also help
• Stiffness problems with large spring constants
• These can cause a lot of numerical error, forcing
  a smaller time step

12
Movie!

• A simple, real-time physically realistic
  spring-mass system

13
Continuum Methods

• We would like to represent the object as an
  entire surface
• This is more realistic, since most real objects
  are continuous surfaces
• More formally, a solid body with mass and
  energies distributed throughout
• Still must represent it as a discrete system,
  which takes this into account

14
Continuum Methods, contd

• Under these methods, we wish to consider the
  equilibrium of a general body acted on by
  external forces
• The object then deforms based on these forces and
  its material properties
• The equilibrium is reached when the bodies
  potential energy is at a minimum

15
Potential Energy

• We can express the potential energy of a
  deformable system as
• where L is the total strain energy (energy stored
  in the body) and W is the work done by loads
  acting on the body (i.e. load at a specific
  point, loads distributed over the entire body,
  loads distributed over just the surface of the
  body)

16
Minimizing P

• We must express L and W in terms of the object
  deformation, i.e. a function of material
  displacement
• We can find the minimum by finding where the
  derivative of the displacement function is zero
• Since it is likely that a closed form of this
  equation either cant be found or would be
  difficult to be found, various numerical methods
  could be applied

17
Numerical Methods

• Believe it or not, mass-spring methods solve this
  by discretizing the equilibrium equation at the
  mesh points
• Finite Element Methods divide the object onto a
  set of simple elements and approximate the
  continuous equation over each element
• Boundary Element Methods is like FEM, but only
  discretizes the boundary (i.e. surface) of the
  object
• Requires reformulating the governing equations
• Many others (Ill mention some later on)

18
Finite Element Methods (FEM)

• FEM is a technique for approximating a continuous
  function that satisfies some equilibrium equation
• As mentioned in the last slide, we divide the
  object with elements that join at discrete node
  points.
• We then find a function that solves the
  appropriate equilibrium equation for each element
• This function has boundary conditions at the node
  points and along the boundaries, which ensures
  continuity between elements

19
More FEM

• On a higher level, this effectively discretizes
  the system by representing the continuous
  function in each element as a finite sum of our
  element interpolation (also called shape)
  functions
• For example, if our equilibrium function is
  scalar function F(x,y,z), then we can approximate
  F as

20
FEM Math Contd

• Where hi are the interpolating (shape) functions
  for the point (x,y,z) and Fi are the values of
  F(x,y,z) at the ith node point.
• From this, solving our equilibrium equation
  reduces to determining the set of node points Fi
  that minimize the total potential energy (P)

21
FEM for Deformations

• Although other uses of FEMs for deformable
  objects exist, a common one is displacement-based
  FEM.
• As built up, we wish to minimize P from before
  with respect to the material properties of the
  object
• Note Since our object is in 3D, the displacement
  function returns a 3D vector

22
FEM Steps

• We can compute object deformations as follows
• Based on material properties, derive the
  potential energy function, P, in terms of
  material displacement
• Select appropriate finite elements and their
  corresponding interpolation (shape) function, and
  subdivide the object into these elements
• Note We can think of the spring-mass system as a
  finite element method in which each element is a
  spring, with the spring equation being the
  interpolating function. But, this does not take
  the entire surface into consideration so it isnt
  a great approximation.

23
FEM Steps

• For each element, convert the components of the
  equilibrium equations into a sum of the
  interpolating functions and the elements node
  displacements
• Combine all of these equations into a single
  system, and solve this system for node
  displacements.
• Using the interpolating function for an element
  and these new node displacements, calculate
  displacements for points within that element.
• Using these values, other quantities about the
  body, such as internal stress, can be computed

24
Step 1 Explained!

• Step 1 Based on material properties, derive the
  potential energy function, P, in terms of
  material displacement
• We need to determine what L and W are in these
  terms
• Strain energy (L), given material stress and
  strain components (s and e, respectively), over a
  volume V is

25
Step 1 Contd

• Where D is a linear matrix relating stress and
  strain for an elastic system, using the
  Generalized Hookes Law
• Each of these are vectors of stress and strain
  components for an elastic system

26
More Step 1

• For an elastic system, the material strain () is
  related to the displacement u(u,v,w) is
  ?
• By putting these into the integral equation, we
  get strain energy in terms of material
  displacement

27
Even more Step 1.

• Fortunately, work W is simpler. We can find the
  work done by an external force f at a point as
  the dot product of the force and the material
  displacement u, integrated over the volume.
• For each external force, we can sum integrate
  over the region they effect and add them together
  for the total work. For instance, forces from
  drag or pressure are applied only to the surface
  instead of the entire volume.
• (We made it through Step 1!)

28
Onto Step 2

• Step 2 Select appropriate finite elements and
  their corresponding interpolation (shape)
  function, and subdivide the object into these
  elements
• Many options are available, each differ in
  accuracy and computational complexity, different
  situations require differetn elements
• Common elements in 2D and 3D

29
Step 2, without too much Math!

• Fortunately, interpolation functions for these
  elements are known, and can be found in many FEM
  texts.
• But, to get an intuitive idea of what it takes
• It is useful to make these functions polynomials.
  So since we must have the correct node values, if
  the element has N nodes, our polynomial must have
  at least N coefficients. We can choose the
  polynomial to be the one of lowest degree which
  fits these constraints.

30
A little more Step 2

• Interpolation function properties
• Only applied to given element, and are zero
  outside the element
• Has a value of one at its corresponding node, the
  function vanishes at the other nodes in the
  element
• The sum of the functions within an element sum to
  one
• They have the same order or form as the
  interpolation equation

31
Interp Fcn Vs. Interp Eqn

• The interpolation equation gives us an
  approximation from which we derive the
  interpolation functions.
• In other words, we use the interpolation equation
  with some constraints to solve for the
  interpolation functions. We use this functions to
  determine how to approximate the values given an
  (x,y,z)

32
Example functions
33
Hooray! Step 3

• Step 3 For each element, convert the components
  of the equilibrium equations into a sum of the
  interpolating functions and the elements node
  displacements
• From our choice of interpolation functions and
  elements, we can express the displacement vector
  as a linear combination of interpolation fcns
  applied to node displacements

34
Step 3 Math

• The linear combination
• Where
• and H is a 3 x 3N matrix, whose form depends on
  the choice of element and interpolation fcn.
• Similarly, we can express the strain e at a point
  in a similar manner, by e BU
• Where B is a 6 x 3N matrix, with the first row
  obtained from differentiating u with respect to x
  and so on, from the equations for e earlier.

35
Step 3 Strain and Work

• By using these in our Strain (L) equation, we can
  get
• And, by using the full Work (W) equation which
  includes forces over the volume, surface, and on
  concentrated places
• The F matrices and P are 3N x 1, and were solved
  by integrating over the entire body or surface as
  needed. These integrals are computed via
  numerical integration techniques.

36
Step 3 Final Energy

• By putting these both into our potential energy
  function, we get
• Which is our potential energy function in terms
  of node displacements!

37
Phew! Step 4 (Easy!)

• Step 4 Combine all of these equations into a
  single system, and solve this system for node
  displacements.
• For each of the results from 3, we can combine
  them into a very, very large linear system
  (though it may be a little tricky)
• We can then solve this system by any linear
  system solver, such as Gaussian Elimination, but
  a sparse matrix solver will probably be
  beneficial

38
Step 5 (Also easy!)

• Step 5 Using the interpolating function for an
  element and these new node displacements,
  calculate displacements for points within that
  element.
• For cases in which the node points are vertices,
  we can displace each of these vertices by that
  amount. Otherwise, well need to have move some
  points that are not on vertices.
• These node displacements can also be used to
  compute other properties, such as internal
  stress.

39
FEM Comments

• All of the integrations can be approximated via
  numerical integrators. However, choice of this is
  important in that bad integrators can introduce a
  lot of error into the computations.
• It may be more intuitive to compute this FEM as
  2,1,3,4,5. This is because step 2 does not depend
  on step 1, but step 3 does depend on step 1.

40
FEM Dynamic Deformation Systems

• These 5 steps mentioned before are to compute the
  minimal energy state through a static analysis.
  However, in many cases we wish to see the object
  actually deform to this state. In this case we
  need to take inertial body forces and
  dampening-like forces into account. We can
  generalize this into a second-order differential
  equation
• Note that this is similar to our equation for
  spring-mass systems.

41
Dynamic Deformations

• In,
• M, C, and K are mass, dampening, and stiffness
  matrices for the entire object
• F is a vector of applied forces, and U is a
  vector of node displacements.
• We get K by taking partial derivatives of P and
  setting these to zero, which yields a linear
  system of the form KUF
• We get M by assembling mass matrices for the
  individual elements, and C assembling dampening
  parameters

42
FEM Pros and Cons

• Advantages
• Much more physically realistic simulation
• In general, requires fewer node points than
  spring-mass systems which leads to solving a
  small system
• Disadvantages
• Very computationally expensive
• Mass and Stiffness matrices can be computed ahead
  of time, but due to size it may mean a large
  preprocessing step
• Large changes require re-evaluation of various
  matrices, but large changes are less likely to
  happen

43
Movie!

• ArtDefo movie (D. James and D. Pai, 99)
• This is a BEM method, but it is similar to FEM
• It requires a change in the equilibrium equation,
  but the process is similar

44
Interactive Skeleton-Driven Dynamic Deformations

• Goal To create a framework for skeleton-driven
  animation of elastically deformation characters.
• Uses a control lattice for which to apply FEM on
• Embeds a bone structure for skeletal controls

45
Notation/Problem Formulation

• The object, W
• The skeleton defines a graph, S (Note S is a
  proper subset of W)
• Motion as a time-dependent fcn
• Goal To solve for the dynamic motion of the
  object given the motion of the skeleton

46
More background

• Since an elastic model is given, p(x,t) is a
  solution of a system of partial differential
  equations, which can be solved via FEM.
• p(x,t) is separated into a rest state r(x), and a
  displacement d(x,t)qa(t)fa(x), giving us
• where fa(x) are elements of a finite basis B
• So, the state at time t is a column vector q
  q(t)

47
The Hierarchical Basis

• Want the simulation to be able to adapt to local
  conditions
• The control lattice is then subdivided several
  times to give a hierarchical basis B
• Note that the lattice should be unstructured, to
  fit the geometry nicely
• Create a linearly independent set of functions as
  the basis, similarly to wavelet decomposition
  (using the lazy wavelet)

48
Control Lattice

• The control lattice is setup as follows
• For all i,j, with i not equal to j, the
  intersection of convex cell i and convex cell j
  is either empty or a face, edge, or vertex shared
  by both
• The edges of the skeleton are edges of cells
• The object domain is contained in the interior of
  the lattice
• Each vertex has valence at least 3
• These conditions allow for a variety of cell
  shapes while maintaining the necessary structure

49
Equations of Motion

• With a separation of p, we can express kinetic
  energy (T) and elastic potential energy (V) as
  functions of q and its time-derivative, given
  by
• where the d terms are gradients with respect to
  qdot and q. Qext is a generalized force, which
  includes external forces such as gravity. The
  last term is a dampening, like friction.

50
More Motion Equations

• To save time, the derivations of some of these
  are omitted. If youd like to see them, ask me
  afterwards.
• But, from generalizations of basic physics
  equations, they found that
• where M is the mass matrix (each element of which
  is an integral over the object of its density at
  that point and each pair of basis functions)
• The other gradient term in the motion equation is
  given in the paper, but is quite long so also
  omitted here. Its very similar to the stiffness
  derivations from the FEM section

51
Numerical Integration

• Many of the terms can be pre-computed, such as
  gravity (in Qext), mass, and stiffness matrices.
• This requires numerical integration.
• They approximate this by subdividing the control
  lattice to a specified level, computing the
  values of the basis functions at each vertex,
  creating another, more regular tetrahedral
  subdivision, and then using these and the new
  values for integration

52
Adding a skeleton and control lattice

• Although some automatic methods are available,
  the authors chose to have the user setup both the
  skeleton and control lattice.

53
Solving the system

• Once the Motion Equation has been updated with
  the derived values, we can solve it
• Nonlinear techniques exist, but are very
  computationally expensive
• Like Baraff and Witkin 98, the authors try to
  linearize the motion equations. This gave great
  results, as long as the time step was not too
  large.

54
Linearizing Motion

• By linearizing the equations like Baraff and
  Witkin 98, they get
• By solving implicitly, they first get (delta)v
  from a Conjugate-Gradient solver, and then
  substitute it in the first equation to get
  (delta)q

55
Adding bone constraints

• In their setup, the skeleton is controlled
  directly by keyframes or some other external
  source
• So, the skeleton is just a complicated
  constraint. Since the control lattice has edges
  on the skeleton and the basis is interpolating,
  new constraints can easily be computed
  algebraically.
• The new constraints cause some values to become
  unknown (certain cases of second formula from
  above)
• This results in

56
Blended Local Linearization

• A big problem for this method (and many others)
  is solving for the stiffness matrix
• They a technique that linearizes the strain
  model, as discusses in many texts (they cite
  Shabana, 98)
• This method makes the computation much faster
  while providing accurate results
• The main drawback is that large deformations will
  cause massive distortions in the object
• By noting that relative to the bones only small
  deformations will occur, they divide the object
  into regions

57
Blended Local Linearization

• This must be done by the user, but it works by
  assigning weights to control vertices
• This lets objects belong to more than one region,
  depending on which control points describe the
  region
• (Refer to Kangaroo image from before)

58
Twist Constraint

• In natural creatures, flesh cannot rotate around
  the axis of the bone without causing the flesh to
  deform with it
• The elastic constraints make this very difficult,
  causing flesh to rotate without deforming
• They solve this by adding a soft constraint to
  penalize all displacements with some radius of
  the bones
• Note This includes more than just deformations!

59
Adaptation

• For easing large deformations, they want to make
  the mesh add detail where needed
• They apply a simple heuristic that they want more
  detail in areas of large deformation
• By tracking the amount of deformation over a
  basis function, then all the basis in the next
  finest level that overlaps with the original
  basis are added
• Similarly, these are also removed when there is
  little deformation
• (This can be seen in the movie, which is next!)

60
Results! Movie!

• Put all of this together and what do we get
• Interactive Skeleton-Driven Dynamic Deformations!

61
Other Methods

• Many other methods exist. Just to name a few,
• Greens Functions
• Hybrid systems
• Snake method
• Global deformation

62
Reference Papers

• Survey of
• Skeleton