Dynamic Modeling with Implicit Surfaces and Polygonal Meshes

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Dynamic Modeling with Implicit Surfaces and
Polygonal Meshes

• Xinlai Ni

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Introduction

• Surfaces in 3D
• Implicit Representation, i.e., isosurfaces
• Explicit Representation, i.e., polygonal meshes

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Mesh Topology

• Manifolds of different topological properties
• Orientable/Non-Orientable
• Genus
• Morse theory reveals the connection between the
  topological structure and the differential
  property of the surface
• Finding the fair Morse function

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Dynamic Surface Modeling

• Dynamic interfaces in simulations such as water,
  fire, etc
• Morphing
• Surface fitting
• Clay modeling

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Moving Surfaces

• For implicit surfaces
• Level Set Method propagates the surface by
  changing voxel values in the simulation grid
• Does not scale well with voxel resolutions
• For polygonal meshes
• Propagate the vertices by the velocity field
• Can have swallowtails or self-intersections

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Moving Surfaces (cont)

• Face Offsetting Methods
• Propagate face and recalculate intersections
• Induces a volume-preserving smoothing method
• Procedural Level Sets
• Projects the geometric motion onto parametric
  representations
• Has nice property for localizable functions, such
  as CSG.

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Outline

• Morse Fairing
• Morse theory, basic fairing algorithm
• Multigrid algorithm
• Applications
• Face Offsetting Method
• Fundamental idea
• Application in volume-preserving mesh smoothing
• Procedural Level Sets
• Mathematical derivation
• Application in surface fitting
• Propagation of localizable surfaces, e.g. CSG

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Morse Fairing

• The Topology Invariance of a space is the
  quantity that is invariant under homotopy
  equivalences (continuous deformations). such
  quantities include etc.,
  but do not include surface area, volume,
  curvature, etc.
• For finite cellular complexes, the Euler
  Characteristic is defined as
• Where cn is the number of n-cells

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Genus and Surface Cutting

• We can construct orientable 2-manifolds by
  identifying pairs of edges in the 2g-gon
• On the contrary, we can give the orientable
  2-manifold a cellular structure of one 0-cell, 2g
  1-cells and one 2-cell.
• Cutting along the 1-cells flattens the surface.
• Polygonal mesh is another cellular structure of
  the surface.

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Morse Theory

• For a real-valued function defined on the
  surface, Morse theory reveals the connection
  between the arrangement of its critical points
  (point with zero gradient) and its genus
• Tracing flowlines from the saddle gives a
  cellular complex structure
• Can perturn such that the flowline from a saddle
  always terminates at a minimum Morse-Smale
  complex

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Morse Theory on Meshes
Banchoff, 1970
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v
1
3
Lk v Link of vertex v
Lk- v Lower link of v, f(v) 5 link
vertices lt f(v) edges


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0
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M


m
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X
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2
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-

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regular vertex Lk- v contractible
maximum Lk- v Lk v index 2
minimum Lk- v ?? index 0
Morse saddle Lk- v 2 pieces index 1
monkey saddle Lk- v 3 pieces index
1 multiplicity 2
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Fair Morse Functions

• Due to perturbation or scanning errors, the
  function on the mesh may have lots of critical
  points
• We are interested in finding fair Morse
  function having the least number of critical
  points, i.e., one minimum, one maximum and two
  saddles per handle

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Morse Fairing

• Define f on mesh by assigning values to vertices
• Algorithm
• Pick vmin and set f (vmin) -1
• Pick vmax and set f (vmax) 1
• Solve Laplacian to define f (v) on remaining
  vertices
• Result
• Every vertex (except vmin,vmax) is average of its
  neighbors
• Cannot be a min or a max
• Can be a saddle
• But there must be only 2g of them

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1
2
1
3
1
2
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7 min 15 saddles 8 max
1 min 2 saddles 1 max
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Multigrid Morse Fairing

• Simplify M repeatedly
• Constructs multires hierarchy M0 Mkwhere M0 M
• Collapses must pass link conditionDey et al.,
  1999
• Base domain Mk M
• Solve Laplacian on base domain Mk
• Reintroduce deleted vertices one at a time
• Approximate value of new vertex with (weighted)
  average of neighbors
• Iterate Laplacian locally until neighborhood
  converges

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Intermediate Value Propagation

• We dont really want a Laplacian smoothing on the
  values
• When reintroducing a vertex, just to make sure it
  doesnt become a min or max
• Intermediate Value Proposition (proved) ensures
  we can always find such a value for the
  reintroduced vertex
• Resulting irregular but correct function

v
v
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Mean-ValueWeights
IntermediateValuePropagation
UniformWeights
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Cutting a Surface into a Disk

• Pick one max correspondingto center of disk
• Pick one (or more) minima which thecuts should
  pass through (e.g. highcurvature points)
• Cut ? circle, Laplacian on coords.Floater, CAGD
  97

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Base Domain Construction

• Base domain end result of greedy simplification
  steps
• Clustering is rather arbitrary
• Morse fairing allows user to pick vertices (min)
  and faces (max) of base domain
• Creates min-saddle-min-saddle quads which can be
  triangulated by ascent paths to max in the middle

MAPS domains Lee et al., S98
Morse Fairing domains
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Fast Developable Patches

• Morse function isGaussian curvature,squared,
  negatedf (p) -kG2(p)
• Maxima at flattest regions
• Minima at feature points

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Visualization
165 tunnels
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Explicit Surface Tracking Method

• Surface painting method, Lawrence Funkhouser
• Local surface deformation
• Adaptively change mesh resolution at painted
  region
• Moving mesh PDE solvers, Li Petzold
• Self-intersection in 2D
• Trimming or delooping
• Preconditioning the curve before propagating it

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Surface Smoothing

• Positional Diffusion
• Laplacian smoothing
• Skin, Markosian et al.
• Curvature flow, Desbrun et al.
• Normal Diffusion
• Gaussian smoothing to surface normals, Ohtake et
  al.
• Volume preservation
• Global volume preservation by scaling, Desbrun et
  al.
• However, local volume preservation is more
  physically meaningful

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Offset Intersections

• For a vertex , its incident faces are moved
  virtually to a new position called offset face
• We want to find the new position
  for such that the quadric error
  to the offset faces is minimized
• Sounds reasonable, but numerically unstable

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Classification of Vertex

• Classify vertex as smooth, ridge or corner by
  looking at the eigensystem of matrix A
• How many large eigenvalues are there ( is
  large
• if )?
• 1 ?? vertex is smooth
• 2 ?? vertex is ridge
• 3 ?? vertex is corner
• Move the vertex only in the space spanned by the
  eigenvectors corresponding to the large
  eigenvalues

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Avoiding Self-Intersections

• Timestep for PDE is restricted by CFL
  (Courant-Friedrichs-Lewy) condition.
• In face offsetting, this is equivalent to
  requiring that the offsets of the disjoint faces
  do not intersect.
• A sufficient condition is that the offset
  intersection projects back to the
  safe region
• If not, take a smaller step along the direction
  of offset intersection

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Local vs. Global Self-Intersection

• Local self-intersection
• Swallowtails, e.g., expanding a concave region
• Can be avoided by ensuring the safe region
  condition
• Global self-intersection
• Topology change, e.g., merging, diverging, etc.
• Need detect global topology change, and indicate
  the location
• So far, can not handle

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Diffusion of Normals

• Naturally induced by the face offsetting method
• Calculate the weighted average normal for a face
• Uniform weight ?? isotropic diffusion
• Non-uniform weight ?? anisotropic diffusion
• Construct the face offset by aligning the
  original face along the averaged normal, fixing
  its center
• Apply the face offsetting method to find the new
  offset intersections

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Volume-Preserving Surface Smoothing

• Diffuse the normals of the faces
• Calculate local volume change and diffuse the
  change back to the face offsets
• Guarantee local volume preservation

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Results
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Procedural Level Sets

• Level set method
• Propagate the surface in normal direction by
  integrating the voxel values
• Automatically accommodate topology change and
  self-intersection
• Does not scale well with resolution of the grid
• Procedural level sets
• Use much fewer parameters than the number of
  voxels in the simulation grid
• Have very nice property for localizable surfaces

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Derivation

• Let be a time-varying scalar field
  function over spatial variable and a vector
  of scalar parameters
• A particle on the surface will satisfy
• Taking the time derivative, we have
• Let be the specified velocity field

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A is a matrix of size n X m n number of
particles m length of q
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Solve for Parameter Speed

• Usually n gt m, i.e., more particles than the
  number of control parameters
• This is an overconstrained system, for which we
  want to find the optimal solution minimizing the
  squared error residual
• Using SVD, decompose AUSVT , then the optimal
  solution is
• A is provably rank-deficient for polynomial
  surfaces, i.e., rank(A) lt m. Ignore the zero
  singular values when taking the inverse S-1

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Result for Surface Fitting

• Scattered point cloud
• Use vector to the nearest point as the velocity
  field

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CSG Surfaces

• Compose new shape from simple primitives
• Using Boolean operation (unioin, inersection,
  subtraction) to organize a tree structure
• Using RFunctions, can guarantee arbitrary
  differentiabilities.
• Composite function F(F1, , Fk)

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Localizable Surfaces

• Composite function F(F1, , Fk), Fi has parameter
  vector qi
• A particle x is said to be on the ith primitive
  if Fi(x) F (x) 0
• The surface F 0 is said to be localizable if
• for all x on primitives other than i
• for
• Think about the union of two spheres

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Decoupling the Propagation

• Decoupling Proposition (proved)
• For localizable surfaces, (e.g CSG surface) the
  propagation solution to for the
  entire surface can be obtained by solving it for
  each primitive independently.

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Preliminary Results
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Preliminary Results
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Future Work

• Mesh adaptation for face offsets
• Global intersection, i.e., topology change

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Future Work

• Convergence condition of the procedural level
  sets
• Create more CSG propagation instances
• Take advantage of other localizable functions
  such as RBF

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• Thank you!