Generierung von Partikeldatenstzen aus Oberflchennetzen mit Hilfe von Distanzfeldern

1
Generierung von Partikeldatensätzenaus
Oberflächennetzenmit Hilfe von Distanzfeldern

• Daniel Fader
• Lehrstuhl für Graphische Datenverarbeitung
• Studienarbeit
• 20. Mai 2008

2
References

• FH04 Distance Transforms of Sampled
  Functions,Pedro F. Felzenszwalb, Daniel P.
  Huttenlocher,Cornell Computing and Information
  Science Technical Report TR2004-1963, 2004
• SWT06 Robust Tetrahedral Meshing of Triangle
  Soups,Jonas Spillmann, Michael Wagner, Matthias
  Teschner,Eurographics / ACM SIGGRAPH Symposium
  on Computer Animation Posters and Demos,
  Vienna, Austria, pp. 1-2, Sep. 2-4, 2006
• BYM05 Particle-Based Simulation of Granular
  Materials,Nathan Bell, Yizhou Yu, Peter J.
  Mucha,ACM SIGGRAPH / Eurographics Symposium on
  Computer Animation, 2005
• WikTI Wikipedia Trilinear Interpolation
  http//en.wikipedia.org/wiki/Trilinear_interpolati
  on

3
Motivation

• Consider a rigid body within a particle-based
  fluid simulation
• Detect collisions of a particle (sphere) and a
  triangle

4
Motivation

• Less involvedCollision detection of a sphere
  and another sphere
• Represent surfacewith particles

lt2r ?
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Outline

• Signed Distance Field
• Unsigned Distance Field
• Sign Computation
• Grid Method
• Floaters Method
• Results

6
Unsigned Distance Field

• Subdivide the objetcs space into a regular grid
• Abstracts from the original surface mesh

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Unsigned Distance Field

• Goal For each gridpoint calculate and save its
  (unsigned) distance to the surface
• First Rasterize the object into the grid

Rasteri- zation
0-Level-Set (surface)
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Unsigned Distance Field

• Calculate the distance from each gridpointto the
  0-Level-Set
• Perform Distance Transform (DT) as in FH04

DT
brightness distance
0-Level-Set
Initially 8
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Signed Distance Field

• The sign distinguishes gridpoints that describe
  the volume
• Gridpoints inside the model have a negative sign

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Signed Distance Field

• Send rays through the grid
• If ray intersects the surface odd times, change
  the sign of the following gridpoints
• One axis aligned ray worksfor completely
  closedvolumes
• Rays from multiple directionsas in SWT06
  detectpseudovolumes
• Average the sign markedby all rays

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Signed Distance Field
Pseudovolume
One Ray Misclassification
surface
outside
/
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Signed Distance Field Summary

• Use linear time DT from FH04 (correctness
  proved)
• Calculate sign by transmitting rays through the
  grid
• Sign correctly determined for most meshes in

outside
inside
model
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Outline

• Signed Distance Field
• Grid Method
• Floaters Method
• Results

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Grid Method

• Signed distance field given
• Walk through the distance field with step size
  radius
• Observed positions in general do not match the
  gridpoints

distance field
?
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Grid Method

• Interpolate the distance value of the surrounding
  gridpoints
• In 3D Use trilinear interpolation
• If the interpolated value is within/below auser
  defined threshold
• Set a surface/volume particle at the observed
  position

WikTI
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Grid Method Summary

• Works completely on the distance field
• Observes positions at fixed intervals
• Depends on the user defined threshold
• Linear runtime

Grid method
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Outline

• Signed Distance Field
• Grid Method
• Floaters Method
• Extended Distance Field
• Results

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Floaters Method

• ?Surface Particle Generation from BYM05
• To more closely match the surface let particles
  ?float on it
• First place particles randomly on the surface
• Then let particles repel each other and constrain
  them to the surface
• ?Floaters are particles with additional velocity
  information

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Floaters Method

• Floating iteration

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Floaters Method

• For each surface triangle t placefloaters on
  it with D User defined density value A
  Area of t r Radiususing random barycentric
  coordinates

radius
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Floaters Method

• Repulsion VR
• For each floater P find neighboring particles S
• Repel P from all Pi ? S
• Determining S takes quadratic runtime

P Examined floater Pi Neighboring
floater S Set of neighboring floaters
K Kernel function
radius
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Floaters Method

• Feedback VF
• Move every floater back on the 0-Level-Set
  (surface)
• Interpolate distance and normal information at P
  from the distance field

distance field
f( P ) Distance at P N Normal at P
0-Level-Set
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Outline

• Signed Distance Field
• Grid Method
• Floaters Method
• Extended Distance Field
• Results

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Extended Distance Field

• Extend the distance field by normal information
• The direction from each gridpoint to the
  0-Level-Set

distance field
0-Level-Set
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Extended Distance Field

• For each gridpoint
• Find nearest gridpoint in the 0-Level-Set
• Very time-consuming

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Extended Distance Field

• Spatial partitioning approach
• Subdivide the distance field into the SPG (spat.
  part. grid)
• Save a list of surface gridpoints for each voxel
  in SPG

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Extended Distance Field

• Compare with surface gridpoints in the same voxel
  in SPG
• Also survey neighboring voxels
• Point to the nearest surface gridpoint(s) found

?
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Extended Distance Field

• Approximate normal if no surface gridpoints lie
  in the neighborhood of a voxel v
• All normals in v point to the nearest voxel that
  contains surface gridpoints
• Usually particles floatnearby the surface

?
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Extended Distance Field Summary

• Preprocessing step
• Reduce the search space speedup about factor 30
• Distance and direction to surface known for every
  gridpoint

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Outline

• Signed Distance Field
• Grid Method
• Floaters Method
• Results

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Floaters Method

• Velocities terms VR and VF can now be evaluated
• Float until surface is evenly covered

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Floaters Method Summary

• Particles are randomly spread and permitted to
  float
• Floaters repel each other
• Floaters are constrained to the surface
• User defines density and number of iterations
• Repulsion calculated in O(n2)

Floaters method
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Outline

• Signed Distance Field
• Grid Method
• Floaters Method
• Results

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Results

• Humanoid, radius 0.48

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Grid, 1271 particles
Floaters, 1602 particles
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Results

• Pyramid, radius 0.03

1
Grid, 2918 particles
Floaters, 3800 particles
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Results

• Cow, radius 0.1

4
Grid, 1382 particles
Floaters, 1359 particles
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Results

• Holey Gourd, radius 0.1

3.2
Grid, 1378 particles
Floaters, 1212 particles
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Results

• Demo Stanford Bunny

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Summary

• Grid method
• Fast
• Only one paramter
• Minimal number of particles on axis aligned
  surfaces
• Floaters method
• Quadratic runtime, several iterations
• User more involved
• High precision
• Future work
• Normals attached to particles
• Faster repulsion calculation

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The end

• Thank you for your interest
• and for paying attention!

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Unsigned Distance Field
Appendum

• Distance Transformby Felzenszwalb and
  Huttenlocher FH04
• Defined on a onedimensional grid G 0, ...,
  n-1
• The distance is determined by a cost function f
• Initially

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Unsigned Distance Field
Appendum

• For each gridpoint pi
• f(pi) is the squared euclidean distance from pi
  to the surface
• Place a standard parabola at (pi, f(pi))
• The final value of f(pi) is determined by the
  lower envelope of parabolas at each surface
  gridpoint

G
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Unsigned Distance Field
Appendum

• Lower envelope calculation is the main part of
  the DT
• Using two arrays it can be done in O(n)
• A higher dimensional DT simply performs the1D
  Transform along each axis
• DT of a three dimensional gridG 0, , n-1
  0, , m-1 0, , k-1 in O(3nmk)

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Unsigned Distance Field
Appendum
Algorithm Onedimensional DTwith linear runtime
FH04
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Signed Distance Field
Appendum

• Use rays to separate inside from outside
• If ray intersects the surface odd times, change
  the sign of the following gridpoints
• One axis aligned ray

surface
inside
outside
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Signed Distance Field
Appendum

• Using only rays with one direction may cause
  misclassification

inside
outside
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Signed Distance Field
Appendum

• Use multiple directions as in SWT06
• Average the sign marked by all rays

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Signed Distance Field
Appendum

• Rays from 26 directions (3D)
• Complexity O(26nmk)

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Signed Distance Field
Appendum

• Misclassification at meshes with intersecting
  volumes

DT outside
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Signed Distance Field
Appendum

• Misclassification at faces with no volume

DT inside
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Floaters Method
Appendum

• Calculate two velocities for each floater in a
  user defined number of floating iterations
• Repulsion VR
• Repels the particles from each other to evenly
  spread them over the surface
• Feedback VF
• Constrains the particles to the surface

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Floaters Method
Appendum

• Velocities terms VR and VF can now be evaluated
• One iteration for Floater P
• First calculate and apply repulsion
• Then interpolate distance and normal and apply
  feedback
• Examine next floater

,
,
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Results
Appendum
DT
Volume
Holey surface mesh
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Grid vs. Floaters
Appendum

• Grid
• O(n)
• Minimal on axis
• aligned surfaces
• Treshold

• Floaters
• O(n2)
• Too many on plain surfaces
• good at round structures
• Density, time steps,
• of iterations

• Runtime
• of Particles
• Parameters