Prsentation PowerPoint

1
Representing Higher Order Vector Fields
Singularities on Piecewise Linear Surfaces
Wan Chiu Li Bruno Vallet Nicolas Ray Bruno Lévy
IEEE Visualization 2006 Baltimore,
USA alice.loria.fr
2
Outline

• Introduction
• Discrete representation
• Singularities
• Encoding an existing vector field
• LIC-based visualization
• Conclusion

3
Introduction

• What is a vector field singularity ?
• It is a 0 of the field
• How can we characterize singularities ?
• By their index

?
d?
-1
1
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Introduction

• What is a vector field singularity ?
• It is a 0 of the field
• How can we characterize singularities ?
• By their index

?
d?
-2
2
5
Introduction
How can we visualize a singularity ? Piecewise
linear methods index?-val/21, val/21
Tricoche00
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Introduction
How can we visualize a singularity ? Piecewise
linear methods index?-val/21, val/21 Higher
order singularities index?Z
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Introduction

• Basic idea
• 2D vectors are complex rei?
• Interpolate r and ?
• Justification
• Singularity (r 0, ? undefined)
• Singularity index depend only on ? in neighborhood

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Discrete representation

• On triangulated meshes
• Dual vertex-edge encoding using polar
  coordinates
• Dual vertices norm r ?R, angle ? ?R
• Dual edges period jump p ? Z
• 3 Step interpolation (0D, 1D, 2D)
• Dual verticesFacet centers (0D)
• Dual edges (1D)
• Subdivision simplex (2D)

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0D
r(v)
?(v)
v
x(v)

• ?(v) measured from a reference vector x(v)
• r(v) vector norm, basis independent

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1D
v
?(v)
P
v
x(v)
e
?(v)
x(v)
Linear interpolation
?(P) ?(v) ??(e)?(e)t
?(e) height ratio, ??(e) angular
variation along e
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1D
v
?(v)
v
P
x(v)
e
?(v)
x(v)
Linear interpolation
?(P) ?(v) ??(e)(1-?(e))t
?(e) height ratio, ??(e) angular
variation along e
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1D
v
Height ratio ?(e) h/H 1-?(e) h/H
e
h
H
h
v
13
1D
Angular variation along e
Period Jump
??(e) ?d? ?(B) - ?(A) 2 p p(e)
e
B
p(e) -1
e
A
B
p(e) 0
e
A
B
p(e) 1
e
A
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1D
15
2D
Subdivision simplex
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2D
Subdivision simplex
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2D

• A variant the side-vertex interpolation
  Nielson79
• Linear along the side
• Constant along a side-vertex path (side value)

side
P
P
vertex
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Singularities

• Singularities may occur only at vertices
• Singularity index depends only on period jumps
• I(v) ?d? I0(v) ? p(e)

e??f
?f
f
v
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Singularities

• Advantages
• Control over placement and index of singularities
• Coherent with Poincare-Hopf index theorem
• Index independent of the valence
• Easy extension to fractional indices

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Extension to fractional indices

• Fractional indices appear in N-symmetry vector
  fields
• Not vectors but equivalence class of vectors by
  N
• uNv ? ?k uR(v, 2kp/N)
• Period jump and Indices are multiples of 1/N

-1/2
1/2
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Extension to fractional indices

• Fractional indices appear in N-symmetry vector
  fields
• Not vectors but equivalence class of vectors by
  N
• uNv ? ?k uR(v, 2kp/N)
• Period jump and Indices are multiples of 1/N

-1/4
1/4
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Encoding an Existing Vector Field

• r(v) norm of the vector at facet center v
• ?(v) choose one of the 3 edges and compute
  angle
• 2pp(e)??(e) - ?(x(v), x(v)) - ?(v)
  ?(v)
• Requires an interpolation or an analytic form

vxdvy vydvx
?? ?d? ?
v2
e
e
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Encoding an Existing Vector Field
?? ?d? ? (vxdvy vydvx) / v2
e
e
d?
v
v
e
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LIC-based Visualization

• GPU accelerated
• Works in image space
• 3 passes

Laramee et. al. 03 Van Wijk 03
Ensure geometric discontinuity (in depth buffer)
Line integral convolution (in image space)
Direction on the surface (in fragment shader)
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Results
Index -3
Index 5
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Results
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Conclusion

• Dual vertex-edge encoding
• 3 steps interpolation
• A very good candidate for visualizing non-linear
  vector fields on piecewise linear surfaces or 2d
  meshes.
• Efficient and simple way to visualize arbitrary
  singularities
• Easy generalization to fractional indices
• Easier particularization to 2d fields

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Future work

• Smooth non singular vertices
• Topological operations
• Trace streamlines
• Extension to 3d vector fields

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Questions ?