Representing Higher OrderRepresenting Higher OrderVector Fields Singularities on Vector Fields Singularities on

Piecewise Linear SurfacesPiecewise Linear Surfaces

Wan Chiu LiWan Chiu LiBruno ValletBruno ValletNicolas RayNicolas RayBruno LévyBruno Lévy

IEEE Visualization 2006IEEE Visualization 2006Baltimore, USABaltimore, USA

alice.loria.fralice.loria.fr

1.1. IntroductionIntroduction

2.2. Discrete representationDiscrete representation

3.3. SingularitiesSingularities

4.4. Encoding an existing vector fieldEncoding an existing vector field

5.5. LIC-based visualizationLIC-based visualization

6.6. ConclusionConclusion

OutlineOutline

• What is a vector field singularity ?What is a vector field singularity ?

• It is a It is a 00 of the field of the field

• How can we characterize singularities ?How can we characterize singularities ?

• By their By their index index ==

IntroductionIntroduction

-1-1+1+1

∫∫ ddθθ

∫∫

• What is a vector field singularity ?What is a vector field singularity ?

• It is a It is a 00 of the field of the field

• How can we characterize singularities ?How can we characterize singularities ?

• By their By their index index ==

IntroductionIntroduction

ddθθ

-2-2+2+2

How can we visualize a singularity ?How can we visualize a singularity ?

Piecewise linear methods: Piecewise linear methods: indexindex [-v∊[-v∊ al/2+1, al/2+1, vval/2+1]al/2+1]

IntroductionIntroduction

[Tricoche00][Tricoche00]

How can we visualize a singularity ?How can we visualize a singularity ?

Piecewise linear methods: Piecewise linear methods: indexindex [-v∊[-v∊ al/2+1, al/2+1, vval/2+1]al/2+1]

Higher order singularities: Higher order singularities: indexindex∊ℤ∊ℤ

IntroductionIntroduction

• Basic idea:Basic idea:

• 2D vectors are complex 2D vectors are complex rreeiiθθ

• Interpolate Interpolate rr and and θθ

• Justification:Justification:

• Singularity = (Singularity = (rr =0, =0, θθ undefined) undefined)

• Singularity index depend only on Singularity index depend only on θθ in neighborhood in neighborhood

IntroductionIntroduction

On triangulated meshesOn triangulated meshes

• Dual Dual vertexvertex--edgeedge encoding using encoding using polar coordinatespolar coordinates::

• Dual verticesDual vertices: norm : norm rr ∊ℝ∊ℝ, angle , angle θθ ∊ℝ∊ℝ

• Dual edgesDual edges: period jump : period jump pp ∊ ℤ ∊ ℤ

• 3 Step interpolation (3 Step interpolation (0D0D, , 1D1D, , 2D2D))

• Dual vertices=Facet centers (0D)Dual vertices=Facet centers (0D)

• Dual edges (1D)Dual edges (1D)

• Subdivision simplex (2D)Subdivision simplex (2D)

Discrete representationDiscrete representation

v*v*x(x(v*v*))

θθ((v*v*))

• θθ((v*v*)) : : measured from a reference vectormeasured from a reference vector x(x(v*v*))• rr((v*v*)) : : vector norm, basis independentvector norm, basis independent

0D0D

rr((v*v*))

e*e*

x(x(v’*v’*))

PP

Linear interpolation:Linear interpolation:

⍺⍺((e*e*)) : height ratio: height ratio, , ∆∆θθ((e*e*)) : angular variation along: angular variation along e*e*

1D1D

θθ((PP)) = = θθ((v*v*)) + + ∆∆θθ((e*e*)) (⍺(⍺ e*e*))tt

v’*v’*x(x(v*v*))

θθ((v*v*)) v*v*

θθ((v’*v’*))

e*e*

x(x(v’*v’*))

PP

Linear interpolation:Linear interpolation:

⍺⍺((e*e*)) : height ratio: height ratio, , ∆∆θθ((e*e*)) : angular variation along: angular variation along e*e*

1D1D

θθ((PP)) = = θθ((v*v*)) + + ∆∆θθ((e*e*))(1-(1- (⍺(⍺ e*e*))))tt

v’*v’*x(x(v*v*))

θθ((v*v*)) v*v*

θθ((v’*v’*))

1D1D

HHe*e*

h’h’

Height ratio:Height ratio:

⍺⍺((e*e*) ) == hh//HH1-1- (⍺(⍺ e*e*)) = = h’h’//HH

v’*v’*

v*v*

hh

1D1D

p(p(e*e*)) = -1 = -1

∆∆θθ((e*e*)) = = ∫d∫dθθ = = θθ((BB) ) -- θθ((AA)) + 2 + 2 ππ p(p(e*e*))

Angular variation alongAngular variation along e*e* : :

e*e*

Period Jump:Period Jump:e*e*

BB

AA

p(p(e*e*)) = 1 = 1

e*e*

BB

AA

e*e*

BB

AA

p(p(e*e*)) = 0 = 0

1D1D

Subdivision simplexSubdivision simplex

2D2D

Subdivision simplexSubdivision simplex

2D2D

A variant the A variant the sideside--vertexvertex interpolation [Nielson79] interpolation [Nielson79]

• Linear along the Linear along the sideside

• Constant along a Constant along a sideside--vertexvertex path (= path (=sideside value) value)

sideside

vertexvertex

2D2D

PPP’P’

• Singularities may occur only at Singularities may occur only at verticesvertices

• Singularity indexSingularity index depends only ondepends only on period jumps period jumps ::

I(I(vv)) = = ∫∫ddθθ = = II00((vv)) + + ∑ ∑ p(p(e*e*))

SingularitiesSingularities

f*f*

e*e*∊∊∂∂f*f*

vv

∂∂f*f*

Advantages:Advantages:

1.1. ControlControl over over placementplacement and and indexindex of singularities of singularities

2.2. Coherent with Coherent with Poincare-HopfPoincare-Hopf index theorem index theorem

3.3. Index Index independentindependent of the valence of the valence

4.4. Easy extension to Easy extension to fractional fractional indicesindices

SingularitiesSingularities

Fractional indices appear in Fractional indices appear in N-symmetryN-symmetry vector fields : vector fields :

• Not vectors but equivalence class of vectors by Not vectors but equivalence class of vectors by ≡≡NN

• uu≡≡NNv ⇔ k | u=R(v, 2k∃v ⇔ k | u=R(v, 2k∃ ππ/N)/N)

• Period jump and Indices are multiples of Period jump and Indices are multiples of 1/N1/N

Extension to fractional Extension to fractional indicesindices

-1/2-1/2+1/2+1/2

Fractional indices appear in Fractional indices appear in N-symmetryN-symmetry vector fields : vector fields :

• Not vectors but equivalence class of vectors by Not vectors but equivalence class of vectors by ≡≡NN

• uu≡≡NNv ⇔ k | u=R(v, 2k∃v ⇔ k | u=R(v, 2k∃ ππ/N)/N)

• Period jump and Indices are multiples of Period jump and Indices are multiples of 1/N1/N

Extension to fractional Extension to fractional indicesindices

-1/4-1/4+1/4+1/4

1.1. rr((v*v*) ) : norm of the vector at facet center : norm of the vector at facet center v*v*

2.2. θθ((v*v*)) : choose one of the 3 edges and compute angle: choose one of the 3 edges and compute angle

3.3. 22ππp(p(e*e*)=)=∆∆θθ((e*e*) ) - - ∠∠((x(x(v’*v’*)),, x( x(v*v*))) - ) - θθ((v’*v’*) ) + + θθ((v*v*))

Requires an interpolation or an analytic form:Requires an interpolation or an analytic form:

Encoding an Existing Encoding an Existing Vector FieldVector Field

e*e* ||v||||v||22e*e*

∆∆θθ = = ∫d∫dθθ = = ∫ ∫ vvxxdvdvyy – v – vyydvdvxx

Encoding an Existing Encoding an Existing Vector FieldVector Field

e*e*v’*v’*

v*v*

ddθθ

e*e*∆∆θθ = = ∫d∫dθθ = = ∫ (v∫ (vxxdvdvyy – v – vyydvdvxx) / ||v||) / ||v||22

e*e*

• GPU acceleratedGPU accelerated

• Works in image spaceWorks in image space

• 3 passes3 passes

Ensure geometricEnsure geometricdiscontinuitydiscontinuity(in depth buffer)(in depth buffer)

Direction on the surfaceDirection on the surface(in fragment shader)(in fragment shader)

Line integral convolutionLine integral convolution(in image space) (in image space)

LIC-based VisualizationLIC-based Visualization

[Laramee et. al. 03][Van Wijk 03]

ResultsResults

Index = -3Index = -3 Index = 5Index = 5

ResultsResults

Dual Dual vertexvertex--edgeedge encoding encoding

+ 3 steps interpolation:+ 3 steps interpolation:

• A very good candidate for visualizing A very good candidate for visualizing non-linearnon-linear vector fields on piecewise vector fields on piecewise linear surfaces or 2d meshes.linear surfaces or 2d meshes.

• Efficient and simple way to visualize Efficient and simple way to visualize arbitraryarbitrary singularities singularities

• Easy Easy generalizationgeneralization to fractional indices to fractional indices

• Easier Easier particularizationparticularization to 2d fields to 2d fields

ConclusionConclusion

• Smooth Smooth non singularnon singular vertices vertices

• Topological operationsTopological operations

• TraceTrace streamlines streamlines

• Extension to Extension to 3d3d vector fields vector fields

Future workFuture work

Questions Questions ??

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