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

1.Introduction 2.Discrete representation 3.Singularities 4.Encoding an existing vector field 5.LIC-based visualization 6.Conclusion Outline

What is a vector field singularity ?What is a vector field singularity ? It is a 0 of the fieldIt is a 0 of the field How can we characterize singularities ?How can we characterize singularities ? By their index =By their index = Introduction +1 dθdθdθdθ

What is a vector field singularity ?What is a vector field singularity ? It is a 0 of the fieldIt is a 0 of the field How can we characterize singularities ?How can we characterize singularities ? By their index =By their index = Introduction dθdθdθdθ -2 +2

How can we visualize a singularity ? Piecewise linear methods: index [-val/2+1, val/2+1] Introduction [Tricoche00]

How can we visualize a singularity ? Piecewise linear methods: index [-val/2+1, val/2+1] Higher order singularities: index Higher order singularities: index Introduction

Basic idea:Basic idea: 2D vectors are complex re iθ2D vectors are complex re iθ Interpolate r and θInterpolate r and θ Justification:Justification: Singularity = ( r =0, θ undefined)Singularity = ( r =0, θ undefined) Singularity index depend only on θ in neighborhoodSingularity index depend only on θ in neighborhood Introduction

On triangulated meshes Dual vertex-edge encoding using polar coordinates:Dual vertex-edge encoding using polar coordinates: Dual vertices: norm r, angle θDual vertices: norm r, angle θ Dual edges: period jump pDual edges: period jump p 3 Step interpolation (0D, 1D, 2D)3 Step interpolation (0D, 1D, 2D) Dual vertices=Facet centers (0D)Dual vertices=Facet centers (0D) Dual edges (1D)Dual edges (1D) Subdivision simplex (2D)Subdivision simplex (2D) Discrete representation

v* x(v*) θ(v*) θ(v*) : measured from a reference vector x(v*) θ(v*) : measured from a reference vector x(v*) r(v*) : vector norm, basis independent r(v*) : vector norm, basis independent 0D r(v*)

e* x(v*) P Linear interpolation: (e*) : height ratio,θ(e*) : angular variation along e* (e*) : height ratio,θ(e*) : angular variation along e* 1D θ(P) = θ(v*) + θ(e*) (e*)t v* x(v*) θ(v*) v*

e* x(v*) P Linear interpolation: (e*) : height ratio,θ(e*) : angular variation along e* (e*) : height ratio,θ(e*) : angular variation along e* 1D θ(P) = θ(v*) + θ(e*)(1- (e*))t v* x(v*) θ(v*) v*

1D H e* h Height ratio: (e*) = h/H (e*) = h/H 1- (e*) = h/H v* v* h

1D p( e* ) = -1 θ(e*) = dθ = θ(B) - θ(A) + 2 π p(e*)θ(e*) = dθ = θ(B) - θ(A) + 2 π p(e*) Angular variation along e* : e* Period Jump: e* B A p( e* ) = 1 e* B A e* B A p( e* ) = 0

1D

Subdivision simplex 2D

2D

A variant the side-vertex interpolation [Nielson79] Linear along the sideLinear along the side Constant along a side-vertex path (=side value)Constant along a side-vertex path (=side value) side vertex 2D P P

Singularities may occur only at verticesSingularities may occur only at vertices Singularity index depends only on period jumps :Singularity index depends only on period jumps : I( v ) = dθ = I 0 ( v ) + p(e*) Singularities f* e*f* v f*f*

Advantages: 1.Control over placement and index of singularities 2.Coherent with Poincare-Hopf index theorem 3.Index independent of the valence 4.Easy extension to fractional indices Singularities

Fractional indices appear in N-symmetry vector fields : Not vectors but equivalence class of vectors by NNot vectors but equivalence class of vectors by N u N v k | u=R(v, 2kπ/N)u N v k | u=R(v, 2kπ/N) Period jump and Indices are multiples of 1/NPeriod jump and Indices are multiples of 1/N Extension to fractional indices -1/2 +1/2

Fractional indices appear in N-symmetry vector fields : Not vectors but equivalence class of vectors by NNot vectors but equivalence class of vectors by N u N v k | u=R(v, 2kπ/N)u N v k | u=R(v, 2kπ/N) Period jump and Indices are multiples of 1/NPeriod jump and Indices are multiples of 1/N Extension to fractional indices -1/4 +1/4

1.r(v*) : norm of the vector at facet center v* 2.θ(v*) : choose one of the 3 edges and compute angle 3.2πp( e* )=θ(e*) - ( x(v*), x(v*) ) - θ( v* ) + θ( v* ) Requires an interpolation or an analytic form: Encoding an Existing Vector Field e* ||v|| 2 e* θ = dθ =θ = dθ = v x dv y – v y dv x

Encoding an Existing Vector Field e* v* v* dθdθdθdθ e* θ = dθ = (v x dv y – v y dv x ) / ||v|| 2θ = dθ = (v x dv y – v y dv x ) / ||v|| 2 e*

GPU acceleratedGPU accelerated Works in image spaceWorks in image space 3 passes3 passes Ensure geometric discontinuity (in depth buffer) Direction on the surface (in fragment shader) Line integral convolution (in image space) LIC-based Visualization [Laramee et. al. 03] [Van Wijk 03]

Results Index = -3 Index = 5

Results

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

Smooth non singular verticesSmooth non singular vertices Topological operationsTopological operations Trace streamlinesTrace streamlines Extension to 3d vector fieldsExtension to 3d vector fields Future work

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