Simplified Representation of Vector Fields Alexandru C. Telea Jarke J. van Wijk Eindhoven University of Technology The Netherlands

Overview Why simple vector field representation? Vector field visualization Simplification Setting the parameters Applications Ongoing research

Problems: how to visualize large 2D/3D fields ? how to convey information to non experts ? how to easily produce visualizations ?

Goal: We desire to obtain an image that: is effective is compact (simple) offers local and global insight is produced automatically

Show a vector field with a few arrows Solution Show a vector field with a few arrows Method But how? dataset visualization

1 Existing Methods Icon-Based Methods (hedgehogs, glyphs, flow icons) visual clutter for large datasets undersampling may cause aliasing arrows are easily interpreted by large public

2 Existing Methods Texture-Based Methods (spot noise, LIC) lack directional information may produce noisy images for complex flows give good global insight easy to generate automatically

3 Existing Methods Advection-Based Methods (stream/streak lines, flow tubes, particles) require strong user input for advection seed point placement results highly dependent on seed point choice give good local insight

4 Existing Methods Feature Extraction / Tracking Methods (vortex detection, selection expressions) highly dependent on seed point choice require user input on which feature to select /track (often in the form of many parameters) too abstract for some users reduces dataset to a compact representation

5 Existing Methods Topological Analysis (vortex / critical point analysis and detection) critical points may be too abstract for some users unstable for fields with many singularities reduces dataset to a compact representation

6 Existing Methods Multiresolution Techniques (Fourier / wavelet analysis) if uniform sampling is used, can not convey both global and local insight mostly used for scalar fields reduced dataset can be easily visualized by other methods

Generate vector field visualizations: Our Goal Generate vector field visualizations: automatically combine local and global insight show directional information convey a simple and intuitive perception of the vector field

A curved arrow is easily understood perception process User sees curved arrow User perceives vector field An arrow carries a clear representation of the direction and curvature of the vector field it suggests

Solution Simplify vector field in zones which are representable by (curved) arrows cluster dataset simplification arrow icon

Algorithm: Initial dataset Cluster = Final root cluster Bottom-up (16 cells, 16 clusters) Cluster = contiguous set of cells Final root cluster Bottom-up clustering

Create clusters of level 0 for all dataset cells Algorithm: Create clusters of level 0 for all dataset cells 1 Seek the two neighboring clusters that are most similar Merge them into new cluster 2 Repeat step 2 until one cluster results. 3

Most Similar: error computation Key operations Most Similar: error computation Merging of clusters A B C Should A be merged with B or with C ? A B AB ? How to compute AB’s vector ?

Similarity computation Clusters are compared by: the direction and length of their vectors the origins of their vectors Cluster shapes are not used

Similarity computation Direction and length comparison (s) elliptic similarity isocontours Origin comparison (t) 2 2 ((x-l)2 + 2)(2-2) + 2 (x-l)2 + (x-l) s = (2-2) xo2 yo2 t = + d2 e2 similarity = As + (1-A)t

How are clusters merged ? Cluster areas are added Cluster vectors are averaged A B A+B AB

How do we use clusters for visualization ? Use cluster tree to visualize the desired simplification level: select all clusters for desired level visualize selected clusters e.g. by an icon

Visualization Options plain arrows curved arrows (obtained by streamline tracing through cluster centers) arrows on a spot noise textured background

More 2D examples...

… and 3D examples

Clustering Parameters Two user-controlled parameters: A: favor clustering along similar directions vs similar origins B: favor clustering along or across the vector field’s direction

Clustering Parameters clusters across streamlines clusters along streamlines uniform sampling

Clustering Parameters Effect of the elliptic similarity function shape Uniform field clustering across (B=0) clusteting along (B=1)

Examples original dataset default settings favor direction favor origin

Examples original dataset hedgehog Toroidal field (vector icons)

Examples favor direction favor origin Toroidal field (curved arrows)

Air convection (kitchen.vtk) Examples Air convection (kitchen.vtk)

Examples Air convection (kitchen.vtk) straight and curved arrows transparent clusters grid shrunk clusters

Implementation The simplification toolkit is implemented in VTK and integrated in the interactive dataflow system VISSION

Possible Extensions ? Convenience Conceptual heuristics for controlling the clustering parameters accelerate cluster pair search introduce perceptual criteria for simplification

The End