Modeling Dendritic Structures Using Path Planning Ling Xu, David Mould

Importance of Dendrites • trees, lichens, coral, lightning, venation, river systems

Man-made dendrites • mazes • networks

Existing Methods L-systems Diffusion-Limited Aggregation

Ontogenetic Modeling • Ontogenetic modeling: approach appearance of model without regard for underlying process • Seek lightweight means of mimicking appearance of dendritic objects • Path planning: – irregular curves – paths from root never cross

Path planned dendrites

Overview • Implementation • Results – timing – model gallery • Augmentations • Future Work

Basic Idea • Geodesics in a weighted graph • Control: – weights in graph influence path shape – endpoint choice affects dendrite’s appearance – generator shape, likewise

Implementation • Dijkstra’s algorithm used to get costs from root to all other nodes in graph • O(N) to cover graph • O(n) for path from arbitrary endpoint to root • endpoints placed by hand or procedurally

Fractal Dendrites • Real objects often exhibit fractal (multiscale) detail • Explicitly introduce hierarchical detail: • Create low-frequency detail • Add structure at higher frequency • Repeat previous step

real DLA imitated DLA

Timing Comparison • Previously reported methods: minutes to hours, depending on complexity • Random walker DLA: 25k sites, 7.5 min • Our method: – simple 2D: about 1 second – simple 3D: about 3 seconds – fractal 2D: about 7.5 seconds

real DLA imitated DLA

“Rocks” • Multi-source path planning partitions space – can be used to produce irregular 3D objects

Model Creation • Extrusion around path • Isosurface within 3D graph – distance values known – choose isovalue, use isosurface extraction to get mesh (marching cubes)

Limitations • Resolution bound to fixed resolution of graph – in 3D, adding diagonal edges costly (26 connected vs. 6-connected) • Solution?

– path smoothing – multiresolution graph

Future Directions • Procedural endpoint placement • Additional phenomena • Path smoothing • Path extrusion

Acknowledgements • Thanks to Jeremy Long for fruitful discussions regarding path planned models • This work was supported by NSERC RGPIN 299070-04 and by the University of Saskatchewan