1. Qualitative Curve Descriptions
(Using Spanners)
Daniel Russel
Leonidas Guibas
Stanford University

2. Goals Qualitative descriptors of curves Selectable granularity
Locality
Applications
Comparison
Matching
Clustering

3. Motivation Sets of simulation data Want to cluster
Need robust partial matching

4. Other Descriptors Local descriptors Embedding based Curvature based
Cartoons
Fragment library based
Embedding based
Distance matrices
Delaunay neighborhoods
Contact maps

5. Proposed Solution Use spanner like structures
Combinatorial structure (edges, more?)
Adjustable descriptiveness
Proximity based
Problems
Degeneracies
Instabilities

6. Teaser

7. What is a Geometric Spanner?
Graph on a set of points
Edge weights are their length
Expansion factor

8. What is a Geometric Spanner?
Graph on a set of points
Edge weights are their length
Expansion factor

9. Which Spanner? Sort all edges by length For each edge
Test if graph path is long
Simple, easily modifiable

10. Spanners of Proteins backbone atom index backbone atom index
Expansion factor is 2
Expansion factor is 2
spanner edges

11. Spanners Another View
Conformation 0
Conformation 1
backbone→

12. Viewing Trajectories Compute spanner for each frame
Match edges from successive frames
Prune short-lived edges
Display edge as pixel (t,start)
Colored by “length”

13. Current Work
Addressing problems
Degeneracies
Scale/noise effects

14. Degeneracies Overview
Parallel lines
Helices
Circles
Edge placements random
Edge densities vary
Factor of 2
Killed edges

15. Degeneracies Examples

16. Another Case of Degeneracies
Cocircular points
Killed edges
vs.

17. Solution? Fuzzy Edges Detect and handle degeneracies Two types
Based on killers
Two types
Nearby killers
Far killers

18. Fuzzy Edges Detect and handle degeneracies Two types Merge first
Based on killers
Two types
Nearby killers
Far killers
Merge first
Add second

19. Fuzzy Edges Examples Find short path For each long edge on path
Check if morph distance is small
Merge edges if small
Representation issues
Currently pair of intervals
Ignores direction
Ignores substructure

20. Fuzzy Edges as Covers

21. Protein Example

22. Achieving Scale Independence
Details of small scale affect large
Distances can be stretched by k
Add all “short” edges
Smooths structure

23. Defining Short Chose so spanner edges unchanged Curve Spanner edge
Short edges
Effect of Noise
Smoothed
Noise free

24. Protein example
Pruned edges
Unpruned edges

25. Other Problems
Matching
Comparisons

26. Matching Revisited Similar to Delaunay case DP matching
Sparser feature set
DP matching
will not work
Edge lengths as features
Pattern of start/end
Strong features
Can be interrupted, need pairs (at least)
Works well in absence of insertions

27. Comparison/Distance Functions
Handing degeneracy
Match covered sets?
Length too?