Qualitative Curve Descriptions

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

• Qualitative descriptors of curves
• Selectable granularity
• Locality
• Applications
• Comparison
• Matching
• Clustering

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Motivation

• Sets of simulation data
• Want to cluster
• Need robust partial matching

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Other Descriptors

• Local descriptors
• Curvature based
• Cartoons
• Fragment library based
• Embedding based
• Distance matrices
• Delaunay neighborhoods
• Contact maps

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Proposed Solution

• Use spanner like structures
• Combinatorial structure (edges, more?)
• Adjustable descriptiveness
• Proximity based
• Problems
• Degeneracies
• Instabilities

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Teaser
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What is a Geometric Spanner?

• Graph on a set of points
• Edge weights are their length
• Expansion factor

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What is a Geometric Spanner?

• Graph on a set of points
• Edge weights are their length
• Expansion factor

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Which Spanner?

• Sort all edges by length
• For each edge
• Test if graph path is long
• Simple, easily modifiable

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Spanners of Proteins
backbone atom index
backbone atom index
Expansion factor is 2
Expansion factor is 2
spanner edges
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Spanners Another View
Conformation 0 Conformation 1
backbone?
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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

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Current Work

• Addressing problems
• Degeneracies
• Scale/noise effects

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Degeneracies Overview

• Parallel lines
• Helices
• Circles
• Edge placements random
• Edge densities vary
• Factor of 2

Killed edges
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Degeneracies Examples
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Another Case of Degeneracies

• Cocircular points
• Killed edges

vs.
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Solution? Fuzzy Edges

• Detect and handle degeneracies
• Based on killers
• Two types
• Nearby killers
• Far killers

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Fuzzy Edges

• Detect and handle degeneracies
• Based on killers
• Two types
• Nearby killers
• Far killers
• Merge first
• Add second

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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

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Fuzzy Edges as Covers
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Protein Example
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Achieving Scale Independence

• Details of small scale affect large
• Distances can be stretched by k
• Add all short edges
• Smooths structure

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Defining Short

• Chose so spanner edgesunchanged

Curve
Spanner edge
Short edges
Noise free
Smoothed
Effect of Noise
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Protein example
Pruned edges Unpruned edges
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Other Problems

• Matching
• Comparisons

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Matching Revisited

• Similar to Delaunay case
• 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

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Comparison/Distance Functions

• Handing degeneracy
• Match covered sets?
• Length too?