Selecting the Aspect Ratio of a Scatter Plot Based on Its Delaunay - - PowerPoint PPT Presentation

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Selecting the Aspect Ratio of a Scatter Plot Based on Its Delaunay Triangulation Martin Fink Lehrstuhl f ur Informatik I Universit at W urzburg Joint work with Jan-Henrik Haunert, Joachim Spoerhase & Alexander Wolff 1 /15

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Selecting the Aspect Ratio

  • f a Scatter Plot

Based on Its Delaunay Triangulation

Martin Fink Lehrstuhl f¨ ur Informatik I Universit¨ at W¨ urzburg

Joint work with Jan-Henrik Haunert, Joachim Spoerhase & Alexander Wolff

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Scatter Plots . . .

. . . reveal trends . . .

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Scatter Plots . . .

. . . reveal trends . . . . . . or clusters.

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Scatter Plots . . .

. . . reveal trends . . . . . . or clusters. . . . are most-frequently used visualizations in scientific publications. [Tufte, 2001]

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Scatter Plots . . .

. . . reveal trends . . . . . . or clusters. . . . are most-frequently used visualizations in scientific publications. [Tufte, 2001] . . . heavily rely on the chosen aspect ratio.

[http://imgs.xkcd.com/comics/aspect ratio.png]

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Scatter Plots . . .

. . . reveal trends . . . . . . or clusters. . . . are most-frequently used visualizations in scientific publications. [Tufte, 2001] . . . heavily rely on the chosen aspect ratio.

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Scatter Plots . . .

. . . reveal trends . . . . . . or clusters. . . . are most-frequently used visualizations in scientific publications. [Tufte, 2001] . . . heavily rely on the chosen aspect ratio.

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Scatter Plots . . .

. . . reveal trends . . . . . . or clusters. . . . are most-frequently used visualizations in scientific publications. [Tufte, 2001] . . . heavily rely on the chosen aspect ratio.

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Scatter Plots . . .

. . . reveal trends . . . . . . or clusters. . . . are most-frequently used visualizations in scientific publications. [Tufte, 2001] . . . heavily rely on the chosen aspect ratio.

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Scatter Plots . . .

. . . reveal trends . . . . . . or clusters. . . . are most-frequently used visualizations in scientific publications. [Tufte, 2001] . . . heavily rely on the chosen aspect ratio.

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Scatter Plots . . .

. . . reveal trends . . . . . . or clusters. . . . are most-frequently used visualizations in scientific publications. [Tufte, 2001] . . . heavily rely on the chosen aspect ratio.

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Scatter Plots . . .

. . . reveal trends . . . . . . or clusters. . . . are most-frequently used visualizations in scientific publications. [Tufte, 2001] . . . heavily rely on the chosen aspect ratio. task: automatically select a good aspect ratio

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Previous Work

aspect-ratio selection for line charts e.g. banking to 45◦ [Heer + Agrawala, 2006]

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Previous Work

aspect-ratio selection for line charts e.g. banking to 45◦ [Heer + Agrawala, 2006] [Cleveland et al., 1988] suggest to use virtual line segments

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Previous Work

aspect-ratio selection for line charts e.g. banking to 45◦ [Heer + Agrawala, 2006] [Cleveland et al., 1988] suggest to use virtual line segments [Talbot et al., 2011]: use contour lines from kernel density estimator

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Previous Work

aspect-ratio selection for line charts e.g. banking to 45◦ [Heer + Agrawala, 2006] [Cleveland et al., 1988] suggest to use virtual line segments [Talbot et al., 2011]: use contour lines from kernel density estimator results depend on initial aspect ratio

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Our Approach

measure quality of different aspect ratios independently

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Our Approach

measure quality of different aspect ratios independently use the Delaunay triangulation

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Our Approach

measure quality of different aspect ratios independently use the Delaunay triangulation

  • ptimization criteria:

– maximize smallest angle – minimize total edge length – optimize compactness of triangles – etc.

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Our Approach

measure quality of different aspect ratios independently use the Delaunay triangulation

  • ptimization criteria:

– maximize smallest angle – minimize total edge length – optimize compactness of triangles – etc.

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Definitions

point Set P = {p1, . . . , pn} point pi = (xi, yi)

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Definitions

point Set P = {p1, . . . , pn} point pi = (xi, yi) scale factor s defines aspect-ratio

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Definitions

point Set P = {p1, . . . , pn} point pi = (xi, yi) scale factor s defines aspect-ratio pi(s) = (1/√s · xi, s · yi)

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Definitions

point Set P = {p1, . . . , pn} point pi = (xi, yi) scale factor s defines aspect-ratio pi(s) = (1/√s · xi, √s · yi) preserve the area

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Definitions

point Set P = {p1, . . . , pn} point pi = (xi, yi) scale factor s defines aspect-ratio pi(s) = (1/√s · xi, √s · yi) preserve the area

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First Idea

aspect ratio s

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First Idea

aspect ratio s

discretize into k aspect ratios

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First Idea

aspect ratio s

discretize into k aspect ratios independently – compute Delaunay triangulation – measure quality

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First Idea

aspect ratio s

discretize into k aspect ratios independently – compute Delaunay triangulation – measure quality select best checked aspect ratio

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First Idea

aspect ratio s

discretize into k aspect ratios independently – compute Delaunay triangulation – measure quality select best checked aspect ratio Θ(n log n) Θ(n)

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First Idea

aspect ratio s

discretize into k aspect ratios independently – compute Delaunay triangulation – measure quality select best checked aspect ratio runtime: Θ(kn log n) Θ(n log n) Θ(n)

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First Idea

aspect ratio s

discretize into k aspect ratios independently – compute Delaunay triangulation – measure quality select best checked aspect ratio runtime: Θ(kn log n) Θ(n log n) Θ(n) approximation? which intermediate ratios?

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Overview

  • 1. Maintaining the Delaunay Triangulation
  • 2. Maximizing the Smallest Angle
  • 3. Minimizing the Total Edge Length
  • 4. Other Optimization Criteria
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  • 1. Maintaining the Delaunay Triangulation

aspect ratio s

start at some s

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  • 1. Maintaining the Delaunay Triangulation

aspect ratio s

start at some s compute Delaunay triangulation

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  • 1. Maintaining the Delaunay Triangulation

aspect ratio s

start at some s compute Delaunay triangulation continuously change s

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  • 1. Maintaining the Delaunay Triangulation

aspect ratio s

start at some s compute Delaunay triangulation continuously change s perform flips if necessary

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  • 1. Maintaining the Delaunay Triangulation

aspect ratio s

start at some s compute Delaunay triangulation continuously change s perform flips if necessary criterion: empty circumcircle of 4 points easy to check

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  • 1. Maintaining the Delaunay Triangulation

aspect ratio s

start at some s compute Delaunay triangulation continuously change s perform flips if necessary

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  • 1. Maintaining the Delaunay Triangulation

aspect ratio s

start at some s compute Delaunay triangulation continuously change s perform flips if necessary go through all flips

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  • 1. Maintaining the Delaunay Triangulation

aspect ratio s

start at some s compute Delaunay triangulation continuously change s perform flips if necessary go through all flips

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  • 1. Maintaining the Delaunay Triangulation f.

aspect ratio s

sweep over possible aspect ratios handle event queue of edge flips

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  • 1. Maintaining the Delaunay Triangulation f.

aspect ratio s

sweep over possible aspect ratios handle event queue of edge flips update takes O(log n) time [Roos, 1993]

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  • 1. Maintaining the Delaunay Triangulation f.

aspect ratio s

sweep over possible aspect ratios handle event queue of edge flips update takes O(log n) time [Roos, 1993] O(n2+ǫ) flips [Rubin, 2012]

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  • 1. Maintaining the Delaunay Triangulation f.

aspect ratio s

sweep over possible aspect ratios handle event queue of edge flips update takes O(log n) time [Roos, 1993] O(n2+ǫ) flips [Rubin, 2012] here: at most 2 flips per possible edge

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  • 1. Maintaining the Delaunay Triangulation f.

aspect ratio s

sweep over possible aspect ratios handle event queue of edge flips update takes O(log n) time [Roos, 1993] O(n2+ǫ) flips [Rubin, 2012] here: at most 2 flips per possible edge

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  • 1. Maintaining the Delaunay Triangulation f.

aspect ratio s

sweep over possible aspect ratios handle event queue of edge flips update takes O(log n) time [Roos, 1993] O(n2+ǫ) flips [Rubin, 2012] here: at most 2 flips per possible edge

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  • 1. Maintaining the Delaunay Triangulation f.

aspect ratio s

sweep over possible aspect ratios handle event queue of edge flips update takes O(log n) time [Roos, 1993] O(n2+ǫ) flips [Rubin, 2012] here: at most 2 flips per possible edge

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  • 1. Maintaining the Delaunay Triangulation f.

aspect ratio s

sweep over possible aspect ratios handle event queue of edge flips update takes O(log n) time [Roos, 1993] O(n2+ǫ) flips [Rubin, 2012] here: at most 2 flips per possible edge total runtime: O(n2 log n) for traversing all topologically different Delaunay triangulations

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  • 2. Maximizing the Smallest Angle

aspect ratio s

  • ptimize between event points
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  • 2. Maximizing the Smallest Angle

aspect ratio s

  • ptimize between event points

angle α describes function α(s) ∠ s

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  • 2. Maximizing the Smallest Angle

aspect ratio s

  • ptimize between event points

angle α describes function α(s) ∠ s put functions together

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  • 2. Maximizing the Smallest Angle

aspect ratio s

  • ptimize between event points

angle α describes function α(s) ∠ s put functions together traverse lower envelope

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  • 2. Maximizing the Smallest Angle

aspect ratio s

  • ptimize between event points

angle α describes function α(s) ∠ s put functions together traverse lower envelope Davenport-Schinzel sequences & [Agarwall + Sharir, 1995]: yields globally optimal aspect ratio in O(n2 log n) time

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  • 3. Minimizing the Total Edge Length

sum of many functions ⇒ previous approach does not work find (1 + ǫ)-approximation

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  • 3. Minimizing the Total Edge Length

sum of many functions ⇒ previous approach does not work find (1 + ǫ)-approximation between flips consider (1 + ǫ)-intermediate steps s (1 + ǫ)s

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  • 3. Minimizing the Total Edge Length

sum of many functions ⇒ previous approach does not work find (1 + ǫ)-approximation between flips consider (1 + ǫ)-intermediate steps length le of edge e within a small intervall: le(s(1 + ǫ)) ≤ (1 + ǫ)le(s) s (1 + ǫ)s

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  • 3. Minimizing the Total Edge Length

sum of many functions ⇒ previous approach does not work find (1 + ǫ)-approximation between flips consider (1 + ǫ)-intermediate steps length le of edge e within a small intervall: le(s(1 + ǫ)) ≤ (1 + ǫ)le(s) s (1 + ǫ)s carries over to sum find (1 + ǫ)-approximation in O(n3+n ·

1 log(1+ǫ)) time

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  • 3. Minimizing the Total Edge Length

sum of many functions ⇒ previous approach does not work find (1 + ǫ)-approximation between flips consider (1 + ǫ)-intermediate steps length le of edge e within a small intervall: le(s(1 + ǫ)) ≤ (1 + ǫ)le(s) s (1 + ǫ)s carries over to sum find (1 + ǫ)-approximation in O(n3+n ·

1 log(1+ǫ)) time

also works for other optimization criteria

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  • 4. Other Optimization Criteria

maximize total compactness of triangles √area perimeter

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  • 4. Other Optimization Criteria

maximize total compactness of triangles minimize total uncompactness of triangles √area perimeter √area perimeter

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  • 4. Other Optimization Criteria

maximize total compactness of triangles minimize total uncompactness of triangles √area perimeter √area perimeter more: – maximize mean inradius – minimize sum of squared angles

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User Study

What do users want? let participants choose

Please participate: www1.informatik.uni-wuerzburg.de/scatterplots

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User Study

What do users want? let participants choose

Please participate: www1.informatik.uni-wuerzburg.de/scatterplots

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User Study

What do users want? let participants choose

Please participate: www1.informatik.uni-wuerzburg.de/scatterplots

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User Study

What do users want? let participants choose

Please participate: www1.informatik.uni-wuerzburg.de/scatterplots

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User Study

What do users want? let participants choose 18 tested instances, e.g. . . .

Please participate: www1.informatik.uni-wuerzburg.de/scatterplots

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Test Results

maximize minimum angle

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Test Results

maximize minimum angle maximize mean inradius

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Test Results

maximize minimum angle maximize mean inradius maximize total compactness of triangles

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Test Results

maximize minimum angle maximize mean inradius maximize total compactness of triangles minimize total uncompactness of triangles

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Test Results

maximize minimum angle maximize mean inradius maximize total compactness of triangles minimize total uncompactness of triangles minimize total edge length

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Test Results

maximize minimum angle maximize mean inradius maximize total compactness of triangles minimize total uncompactness of triangles minimize total edge length

✗ ✗ ✗

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Test Results

maximize minimum angle maximize mean inradius maximize total compactness of triangles minimize total uncompactness of triangles minimize total edge length

✗ ✗ ✗

  • preliminary results
  • f the user study

support this

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Conclusion

Delaunay triangulation helps to optimize scatter plots Please participate in our user study! www1.informatik.uni-wuerzburg.de/scatterplots

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Conclusion

Delaunay triangulation helps to optimize scatter plots maintaining the Delaunay triangulation is fast Please participate in our user study! www1.informatik.uni-wuerzburg.de/scatterplots

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Conclusion

Delaunay triangulation helps to optimize scatter plots maintaining the Delaunay triangulation is fast more than one good quality measure Please participate in our user study! www1.informatik.uni-wuerzburg.de/scatterplots