MLSEB: Edge Bundling using Moving Least Squares ...MLSEB: Edge Bundling using Moving Least Squares...
Transcript of MLSEB: Edge Bundling using Moving Least Squares ...MLSEB: Edge Bundling using Moving Least Squares...
MLSEB: Edge Bundling using Moving Least Squares Approximation
Jieting Wu, Jianping Zeng, Feiyu Zhu, Hongfeng Yu
Department of Computer Science & Engineering
University of Nebraska-Lincoln, Lincoln, Nebraska
Outline
β’ Motivation
β’ Background
β’ Approach
β’ Results
β’ Conclusion
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Motivation
β’ State-of-the-art graph visualization
β Node-Link diagram β’ Pro
β Simple and intuitive β’ Con
β Easily incur visual clutter
β Edge bundlingβ’ Pros
β Effectively remedy visual clutter β Reveal high-level graph structures
β’ Consβ High complexityβ Non-trivial quality evaluation
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FFTEB [Lhuillier2017]
Background
β’ Edge bundling algorithms
β Visually merge edges based on similarity measurements
β’ Iterative refinement
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Node-link Diagram
Edge Bundling
Rendering
Sampling Edges
Measuring Similarity
Moving Sample Points
(Resampling)
Background
β’ Force-directed edge bundling [Holten2010]
β’ Kernel density estimation (KDE) based methods
β KDEEB: Graph Bundling by Kernel Density Estimation [Hurter2012]
β CUBu: CUDA Universal Bundling [Matthew van der Zwan2016]
β FFTEB: Fast Fourier Transform Edge Bundling [Lhuillier2017]
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Background
β’ Kernel density estimation (KDE) based methods
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Background
β’ Kernel density estimation (KDE) based methods
β Image-based sampling
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Background
β’ Kernel density estimation (KDE) based methods
β Mean-shift clustering
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Background
β’ Kernel density estimation (KDE) based methods
β Mean-shift clustering
β’ Kernel density estimation
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Background
β’ Kernel density estimation (KDE) based methods
β Mean-shift clustering
β’ Kernel density estimation
β’ Gradient-based advection
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Background
β’ Kernel density estimation (KDE) based methods
β Incur excessive convergence artifact
β’ Require resampling to avoid excessive convergence
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Background (Complexity)
β’ Kernel density estimation (KDE) based methods
β Image-based sampling
β Mean-shift clustering
β Iterative refinement (resampling)
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Complexity: O(SNI + IE)
S: sample points
N: image pixel number
I: iteration number
E: edge number
Graph Bundling by Kernel Density Estimation [Hurter2012]
Examples of Existing Edge Bundling Methods
Original graph GBEB [Cui2008] FDEB [Holten2010]
SBEB [Ersoy2012]WR [Lambert2010]
KDEEB [Hurter2012] CUBu [Matthew van der Zwan2016] FFTEB [Lhuillier2017]
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MINGLE [Ganser2011]
Evaluation?
GBEB [Cui2008] FDEB [Holten2010]
SBEB [Ersoy2012]WR [Lambert2010]
KDEEB [Hurter2012] CUBu [Matthew van der Zwan2016] FFTEB [Lhuillier2017]
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MINGLE [Ganser2011]
Evaluation
β’ Quality of edge bundling
β Lhuillier et al. [Lhuillier2017] suggested to use the ratio of clutter reduction to amount of distortion to quantify the quality of a bundled graph
π =πΆ
π
β’ C: clutter reduction
β’ T: amount of distortion
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Evaluation
β’ Quality of edge bundling
β T: The distortion is measured by computing the distance between original edge drawings and the bundled edge drawings
β C: The calculation of clutter reduction has not been fully concluded in the existing work
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Evaluation
β’ Quality of edge bundling
β We propose to employ the reduction of the used pixel number in a graph drawing to measure πΆ
πΆ = βπ = π β πβ²
β We also propose to use the average distortion ΰ΄€π, instead of the total distortion of all the sample points
ΰ΄€π =π
π
T is the total distortion generated
S is the number of sample points
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Evaluation
β’ Quality of edge bundling
β We have a quality metric to quantify the quality of edge bundling
π =βπ
ΰ΄€π
β’ βπ: reduced pixels β
β’ ΰ΄€π: average distortion β
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Evaluation
β’ Quality of edge bundling
β The pros and cons of the existing methods
β’ Pros
β Create visually appealing edge bundles that reduce clutter
β’ Cons
β Resampling is required in iterative refinement
β Does not take distortion into their methods
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Contribution
β’ We present MLSEB, a novel method to generate edge bundles based on moving least squares (MLS) approximation
β Introduce MLS into edge bundling
β’ Simplify the edge bundling pipeline
β’ Generate better quality results compared to other methods
β Based on the aforementioned quality metric
β Ensure scalability and efficiency
β’ A set of graphs that range from ten thousand to a half million edges
β’ A GPU implementation
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Approach
β’ The pipeline of moving least squares edge bundling
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Node-link Diagram
Edge Bundling
Rendering
Sampling Edges
Measuring Similarity
Moving Sample Points
(Resampling)
Approach
β’ The pipeline of moving least squares edge bundling
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Node-link Diagram
Edge Bundling
Rendering
Sampling Edges
Measuring Similarity
Moving Sample Points
Moving Least Squares
(Resampling)
Approach
β’ The pipeline of moving least squares edge bundling
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Node-link Diagram
Edge Bundling
Rendering
Sampling Edges
Measuring Similarity
Moving Sample Points
Moving Least Squares
Approach
β’ Moving least squares application
β Reconstructing continuous functions from a set of unorganized point samples
β’ 2D curve reconstruction
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Curve Reconstruction from Unorganized Points [Lee00]
Approach
β’ Moving least squares application
β Reconstructing continuous functions from a set of unorganized point samples
β’ 3D surface reconstruction
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Moving Least Squares Multiresolution Surface Approximation [Mederos03]
Approach
β’ MLSEBβ Image-based sampling
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Approach
β’ MLSEB
β Assume there is an implicit skeleton that is a suitable place to gather sample points and form bundles
β’ Skeleton can be interpreted as a curve
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β’ MLSEB
β Skeleton can be interpreted as a piece-wise polynomial curve
β’ Calculate ππ by minimizing a weighted least squares error π
β Within a radial neighborhood βπ of π₯π
Approach
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π₯π
Least squares approximation
ππ
βπ
π =
π=1
βπ
( π₯π β ππ )2π |π₯π β π₯π|
β’ MLSEB
β Skeleton can be interpreted as a piece-wise polynomial curve
β’ Calculate ππ by minimizing a weighted least squares error π
β Within a radial neighborhood βπ of π₯π
Approach
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Weighting function:Gaussian function
π =
π=1
βπ
( π₯π β ππ )2π |π₯π β π₯π|
π₯π
ππ
βπ
Approach
β’ MLSEB
β Skeleton can be interpreted as a piece-wise polynomial curve
β’ Calculate ππ by minimizing a weighted least squares error π
β Within a radial neighborhood βπ of π₯πβ Project π₯π into ππ
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π =
π=1
βπ
( π₯π β ππ )2π |π₯π β π₯π|
Approach
β’ MLS vs. KDE
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MLSKDE
Approach
β’ MLS vs. KDE
β KDE-based methods incur excessive convergence
β’ Resampling is required to generate better bundling results
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KDE
Approach
β’ MLS vs. KDE
β MLS method only samples edges in the initial step, and it doesnβtincur excessive convergence in the following iterations
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MLS
Iteration 0 Iteration 2 Iteration 5 Iteration 10
Sample edges only once
Approach
β’ Moving least squares edge bundling
β Project a sample point π₯π into its local regression curve ππβ’ ππ is locally approximated
β Within a radial neighborhood of π₯πβ’ The distortion of π₯π is locally minimized
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DistortionEuclidean distance
π =
π=1
βπ
( π₯π β ππ )2π |π₯π β π₯π|
Approach
β’ Moving least squares edge bundling
β Image-based sampling (sample edges in the initial step)
β Moving least squares approximation and projection
β Iterative refinement
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Complexity: O(SNI + E)
S: sample points
N: image pixel number
I: iteration number
E: edge number
Results
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β’ Dataset 1: a small US migrations graph (9780 edges)
FDEB
FFTEBMLSEB (our method)
SamplesTime (ms)
/ iterationIterations Quality
FDEB 3785K 80 300 8.9
FFTEB 489K 48 262 7.60
MLSEB 207K 38 10 9.20
Results
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β’ Dataset 2: a France airlines graph (17274 edges)
FDEB
MLSEB (our method) FFTEB
SamplesTime (ms)
/ iterationIterations Quality
FDEB 6685K 110 300 3.7
FFTEB 864K 70 244 21.3
MLSEB 990K 94 10 26.0
Results
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β’ Dataset 3: a large US migrations graph (545881 edges)
FFTEB MLSEB (our method)
SamplesTime (ms)
/ iterationIterations Quality
FFTEB 6.4M 123 390 13.28
MLSEB 5.8M 554 20 13.30
Conclusion
β’ Moving Least Squares Edge Bundling (MLSEB)
β A simple and efficient method for constructing edge bundles of large graphs using MLS projection
β’ Only sample edges once, and avoid resampling in the following iterations
β’ Achieve better visualization results based on a quality metric
β’ Ensure scalability and efficiency
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Acknowledgement
β’ This research has been sponsored by the National Science Foundation through grants IIS-1652846, IIS-1423487, and ICER-1541043.
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Thank You!
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