An Information-Theoretic Framework for

Flow Visualization

Lijie Xu, Teng-Yok Lee, & Han-Wei Shen

The Ohio State University

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Introduction: Visualization vs Communication

Data

Visualization Algorithm

OutputInput

Communication Channel

Visualization

The effectiveness of a visualization algorithm depends

on the amount of information that can be transmitted

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Introduction: Information-theoretic Visualization Framework

Can more information be shown?

Visualization Algorithm

Yes

VisualizationData

Visualization should be information-dependente.g. for flow visualization, the flow near the salient flow features shouldn’t be missed

Information

Information in Visualization

No

Information in Data

Stop

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Outline

1. Realization of this visualization framework for static flow

a. Measurement of information in data (vector field)

b. Measurement for information in visualization (streamlines)

c. Information comparison between data & visualization

2. An information-aware streamline placement algorithm• Place streamlines to highlight salient flow features• Automatically stop when no more information can be

shown

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Review: Information and Entropy

Information theory: quantitatively measures the amount of information contained in a data source

Data SourceRandom variable X

x1 x2 x3 x4

p(xi): Probability X = xi

p(xi)

x1 x2 x3 x4

Less information

Only one possible outcome

p(xi)

x1 x2 x3 x4

More information

More equally possible outcomes

Maximal EntropyMinimal Entropy

Entropy of X H(X) = -∑p(xi) log2p(xi)

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Information in Vector Fields

• Concept• Treat the vector field as a data source that generates vector orientation

as outcome• The more diverse the vector orientations, the more information is

contained in the vector field

• Measurement • Estimate the distribution of the vector orientation• Compute the entropy of this distribution as the measurement

Vector field Polar Histogram

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Information in Vector Fields (contd’)

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Entropy Field and Seeding

Measure the entropy around each point’s neighborhood

Vector Field Entropy field: higher value means more information in the corresponding region

Entropy-based seeding: Places streamlines on the region with high entropy

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Information in Streamlines

Given information: estimate the vector along the streamlines by their trajectory

Reconstructed information: synthesize the vectors of the empty regions

What kind of information is represented by the streamlines?

A synthesized vector field

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The Information Comparison between Data/Visualization

Conditional entropy H(X|Y):The information in X not represented by Y

An effective visualization should represent most information in the data, i.e.

H(X|Y) should be small

Vector Field X

H(X) H(Y)

Streamlines Y

H(X|Y)

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H(X)

Conditional Entropy and Joint Entropy

H(X|Y) H(Y) H(X, Y) H(Y) – =

Vector field from the streamlines

H(Y): entropy of the distribution of only vectors Y on all points

Input vector field

H(X, Y): entropy of the joint distribution of both vectors X and Y on all points

H(X, Y): Joint Entropy of

both X and Y

H(Y): Entropy of Y

streamlines

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Conditional Entropy Field and Seeding

Measure the under-represented information in each region

Streamlines Conditional entropy field

Conditional-entropy-based seeding: Place more seeds on regions with higher under-represented information

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Removal of Redundant Streamlines

• Remove the streamlines that cannot reduce the conditional entropy

• Computing conditional entropy per streamline is slow

• Approximation• Discard a streamline if all its points are too

close to existing streamlines• Use a smaller distance threshold for

regions with higher entropy

After the removal of redundant streamlines

Initially placed seeds

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Information-theoretic Streamline Placement Algorithm

Can more information be shown?

Visualization Algorithm

Yes

(Streamlines)(Vector Field)

Information

Information in Visualization

No

Information in Data

Stop

(Entropy Field)Iteration 1

Entropy-based Seeding

Iteration > 1 Conditional-

entropy-based Seeding

Can H(X|Y) be further reduced?

(Synthesized Vector Y)(Original Vector X)

Data Visualization

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Result: 2D Vector Fields

1st iteration: Entropy-based seeding

2nd iteration: Cond.-entropy-based seeding

When conditional entropy converges

Conditional entropy

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Comparison with Distance-based Methods

Evenly-space streamline placement (Jobard & Lefer, 1997)1

Farthest-point streamline placement (Mebarki et al., 2005)2

Information-theoretic streamline placement

1. B. Jobard and W. Lefer. Creating evenly-spaced streamlines of arbitrary density. In Eurographics Workshop on Visualization in Scientific Computing, 1997.

2. A. Mebarki, et al. Farthest point seeding for efficient placement of streamlines. In IEEE Vis ’05, 2005.

Emphasizes regions with more information

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Comparison with Distance-based Methods (contd’)

• The first 54 streamlines placed by different methods. • Distance-based methods need more

streamlines to capture the salient flow features

Entropy-based streamline placement

Evenly-space streamline placement (Jobard & Lefer, 1997)1

Farthest-point streamline placement (Mebarki et al., 2005)2

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Result: 3D Vector Fields

Entropy field1st iteration: Entropy-

based seeding2nd iteration: Cond.-

entropy-based seeding

Conditional entropy

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Entropy-based Occlusion-reduction

• Reduce the occlusion to the salient features in 3D flow field• Use a transfer function to map the entropy of each

streamline fragment to its opacity

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Conclusion

• Contributions• An information-theoretic visualization framework• Information measurement for flow field visualization• An information-aware streamline placement algorithm

• Future work• Time-varying flow fields• Other types of visualization

• Isosurface• Direct volume rendering

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Acknowledgement

• The authors would like to thank Torsten Möller, Carrie Stein, and the anonymous reviewers for their comments

• This work was supported in part by • NSF ITR Grant ACI-0325934• NSF RI Grant CNS-0403342• NSF Career Award CCF-0346883• DOE SciDAC grant DE-FC02-06ER25779

• Questions?