Evaluating I sosurfaces with Level-set b ased I nformation M aps

EuroVis 2013The Eurographics Conference on Visualization

Evaluating Isosurfaces with Level-set based Information Maps

Tzu-Hsuan Wei, Teng-Yok Lee, and Han-Wei Shen

Department of Computer Science & EngineeringThe Ohio State University, USA

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Introduction

• Choosing salient isosurfaces is non-trivial– Based on distribution/topology/geometry, multiple

techniques have been designed

• Two relevant questions still to be addressed1. Given a set of isosurfaces, how much information

from the scalar field is represented?

2. Which isosurfaces should be added to fill the missing information?

T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps

3

The Idea


Isocontour x0

Isocontour x1

An enclosed isocontour is expected to be a circle too;

If a true isocontour has a very different shape, it

should be displayed

Given two isosurfaces x0 and x1, examine whether other isosurfaces in (x0, x1) CANNOT be inferred from them

x0

x1

x0

x1

Given two isocontours as circles

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Overview & Contributions


Isosurface evaluation via surface morphing Isosurface information map

Information-theoretic isosurface selectionRefine the visualization by adding the most

under-represented isosurface.

Given an interval volume, check if the boundary isosurfaces can be used to infer

the intermediate isosurfaces

A distribution-based approach to measure the information in the interval volume that is not

represented by the morphed surfaces

Contribution: A quantitative approach to evaluate and refine isosurfaces

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Level-set-based Surface Morphing

• How to morph isosurfaces with different topologies?– Non-trivial if the morphing is done by interpolating the surface vertices

• Level-set method: A volumetric approach without surface vertex mapping– Compute and update a scalar field where the isosurface of value 0 is the morphed

surface


To morph from the blue rectangle to the red circle…

Compute the distance to the initial surface.

At each step, update the scalar field based on the distance to the target

surface.

Initial isosurface

Target isosurface

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Distribution-based Surface Evaluation


A sample scalar field where the isosurfaces

are layers of boxes

If a surface aligns with an isosurface h, only h will be sampled on the surface

If the surface intersects with multiple isosurfaces, a wide span of isovalues

will have non-zero probability.

Surfaces in the scalar fields

Sampled values on the surface

Use the value distribution to evaluate whether an intermediate morph-surface is aligned with any of the true isosurfaces

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Under-represented Isosurface Detection


Isovalue 0.2: Only few histograms have non-zero probability

Conversely, if an isovalue is found on the samples of multiple morphed surfaces, this isosurface intersects with those morphed surfaces and thus is not well represented.

Isovalue 0.5: More histograms have non-zero probability

4 morphed surfaces and the sampled histograms

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Isosurface Information MapsBy stacking the histograms collected from the morphed

surfaces, a 2D map is formed


Isosurface Information Map P(X, Y) X: The isovalue; Y: The morphed surface

The 2D map is normalized as a joint probability distribution function pdf to form the isosurface information map

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Specific Conditional Entropy

Initial Isosurface 1

Target Isosurface 100

Isosurface 19 has the highest H(Y|

X=Xi)


Specific conditional entropy of isovalue xi: H(Y | X = xi) = - Σy p(y|xi)log2 p(y|xi)

If an isosurface intersects multiple isosurfaces, its conditional probability function will have a wide value range with non-zero probability and the entropy will be high.

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Isovalue Interval Evaluation

Mutual InformationI(X, Y) = ΣxΣy p(x, y)[log2 p(x, y) – log2 p(x) – log2 p(y)

Normalized Conditional Entropy (Nx: #bins)

H’(X|Y) = (H(X) – I(X, Y))/log2 Nx


High I(X,Y) & Low H’(X|Y): The morphed surfaces are aligned

with the true isosurfaces

Low I(X,Y) & High H’(X|Y): The morphed surfaces and the true isosurfaces do

not align and more isosurfaces are needed.

Init Target Init Target

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Isosurface Selection Algorithm

For each pair of consecutive isovalues (xi, xi+1) in a given set of isosurfaces:

1. Compute the isosurface information map for the value interval (xi, xi+1),

2. Stop if the derived H’(X|Y) is smaller than a threshold or the isosurfaces of xi and xi+1 are too close

3. Select the next isovalue x* in (xi, xi+1) with the maximal specific conditional entropy

4. Recursively evaluate and refine (xi, x*) and (x*, xi+1)


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Case Study: HydrogenAtom

Isosurface 6


Init Isosurface 1


Isosurface 34the left sphere and right sphere start to close when sweeping through isovalue

The ring starts to disappear.

Isosurface 19

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Case Study: Tooth


Regular Selection

Recursive Isosurface Selections with Isosurface Information Maps

600 1100

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Case Study: Plume


Regular Sampled Isosuiface

Isosurface-Information-Maps-based Selection

0.5 2.5 3.7 8.8 12.0 14.5 15.8

2.3 4.5 6.7 8.9 11.2 13.4 15.6 17.8

0.1 20.0

16.6

Isosurface 0.5: The inner turbulent flow and the outer smooth flow are mixed.

As the Isosurface is changed from smooth (0.1) to turbulent (2.3), more isosurfaces are needed between

them to sample the change.

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Implementation

Both performance bottlenecks are related to the level set method


The distance computation from each voxel to all vertices of the

initial and target surfaces

At each iteration• Update the entire scalar field• Histogram computation on the

morphed surface

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Performance Optimization

• Solutions– Narrow-band-based level set method [Adalsteinsson and Sethian, 1995]– GPU-based distance computation with the cached constant memory

• Performance – With GPUs, distance computation can be accelerated by 33 – 50 time


Data Set Value Interval

# level set step

Timing (seconds) per Iteration Distance Computation (seconds)

Level Set Func. Update

Marching Cube Dist (CPU) Dist (GPU)

HydrogenAtom 1-100 170 8.9 4.2 401.6 8.0

Plume 0.1-20 241 11.0 2.7 248.6 7.5

Tooth 600-1100 62 0.4 0.9 128.3 2.8

Intel(R) Core 2 Duo E6750 CPU, 8 GB memory, and nVidia GeForce GTX 460 GPU with 1GB of texture memory

Adalsteinsson and Sethian, A Fast Level Set Method for Propagating Interfaces. Journal of Computational Physics, 118(2):269-227, May 1995

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Conclusion

• Summary– Quantitatively evaluate how well the scalar field is

represented by the given isosurfaces via• Surface morphing via level set methods• Information theory

– Present an information-theoretic isosurface selection algorithm as the application

• Future Works: Integrate with other isosurface selection algorithms


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Acknowledgements

• Thank the anonymous reviewers for their comments.

• Supported in part by NSF grant IIS-1017635, US Department of Energy DOESC0005036, Battelle Contract No. 137365, and Department of Energy SciDAC grant DE-FC02-06ER25779, program manager Lucy Nowell.

• Data sources– Plume was released by NCAR (National Center for Atmospheric Research)– HydrogenAtom was released by German Research Council (DFG) – Tooth was released by GE Aircraft Engines, Evendale, Ohio, USA HydrogenAtom and Tooth were downloaded from Carlos Scheidegger’s

website QUESTIONS?T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps

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Appendix – 1: Comparison with Contour Trees

• What’s the main difference between our approach and contour trees?

• Contour tree mainly considers the topology change, while our method considers other differences (e.g geometry)


When sweeping through the isosurface 6, the outer spheres are changed from open to closed, which is not a change of connected components.

Init Isosurface 1


Isosurface 6, the one selected in [1, 19]

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Appendix – 2: Comparison with Isosurface Similarity Maps

• What’s the main difference between our approach and Bruckner and Möller’s Isosurface Similarity Maps (EuroVis ’10)?

• Similarity– Evaluate/compare isosurfaces shape with information theories

• Differences– Compare isosurfaces vs. Evaluate Interval volumes– Degree of sensitivity to scaling The metric is originally for shape registration* and designed to be

sensitive to rotation/translation/scaling


*Huang et al., Shape Registration in Implicit Spaces Using Information Theory and Free Form Deformations. PAMI 28(8), 2006

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An Emprical Comparison


Target Isosurface

Initial Isosurface

isosurface similarity map: Sample N isosurfaces within them

The normalized mutual information of the sampled isosurfaces from the

isosurface information map

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Image source: Huang et al., Shape Registration in Implicit Spaces Using Information Theory and Free Form Deformations. PAMI 28(8), 2006

Use distance transforms to register two shapes with the optimal transform.

Translation Rotation + Scaling

Translation+ Scaling

Translation+ Rotation

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Evaluating I sosurfaces with Level-set b ased I nformation M aps