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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
2
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
T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps
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
4
Overview & Contributions
T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps
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
T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps
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
T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps
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
T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps
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
T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps
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)
T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps
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
T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps
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)
T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps
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Case Study: HydrogenAtom
Isosurface 6
T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps
Init Isosurface 1
Target Isosurface 100
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
T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps
Regular Selection
Recursive Isosurface Selections with Isosurface Information Maps
600 1100
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Case Study: Plume
T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps
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.
15
Implementation
Both performance bottlenecks are related to the level set method
T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps
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
T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps
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
17
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
T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps
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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)
T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps
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
Target Isosurface 19
Isosurface 6, the one selected in [1, 19]
20
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
T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps
*Huang et al., Shape Registration in Implicit Spaces Using Information Theory and Free Form Deformations. PAMI 28(8), 2006
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Appendix – 2: Comparison with Isosurface Similarity Maps
An Emprical Comparison
T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps
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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Appendix – 2: Comparison with Isosurface Similarity Maps
T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps
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