1. EuroVis 2013 The 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 & Engineering The 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 addressed 1.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? 2 T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps

3. The Idea 3 T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps Isocontour x 0 Isocontour x 1 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 x 0 and x 1, examine whether other isosurfaces in (x 0, x 1 ) CANNOT be inferred from them x0x0 x1x1 x0x0 x1x1 Given two isocontours as circles

4. Overview & Contributions 4 T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps Isosurface evaluation via surface morphing Isosurface information map Information-theoretic isosurface selection Refine 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

5. 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 5 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

6. Distribution-based Surface Evaluation 6 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

7. Under-represented Isosurface Detection 7 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

8. Isosurface Information Maps By stacking the histograms collected from the morphed surfaces, a 2D map is formed 8 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

9. Specific Conditional Entropy 9 Initial Isosurface 1 Target Isosurface 100 Isosurface 19 has the highest H(Y|X=X i ) T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps Specific conditional entropy of isovalue x i : H(Y | X = x i ) = - Σ y p(y|x i )log 2 p(y|x i ) 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.

10. Isovalue Interval Evaluation Mutual Information I(X, Y) = Σ x Σ y p(x, y)[log 2 p(x, y) – log 2 p(x) – log 2 p(y) Normalized Conditional Entropy (N x : #bins) H’(X|Y) = (H(X) – I(X, Y))/log 2 N x 10 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

11. Isosurface Selection Algorithm For each pair of consecutive isovalues (x i, x i+1 ) in a given set of isosurfaces: 1.Compute the isosurface information map for the value interval (x i, x i+1 ), 2.Stop if the derived H’(X|Y) is smaller than a threshold or the isosurfaces of x i and x i+1 are too close 3.Select the next isovalue x* in (x i, x i+1 ) with the maximal specific conditional entropy 4.Recursively evaluate and refine (x i, x*) and (x*, x i+1 ) 11 T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps

12. Case Study: HydrogenAtom 12 Isosurface 6 T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps Init Isosurface 1 Target Isosurface 100 Isosurface 34 the left sphere and right sphere start to close when sweeping through isovalue The ring starts to disappear. Isosurface 19

13. Case Study: Tooth 13 T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps Regular Selection Recursive Isosurface Selections with Isosurface Information Maps 600 1100

14. Case Study: Plume 14 T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps Regular Sampled Isosuiface Isosurface-Information-Maps-based Selection 0.52.53.78.812.014.515.8 2.34.56.78.911.213.415.6 17.8 0.120.0 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. 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 15 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

16. 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 16 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) HydrogenAtom1-1001708.94.2401.68.0 Plume0.1-2024111.02.7248.67.5 Tooth600-1100620.40.9128.32.8 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 17 T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps

18. 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? 18 T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps

19. 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) 19 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 20 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

21. Appendix – 2: Comparison with Isosurface Similarity Maps An Emprical Comparison 21 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

22. Appendix – 2: Comparison with Isosurface Similarity Maps 22 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. TranslationRotation + Scaling Translation + Scaling Translation + Rotation