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

1. 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 • Given a set of isosurfaces, how much information from the scalar field is represented? • Which isosurfaces should be added to fill the missing information? T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps

3. The Idea Given two isosurfacesx0 and x1, examine whether other isosurfaces in (x0, x1) CANNOT be inferred from them Isocontourx1 x1 x1 Isocontourx0 x0 x0 If a true isocontour has a very different shape, it should be displayed Given two isocontours as circles An enclosed isocontour is expected to be a circle too; T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps

4. Overview & Contributions Isosurface evaluation via surface morphing Isosurface information map 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 Information-theoretic isosurface selection Refine the visualization by adding the most under-represented isosurface. Contribution: A quantitative approach to evaluate and refine isosurfaces T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps

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 Initial isosurface Target isosurface Compute the distance to the initial surface. At each step, update the scalar field based on the distance to the target surface. To morph from the blue rectangle to the red circle… T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps

6. Distribution-based Surface Evaluation Use the value distribution to evaluate whether an intermediate morph-surface is aligned with any of the true isosurfaces If a surface aligns with an isosurfaceh, 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. A sample scalar field where the isosurfaces are layers of boxes Surfaces in the scalar fields Sampled values on the surface T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps

7. Under-represented Isosurface Detection 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 0.2: Only few histograms have non-zero probability Isovalue 0.5: More histograms have non-zero probability 4 morphed surfaces and the sampled histograms T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps

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

9. Specific Conditional Entropy 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. Initial Isosurface 1 Isosurface 19 has the highest H(Y|X=Xi) Specific conditional entropy of isovaluexi: H(Y | X = xi) = - Σyp(y|xi)log2p(y|xi) Target Isosurface 100 T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps

10. Isovalue Interval Evaluation Mutual Information I(X, Y) = ΣxΣyp(x, y)[log2p(x, y) – log2p(x) – log2p(y) Normalized Conditional Entropy (Nx: #bins) H’(X|Y) = (H(X) – I(X, Y))/log2Nx Init Init Target Target 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. T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps

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

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

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

14. Case Study: Plume Regular Sampled Isosuiface 17.8 2.3 4.5 6.7 8.9 11.2 13.4 15.6 Isosurface-Information-Maps-based Selection 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. 0.1 20.0 16.6 0.5 2.5 3.7 8.8 12.0 14.5 15.8 T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps

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

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 Intel(R) Core 2 Duo E6750 CPU, 8 GB memory, and nVidiaGeForce 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 T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps

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

18. QUESTIONS? 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 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) Target Isosurface 19 Init Isosurface 1 Isosurface 6, the one selected in [1, 19] When sweeping through the isosurface 6, the outer spheres are changed from open to closed, which is not a change of connected components. T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps

20. Appendix – 2: Comparison with Isosurface Similarity Maps • What’s the main difference between our approach and Bruckner and Möller’sIsosurface 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 T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps

21. Appendix – 2: Comparison with Isosurface Similarity Maps An Emprical Comparison isosurface similarity map: Sample Nisosurfaces within them Initial Isosurface The normalized mutual information of the sampled isosurfaces from the isosurface information map Target Isosurface T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps

22. Appendix – 2: Comparison with Isosurface Similarity Maps Use distance transforms to register two shapes with the optimal transform. Translation Rotation + Scaling Translation + Scaling Translation + Rotation Image source: Huang et al., Shape Registration in Implicit Spaces Using Information Theory and Free Form Deformations. PAMI 28(8), 2006 T.-Y. Lee: Evaluating Isosurfaces with Level-set based Information Maps