The document introduces a framework for verifying isosurface extraction algorithms using the Method of Manufactured Solutions (MMS). MMS establishes theoretical orders of accuracy for metrics like algebraic distance, normal, area, and curvature. Implementations are tested against manufactured solutions and orders of accuracy are observed, exposing bugs where observed accuracy did not match theory. The framework allows rigorous verification of isosurface extraction codes and building confidence in their results.

1. Verifiable Visualization for Isosurface Extraction Tiago EtieneCarlos Scheidegger Luis Gustavo Nonato Mike Kirby Cláudio Silva

2. Introduction “The need for verifiable visualization”

3. Introduction Visualization is critical part of the scientific pipeline How can the visualization community build confidence on Algorithms & Implementations being used?

4. How to build confidence Consider the problem of Isosurface extraction. Common correctness tests: Expert analysis Visual comparisons Benchmark suites Comparison with “trustworthy” codes Simple vs. Real world data ….

5. Related work Visualization: “Evaluation of Visualization Software” [Globus and Uselton, 1995] “The Need for Verifiable Visualization” [Kirby and Silva, 2008] Computational Simulation Community: Validation & Verification [I. Babuska and J. Oden, 2004]

6. Traditional Scientific Simulation Pipeline Validation – Does the mathematical model represent the physical phenomena correctly? Verification – Does the computational model and its implementation represent the mathematical model accurately?

7. Scientific Simulation Pipeline by Kirby and Silva Physical Process Validation – Does the mathematical model represent the physical phenomena correctly? Verification – Does the computational model and its implementation represent the mathematical model accurately?

8. Goal Introduce to the visualization community a practical framework for code and algorithm verification in the context of isosurface extraction

9. Verification & visualization Building confidence through verification

10. Verification Verification in order of increase rigor [C. J. Roy, 2005]: Expert judgment Error quantification Consistency/convergence Order of accuracy

11. Formal Order of Accuracy Truncation error We want to write errors as: Example: Linear Interpolation f(a) a a+h f(a+h)

12. Observed Order of Accuracy In practice:

13. Method of Manufactured Solution (MMS) Establish assumption on input Determine theoretical behavior for different inputs Develop manufactured solutions that: meet the assumptions: testing against theoretical results violate assumptions: testing the tightness of the theory Systematically test the method against manufactured solutions Compare formal and observed order of accuracy

14. Method of Manufactured Solution – Pipeline for Isosurface Extraction h i i (h , E ) E = ||u( ) – u( )|| h L i h i

15. THEORETICAL RESULTS Formalizing algorithm behavior

16. Formal Order of Accuracy Given a level set f = λ and assuming linear interpolation we know that: Algebraic distance Normal (cross product of edges) Gaussian curvature (angle deficit method) Surface area:

17. Practical results Observed implementation behavior

18. Implementations under verification VTK Marching Cubes [W. Lorensen and H. Cline, 1987] Macet[C. A. Dietrich et. al., 2008] Dual Contouring[T. Ju et. al., 2002] SnapMC[S. Raman and R. Wenger, 2008] Afront[J. Schreiner et. al., 2006] DelIso[T. Dey and J. Levine, 2007] Manufactured Solution:

19. Observed order of accuracy Algebraic distance h

20. Observed order of accuracy Algebraic distance h

21. Observed order of accuracy Algebraic distance h

22. Observed order of accuracy Algebraic distance h

23. Observed order of accuracy Algebraic distance h

24. Observed order of accuracy Algebraic distance h

25. Observed order of accuracy Algebraic distance - Bug h

26. Observed order of accuracy Algebraic distance - Fixed h

27. Observed order of accuracy Normal - Bug h

28. Observed order of accuracy Normal - Fixed h

29. Qualitatively

30. Qualitatively

31. Conclusion & future WORK What we’ve learned

32. Conclusion & Future Work Implementation of MMS: Easy to code test: implementation is a black box Future work: Isosurface extraction: Consider topology Build a more complete set of manufactured solutions MMS in Visualization Streamlines computation Volume rendering Mesh simplification

33. Conclusion & Future Work Use of MMS: Verifiable Visualization Development of a culture of verification Algorithm analysis Global effort to better understand algorithm properties/limitations Manufactured solution Development of a database of strong solutions

34. Acknowledgements We thank LauroLins and Tom Peters for help with the paper JoãoComba for help with the Macet code TamalDey and Joshua Levine for the customized version of DelIso used in this work This work was supported in part by grants from NSF, DOE, IBM Faculty Awards and PhD Fellowship, the US ARO, ExxonMobil and Fapesp-Brazil.

35. Thank you. Questions?

36. References I. Babuska and J. Oden. Verification and validation in computational engineering and science: basic concepts. Computer Methods in Applied Mechanics and Engineering, 193(36-38):4057–4066, 2004. J. Schreiner, C. Scheidegger, and C. Silva. High-quality extraction of isosurfaces from regular and irregular grids. IEEE TVCG, 12(5):1205– 1212, 2006. T. K. Dey and J. A. Levine. Delaunay meshing of isosurfaces. In SMI ’07: Proceedings of the IEEE International Conference on Shape Modeling and Applications 2007, pages 241–250. IEEE Computer Society, 2007. T. Ju, F. Losasso, S. Schaefer, and J. Warren. Dual contouring of hermite data. In SIGGRAPH’02, pages 339–346. ACM, 2002.

37. References C. A. Dietrich, C. Scheidegger, J. Schreiner, J. L. D. Comba, L. P. Nedel, and C. Silva. Edge transformations for improving mesh quality of marching cubes. IEEE TVCG, 15(1):150–159, 2008. W. Lorensen and H. Cline. Marching cubes: A high resolution 3d surface construction algorithm. SIGGRAPH Comp. Graph., 21:163–169, 1987. C. J. Roy. Review of code and solution verification procedures for computational simulation. J. Comput. Phys., 205(1):131–156, 2005. S. Raman and R. Wenger. Quality isosurface mesh generation using an extended marching cubes lookup table. Computer Graphics Forum, 27(3):791–798, 2008.

38. References A. Globus and S. Uselton. Evaluation of visualization software. SIGGRAPH Comp. Graph., 29(2):41–44, 1995 R. Kirby and C. Silva. The need for verifiable visualization. IEEE Computer Graphics and Applications, 28(5):78–83, 2008

39. Observed order of accuracy Algebraic distance (bug) (fixed)

40. Observed order of accuracyNormal (bug) (fixed)

41. Observed order of accuracyArea (bug) (fixed)

42. Observed order of accuracyCurvature (bug) (fixed)

43. Observed order of accuracyTranscendental function