Edge Groups: An Approach to Understanding the Mesh Quality of Marching Methods

Transcription

1 Edge Groups: An Approach to Understanding the Mesh Quality of Marching Methods Carlos A. Dietrich, Carlos Scheidegger João L. D. Comba, Luciana Porcher Nedel, Cláudio T. Silva

2 Triangle Quality of Marching Cubes good Radii ratio (2r/R) bad

3 Triangle Quality of Marching Cubes Mostly good quality triangles. But some very poor quality triangles good bad

4 Triangle Quality of Marching Cubes In this work we propose a way to understand why this happens.

5 Triangle Quality of Marching Cubes Look at each triangle individually Important Observation: Each triangle generated by MC has vertices only over active edges. Therefore, it does not generate arbitrary triangles

6 Triangle Quality of Marching Cubes Look at each triangle individually Important Observation: Each triangle generated by MC has vertices only over active edges. Therefore, it does not generate arbitrary triangles

7 Triangle Quality of Marching Cubes Look at each triangle individually Important Observation: Each triangle generated by MC has vertices only over active edges. Therefore, it does not generate arbitrary triangles

8 Triangle Quality of Marching Cubes Look at each triangle individually Idea Since three active edges define a triangle, we could evaluate all possible triangles generated by such configuration

9 Triangle Quality of Marching Cubes Look at each triangle individually Idea Since three active edges define a triangle, we could evaluate all possible triangles generated by such configuration

10 Triangle Quality of Marching Cubes Look at each triangle individually Bad Good histogram of triangle quality Idea Since three active edges define a triangle, we could evaluate all possible triangles generated by such configuration

11 Triangle Quality of Marching Cubes Several conclusions can be derived Bad histogram of triangle quality Good Interesting Observation This configuration is responsible for the worst triangles of MC!!!

12 Triangle Quality of Marching Cubes Other Questions to answer: How many cases of edge configurations?

13 Triangle Quality of Marching Cubes Other Questions to answer: How many cases of edge configurations? Is it similar to Marching Cubes Cases Enumeration?

14 Triangle Quality of Marching Cubes Other Questions to answer: How many cases of edge configurations? Is it similar to Marching Cubes Cases Enumeration? YES, but at edge level

15 Outline Motivation Edge Groups in Cubic Cells Improving Triangle Quality Based on Edge Groups Results Conclusions

16 Edge Groups in Cubic Cells Counting Cases in MC [Lorensen and Cline 87] 15 cases based on inspection [Nielson et al 2002] 23 cases by rotation only [Banks et al 2004] formal proof using group theory

17 Edge Groups in Cubic Cells Counting Cases in MC [Lorensen and Cline 87] 15 cases based on inspection [Nielson et al 2002] 23 cases by rotation only [Banks et al 2004] formal proof using group theory Counting Cases in Edge Configurations: Edge configuration (e0, e1, e2): 3 different edges of the cube Edge (Symmetry) Group: collection of edge configurations that can be transformed into another by symmetry transformations (e.g. rotations and improper reflections)

18 Edge Groups in Cubic Cells Counting Cases in MC [Lorensen and Cline 87] 15 cases based on inspection [Nielson et al 2002] 23 cases by rotation only [Banks et al 2004] formal proof using group theory Counting Cases in Edge Configurations: Edge configuration (e0, e1, e2): 3 different edges of the cube Edge (Symmetry) Group: collection of edge configurations that can be transformed into another by symmetry transformations (e.g. rotations and improper reflections) (0,5,9) (0,5,11) (0,9,11)

19 Counting Cases in Edge Groups True to the original Marching Cubes paper, we first did it by inspection too How?

20 Edge Groups in the MC table

21 Edge Groups in the MC table

22 Edge Groups in the MC table

23 Edge Groups in the MC table

24 Edge Groups in the MC table

25 Edge Groups in the MC table

26 Edge Groups in the MC table

27 Edge Groups in the MC table

28 Edge Groups in the MC table

29 Edge Groups in the MC table

30 Edge Groups in the MC table

31 Edge Groups in the MC table

32 Edge Groups in the MC table

33 Edge Groups in the MC table

34 Edge Groups proof (paper)

35 8 Edge Groups in Cubic Cells

36 Triangle quality analysis: Worst triangle of case 4 4

37 Triangle quality analysis: Worst triangle of case 6 6

38 Triangle quality analysis: Worst triangle of case 8 8

39 Triangle quality analysis: Best triangle of case 2 2

40 Triangle Quality Analysis: Edge Groups

41 Edge Groups Occurrence 30 datasets

42 Interesting Observation: Edge Group 2 dominates 30 datasets

43 Outline Motivation Edge Groups in Cubic Cells Improving Triangle Quality Based on Edge Groups Results Conclusions

44 Improving Triangle Quality based on Edge Groups Observation 1 : Edge Group 2 generates the worst triangles Changing the MC table Observation 2: Bad triangles when isosurface is parallel to the edge Edge transformations using Macet [Dietrich et al 2009]

45 Improvement 1: Changing MC Table

46 Improvement 1: Changing MC Table

47 Improvement 2: Edge Transformations using Macet Macet [Dietrich et al 2009] Edge transformations alter positions of grid points in MC Gradient Transformation Tangent Transformation

48 Improvement 2: Edge Transformations using Macet Displacement of Intersection Vertices

49 Improvement 2: Edge Transformations using Macet

50 Improvement 2: Edge Transformations using Macet

51 Summary of Changes to Marching Cubes Code triangulation table compute vertex polarity n New Table? identify cell case y new triangulation table identify active edges Modify grid? n generate intersection vertices y edge transformation Modify vertices? vertex displacement generate triangulation

52 Outline Motivation Edge Groups in Cubic Cells Improving Triangle Quality Based on Edge Groups Results Conclusions

53 Results Original MC Macet Original MC Macet Old Table Old Table New Table New Table Displacements

54 Results Original MC Macet Original MC Macet Old Table Old Table New Table New Table Displacements

55 Results Original MC Macet Original MC Macet Old Table Old Table New Table New Table Displacements

56 Results Original MC Macet Original MC Macet Old Table Old Table New Table New Table Displacements

57 Edge Groups in Tetrahedra Cells Extensions to tetrahedra cells BCC Tetrahedra (4 Edge Groups)

58 Edge Groups in Tetrahedra Cells Kuhn Tetrahedra (9 Edge Groups)

59 Ongoing Work Unified Edge Transformations Macet (Technical Report coming soon) Single Edge Transformation Optional Vertex Displacement More efficient, more quality Group 2 Group 3 λ λ p i - - pi p j + + p j

60 60 Latest Quality and Performance Results Unified Edge Transformations Macet MC Unified Macet MC Macet 3 time quality time quality Bonsai Engine Hydrogen Laçador e Neghip Siliicium

61 Demo Side-by-side comparison: MC x Unified Macet Marching Cubes Macet

62 Conclusions Main contributions: Edge groups proposal and analysis in MC: Histogram analysis of each individual case Case 2 generates the worst triangles Modifications in MC Change of MC table Improvements and Validation of Macet Reproducibility (open source): (Vistrails)

63 Acknowledegments CNPq (Brazil) scholarship and grants /2007-0, / IBM Ph.D. Student Fellowship. Department of Energy SciDAC (VACET and SDM centers) National Science Foundation (grants CNS , CCF , OCE , CNS , IIS , CCF , OISE , CNS ) IBM Faculty Awards (2005, 2006, and 2007) Reviewers for feedback

64 Questions?