1. ECS 289L A Survey of Mesh-Based Multiresolution Representations Ken Joy Center for Image Processing and Integrated Computing Computer Science Department University of California, Davis joy@cs.ucdavis.edu

2. ECS 289Multiresolution MethodsSlide 2 Multiresolution Methods

3. ECS 289Multiresolution MethodsSlide 3 Bottom-up Methods –Start with the finest resolution. –Iteratively remove elements to generate lower- resolution versions of the data Top-down Methods –Start with a simplified mesh that coarsely approximates the given data set. –Add elements to this mesh, generating better approximations at each step. The Basic Methods

4. ECS 289Multiresolution MethodsSlide 4 Classification of bottom-up methods Select an element of the mesh –A vertex –An edge –A triangle –A tetrahedron Remove the element and all mesh edges that connect to the element. Replace the resulting hole in the mesh with fewer triangular or tetrahedral elements.

5. ECS 289Multiresolution MethodsSlide 5 Bottom-up methods

6. ECS 289Multiresolution MethodsSlide 6 Bottom-up methods Remove elements, creating a hole in the mesh.

7. ECS 289Multiresolution MethodsSlide 7 Bottom-up methods Re-triangulate the hole using fewer elements

8. ECS 289Multiresolution MethodsSlide 8 Bottom-up Methods We would like to fill the hole “optimally” –The error between the two triangulations should be minimized. Hausdorf error is best, but most errors are approximated (L 1,L 2, L ∞ ) –The number of triangles filling the hole should be minimized. “reduced” is OK

9. ECS 289Multiresolution MethodsSlide 9 Bottom-up Methods Most methods operate from a priority queue of elements. –Vertices, edges, or triangles are ordered in a priority queue and processed one-by-one as they come out of the queue. –The “cost function,” which assigns the priority, determines the ordering.

10. ECS 289Multiresolution MethodsSlide 10 Bottom-up Methods Vertex Removal Methods Edge Removal Methods Triangle Removal Methods Tetrahedra Removal Methods

11. ECS 289Multiresolution MethodsSlide 11 Vertex removal methods Select an element to be eliminated.

12. ECS 289Multiresolution MethodsSlide 12 Vertex removal methods Select all triangles sharing this vertex.

13. ECS 289Multiresolution MethodsSlide 13 Vertex removal methods Remove the selected triangles, creating the hole.

14. ECS 289Multiresolution MethodsSlide 14 Vertex removal methods Fill the hole with triangles.

15. ECS 289Multiresolution MethodsSlide 15 Vertex removal methods Most frequently used method of triangulations: –Use a constrained Delaunay triangulation to fill the hole (Schroeder, et al.). –Works best on height fields –Must alter the “Delaunay” criterion slightly to work with data sets that are not height fields

16. ECS 289Multiresolution MethodsSlide 16 Prioritizing Vertices for Removal Using a point-to-plane distance test. –Fit a plane through the points neighboring each vertex. –Measure the distance of each vertex from its respective best-fit plane. –Higher priority to those points that have associated the smaller distances.

17. ECS 289Multiresolution MethodsSlide 17 Edge removal methods Select an edge to be eliminated.

18. ECS 289Multiresolution MethodsSlide 18 Edge removal methods Select all triangles adjacent to this edge.

19. ECS 289Multiresolution MethodsSlide 19 Edge removal methods Remove these triangles, creating a hole.

20. ECS 289Multiresolution MethodsSlide 20 Edge removal methods Fill the hole with triangles.

21. ECS 289Multiresolution MethodsSlide 21 Triangle removal methods Select a triangle for removal

22. ECS 289Multiresolution MethodsSlide 22 Triangle removal methods Select all triangles that are adjacent to the selected one.

23. ECS 289Multiresolution MethodsSlide 23 Triangle removal methods Remove the selected triangles

24. ECS 289Multiresolution MethodsSlide 24 Triangle removal methods Re-triangulate the region

25. ECS 289Multiresolution MethodsSlide 25 Bottom-up methods Mathematical conditions must be established that select the element to be removed. Methods must be established that “optimally” fill the hole, minimizing the “error” between the resulting data sets.

26. ECS 289Multiresolution MethodsSlide 26 Vertex removal methods

27. ECS 289Multiresolution MethodsSlide 27 Vertex removal methods Use a constrained Delaunay triangulation to fill the hole. (Schroeder, et al., 1992).

28. ECS 289Multiresolution MethodsSlide 28 Edge collapse methods

29. ECS 289Multiresolution MethodsSlide 29 Edge collapse methods

30. ECS 289Multiresolution MethodsSlide 30 Edge collapse methods

31. ECS 289Multiresolution MethodsSlide 31 Edge collapse methods

32. ECS 289Multiresolution MethodsSlide 32 Edge collapse methods

33. ECS 289Multiresolution MethodsSlide 33 Edge collapse methods

34. ECS 289Multiresolution MethodsSlide 34 Edge collapse methods

35. ECS 289Multiresolution MethodsSlide 35 Edge collapse methods

36. ECS 289Multiresolution MethodsSlide 36 Edge collapse methods

37. ECS 289Multiresolution MethodsSlide 37 Edge collapse methods

38. ECS 289Multiresolution MethodsSlide 38 Edge collapse methods

39. ECS 289Multiresolution MethodsSlide 39 Triangle collapse methods

40. ECS 289Multiresolution MethodsSlide 40 Triangle collapse methods

41. ECS 289Multiresolution MethodsSlide 41 Triangle collapse methods

42. ECS 289Multiresolution MethodsSlide 42 Triangle collapse methods

43. ECS 289Multiresolution MethodsSlide 43 Triangle collapse methods

44. ECS 289Multiresolution MethodsSlide 44 Prioritizing Elements Priority of an element. Prioritizes for the characteristic you want to select Prioritizes for topology characteristics you wish to select Prioritizes inversely for artifacts that your algorithm produces that you would like to avoid. P(E) = F(E) + T(E) + A(E)

45. ECS 289Multiresolution MethodsSlide 45 Priority-Queue Methods Using the priority (cost) function, prioritize all elements in the mesh. Collapse the element with the lowest priority. Re-adjust the priorities of the elements of the queue affected by the collapse.

46. ECS 289Multiresolution MethodsSlide 46 Progressive Meshes Hoppe, SIGGRAPH 97 Uses edge collapse –Benefit: Uses only the points of the original mesh. Adds no points to the mesh. Prioritizes edge collapse operations by using strategies from “mesh optimizations”

47. ECS 289Multiresolution MethodsSlide 47 Progressive Meshes Priority of an element. Distance of V from the original mesh. Energy if a spring is placed on each edge. Scalar attribute deviation

48. ECS 289Multiresolution MethodsSlide 48 QEM Simplification Garland and Heckbert, SIGGRAPH 97 An edge-collapse method. Introduces the Quadric Error Metric to prioritize collapses –A 4x4 matrix Q is associated with each vertex.

49. ECS 289Multiresolution MethodsSlide 49 QEM Simplification Error of a vertex v is v T Qv Compute matrices Q for all vertices Compute the collapse cost an edge by summing the quadric error metrics of its vertices, i.e., of the edge joining v 1 and v 2 by Q 1 +Q 2. The priority queue is constructed on the “collapse cost”

50. ECS 289Multiresolution MethodsSlide 50 QEM Simplification (Courtesy: M.Garland, UIUC)

51. ECS 289Multiresolution MethodsSlide 51 QEM Simplification Full Resolution60,000 triangles1000 triangles (Courtesy: M.Garland, UIUC)

52. ECS 289Multiresolution MethodsSlide 52 Out-of-Core Methods Lindstrom, SIGGRAPH 2000 Develop clusters of vertices to be removed. Use quadrics, developing one quadric per cluster Adding the quadrics for each triangle in a cluster, use the block decomposition of the quadric matrix to solve Ax=b for the optimal vertex position representing the cluster

53. ECS 289Multiresolution MethodsSlide 53 Out-of-Core Methods Predetermine the cluster grid – usually a uniform grid Input the triangles, computing quadrics for each cluster cell Use the resulting quadrics to calculate representative points for each cell.

54. ECS 289Multiresolution MethodsSlide 54 Out-of-Core Methods (1,087,716 triangles)(208,750 triangles)(62,354 triangles)

55. ECS 289Multiresolution MethodsSlide 55 Out-of-Core Methods Advantages –Single pass through the data –Straightforward calculations Disadvantages –a priori generation of the initial grid –Small details could be missed –Possible topology changes

56. ECS 289Multiresolution MethodsSlide 56 Triangle-Collapse Algorithms Gieng, et al., Visualization ’98 Proceedings Collapses individual triangles to a point. Collapse point is chosen by a best-fit quadric to the surface about a triangle.

57. ECS 289Multiresolution MethodsSlide 57 Prioritizing for Triangle Collapse Priority of an element. Prioritizes high for low absolute curvature Prioritizes high for equilateral triangles, low for skinny triangles. Prioritizes high for collapses that do not generate high-valence vertices Prioritizes high for low area

58. ECS 289Multiresolution MethodsSlide 58 Choosing the collapse point

59. ECS 289Multiresolution MethodsSlide 59 Triangle collapse

60. ECS 289Multiresolution MethodsSlide 60 Collapsing the bunny

61. ECS 289Multiresolution MethodsSlide 61 Collapsing the bunny

62. ECS 289Multiresolution MethodsSlide 62 Collapsing the bunny

63. ECS 289Multiresolution MethodsSlide 63 Collapsing the bunny

64. ECS 289Multiresolution MethodsSlide 64 Collapsing the bunny

65. ECS 289Multiresolution MethodsSlide 65 Collapsing the bunny

66. ECS 289Multiresolution MethodsSlide 66 Collapsing the bunny

67. ECS 289Multiresolution MethodsSlide 67 The error between meshes

68. ECS 289Multiresolution MethodsSlide 68 Crater Lake

69. ECS 289Multiresolution MethodsSlide 69 Crater Lake

70. ECS 289Multiresolution MethodsSlide 70 Tetrahedral Meshes Topological problems of tetrahedron collapse algorithms are overwhelming. Resorting to edge collapse algorithms –Even then, the topological problems are difficult to overcome.

71. ECS 289Multiresolution MethodsSlide 71 Tetrahedral collapse algorithms Consider a triangle mesh on a regular grid. Collapse this edge

72. ECS 289Multiresolution MethodsSlide 72 Tetrahedral collapse algorithms

73. ECS 289Multiresolution MethodsSlide 73 Tetrahedral collapse algorithms

74. ECS 289Multiresolution MethodsSlide 74 Tetrahedral collapse algorithms

75. ECS 289Multiresolution MethodsSlide 75 Tetrahedral collapse algorithms

76. ECS 289Multiresolution MethodsSlide 76 Tetrahedral collapse algorithms

77. ECS 289Multiresolution MethodsSlide 77 Tetrahedral collapse algorithms Two triangles will become degenerate.

78. ECS 289Multiresolution MethodsSlide 78 Tetrahedral collapse algorithms (Courtesy: I. Trotts, UC Davis)

79. ECS 289Multiresolution MethodsSlide 79 Tetrahedral collapse algorithms (Courtesy: I. Trotts, UC Davis)

80. ECS 289Multiresolution MethodsSlide 80 Tetrahedral collapse algorithms (Courtesy: I. Trotts, UC Davis)

81. ECS 289Multiresolution MethodsSlide 81 Tetrahedral collapse algorithms (Courtesy: I. Trotts, UC Davis)

82. ECS 289Multiresolution MethodsSlide 82 Tetrahedral collapse algorithms (Courtesy: I. Trotts, UC Davis)

83. ECS 289Multiresolution MethodsSlide 83 Summary of bottom-up methods All methods start with the original mesh. All methods remove elements from the mesh, replacing them by fewer elements. Most work with a “priority queue”

84. ECS 289Multiresolution MethodsSlide 84 Top-down methods These methods start with a very coarse approximation to the original mesh. New points are inserted into the coarse mesh depending on an error estimate.

85. ECS 289Multiresolution MethodsSlide 85 Delaunay Methods Cignoni, deFloriani, Scopigno papers Start with a coarse mesh. Insert the point of the original data set that has the maximum error. Re-triangulate to insure a Delaunay triangulation. “Delaunay” definition may have to be changed for non-height fields.

86. ECS 289Multiresolution MethodsSlide 86 Voronoi Methods For each point, use a Voronoi diagram to establish the “grid.” Use Sibson’s interpolant on Voronoi Cells for approximation.

87. ECS 289Multiresolution MethodsSlide 87 “Dying Sun” 100 tiles500 tiles2000 tiles

88. ECS 289Multiresolution MethodsSlide 88 Summary of top-down methods All methods refine a coarse approximation, creating successively finer mesh approximations. This area is not as well developed as the bottom-up methods.

89. ECS 289Multiresolution MethodsSlide 89 Challenges Complex Surfaces Isosurface taken from a 2000x2000x2000 data set Isosurface contains 470 million triangles With (x,y,z) coordinates, normals, and scalar values, the isosurface takes up more space than the original data set.

90. ECS 289Multiresolution MethodsSlide 90 Most of the mesh-based algorithms do not scale well! They require the data set to be loaded into memory. They lose small details. They change the topology of the isosurface. They create artifacts Challenges Complex Surfaces

91. ECS 289Multiresolution MethodsSlide 91 Challenges Complex Surfaces The isosurface requires at least a 1000 fold reduction in triangle count to be displayed at “interactive rates.

92. ECS 289Multiresolution MethodsSlide 92 Challenges 100%15% - smoothed