1. Implicit Representations of Surfaces and Polygonalization Algorithms Dr. Scott Schaefer

2. 2/47 Polygon Models  Advantages  Explicit connectivity information  Easy to render  (Relatively) small storage  Disadvantages  Topology changes difficult  Inside/Outside test hard

3. 3/47 Implicit Representations of Shape  Shape described by solution to f(x)=c

4. 4/47 Implicit Representations of Shape  Shape described by solution to f(x)=c

5. 5/47 Implicit Representations of Shape  Shape described by solution to f(x)=c - - - - - - - - - - - - - - - -

6. 6/47 Implicit Representations of Shape  Shape described by solution to f(x)=c - - - - - - - - + + + + + ++ + - - - - - - - -

7. 7/47 Advantages  No topology to maintain  Always defines a closed surface!  Inside/Outside test  CSG operations

8. 8/47 Advantages  No topology to maintain  Always defines a closed surface!  Inside/Outside test  CSG operations - - - - - - - - + + + + + ++ + - - - - - - - -

9. 9/47 Advantages  No topology to maintain  Always defines a closed surface!  Inside/Outside test  CSG operations

10. 10/47 Advantages  No topology to maintain  Always defines a closed surface!  Inside/Outside test  CSG operations  Union

11. 11/47 Advantages  No topology to maintain  Always defines a closed surface!  Inside/Outside test  CSG operations  Union

12. 12/47 Advantages  No topology to maintain  Always defines a closed surface!  Inside/Outside test  CSG operations  Union - - - - - + + + + + + - - - - -

13. 13/47 Advantages  No topology to maintain  Always defines a closed surface!  Inside/Outside test  CSG operations  Union - - - - - + + + + + + - - - - -

14. 14/47 Advantages  No topology to maintain  Always defines a closed surface!  Inside/Outside test  CSG operations  Union - - - - - + + + + + + - - - - - + + + + + + - - - - - - - - - - + + + + + + - - - - -

15. 15/47 Advantages  No topology to maintain  Always defines a closed surface!  Inside/Outside test  CSG operations  Union - + + + + + + --- - - - -

16. 16/47 Advantages  No topology to maintain  Always defines a closed surface!  Inside/Outside test  CSG operations  Union  Intersection - - - - - + + + + + + - - - - - + + + + + + - - - - - - - - - - + + + + + + - - - - -

17. 17/47 Advantages  No topology to maintain  Always defines a closed surface!  Inside/Outside test  CSG operations  Union  Intersection - - + + + + - -

18. 18/47 Advantages  No topology to maintain  Always defines a closed surface!  Inside/Outside test  CSG operations  Union  Intersection  Subtraction - - - - - + + + + + + - - - - - + + + + + + - - - - - - - - - - + + + + + + - - - - -

19. 19/47 Advantages  No topology to maintain  Always defines a closed surface!  Inside/Outside test  CSG operations  Union  Intersection  Subtraction - - - - - + + + + + + - - - - - + + + + + + - - - - - - - - - -

20. 20/47 Advantages  No topology to maintain  Always defines a closed surface!  Inside/Outside test  CSG operations  Union  Intersection  Subtraction - - - + + + + + - - -

21. 21/47 Advantages  No topology to maintain  Always defines a closed surface!  Inside/Outside test  CSG operations  Union  Intersection  Subtraction

22. 22/47 Disadvantages  Hard to render - no polygons  Creating polygons amounts to root finding  Arbitrary shapes hard to represent as an analytic function  Certain operations (like simplification) can be difficult

23. 23/47 Non-Analytic Implicit Functions  Sample functions over grids

24. 24/47 Non-Analytic Implicit Functions  Sample functions over grids

25. 25/47 Data Sources

26. 26/47 Data Sources

27. 27/47 Data Sources

28. 28/47 Data Sources

29. 29/47 Data Sources

30. 30/47 Data Sources

31. 31/47 2D Surface Reconstruction

32. 32/47 2D Surface Reconstruction

33. 33/47 2D Surface Reconstruction

34. 34/47 2D Surface Reconstruction

35. 35/47 2D Surface Reconstruction

36. 36/47 Marching Cubes

37. 37/47 Marching Cubes

38. 38/47 Dual Contouring  Place vertices inside of square  Generate segments across edges with zero  Dual to polygons produced by MC

39. 39/47 Comparison of Primal/Dual  Produces well-shaped quads  Allows more freedom in positioning vertices Marching Cubes (Primal) Dual Contouring (Dual)

40. 40/47 Dual Contouring With Hermite Data  Place vertices at minimizer of QEFs  Generate segments across edges with zeros

41. 41/47 Comparison Marching CubesDual Contouring

42. 42/47 Contouring Signed Octrees  For each minimal edge with zero,  Connect vertices of cubes containing edge  Constructs closed surface mesh for any octree

43. 43/47 Fast Polygon Generation  Recursive octree traversal  Linear time in size of octree

44. 44/47 Extensions  Multiple materials  CSG operations  Simplification via QEFs  Topological safety

45. Dual Marching Cubes  Generate cells for contouring using the dual of the octree  Creates adaptive, crack-free partitioning of space  Use Marching Cubes on dual cells to construct polygons 45/47

46. Dual Marching Cubes  Enumerate dual grid using recursive walk  Three types of recursive calls 46/47

47. 47/47 Dual Marching Cubes  Advantages  Always creates a manifold surface  Same as Marching Cubes over uniform grids  Works well for data centered in cells  Disadvantages  Octrees with data at vertices instead of cells  ?…