Implicit Representations of Surfaces and Polygonalization Algorithms Dr. Scott Schaefer

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

25/47 Data Sources

26/47 Data Sources

27/47 Data Sources

28/47 Data Sources

29/47 Data Sources

30/47 Data Sources

31/47 2D Surface Reconstruction

32/47 2D Surface Reconstruction

33/47 2D Surface Reconstruction

34/47 2D Surface Reconstruction

35/47 2D Surface Reconstruction

36/47 Marching Cubes

37/47 Marching Cubes

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

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

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

41/47 Comparison Marching CubesDual Contouring

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

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

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

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

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

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  ?…