1. 3D ModelingSubdivision Surfaces

2. Modeling How do we ... Represent 3D objects in a computer? Construct such representations quickly and/or automatically with a computer? Manipulate 3D objects with a computer? 3D Representations provide the foundations for Computer Graphics, Computer-Aided Geometric Design, Visualization, and Robotics Different methods for different object representations

3. 3D Object Representations Raw data Point cloud Range image Polygon soup Surfaces Mesh Subdivision Parametric Implicit Solids Voxels BSP tree CSG

4. Point Cloud Unstructured set of 3D point samples

5. Range Image Set of 3D points mapping to pixels of depth image Acquired from range scanner

6. Polygon Soup Unstructured set of polygons

7. 3D Object Representations Raw data Point cloud Range image Polygon soup Surfaces Mesh Subdivision Parametric Implicit Solids Voxels BSP tree CSG

8. Mesh Connected set of polygons (usually triangles) May not be closed

9. Parametric Surface Tensor product spline patchs Careful constraints to maintain continuity

10. Subdivision Surface Coarse mesh & subdivision rule Define smooth surface as limit of sequence of refinements

11. Implicit Surface Points satisfying: F(x,y,z) = 0

12. 3D Object Representations Raw data Point cloud Range image Polygon soup Surfaces Mesh Subdivision Parametric Implicit Solids Voxels BSP tree CSG

13. Voxels Uniform grid of volumetric samples Acquired from CAT, MRI, etc.

14. BSP Tree Binary space partition with solid cells labeled Constructed from polygonal representations

15. CSG Hierarchy of boolean set operations (union, difference, intersect) applied to simple shapes

16. Subdivision Surface Subdivision methods refine a (simple) control polygon such that in the limit it converges to a smooth curve. A multitude of different curve and surface types can be described by subdivision.

17. Subdivision Surfaces Approach Limit Curve Surface through an Iterative Refinement Process.

18. Subdivision in 3D Same approach works in 3D

19. Problems with Splines Traditionally spline patches (NURBS) have been used in production for character animation. Difficult to stitch together Maintaining continuity is hard Trimming boundaries is hard Difficult to model objects with complex topology

20. Example: Geri�s Game Subdivision surfaces are used for: Geri�s hands and head. Clothes: Jacket, Pants, Shirt. Tie and Shoes. Movie

21. Why Subdivision Surfaces? Subdivision methods have a series of interesting properties: Applicable to meshes of arbitrary topology (non-manifold meshes). Scalability, level-of-detail. Numerical stability. Simple implementation. Compact support. Affine invariance. Continuity.

22. Types of Subdivision Interpolating Schemes Limit Surfaces/Curve will pass through original set of data points. Approximating Schemes Limit Surface will not necessarily pass through the original set of data points.

23. Subdivision in 1-D Simplest example: piecewise linear subdivision

24. Chaiken�s Algorithm

25. Key Questions in Subdivision How refine mesh? Topology change Where to place new vertices Provable properties about limit surface

26. Loop Subdivision Scheme How refine mesh? Refine each triangle into 4 triangles by splitting each edge and connecting new vertices

27. Loop Subdivision Scheme How position new vertices? Choose locations for new vertices as weighted average of original vertices in local neighborhood What if vertex does not have degree 6?

28. Loop Subdivision Scheme:Rule for extraordinary vertices

29. Loop Subdivision Scheme How to choose Analyze properties of limit surface Interested in continuity of surface and smoothness Involves calculating eigenvalues of matrices Original Warren

30. Loop Subdivision Boundaries Subdivision Mask for Boundary Conditions

31. Subdivision as Matrices Subdivision can be expressed as a matrix Smask of weights w. Smask is very sparse Never Implement this way! Allows for analysis Curvature Limit Surface

32. Loop Subdivision Scheme Works on triangular meshes Is an Approximating Scheme Guaranteed to be smooth everywhere except at extraordinary vertices.

33. Subdivision Schemes There are different subdivision schemes Different methods for refining topology Different rules for positioning vertices Interpolating versus approximating

34. Catmull-Clark Subdivision

35. Adaptive Subdivision Not all regions of a model need to be subdivided. Idea: Use some criteria and adaptively subdivide mesh where needed. Curvature Screen size ( make triangles < size of pixel ) View dependence Distance from viewer Silhouettes In view frustum Careful! Must ensure that �cracks� aren�t made

36. Subdivision Surfaces for Compression

37. Subdivision Surface Summary Advantages Simple method for describing complex surfaces Relatively easy to implement Arbitrary topology Local support Guaranteed continuity Multi-resolution Difficulties Intuitive specification Parameterization Intersections