Subdivision Surfaces in Character Animation

1. Subdivision Surfaces in Character Animation Chris DeCoro Presentation of an article by DeRose, et. Al.

2. Talk outline • Introduction • Motivation and applications • Previous work • Additional requirements of animation • Semi-sharp Creases • Rendering • Physically-based Modeling • Conclusion

3. Motivation and Background • Animation requires smooth surfaces for visual appeal • Traditional Approach: NURBS • However, single patches have limited topology • Difficult to maintain smoothness during animation • Often require expensive and inaccurate trimming • Proposed Solution: Subdivision Surfaces • Able to represent arbitrary topology • No trimming • No seams; always smooth

4. Previous work • Catmull-Clark Subdivision • Input is a quadrilateral mesh, M0 • From mesh Mn, reaches Mn+1 through subdivision • Adds additional vertices to make each quad into 4 subdivided quads • Limit surface (infinite subdivisions) is smooth • Infinitely-sharp Creases • In some locations, the ideal surface is not smooth • For these applications, we require creases • Sharp edges are manually marked as such • Algorithm uses special subdivision rules

5. Additional requirements of animation • Semi-sharp Creases • An adjustable intermediate between smooth surfaces and infinitely sharp creases • Physical Modeling • Surfaces pose challenges in representing real world movement and collisions • The article will focus on physical modeling of cloth • Rendering • NURBS have a direct parameterization for texture mapping • Subdivision surfaces do not

6. Talk outline • Introduction • Semi-sharp Creases • The need for Semi-sharp Creases • Standard Subdivision • Infinitely-sharp Creases • Overall Algorithm • Rendering • Physically-based Modeling • Conclusion

7. The Need for Semi-sharp Creases • Most “sharp” edges in nature are smooth at a sufficiently close distance • E.g. the edge of a table top • Often we wish to control the degree of sharpness • Shown in the illustrations below

8. V E F 1/4 1/4 1/16 3/8 1/16 1/4 1/4 3/8 1/16 1/16 Standard Subdivision • Standard Catmull-Clark Subdivision has three types of points in each refined mesh • Face points • Vertex points • Edge points • Face points = centroid of each quad region • Edge points = 0.25*(vi + ei + fJ-1 + fJ ) • Vertex points = (n-2)/n*vi + 1/n2*sum(ei) + 1/n2*sum(fi+1)

9. 1/64 3/32 1/64 9/16 3/32 3/32 1/64 3/32 1/64

10. Infinitely-Sharp Creases • Control vertices and edges are manually tagged as sharp • Face points: same as smooth rule • Edge points: place at midpoint of edge • Vertex points • One sharp incident edge (dart): same as smooth rule • Two sharp edges (crease): (e1 + 6vi + e2) / 8 • Three or more sharp edges (corner): do not modify point

11. Overall Algorithm • For creases with integer sharpness “s” • Subdivide using infinitely-sharp rules “s” times • For creases with non-integer sharpness “s” • Assume creases with sharpness floor(s) and ceil(s) • Determine subdivision points for both cases • Linearly interpolate to compute points for “s” • After previous steps, and for all elements • Perform standard (smooth) subdivision to the limit

12. Talk outline • Introduction • Semi-sharp Creases • Rendering • Surface texture mapping • Implementation issues • Physically-based Modeling • Conclusion

13. Texture Mapping • Textures parameterized onto a 2d plane • Ideal for NURBS • Not so ideal for Subdivision Surfaces • Solution: Subdivide texture coordinates • Manually apply texture coordinates to the base mesh • Treat vertex as (x,y,z,s,t) five-tuple • The texture coordinates are subdivided along with the coordinates • Can be applied to arbitrary attributes • Normals, arbitrary shader parameters

14. Implementation Issues • The author is working with the Pixar RenderMan system • Requires geometry be subdivided to sub-pixel sized micropolygons • Each primitive must bound itself, dice into micropolygons • Bounding takes advantage of convex hull property • Each primitive must dice itself into micropolygons • Locally, surface represents bicubic B-spline patch • Surface is converted into patch, uses less space (4x4 grid) • Regular structure makes easier to dice into micropolygons

15. Talk outline • Introduction • Semi-sharp Creases • Rendering • Physically-based Modeling • Computing cloth energy functional • Collision determination • Conclusion

16. Determining Energy Functional • Basic properties are specified by energy functional • Represents attraction or resistance of material to deformation • Catmull-Clark methods become regular grids • Ideal for representing woven fabrics • Energy term represented by: • E(p1,p2) = 0.5*(|p1-p2|/|p1*-p2*|)2 • p1*, p2* are the positions at rest • This can be used to represent the motion of cloth • Limited motion along thread directions • Folds freely along the diagonals

17. Collision Determination • Several approaches stand out • Test every object against each other (impractical) • Three-D spatial partitioning (not ideal) • Two-D surface partitioning • Advantages of surface partitioning • Hierarchy fixed if connectivity does not change • All data statically allocated, compute in preprocess • Hierarchy generation through “un-subdividing” • Pick a candidate edge “e” • Remove faces adjacent to “e”, merge to “super-face” • Mark edges adjacent to super-face as non-candidates for one iteration (for better balancing) • Store bounding box • At run time, use precomputed hierarchy to compare bounding boxes between object and obstacles

18. Conclusion • Subdivision surfaces are a better alternative to NURBS • Always smooth, no seams • No trimming, greater range of topology • Semi-sharp Creases allow for enhanced control • Sharpness can range smooth from none to infinite • Scalar-field texture maps allow for simplified texturing • Directly generated from the base mesh • Successfully integrated into RenderMan • Thus has shown to be successful in real-world applications • Collision detection model / Energy functional • Allows for realistic physics