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Subdivision Schemes

Subdivision Schemes. What is Subdivision?. Subdivision is a process in which a poly-line/mesh is recursively refined in order to achieve a smooth curve/surface. Two main groups of schemes: Approximating - original vertices are moved Interpolating – original vertices are unaffected.

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Subdivision Schemes

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  1. Subdivision Schemes

  2. What is Subdivision? • Subdivision is a process in which a poly-line/mesh is recursively refined in order to achieve a smooth curve/surface. • Two main groups of schemes: • Approximating - original vertices are moved • Interpolating – original vertices are unaffected Is the scheme used here interpolating or approximating? Center for Graphics and Geometric Computing, Technion

  3. Why Subdivision? Frame from “Geri’s Game” by Pixar Center for Graphics and Geometric Computing, Technion

  4. Why Subdivision? • LOD • Compression • Smoothing 13.3Mb 424Kb 1Kb 52Kb Center for Graphics and Geometric Computing, Technion

  5. Corner Cutting Center for Graphics and Geometric Computing, Technion

  6. Corner Cutting 3 : 1 1 : 3 Center for Graphics and Geometric Computing, Technion

  7. Corner Cutting Center for Graphics and Geometric Computing, Technion

  8. Corner Cutting Center for Graphics and Geometric Computing, Technion

  9. Corner Cutting Center for Graphics and Geometric Computing, Technion

  10. Corner Cutting – Limit Curve Center for Graphics and Geometric Computing, Technion

  11. Corner Cutting The limit curve – Quadratic B-Spline Curve A control point The control polygon Center for Graphics and Geometric Computing, Technion

  12. 4-Point Scheme Center for Graphics and Geometric Computing, Technion

  13. 4-Point Scheme Center for Graphics and Geometric Computing, Technion

  14. 4-Point Scheme 1 : 1 1 : 1 Center for Graphics and Geometric Computing, Technion

  15. 4-Point Scheme 1 : 8 Center for Graphics and Geometric Computing, Technion

  16. 4-Point Scheme Center for Graphics and Geometric Computing, Technion

  17. 4-Point Scheme Center for Graphics and Geometric Computing, Technion

  18. 4-Point Scheme Center for Graphics and Geometric Computing, Technion

  19. 4-Point Scheme Center for Graphics and Geometric Computing, Technion

  20. 4-Point Scheme Center for Graphics and Geometric Computing, Technion

  21. 4-Point Scheme A control point The limit curve The control polygon Center for Graphics and Geometric Computing, Technion

  22. Comparison Non interpolatory subdivision schemes • Corner Cutting Interpolatory subdivision schemes • The 4-point scheme Center for Graphics and Geometric Computing, Technion

  23. Theoretical Questions • Given a Subdivision scheme, does it converge for all polygons? • If so, does it converge to a smooth curve? • Better? • Does the limit surface have any singular points? • How do we compute the derivative of the limit surface? Center for Graphics and Geometric Computing, Technion

  24. Surface subdivision • A surface subdivision scheme starts with a control net (i.e. vertices, edges and faces) • In each iteration, the scheme constructs a refined net, increasing the number of vertices by some factor. • The limit of the control vertices should be a limit surface. • a scheme always consists of 2 main parts: • A method to generate the topology of the new net. • Rules to determine the geometry of the vertices in the new net. Center for Graphics and Geometric Computing, Technion

  25. General Notations • There are 3 types of new control points: • Vertex points - vertices that are created in place of an old vertex. • Edge points - vertices that are created on an old edge. • Face points – vertices that are created inside an old face. • Every scheme has rules on how (if) to create any of the above. • If a scheme does not change old vertices (for example - interpolating), then it is viewed simply as if Center for Graphics and Geometric Computing, Technion

  26. Loop’s Subdivision - topology • Based on a triangular mesh • Loop’s scheme does not create face points New face Old face Vertex points Edge points Center for Graphics and Geometric Computing, Technion

  27. 1 3 3 1 Loop’s subdivision – stencil • Every new vertex is a weighted average of old ones. • The list of weights is called a Stencil • Is this scheme approximating or interpolating? The rule for vertex points The rule for edge points 1 1 1 1 1 n– vertex point’s valence Center for Graphics and Geometric Computing, Technion

  28. Loop - Results Center for Graphics and Geometric Computing, Technion

  29. Loop - Results Center for Graphics and Geometric Computing, Technion

  30. Loop - Results Center for Graphics and Geometric Computing, Technion

  31. Loop - Results Center for Graphics and Geometric Computing, Technion

  32. Loop - Results • Behavior of the subdivision along edges Loop’s scheme results in a limit surface which is of continuity everywhere except for a finite number of singular points, in which it is . Center for Graphics and Geometric Computing, Technion

  33. -1 -1 2 8 8 -1 2 -1 Butterfly Scheme • Butterfly is an interpolatory scheme. • Topology is the same as in Loop’s scheme. • Vertex points use the location of the old vertex. • Edge points use the following stencil: Center for Graphics and Geometric Computing, Technion

  34. Butterfly - results Center for Graphics and Geometric Computing, Technion

  35. Butterfly - results Center for Graphics and Geometric Computing, Technion

  36. Butterfly - results Center for Graphics and Geometric Computing, Technion

  37. Butterfly - results Center for Graphics and Geometric Computing, Technion

  38. Butterfly - results The Butterfly Scheme results in a surface which is but is not differentiable twice anywhere. Center for Graphics and Geometric Computing, Technion

  39. Catmull-Clark • The mesh is the control net of a tensor product B-Spline surface. • The refined mesh is also a control net, and the scheme was devised so that both nets create the same B-Spline surface. • Uses face points, edge points and vertex points. • The construction is incremental – • First the face points are calculated, • Then using the face points, the edge points are computed. • Finally using both face and edge points, we calculate the vertex points. Center for Graphics and Geometric Computing, Technion

  40. 1 1 1 1 1 1 1 1 1 Catmull-Clark Step 1 Step 2 Step 3 First, all the face points are calculated Then the edge points are calculated using the values of the face points and the original vertices Last, the vertex points are calculated using the values of the face and edge points and the original vertex 2 1 1 2 2 1 1 1 2 2 n- the vertex valence Face points Edge points Vertex points Center for Graphics and Geometric Computing, Technion

  41. After Computing the new points, new edges are formed by: connecting each new face point to the new edge points of the edges defining the old face. Connecting each new vertex point to the new edge points of all old edges incident on the old vertex point. Connecting The Dots Gone Center for Graphics and Geometric Computing, Technion

  42. Catmull-Clark - results Center for Graphics and Geometric Computing, Technion

  43. Catmull-Clark - results Center for Graphics and Geometric Computing, Technion

  44. Catmull-Clark - results Center for Graphics and Geometric Computing, Technion

  45. Catmull-Clark - results Center for Graphics and Geometric Computing, Technion

  46. Catmull-Clark - results Catmull-Clark Scheme results in a surface which is almost everywhere Center for Graphics and Geometric Computing, Technion

  47. Comparison Butterfly Loop Catmull-Clark Center for Graphics and Geometric Computing, Technion

  48. Pros and Cons • Pros • A single mesh defines the whole model • Simple local rules • Easy to implement. • Numerical Stability • Easy to generate sharp feature with • Lod, Compression etc • Cons • Evaluating a single point on the surface is hard • Not suitable for CAGD • Mesh topology has great influence on the over all shape. • May become expensive in term of rendering • Global subdivision Center for Graphics and Geometric Computing, Technion

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