1. Lee Byung-Gook Dongseo Univ. http://kowon.dongseo.ac.kr/~lbg/
Subdivision Schemes
Lee Byung-Gook
Dongseo Univ.
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2. Graphics Programming, Lee Byung-Gook, Dongseo Univ
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3. Is the scheme used here interpolating or approximating?
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?
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4. Frame from “Geri’s Game” by Pixar
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5. Subdivision Curves How do You Make a Smooth Curve?
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6. Chaikin’s Algorithm
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7. Corner Cutting
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8. Corner Cutting
: 1
1 :
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9. Corner Cutting
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10. Corner Cutting
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11. Corner Cutting
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12. Corner Cutting – Limit Curve
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13. Corner Cutting The limit curve – Quadratic B-Spline Curve
A control point
The control polygon
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14. Linear B-spline Subdivision Scheme
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15. Quadratic B-spline Subdivision Scheme
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16. Cubic B-spline Subdivision
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17. N. Dyn 4-points Subdivision Scheme
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18. 4-Point Scheme
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19. 4-Point Scheme
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20. 4-Point Scheme
1 : 1
1 : 1
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21. 4-Point Scheme
1 :
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22. 4-Point Scheme
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23. 4-Point Scheme
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24. 4-Point Scheme
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25. 4-Point Scheme
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26. 4-Point Scheme
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27. 4-Point Scheme The limit curve The control polygon A control point
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28. Comparison Non interpolatory subdivision schemes Corner Cutting
The 4-point scheme
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29. 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?
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30. Subdivision Surfaces
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31. Subdivision Surfaces
Geri’s hand as a piecewise smooth Catmull-Clark surface
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32. Subdivision Surfaces
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.
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33. 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
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34. Doo-Sabin Subdivision
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35. Doo-Sabin Subdivision
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36. Doo-Sabin Subdivision
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37. Doo-Sabin Subdivision
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38. Doo-Sabin subdivision
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39. Catmull-Clark Subdivision
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40. Catmull-Clark Subdivision
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41. Catmull-Clark Subdivision
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42. Catmull-Clark Subdivision
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43. Catmull-Clark Subdivision
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.
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44. Graphics Programming, Lee Byung-Gook, Dongseo Univ
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45. Catmull-Clark - results
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46. Catmull-Clark - results
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47. Catmull-Clark - results
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48. Catmull-Clark - results
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49. Catmull-Clark - results
Catmull-Clark Scheme results in a surface which is almost everywhere
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50. Loop Subdivision Based on a triangular mesh
Loop’s scheme does not create face points
Vertex points
New face
Old face
Edge points
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51. Loop Subdivision How Position New Vertices?
Choose locations for new vertices as weighted average of original vertices in local neighborhood
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52. The rule for edge points
Loop Subdivision
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 3 1 1 1
The rule for edge points 1 1 1
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53. Loop Subdivision How Refine Mesh?
Refine each triangle into 4 triangles by splitting each edge and connecting new vertices
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54. Box Spline
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55. Loop - Results
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56. Loop - Results
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57. Loop - Results
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58. Loop - Results
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59. Loop - Results
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
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60. 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: -w
1/2 2w
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61. Butterfly - results
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62. Butterfly - results
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63. Butterfly - results
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64. Butterfly - results
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65. Butterfly - results
The Butterfly Scheme results in a surface which is but is not differentiable twice anywhere.
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66. Comparison Butterfly Loop Catmull-Clark
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67. Kobblet sqrt(3) Subdivision
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68. Kobblet sqrt(3) Subdivision
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69. Kobblet sqrt(3) Subdivision
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70. Graphics Programming, Lee Byung-Gook, Dongseo Univ
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71. Subdivision Schemes Loop Butterfly Catmull-Clark Doo-Sabin
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72. Subdivision Schemes Loop Butterfly Catmull-Clark Doo-Sabin
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73. Advantages & Difficulties
Simple method for describing complex surfaces
Relatively easy to implement
Arbitrary topology
Smoothness guarantees
Multiresolution
Difficulties:
Intuitive specification
Parameterization
Intersections
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74. Subdivision Schemes There are different subdivision schemes
Different methods for refining topology
Different rules for positioning vertices
Interpolating versus approximating
Face Split
Vertex Split
Triangular Meshes
Quad. Meshes
Doo-Sabin, Midege (C1)
Approximating
Loop (C2)
Catmull-Clark (C2)
Biquartic (C2)
Interpolating
Butterfly (C1)
Kobbelt (C1)
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