1. T-splines Speaker : 周 联 2007.12.12

2. Mian works Sederberg,T.W., Zheng,J.M., Bakenov,A., Nasri,A., T-splines and T-NURCCS. SIGGRAPH 2003. Sederberg,T.W., David L. C., Zheng, J.M., Lyche,T., T-spline Simplication and Local Refinement. SIGGRAPH 2004

3. Authors Thomas W. Sederberg, Brigham Young University Jianmin Zheng, Nanyang Technological University Almaz Bakenov, Embassy of Kyrgyz Republic Washington, D.C. Ahmad Nasri, American University of Beirut Tom Lyche, University of Oslo David L. Cardon, Brigham Young University

4. Other works Song W.H., Yang X.N., Free-form deformation with weighted T-spline, The Visual Computer 2005 Xin Li, Jiansong Deng, Falai Chen, Dimensions of Spline Spaces Over 3D Hierarchical T-Meshes, Journal of Information and Computational Science, Vol.3, No.3, 487--501, 2006. (EI) Zhangjing Huang, Jiansong Deng, Yuyu Feng, and Falai Chen, New Proof of Dimension Formula of Spline Spaces over T-meshes via Smoothing Cofactors, Journal of Computational Mathematics, Vol.24, No.4, 501--514, 2006 Jiansong Deng, Falai Chen, Yuyu Feng, Dimensions of spline spaces over T- meshes, Journal of Computational and Applied Mathematics, Vol.194, No.2, 267-- 283, 2006. Xin Li, Jiansong Deng, Falai Chen, Surface Modeling with Polynomial Splines over Hierarchical T-meshes, accepted by CAD/CG'2007, and published on The Visual Computer, 2007. Jiansong Deng, Falai Chen, etal., Polynomial splines over hierarchical T-meshes, submitted to Graphical Models, 2006.10

5. What are T-Splines? T-Splines: a generalization of non-uniform B-spline surfaces "T-Splines are the next thing...They have opened up possibilities to work with surfaces that were simply impossible before." -- Eric Allen, Production Manager, DAZ http://www.daz3d.com/

6. Why use T-Splines? Add detail only where you need it Create even the most complex shapes as a single, editable surface Create natural edge flow and non- rectangular topology www.tsplines.com

7. Why use T-Splines? Fits into your Workflow

8. T-Splines vs. NURBS Reduce the number of superfluous control points.

9. T-Splines vs. NURBS Remove unwanted ripples.

10. T-Splines vs. NURBS Remove gap

11. Other methods hierarchical B splines  D. Forsey, R.H. Bartels, Hierarchical B-spline refinement, Comput. Graphics 22 (4) (1988) 205–212 a spline space over a more general T-mesh, where crossing, T-junctional, and L-junctional vertices are allowed.  F. Weller, H. Hagen, Tensor-product spline spaces with knot segments, in: M. Dalen, T. Lyche, L.L. Schumaker (Eds.), Mathematical Methods for Curves and Surfaces, Vanderbilt University Press, Nashville, TN, 1995, pp. 563–572.

12. Polar Form Definition: Examples:

13. Polar Form Definition: Ramshaw, L. 1989. Blossoms are polar forms. Computer Aided Geometric Design 6, 323-358.

14. Polar Form

15. Knot Intervals

17. PB-splines whose control points have no topological relationship with each other it is point based instead of grid based

18. PB-splines

19. T-mesh A T-spline is a PB-spline by means of a control grid called a T-mesh.

20. T-mesh Infer knot vectors from T-grid

21. Two rules for T-mesh

22. Control Point Insertion (2003)

23. Local knot insertion

28. Create features

29. Extract Bezier Patches

30. Merge B-splines into a T-spline Traditional way Use cubic NURSSes (SIGGRAPH 98)

31. Merge B-splines into a T-spline

33. Examples

35. A problem A local knot insertion sometimes requires that other local knot insertions must be performed. Are there cases in which these prerequisites cannot all be satisfied?

36. T-spline Local Refinement (2004)

37. Blending Function Refinement

39. T-spline Spaces T-spline Spaces : the set of all T-splines that have the same T-mesh topology, knot intervals, and knot coordinate system.

40. T-spline Spaces

42. Local Refinement Algorithm

44. = +

46. Compared with old one always work requires far fewer unrequested control point insertions

47. T-spline Simplication 1234512345

49. Compared with B-spline wavelet decomposition

50. Create a T-spline model

51. Convert a T-spline into a B-spline surface

52. Standard T-splines

53. Semi-Standard T-splines

54. Some open questions What T-mesh configurations yield a standard T-spline? Are T-spline blending functions always linearly independent? What are the fairness properties of PB-splines?

55. Free-form deformation with weighted T-spline Wenhao Song, XunnianYang The Visual Computer (2005)

56. Weighted 3D PB-splines

57. T-lattice A T-lattice is a rectangular parallelepiped, that allows T-junctions.  Rule 1. In each minimal cell, the sum of knot intervals in the same direction must be equal.  Rule 2. Any edge must be a cell edge.  Rule 3. There are no zero edges in T lattices.

59. Automatic generation of T-lattice 1. Define the initial lattice to be deformed. 2. If the cell contains any vertex of the model, subdivide it by applying the octree subdivision. 3. Repeat steps 2 and 3 until a user-specified threshold is reached.

60. Automatic generation of T-lattice

63. Parametrization Nonlinear conjugate gradient method

64. Deformation algorithm Step 1. Define the initial region of the model to be deformed. Step 2. Generate the multiresolution lattices and set initial weights for T-spline volumes. Step 3. Calculate the parametric coordinates (u, v, w) for each point. Step 4. Modify the control points or weights and evaluate the new locations of the points.

67. Some open questions To generate control lattices with arbitrary topology that is better than octree subdivision lattices; To support hierarchical deformation; To import T-spline surfaces by creating a T-spline volume in which the surface is an isosurface; To implement w-TFFD by hardware acceleration

68. Dimensions of spline spaces over T-meshes Jiansong Deng, Falai Chen,Yuyu Feng Journal of Computational and Applied Mathematics 194 (2006) 267–283

69. The spline space over the given T-mesh

70. Dimension formula

71. Some open questions construction of a set of basis functions with good properties do not know whether the dimension relies on the geometry of the T-mesh or not