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T-splines

T-splines. Speaker : 周 联 2007.12.12. 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. Authors. Tom Lyche ,

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T-splines

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  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 • Tom Lyche, University of Oslo • David L. Cardon, Brigham Young University • 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

  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

  16. 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

  24. Create features

  25. Extract Bezier Patches

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

  27. Merge B-splines into a T-spline

  28. Merge B-splines into a T-spline

  29. Examples

  30. Examples

  31. 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?

  32. T-spline Local Refinement (2004)

  33. Blending Function Refinement

  34. Blending Function Refinement

  35. 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.

  36. T-spline Spaces

  37. T-spline Spaces

  38. Local Refinement Algorithm

  39. Local Refinement Algorithm

  40. Local Refinement Algorithm = +

  41. Local Refinement Algorithm

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

  43. T-spline Simplication 1 2 3 4 5

  44. Compared with B-spline wavelet decomposition

  45. Create a T-spline model

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