1. Seminar on B-Spline over triangular domain Reporter: Gang Xu Institute of Computer Images and Graphics, Math Dept. ZJU October 26

2. Outline 1.Introduction 2.Mathmatic Preliminaries 3.B-Patches and Simplex Splines 4.DMS-Splines and its application 5.G-Patches 6.Future Work

3. Introduction Ramshaw’s “juiciest” challenge(1987) “Find a natural way to construct a triangular patch surface that builds in the appropriate continuity conditions, similar to what is done with the B-Spline.”

4. Introduction Bézier Curves B-Spline Curves Triangular Bézier patches What?

5. Desirable Control Scheme Attributes  Piecewise polynomial of a fixed degree  Individual piecewise polynomials are associated to regions of the domain  Control Points and Interactivity  Local Control  Automatic Continuity Maintenance  Simplifies to Univariate Splines  Numerical Stability

6. A Bleak Property -continuity, where Triangular Bézier Patch Continuity Constraints For a surface consisting of degree n≥1 triangular Bézier patches the highest degree of continuity possible, while still providing local flexibility, is

7. Examples  Cubic  Quartic The examples are from (Zhang et.al, 2005)

8. Current Situation A lot of work focus on this problem Each has its own specialized use, but Inevitably each has its own fundamental limits None is the true generalization of the B-spline!

9. Mathematical Preliminaries  Barycentric Coordinates on line

10.  Barycentric coordinates on plane Mathematical Preliminaries

11.  Fundmental Idea Represent the univariate complex function by the multivariate simple function  Related to Polar Forms Blossom

12. Blossom Principle  Symmetric  Multi-affine  Diagonal

13. Some Terminologies Multi-affine blossom of F Blossom argument Blossom value

14. Some Examples

15. Blossom form of CAGD  Most of curves and surfaces in CAGD have a blossom form  Bézier Curves  B-Spline Curves  Tensor product surfaces  Triangular Bézier patches  C- Bézier curves and H- Bézier curves,  Also their tensor product surfaces

16. Blossom of Bézier curves

17. de Casteljau algorithm in Blossom

18. Example

19. Blossom of B-spline curves

20. Blossom of Triangular Bézier patches

21. Pyramid Algorithms of B-B Surface

22. Shortcomings of B-B Surfaces  Modeling sufficiently complex surfaces requires the surfaces to have an extremely high degree Divide the domain into small triangular regions, define a B-B surfaces for each region, as B-spline curves. How can we get it?

23. B-Patches  Motivation Bézier curves B-spline curves

24. B-Patches Triangular Bézier patches

25. B-Patch’s control net

26. de Boor style algorithm of B-Patches

27. Shortcomings of B-Patches In order to be continuity, the knots along the shared domain edge must be Collinear. (Seidel,1991) Not extend well to a network of patches!

28. Example It is useless for surface modeling!

29. Simplex Splines The major problem with B-Patches is that the underlying basis functions don’t automatically provide the required degrees of continuity The simplex splines overcome it!

30. Simplex Splines Piecewise polynomial functions defined using a set of points in.The set of these points is called knot set (knot clouds). The simplex splines defined using knots has degree The simplex splines has overall continuity provided that the knot set does not contain a collinear subset of three knots. The simplex spline does not have control points.

31. Half-open Convex Hull x belongs to [v) if and only if there exists a small triangle that lies entirely within the [v] x belongs to exactly one triangle

32. Examples

33. Definition of simplex splines A degree simplex spline with knots is defined recursively as follows are barycentric coordinates with respect to

34. Examples 1

35. Examples 2

36. Examples 3

37. Examples 4

38. Shortcomings of Simplex Splines The choice of the knots to place in W during each recursive evaluation can effect the results of the computation if not chosen carefully. Plagued with numerical stability issues Computationally expensive Have no control points It is useless for surface modeling!

39. DMS-Splines Motivation B-Patches nice labelling of control points Simplex splines Smooth basis functions DMS-Splines Take the advantage of them!

40. The inventor Dahmen, Micchelli, Seidel, 1992 TVCG, IJSM,CAGD, TVC,GMOD,CGF

41. Definition of DMS-splines Triangulate the domain A knot cloud is arranged with each corner of the domain. For a degree n triangular domain, n knots are pulled out. quadratic

42. Definition of DMS-splines For a domain region control points Similar with simplex splines, define a set To be normalized, define

43. Definition of DMS-splines

44. Examples 1

45. Examples 2

46. Examples 3

47. Properties of DMS-splines Convex hull property Local control Smoothness Parametric affine invariance

48. Continuity Control by Placing Knots Make several knots collinear to decrease continuity quadratic Three knots collinear discontinuity Four knots collinear

49. Examples

50. Application(1) Filling Holes

52. Application(2) Fit Scattered Data The problem Fitting of a functional surface to a collection of scattered functional data Our goal Find a smooth surface F that is a reasonable approximation to the data

53. Application(2) Fit Scattered Data Why we choose DMS splines? Automatic smoothness properties Ability to define a surface over an arbitrary triangulation (which can be adapted to the local density of sampled data)

54. Finding a Triangulation Properties of a good triangulation All sample points must be contained in some triangle of the triangulation Points within each triangle are distributed as uniformly as possible Triangles are not too elongated Neighbouring triangles are roughly comparable in size

55. Finding a Triangulation Delaunay triangulation explosion of triangles! Quadtree division of the domain Require that the quadtree be balanced The depth of two adjacent leaf nodes differ by at most one

56. Finding a Triangulation

57. Assigning Knot Clouds Avoid collinearity of knots associated with a particular triangle k+2 of the knots are placed collinearity, the continuity of the surface along that parametric line will be reduced by k

58. Least Square Fitting A linear system

59. Least Square Fitting Advantages Simple to understand Easy to implement Disadvantages Sensitive to the location of data points with respect to the given set of basis functions Lie close to data points, not be very smooth

60. Combining Least Squares and Smoothing Localizing the smoothing effect

61. Examples 1

62. Examples 2

63. Examples 3

64. Examples 4

66. Examples 5

68. Application(3) Surface Reconstruction The Problem Given a set of points,find a parametric surface that approximates Existing approach Polygonal meshes Splines Zero-set surface

69. Application(3) Surface Reconstruction Why use DMS-splines? Arbitrary topological type Be able to model discontinuities like sharp edges or corners as well( tensor product B-spline will produce a discontinuity curve across the whole patch)

70. Application(3) Surface Reconstruction Constructing an initial domain triangulation Feature detection Domain partition Constrained Delaunay triangulations

71. Application(3) Surface Reconstruction Fitting with triangular B-splines Solve control points Solve knots

72. Application(3) Surface Reconstruction Adaptive refinement Repeat Subdivide the domain triangles with large fitting error Solve the control points sub-problem for affected triangles Solve the knots for new vertices Until

73. Experimental Results(1)

75. Experimental Results(2)

77. Application(4) Image Registration The problem Given source image, and target image, defined on the domain, the problem of registration is to find an optimal geometrical transformation such that the pixels in both images are matched properly

78. Application(4) Image Registration Development Rigid global Non-rigid local Rigid and non-rigid continuity Tensor-product B-splines DMS-Splines Sharp features can not lie in arbitrary directions

79. Why choose DMS splines? flexible domain local control space-varying smoothness modeling Application(4) Image Registration

80. Steps Transformation Model Point-based Constraints Optimization

81. Application(4) Image Registration

84. Application(5) Triangular NURBS Similar with NURBS! Dynamic Generalization!

85. Application(5) Triangular NURBS Modeling Applications Rounding (filet) Scattered Data Fitting Dynamic Interactive Sculpting

86. Experimental Results(1)

87. Experimental Results(2)

88. Experimental Results(3)

89. Experimental Results(4)

90. Experimental Results(5)

91. Application(6) Solid Modeling 2D3D triangular tetrahedra triangulation terahedralization Increment Flip Algorithm

92. Application(6) Solid Modeling

93. Geometric editing using control points

94. Application(6) Solid Modeling Attribute editing using control coefficient or control points

95. Application(6) Solid Modeling Feature Sensitive Data Fitting Similar with DMS spline but need to preprocess the dataset!

96. Experimental Results(1)

97. Experimental Results(2)

98. APP(7) Rational Spherical DMS-splines Spherical DMS-splines (Pfeifle, Seidel,1995) No convex hull property!

99. APP(7) Rational Spherical DMS-splines Rational Spherical DMS-splines Convex hull property

100. APP(7) Rational Spherical DMS-splines Genus zero surface reconstruction Similar with Application (3)!

101. APP(7) Rational Spherical DMS-splines Editing the details

102. APP(7) Rational Spherical DMS-splines Editing the control net

103. APP(7) Rational Spherical DMS-splines Computing the differential properties Modeling features

104. APP(7) Rational Spherical DMS-splines Brain image analysis using spherical DMS-splines

105. APP(7) Rational Spherical DMS-splines Segmentation by mean curvature

106. APP(7) Rational Spherical DMS-splines Sulci and Gyri tracing

107. Application(8) Manifold DMS-splines X.Gu,Y.He, and H.Qin, Manifold splines, in Proceedings of ACM SPM’05, pp27-38,2005 Planar Domain Manifold Domain

108. Application(8) Manifold DMS-splines Corollary1(Existence of Singular Points) The manifold splines must have singular points if the domain manifold is closed and not a torus. Corollary2(Minimal Number of Singular Points) Given a closed domain 2-manifold, if its Euler number is not zero, a manifold spline can be constructed such that the spline has only one singular point.

109. Application(8) Manifold DMS-splines

112. Comparison of Various DMS-spline

113. Manifold “other” splines

114. Fairing Manifold DMS-splines Motivation High curvature concentration along the edges of adjacent spline patches Knot line

115. Fairing Manifold DMS-splines Method (Ying.H, Xianfeng.G, Hong.Oin, 2005) Inspired by the knot-line elimination work of (Gormaz,1994).

116. Fairing Manifold DMS-splines Least square problem Lagrange multipliers method

117. Fairing Manifold DMS-splines

120. Valuable properties Applied extensively, from graphics to image Is it the true generalization of the B-spline?! Conclusion of DMS-spline

121. Not correlate to the Bézier patches Computational cost is so big Not present an elegant user interface Moving the knots has unexpected results Prevent too many knots from being collinear

122. Multiresolution triangular B-splines

123. G-Patches Main idea (Christopher K,2003) Generalize the geometry of a uniform B-spline curve over triangular domain Generalization of the blending fucntions used in the uniform B-splines

124. G-Patches

125. Continuity

126. G-Patches

127. Reduce to the classic univariate B-splines Local control Evaluation is very fast Manipulation is extremely intuitive continuityOnly The only fatal disadvantage Remove it from being a viable modeling tool!

128. Future Work Create the true generalization of B-spline over triangular domain Circular C- Bézier or H- Bézier, Spherical is also Manifold C-B-spline or H-B-spline Other application of DMS-splines C- Bézier over triangular domain New method of surface reconstruction

129. Questions

130. Thank you!

131. Main References