1. Curve fitting to point clouds Reporter: Lincong Fang Oct 18, 2006

2. Curve fitting The data points are ordered.

3. Curve fitting to point clouds The data points are unorganized.

4. Applications Some applications: Reverse engineering Curve design Surface reconstruction Etc.

5. Approaches overview Preprocess the point clouds Thin the point clouds (Levin98, Lee00) Point clusters (Lin 04) Map into a digital image (Goshtasby00) Mathematical model Parameteric curves Implicit curves Other methods

6. Curve Reconstruction from unorganized points In-Kwon Lee CAGD 2000

7. Least squares

8. Moving Least Squares

12. The choice of H

13. Improved Moving Least Squares Delaunay triangulation Euclidean minimum spanning tree (EMSP)

15. Correlation

19. Refining

20. Compare with and without EMST

21. Ordering points

22. Example

23. Approaches overview Preprocess the point clouds Thin the point clouds (Levin98, Lee00) Point clusters (Lin 04) Map into a digital image (Goshtasby00) Mathematical model Parameteric curves Implicit curves Other methods

24. Curve reconstruction based on interval B-spline curve Hongwei Lin, Wei Chen, Guojin Wang The Visual Computer,21(6), 418- 427,2005

25. Overview

28. Shape-based joining scheme

33. Sequence Joining Method

34. Boundary sequence

35. Example

37. Approaches overview Preprocess the point clouds Thin the point clouds (Levin98, Lee00) Point clusters (Lin 04) Map into a digital image (Goshtasby00) Mathematical model Parameteric curves Implicit curves Other methods

38. Grouping and parameterizing irregularly spaced points for curve fitting Ardeshir Goshtasby ACM Transactions on Graphics, 19:185--203, 2000

40. Minor and major ridges

41. Map into a digital image

42. Minor and major ridges

43. Example

45. Approaches overview Preprocess the point clouds Thin the point clouds (Levin98, Lee00) Point clusters (Lin 04) Map into a digital image (Goshtasby00) Mathematical model Parameteric curves Implicit curves Other methods

46. Multidimensional curve fitting to unorganized data points by nonlinear minimization Lian Fang, David C Gossard CAD 95

48. Physical analogy

50. Error term

51. Example

52. Fitting B-spline curves to point clouds by curvature-based squared distance minimization Wenping Wang, Helmut Pottmann, Yang Liu ToG 2006

55. Point distance minimization

56. Tangent distance minimization

57. Squared distance minimization Pottman 2003

58. Squared distance minimization

60. Comparison PDM has slow convergence TDM has fast but unstable convergence SDM yields a more balanced performance between efficiency and stability

61. Comparison Initial curve The fitting curve generated by PDM, TDM, SDM in 50 iterations

62. Foot point computation

63. Open curves

64. Initial curves and control points Specify by user Compute a quadtree partition of the data points Automatic or specify by user, and adjustment (Yang 2004)

65. Example

66. Reconstructing B-spline curves from point clouds — A tangential flow approach using least squares minimization Yang Liu, Huaiping Yang, Wenping Wang Shape Modeling and Applications, 2005 International Conference

68. Input Unacceptable point clouds.

69. Data Analysis

70. Initialization and approximation Random point S I Fitting line L B-spline curve

71. Growing

72. Knot insertion All points are handled, add a knot where the maximum error occurs Else insert a knot and redistribute all the knots and make them equally spaced

73. Finding projection points Sharp corners

74. Filtering points T

75. Other cases Less control points EMST with wrong topologyVery sharp corner

76. Example

78. Approaches overview Preprocess the point clouds Thin the point clouds (Levin98, Lee00) Point clusters (Lin 04) Map into a digital image (Goshtasby00) Mathematical model Parameteric curves Implicit curves Other methods

79. Fitting unorganized point clouds with active implicit B-spline curves Zhouwang Yang, Jiansong Deng, Falai Chen Visual Computer 2005

84. Example

87. Approaches overview Preprocess the point clouds Thin the point clouds (Levin98, Lee00) Point clusters (Lin 04) Map into a digital image (Goshtasby00) Mathematical model Parameteric curves Implicit curves Other methods

88. Conclusion Complex topology Digital image Implicit curves Tangential flow Initial curves Parameteric curves Implicit curves

89. Problems and future work Knot insertion Foot point compute Singular points Surface reconstruction