## Surface Fitting with Hierarchical Splines (1995)

Venue: | ACM Transactions on Graphics |

Citations: | 44 - 1 self |

### BibTeX

@ARTICLE{Forsey95surfacefitting,

author = {David R. Forsey and Richard H. Bartels},

title = {Surface Fitting with Hierarchical Splines},

journal = {ACM Transactions on Graphics},

year = {1995},

volume = {14},

pages = {134--161}

}

### Years of Citing Articles

### OpenURL

### Abstract

We consider the fitting of tensor product parametric spline surfaces to gridded data. The continuity of the surface is provided by the basis chosen. When tensor product splines are used with gridded data, the surface fitting problem decomposes into a sequence of curve fitting processes, making the computations particularly e#cient. The use of a hierarchical representation for the surface adds further e#ciency by adaptively decomposing the fitting process into subproblems involving only a portion of the data. Hierarchy also provides a means of storing the resulting surface in a compressed format. Our approach is compared to multiresolution analysis and the use of wavelets. 1 Introduction In [9] an adaptive process was presented for fitting surface data with a geometrically continuous collection of rectangular Bezier patches. The adaptivity resulted from fitting a portion of the data with a patch, testing the fit for satisfaction within a given tolerance, and subdividing the patch if th...

### Citations

1093 |
A Practical Guide to Splines
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- 1978
(Show Context)
Citation Context ... such as laser rangers, CAT imagery systems, and optical scanners. Advantage can be taken of this format by reducing the surface fitting problem into a sequence of much smaller curve fitting problems =-=[1, 3]-=-. Bezier patches require that a significant number of constraints be imposed on the control vertices to piece patches together in a continuous composite surface. The broader class of multi-patch tenso... |

696 |
An introduction to wavelets
- Chui
- 1992
(Show Context)
Citation Context ...rtion of exactly one knot at the midpoint between every two existing knots. In this restricted setting, the fitting process takes place in the setting usually encountered for multiresolution analysis =-=[2]-=-. Letting B 0,i (u) = B i (u) stand for the canonical B-splines, we note that B 0,i (u) = B 0,0 (u + i), where B 0,0 (u) is the canonical B-spline with support on the interval [0, #] (knots 0, 1, . . ... |

238 |
Kronecker Products and Matrix Calculus With Applications
- Graham
- 1981
(Show Context)
Citation Context ...ows and n columns, each block being a copy of B multiplied by an element of C: 2 6 6 4 C 0 (v 0 )B C 1 (v 0 )B C 0 (v 1 )B . . . . . . . . . C n (v N )B 3 7 7 5 3 Kronecker Products A useful notation =-=[6]-=- is that of the Kronecker product. If F = 2 6 6 6 4 f 0,0 f 0,1 f 0,# f 1,0 f 1,1 f 1,# . . . . . . . . . f #,0 f #,1 f #,# 3 7 7 7 5 (8) is any matrix and G is any other matrix, then F #G = 2 6 6 4 f... |

41 |
An adaptive subdivision method for surface-fitting from sampled data
- Schmitt, Barsky, et al.
- 1986
(Show Context)
Citation Context ...n of the data. Hierarchy also provides a means of storing the resulting surface in a compressed format. Our approach is compared to multiresolution analysis and the use of wavelets. 1 Introduction In =-=[9]-=- an adaptive process was presented for fitting surface data with a geometrically continuous collection of rectangular Bezier patches. The adaptivity resulted from fitting a portion of the data with a ... |

18 |
rken, Spline-wavelets of minimal support
- Lyche, M
- 1992
(Show Context)
Citation Context ...L (u), f(u) - g J-1 (u) -s- g J-L (u)#|s2 0 for a given tolerance # (using the coe#cients d), or an equally useful approximation |#f(u) - f J (u), f(u) - f J (u)#|s2 1 (using the coe#cients c). 15 In =-=[8]-=- the Multiresolution Analysis Setting has been extended fully to B-splines. The nested spaces V # j consist of any that can be composed by arbitrary knot insertions. Algorithms are given for computing... |

1 |
An Algorithm for Least-Squares Fitting of Cubic Spline Surfaces to Functions on a Rectilinear Mesh over a Rectangle
- Dierckx
- 1977
(Show Context)
Citation Context ... such as laser rangers, CAT imagery systems, and optical scanners. Advantage can be taken of this format by reducing the surface fitting problem into a sequence of much smaller curve fitting problems =-=[1, 3]-=-. Bezier patches require that a significant number of constraints be imposed on the control vertices to piece patches together in a continuous composite surface. The broader class of multi-patch tenso... |

1 |
Hierarchical B-Spline Refinement. Proceeding of SIGGRAPH '88
- Forsey, Bartels
(Show Context)
Citation Context ... surfaces; e.g. B-splines, #-splines, or their rational counterparts, provide continuity without the imposition of constraints in the least squares fitting process. The hierarchical representation of =-=[4]-=- allows an adaptive approach to the fitting process. When areas of large scale data have been fit within a specified tolerance by a surface having a certain level of refinement, there may remain isola... |

1 | Matrix-nullspace Wavelet Construction
- Sivalingam, Bartels
- 1994
(Show Context)
Citation Context ...ted B-wavelets in this setting. A further construction method for computing spline wavelets that extends to discrete inner products; i.e., P i f(u i )g(u i ) instead of R +# -# f(u)g(u)du is given in =-=[10]-=-. This is the setting that corresponds to the surface fitting approach we are using. However, three obstacles stand in our way of obtaining our surface fits via B-wavelets: . In our setting we wish to... |

1 |
Adaptive Hierarchical Fitting of Curves and Surfaces
- Sreckovic
- 1992
(Show Context)
Citation Context ...e surfaces produced from midpoint refinements, and Plate 19 shows an enlarged view of the final fit in Plate 18. Plate 20 shows, in wire frame, a set of data used for adaptive, nonmidpoint refinement =-=[12]-=-. The surface to be fit is flat save for a spike in the center. Plate 21 shows the patch structure of the resulting fit, Plate 22 shows normal vectors for the resulting fit, and Plate 23 shows the fit... |