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## Interpolation by Regularized Spline with Tension: I. Theory and Implementation ~

### Citations

3981 |
Handbook of Mathematical Functions
- Abramowitz, Stegun
- 1970
(Show Context)
Citation Context ... integration, term by term, and subsequent resummationsRegularized Spline with Tension: I 653 R(x, x Ijl) = R(rj) - '~ In 4~r + E~ k + Ce , d = 2 (A7) 1 where E~ is the exponential integral function (=-=Abramowitz and Stegun, 1964-=-) and Ce is the Euler constant (CE = 0.577215...). The multiplicative constant ~2/4~r can be omitted because it can always be absorbed into coefficients { ),: } in Eq. (5) so that the basis function i... |

350 |
Scattered data interpolation: Tests of some method.
- Franke
- 1982
(Show Context)
Citation Context ... accuracy of CRS was compared with other methods using the bivariate function proposed by Franke (1982a). This function was chosen because it was used in tests of several interpolation methods (e.g., =-=Franke, 1982-=-a; Renka andsRegularized Spline with Tension: I 647 60 I I [ i iii~ ] 4.0 , • [ • k ,F ~ ~ r~ I ] ¸¸ :!i 2o • ,)ili i :• ii! 0.0 0.0 2.0 4.0 6.0 Fig. 1. Segmented processing. Interpolation function in... |

280 |
Multiquadric Equations for Topography and Other Irregular Surfaces
- Hardy
- 1971
(Show Context)
Citation Context ...ies in application to large datasets. An approach to overcome this problem was suggested by Franke (1982b), who has proposed segmented processing with smooth overlaps. Hardy's multiquadric functions (=-=Hardy 1971-=-; Hardy, 1990) belong to the same class of methods and a connection of TPS to kriging has been proved (e.g., Wahba, 1990). In this paper, further development of this approach is presented which widens... |

117 |
Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces,
- Duchon
- 1976
(Show Context)
Citation Context ...ce the computational cost of the numerical solution. Numerical solution of the variational condition is not needed when an explicit solution is known, as is the case with the thin plate spline (TPS) (=-=Duchon, 1976-=-; Dubrule, 1984); however, this function has not been used widely in geosciences because of its global character and consequent difficulties in application to large datasets. An approach to overcome t... |

98 | Interpolation by regularized spline with tension: I. theory and implementation. Theory and implementation - Mitasova, Mitas - 1993 |

94 | Machine contouring using minimum curvature:
- Briggs
- 1974
(Show Context)
Citation Context ...othness of this function. Probably the best known method obtained within this approach is minimum curvature (MINQ) interpolation with numerical finitedifference solution of the variational condition (=-=Briggs, 1974-=-). The main drawback of this method is the generation of overshoots, which appear in regions with rapid change of gradient; in addition, the numerical solution can be computationally costly. The metho... |

81 |
Theory and Applications of the Multiquadric-Biharmonic Method
- Hardy
(Show Context)
Citation Context ...cation to large datasets. An approach to overcome this problem was suggested by Franke (1982b), who has proposed segmented processing with smooth overlaps. Hardy's multiquadric functions (Hardy 1971; =-=Hardy, 1990-=-) belong to the same class of methods and a connection of TPS to kriging has been proved (e.g., Wahba, 1990). In this paper, further development of this approach is presented which widens the range of... |

39 |
Smooth interpolation of scattered data by local thin plate splines.
- Franke
- 1982
(Show Context)
Citation Context ... accuracy of CRS was compared with other methods using the bivariate function proposed by Franke (1982a). This function was chosen because it was used in tests of several interpolation methods (e.g., =-=Franke, 1982-=-a; Renka andsRegularized Spline with Tension: I 647 60 I I [ i iii~ ] 4.0 , • [ • k ,F ~ ~ r~ I ] ¸¸ :!i 2o • ,)ili i :• ii! 0.0 0.0 2.0 4.0 6.0 Fig. 1. Segmented processing. Interpolation function in... |

18 |
Terrain and surface modelling systems: theory and practice
- McCullagh
- 1988
(Show Context)
Citation Context ...he coordinate transformation for d = 3 is straightforward. Segmentation for Large Datasets Processing The quality of global interpolation methods (as TPS) has been recognized in several papers (e.g., =-=McCullagh, 1988-=-; Hutchinson, 1989); however, they were considered to have limited practical use. The reason is that the computa646 Mitfi~ofi and Mitfi~ tional demands are proportional to N 3 (N is number of data po... |

15 |
R.: Digital Terrain Modelling: A
- Moore, Grayson, et al.
- 1991
(Show Context)
Citation Context ...ibravsk~i cesta 9, Bratislava, Czechoslovakia. 641 0882 8121/93/0800-0641 $07.00/I © 1993 lntemationai Association for Mathematical Geologys642 Mit,'i~,o,'t and Mitfi~, for dense grids are necessary (=-=Moore et al., 1991-=-). Many traditional methods are not sufficient or require considerable additional manual work and empirical adjustments for these types of applications. In an effort to handle these problems, growing ... |

15 |
Method for smooth approximation of data,
- Talmi, Gilat
- 1977
(Show Context)
Citation Context ... a d-dimensional space, find a function z (x) which fulfills the conditions Z(X [jl) = g [j], j = 1 ..... N (3) E2(Z) = minimum (4) Under fairly general conditions, the problem has a unique solution (=-=Talmi and Gilat, 1977-=-), which can be expressed as the sum of two components The "trend" function T(x) is given by N z(x) = T(x) + ~] XjR(x, x Ijl) (5) j=l M T(x) = ~ ajl(x) (6) l=t where { fi(x)} is a set of linearly inde... |

13 |
Comparing Splines and Kriging
- Dubrule
- 1984
(Show Context)
Citation Context ...tional cost of the numerical solution. Numerical solution of the variational condition is not needed when an explicit solution is known, as is the case with the thin plate spline (TPS) (Duchon, 1976; =-=Dubrule, 1984-=-); however, this function has not been used widely in geosciences because of its global character and consequent difficulties in application to large datasets. An approach to overcome this problem was... |

12 |
Thin plate splines with tension
- Franke
- 1985
(Show Context)
Citation Context ...ients. In this case, overshoots appear due to the plate's stiffness. The stiffness of the plate can be suppressed by including first derivatives to the smooth seminorm that leads to TPS with tension (=-=Franke, 1985-=-; Mit~i~ and Mit~i~ovfi, 1988), which enables the character of the interpolation surface to be tuned from thin plate to membrane. However, for various applications, especially where analysis of the su... |

11 |
A method for construction of surfaces under tension
- Nielson, Franke
- 1984
(Show Context)
Citation Context ...ted on a bivariate function with fault, originally presented by Nielson and Franke (1984). The function was sampled over 33 scattered data points within the unit square (data were taken from Table 1; =-=Nielson and Franke, 1984-=-). The result of applying the thin plate spline to these data (Fig. 2) is similar to that obtained by several other standard methods (Nielson and Franke, 1984). The resulting surface exhibits a strong... |

5 |
US Army Corps of Engineers. Construction Engineering Research Laboratories
- Manual
- 1993
(Show Context)
Citation Context ....). Presented method was incorporated into the public domain software for geographic information systems GRASS (Geographic Resources Analysis Support System) as command s.surf.tps (GRASS4.1 Reference =-=Manual, 1993-=-). It has been already successfully used for many applications, e.g., terrain modeling and analysis for the erosion risk assessment (Mit~ov~i and Hofierka, 1993), interpolation of climatic data and cl... |

3 | Surface modeling and analysis using global spline functions: Student scientific projects competition work, Comenius - Hofierka, S'uri - 1989 |

1 |
Regularized Spline with Tension: I 655
- Gradshteyn, Ryzhik
- 1980
(Show Context)
Citation Context ... basis function is given by Eq. (10). The trivariate case is even more simple. After transformation to spherical variables and integration over angular variables, the remaining integral is tabulated (=-=Gradshteyn and Ryzhik, 1980-=-) and the basis function is given by R(x, x Ijl) = R(rj) = ~ erf - , d = 3 (A8) where eft(-) is the error function. The multiplicative constant ~o 3/4~r can again be omitted and the basis function is ... |

1 |
A New Procedure tbr Gridding Elevation and Stream Line Data with Automatic Removal of Spurious Pits
- Hutchinson
- 1989
(Show Context)
Citation Context ...nsformation for d = 3 is straightforward. Segmentation for Large Datasets Processing The quality of global interpolation methods (as TPS) has been recognized in several papers (e.g., McCullagh, 1988; =-=Hutchinson, 1989-=-); however, they were considered to have limited practical use. The reason is that the computa646 Mitfi~ofi and Mitfi~ tional demands are proportional to N 3 (N is number of data points). This scalin... |

1 | General Variational Approach to the Interpolation Problem - Miti, Mitiov - 1988 |

1 | Interpolation Methods for Geological Information System - Mitfig, Novik - 1985 |

1 | Cartographic Aspects of Computer Surface Modeling - Mitfigowl - 1987 |

1 | Cartographic Surface and Volume Modeling Using Splines with Tension: Geodetick~ a kartograficks~ obzor, v - Mitigovi, Hofierka, et al. - 1990 |

1 |
A Triangle-Based C
- Renka, Cline
- 1984
(Show Context)
Citation Context ...s for interpolation is larger than the given K,n~, (see text). Cline, 1984). The function is sampled by 100 scattered data points within the unit square (their coordinates were digitized from Fig. 5; =-=Renka and Cline, 1984-=-). The accuracy of the interpolation was tested at 33 x 33 regular grid points and compared with results obtained from other methods (Table 1). From these results, at least for this type of surface, C... |

1 |
Spline Models tbr Observational Data
- Wahba
- 1990
(Show Context)
Citation Context ...posed segmented processing with smooth overlaps. Hardy's multiquadric functions (Hardy 1971; Hardy, 1990) belong to the same class of methods and a connection of TPS to kriging has been proved (e.g., =-=Wahba, 1990-=-). In this paper, further development of this approach is presented which widens the range of its application to bivariate and trivariate interpolation with direct estimation of first and second order... |