Thin plate spline

There are many choices of radial basis function. Past research has demonstrated that polyharmhonic thin plate splines of various order a work best to represent systems governed by the balance equations [4], A polyharmhonic thin plate spline is given by... 

The RFM solution was compiled using 1029 collocation points (Fig. 11.2) and second order (a = 2) thin-plate splines. Figure 11.4 presents a comparison between... 

In order to solve the coupled equation system, Estrada used RFM with a third order thin plate spline function and 740 collocation points arranged in a grid, while Lopez and Osswald also used 740 collocation points randomly arranged throughout the domain5 (Fig. 11.14). The RFM solution was compared to an FDM solution. Four different cases were analyzed ... 

All available ah initio data points are incorporated in our studies and additional calculations were performed to complete the reactive surfaces. After the projection onto the (r, j-coordinate space, the Thin-Plate-Spline method is used to interpolate the 2D-surfaces. This allows their representation on arbitrary grids, which is necessary for the wavepacket propagations. 

Bookstein FL. Prinicpal warps—thin-plate splines and the decomposition of deformations. IEEE Trans Pattern Analysis Mach InteU 1989 ll(6) 567-85. 

So we have a noise model. What kinds of scientific descriptions can we pursue with its aid By analogy with the bending energy of the thin-plate spline, which led to useful formalisms such as time partial warps, the gauge metric of these self-similar processes likewise has an eigenstructure, which 1 have tentatively named the intrinsic warps (IWs) of the mean point configuration. Keep in mind that these exist only in the non-uniform subspace of the overall shape space, though they apply at all scales. 

It is one implication of this approach that more care needs to be taken with the selection of landmark points than is presently the case. The allocation of a landmark halfway between two others, or in the middle of a surface path bounded by some curves, adds information to a data-set even if the landmark is located exactly where a spline driven by the remaining information would place it. It adds, precisely, the information that that location was observed with only digitizing error, so points nearby have a prediction error that is lower simply by virtue of the additional data. Where landmarks are widely spaced, the deformations predicted by, say, a thin-plate spline are far less reliable than where landmarks are precise regardless of the statistics of the landmark locations themselves. 

Bookstein, F.L. (1999) Linear methods for nonlinear maps proemstes fits, thin-plate splines, and the biometric analysis of shape variability. In Brain Warping (ed A. Toga), Academic Press, London, pp. 157-181. Bookstein, F.L. (2000) Creases as local features of deformation grids. Medical Image Analysis, 4 93-110. Bookstein, F.L. (2002) Creases as morphometric characters. In Morphology. Shape, and Phylogeny (eds N. MacLeod and P. Forey), Systematic Association Special Volume Series 64, Taylor Francis, London, pp. 139-174. 

Kent, J.T. and Mardia, K.V. (1994) The link between kriging and thin-plate splines. In Probability, Statistics,... 

Rohlf, F.J. (1994) Thin Plate Spline Relative Warps Analysis (TPS), Version 1.22. Department of Ecology and Evolution, State University of New York at Stony Brook, New York. 

Here, p is a linear polynomial in two variables and - denotes the Euclidean norm. For more details concerning thin plate spline interpolation, due to Duchon [10], and related interpolation methods, we refer to the recent tutorial [17]. 

Hence, the thin plate spline interpolant s in (23) matches current cell average values of Ui, in the Moore neighbourhood of the cell Tg, but not at Tg itself, i.e., we have u(f,g) 7 s g) in general. Now the error indication r]g for the cell Tg is small whenever the reproduction quality of ug by s around the... 

Duchon has shown that possible kernel functions are the radial thin plate splines... 

The name thin plate splines refers to the problem of elasticity as discussed in the introduction. Duchon s idea to identify these functions is based on the concept of fundamental solutions of the iterated Laplace operator... 

Indeed, up to a constant, radial thin plate splines are fundamental solutions of this operator. 

We now turn our attention to the dual formulation of kriging which leads us to linear equations of the same type as those that arise from interpolation with radial basic functions such as thin plate splines. Some of the basic ideas can be found in [4]. The first studies go back to [8],... 

RFM is a collocation method that allows the description of a field of values in a domain and its boundary by a linear interpolation of several coefficients using basic functions. This method makes it possible to interpolate values in the domain and calculate differential operators in the field by knowing some of the values, and hence, solve partial differential equations [2-4], In particular, the basic function that was used is the Thin Plate Spline (TPS) ... 

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