Reconstruction of B-spline surfaces from scattered data points

Abstract

We present a new approach for reconstructing a smooth surface from a set of scattered points in three-dimensional (3D) space. Our algorithm first decomposes a given point set into a quadtree-like data structure known as a strip tree. The strip tree is used to fit a set of least squares quadratic surfaces to the data points. These quadratic surfaces are then degree-elevated to bi-cubic surfaces and blended together to form a set of B-spline surfaces that approximates the given point set.

Melting of tantalum at high pressure determined by angle dispersive x-ray diffraction in a double-sided laser-heated diamond-anvil cell

“…[43] and [44] by using a B-spline approximation [45]. Figure 4(b) shows the obta ined radial temperature profile in the same sample in those directions that minimize (y-axis of Fig.…”

Pattern of recent vertical crustal movements in Canada

“…The method of SPAA avoids many of the limitations associated with traditional approaches of data fitting such as the requirement that the data be of point values, as it is seen in MQ method (Holdahl and Hardy, 1978) and in B-splines (Gregorski et al, 2000;Greiner and Hormann, 1996); or that they should be on grid or at least well distributed (Zhou et al, 1997). SPAA is not restricted to low degree polynomial (as it is seen in Carrera et al, 1991;Vaníček and Nagy, 1981) and the smoothness of the resulting function is guaranteed along the patch boundaries by imposing the continuity and smoothness (zero and first derivative) constraints and the degree of smoothness can be simply controlled by the number and degree of differentiability constraints in the model, which results in a smooth surface.…”

Multilevel regularization of wavelet based fitting of scattered data ? some experiments

“…There is no explicit grid construction: the data representation is in this sense independent of the original data points. These benefits have made approaches on structured grids a well established, much followed and universally applicable procedure, covering the gamut from strictly mathematical analysis to applications in various areas of sciences, see, e.g., [15,[18][19][20]23,24,29,30,33,34,37,44]. Employing wavelets as basis functions for the data representation provides additional features in the problem formulation regarding computational efficiency and sparseness of the representation, such as good conditioning and a natural built-in potential for adaptivity (see, e.g., [7] for an introduction to basic wavelet theory).…”