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

Abstract

In [6], an adaptive method to approximate unorganized clouds of points by smooth surfaces based on wavelets has been described. The general fitting algorithm operates on a coarse-tofine basis. It selects on each refinement level in a first step a reduced number of wavelets which are appropriate to represent the features of the data set. In a second step, the fitting surface is constructed as the linear combination of the wavelets which minimizes the distance to the data in a least squares sense. This is followed by a thresholding procedure on the wavelet coefficients to discard those which are too small to contribute much to the surface representation.In this paper, we firstly generalize this strategy to a classically regularized least squares functional by adding a Sobolev norm, taking advantage of the capability of wavelets to characterize Sobolev spaces of even fractional order. After recalling the usual cross-validation technique to determine the involved smoothing parameters, some examples of fitting severely irregularly distributed data, synthetically produced and of geophysical origin, are presented. In order to reduce computational costs, we then introduce a multilevel generalized crossvalidation technique which goes beyond the Sobolev formulation and exploits the hierarchical setting based on wavelets. We illustrate the performance of the new strategy on some geophysical data.

Recognition of Planar Segments in Point Cloud based on Wavelet Transform

“…However, the use of DWT in 3D point cloud processing is relatively new and thus far very limited. Some recent papers reported the utilization of wavelet transform in analysis of 3D geometric texture [26], scattered point cloud structuring [27] and compression [28], as well as in 3D point clouds integration [29]. Nevertheless, to the best of our knowledge, except in our research, wavelet transform has not been previously employed in recognition of planar segments in a point cloud.…”

Spherical Harmonics for Surface Parametrisation and Remeshing

“…where A is the basis matrix, x is the radii vector, b is the coefficient vector, ] is a tuning parameter used to balance the fidelity of the data and the smoothness requirement [30], and Γ is the Tikhonov matrix. Minimising this newly formed functional results in an equivalent solution to the following equation:…”

Scattered data reconstruction by regularization in B-spline and associated wavelet spaces

“…The two methods are used to solve the problem at different scales h = 1/32, 1/64, 1/128, 1/256 (i.e. J = 5, 6,7,8). In the experiment, the starting level J 0 = 3 (for the wavelet method) and the smoothing parameter α = 10 −2 are fixed at all scales.…”

“…Several approximation methods employ a multilevel structure to approximate data efficiently. In particular, a multilevel scheme based on B-splines is proposed in [34] to approximate scattered data; a wavelet-based smoothing method which operates in a coarse-tofine manner to get the fitting function efficiently is suggested in [6]. We mention that the use of uniform B-splines as basis functions for scattered data approximation is not new.…”