Evelyne Vanraes

Stabilised wavelet transforms for non-equispaced data smoothing

This paper discusses wavelet thresholding in smoothing from non-equispaced, noisy data in one dimension. To deal with the irregularity of the grid we use the so-called second generation wavelets, based on the lifting scheme. The lifting scheme itself leads to a grid-adaptive wavelet transform. We explain that a good numerical condition is an absolute requisite for successful thresholding. If this condition is not satisfied the output signal can show an arbitrary bias. We examine the nature and origin of stability problems in second generation wavelet transforms. The investigation concentrates on lifting with interpolating prediction, but the conclusions are extendible. The stability problem is a cumulated effect of the three successive steps in a lifting scheme: split, predict and update. The paper proposes three ways to stabilise the second generation wavelet transform. The first is a change in update and reduces the influence of the previous steps. The second is a change in prediction and operates on the interval boundaries. The third is a change in splitting procedure and concentrates on the irregularity of the data points. Illustrations show that reconstruction from thresholded coefficients with this stabilised second generation wavelet transform leads to smooth and close fits.

POWELL–SABIN SPLINE WAVELETS

Recently we developed a subdivision scheme for Powell–Sabin splines. It is a triadic scheme and it is general in the sense that it is not restricted to uniform triangles, the vertices must not have valence six and there are no restrictions on the initial triangulation. A sequence of nested spaces or multiresolution analysis can be associated with the base triangulation. In this paper we use the lifting scheme to construct basis functions for the complement space that captures the details that are lost when going to a coarser resolution. The subdivision scheme appears as the first lifting step or prediction step. A second lifting step, the update, is used to achieve certain properties for the complement spaces and the wavelet functions such as orthogonality and vanishing moments. The design of the update step is based on stability considerations. We prove stability for both the scaling functions and the wavelet functions.

Automatic construction of control triangles for subdivided Powell-Sabin splines

In this paper we present an algorithm for calculating the B-spline representation of a Powell-Sabin spline surface on a refinement of the given triangulation. The resulting subdivision scheme is a √3 scheme; a new vertex is added inside every original triangle. Applying the √3 scheme twice yields a triadic scheme, every original edge is split into three new edges, but special care is needed at the boundaries. The scheme is numerically stable and generally applicable, there are no restrictions on the initial triangulation.

Uniform Powell--Sabin spline wavelets

This paper discusses how the subdivision scheme for uniform Powell-Sabin spline surfaces makes it possible to place those surfaces in a multiresolution context. We first show that the basis functions are translates and dilates of one vector of scaling functions. This defines a sequence of nested spaces. We then use the subdivision scheme as the prediction step in the lifting scheme and add an update step to construct wavelets that describe a sequence of complement spaces. Finally, as an example application, we use the new wavelet transform to reduce noise on a uniform Powell-Sabin spline surface.

A tangent subdivision scheme

In this article, we propose a new subdivision scheme based on uniform Powell-Sabin spline subdivision. It belongs to the class of vector subdivision schemes; for each vertex, we have three control points that form a control triangle tangent to the surface instead of one control point. The main advantage of this scheme is that we can choose the values of the normals in the initial vertices which results in more design possibilities. At first sight, it is an approximating scheme because the control points change each iteration. However, the point where the control triangle is tangent to the surface remains the same. Therefore, it is an interpolating scheme. In the regular regions, we use the uniform Powell-Sabin rules, and we develop additional subdivision rules for the new vertices in the neighborhood of extraordinary vertices. The scheme yields C1 continuous surfaces. We also do the convergence analysis based on the eigenproperties of the subdivision matrix and the properties of the characteristic map.

Dyadic and /spl radic/3-subdivision for uniform Powell-Sabin splines

We give two different possibilities for subdivision of Powell-Sabin spline surfaces on uniform triangulations. In the first case, dyadic subdivision, a new vertex is introduced on each edge between two old vertices. In the second case, /spl radic/3-subdivision, a new vertex is introduced in the center of each triangle of the triangulation. We give subdivision rules to find the new control points of the refined surface for both cases.

Stabilized lifting steps in noise reduction for non-equispaced samples

This paper discusses wavelet thresholding in smoothing from non-equispaced, noisy data in one dimension. To deal with the irregularity of the grid we use so called second generation wavelets, based on the lifting scheme. We explain that a good numerical condition is an absolute requisite for successful thresholding. If this condition is not satisfied the output signal can show an arbitrary bias. We examine the nature and origin of stability problems in second generation wavelet transforms. The investigation concentrates on lifting with interpolating prediction, but the conclusions are extendible. The stability problem is a cumulated effect of the three successive steps in a lifting scheme: split, predict and update. The paper proposes three ways to stabilize the second generation wavelet transform. The first is a change in update and reduces the influence of the previous steps. The second is a change in prediction and operates on the interval boundaries. The third is a change in splitting procedure and concentrates on the irregularity of the data points. Illustrations show that reconstruction from thresholded coefficients with this stabilized second generation wavelet transform leads to smooth and close fits.