Powell-Sabin splines and multiresolution techniques

nrpages: xliii+146status: publishe

Bases for C¹ quadratic splines on Powell-Sabin triangle splits

In this paper we look at different bases for the space S^1_2(Δ_{PS}) of C^1 continuous quadratic splines on Powell-Sabin triangle splits. First, we describe a general framework that leads to the notion of control points and control triangles. This is a useful property that gives insight in the shape of the surface. Then we consider two bases in particular: a normalised B-spline basis, and one based on minimal determining sets. In the normalised B-spline representation the basis functions are all positive, have local support and form a partition of unity. Geometrically these requirements can be interpreted as finding a triangle containing a specific set of Bézier points. We show why the triangle with minimal area is a suitable choice. The approach with minimal determining sets leads to bases that are stable as a function of the smallest angle in the triangulation. The basis functions are not only linearly independent but also local linearly independent. Compared to the normalised B-spline representation, we have given up positivity for other properties. In between these two are other possibilities that depend on the requested properties of the basis functions.status: publishe

Tangent subdivision scheme

In this paper we propose a new subdivision scheme based on uniform Powell-Sabin spline subdivision. For each vertex we have a control triangle tangent to the surface instead of a control point. The main advantage of this scheme is that we can choose the values of the tangent vectors in the initial vertices, which gives more design possibilities. Strictly speaking it is an approximating scheme, the control points for the vertices change each iteration. However, the point where the control triangle is tangent to the surface remains the same. Therefore in practice it is an interpolating scheme. In the regular regions we use the uniform Powell-Sabin rules and we develop subdivision rules for the new vertices in the neighbourhood 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.nrpages: 15status: publishe

Powell-Sabin spline wavelets

Recently we developped 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 triangles. 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 complementary space that captures the details that are lost when going to a coarser resolution. The subdivision scheame 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. Te design of the update step is based on stability considerations. We prove stability for both the scaling functions and the wavelet functions.nrpages: 18status: publishe

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. (C) 2002 Elsevier Science B.V. All rights reserved.status: publishe

Stabilized lifting steps in noise reduction for nonequispaced 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.Proceedings SPIE, vol. 4478status: publishe