Non-equispaced B-spline wavelets

This paper has three main contributions. The first is the construction of wavelet transforms from B-spline scaling functions defined on a grid of non-equispaced knots. The new construction extends the equispaced, biorthogonal, compactly supported Cohen-Daubechies-Feauveau wavelets. The new construction is based on the factorisation of wavelet transforms into lifting steps. The second and third contributions are new insights on how to use these and other wavelets in statistical applications. The second contribution is related to the bias of a wavelet representation. It is investigated how the fine scaling coefficients should be derived from the observations. In the context of equispaced data, it is common practice to simply take the observations as fine scale coefficients. It is argued in this paper that this is not acceptable for non-interpolating wavelets on non-equidistant data. Finally, the third contribution is the study of the variance in a non-orthogonal wavelet transform in a new framework, replacing the numerical condition as a measure for non-orthogonality. By controlling the variances of the reconstruction from the wavelet coefficients, the new framework allows us to design wavelet transforms on irregular point sets with a focus on their use for smoothing or other applications in statistics.Comment: 42 pages, 2 figure

Nondyadic and nonlinear multiresolution image approximations

This thesis focuses on the development of novel multiresolution image approximations. Specifically, we present two kinds of generalization of multiresolution techniques: image reduction for arbitrary scales, and nonlinear approximations using other metrics than the standard Euclidean one. Traditional multiresolution decompositions are restricted to dyadic scales. As first contribution of this thesis, we develop a method that goes beyond this restriction and that is well suited to arbitrary scale-change computations. The key component is a new and numerically exact algorithm for computing inner products between a continuously defined signal and B-splines of any order and of arbitrary sizes. The technique can also be applied for non-uniform to uniform grid conversion, which is another approximation problem where our method excels. Main applications are resampling and signal reconstruction. Although simple to implement, least-squares approximations lead to artifacts that could be reduced if nonlinear methods would be used instead. The second contribution of the thesis is the development of nonlinear spline pyramids that are optimal for lp-norms. First, we introduce a Banach-space formulation of the problem and show that the solution is well defined. Second, we compute the lp-approximation thanks to an iterative optimization algorithm based on digital filtering. We conclude that l1-approximations reduce the artifacts that are inherent to least-squares methods; in particular, edge blurring and ringing. In addition, we observe that the error of l1-approximations is sparser. Finally, we derive an exact formula for the asymptotic Lp-error; this result justifies using the least-squares approximation as initial solution for the iterative optimization algorithm when the degree of the spline is even; otherwise, one has to include an appropriate correction term. The theoretical background of the thesis includes the modelisation of images in a continuous/discrete formalism and takes advantage of the approximation theory of linear shift-invariant operators. We have chosen B-splines as basis functions because of their nice properties. We also propose a new graphical formalism that links B-splines, finite differences, differential operators, and arbitrary scale changes

Multiresolution analysis as an approach for tool path planning in NC machining

Wavelets permit multiresolution analysis of curves and surfaces. A complex curve can be decomposed using wavelet theory into lower resolution curves. The low-resolution (coarse) curves are similar to rough-cuts and high-resolution (fine) curves to finish-cuts in numerical controlled (NC) machining.;In this project, we investigate the applicability of multiresolution analysis using B-spline wavelets to NC machining of contoured 2D objects. High-resolution curves are used close to the object boundary similar to conventional offsetting, while lower resolution curves, straight lines and circular arcs are used farther away from the object boundary.;Experimental results indicate that wavelet-based multiresolution tool path planning improves machining efficiency. Tool path length is reduced, sharp corners are smoothed out thereby reducing uncut areas and larger tools can be selected for rough-cuts

Splines Are Universal Solutions of Linear Inverse Problems with Generalized TV Regularization

Splines come in a variety of flavors that can be characterized in terms of some differential operator L. The simplest piecewise-constant model corresponds to the derivative operator. Likewise, one can extend the traditional notion of total variation by considering more general operators than the derivative. This results in the definitions of a generalized total variation seminorm and its corresponding native space, which is further identified as the direct sum of two Banach spaces. We then prove that the minimization of the generalized total variation (gTV), subject to some arbitrary (convex) consistency constraints on the linear measurements of the signal, admits nonuniform L-spline solutions with fewer knots than the number of measurements. This shows that nonuniform splines are universal solutions of continuous-domain linear inverse problems with LASSO, $L _{ 1 }$ , or total-variationlike regularization constraints. Remarkably, the type of spline is fully determined by the choice of L and does not depend on the actual nature of the measurements

A multiscale method for the double layer potential equation on a polyhedron

This paper is concerned with the numerical solution of the double layer potential equation on polyhedra. Specifically, we consider collocation schemes based on multiscale decompositions of piecewise linear finite element spaces defined on polyhedra. An essential difficulty is that the resulting linear systems are not sparse. However, for uniform grids and periodic problems one can show that the use of multiscale bases gives rise to matrices that can be well approximated by sparse matrices in such a way that the solutions to the perturbed equations exhibits still sufficient accuracy. Our objective is to explore to what extent the presence of corners and edges in the domain as well as the lack of uniform discretizations affects the performance of such schemes. Here we propose a concrete algorithm, describe its ingredients, discuss some consequences, future perspectives, and open questions, and present the results of numerical experiments for several test domains including non-convex domains

Adaptive Scattered Data Fitting with Tensor Product Spline-Wavelets

The core of the work we present here is an algorithm that constructs a least squares approximation to a given set of unorganized points. The approximation is expressed as a linear combination of particular B-spline wavelets. It implies a multiresolution setting which constructs a hierarchy of approximations to the data with increasing level of detail, proceeding from coarsest to finest scales. It allows for an efficient selection of the degrees of freedom of the problem and avoids the introduction of an artificial uniform grid. In fact, an analysis of the data can be done at each of the scales of the hierarchy, which can be used to select adaptively a set of wavelets that can represent economically the characteristics of the cloud of points in the next level of detail. The data adaption of our method is twofold, as it takes into account both horizontal distribution and vertical irregularities of data. This strategy can lead to a striking reduction of the problem complexity. Furthermore, among the possible ways to achieve a multiscale formulation, the wavelet approach shows additional advantages, based on good conditioning properties and level-wise orthogonality. We exploit these features to enhance the efficiency of iterative solution methods for the system of normal equations of the problem. The combination of multiresolution adaptivity with the numerical properties of the wavelet basis gives rise to an algorithm well suited to cope with problems requiring fast solution methods. We illustrate this by means of numerical experiments that compare the performance of the method on various data sets working with different multi-resolution bases. Afterwards, we use the equivalence relation between wavelets and Besov spaces to formulate the problem of data fitting with regularization. We find that the multiscale formulation allows for a flexible and efficient treatment of some aspects of this problem. Moreover, we study the problem known as robust fitting, in which the data is assumed to be corrupted by wrong measurements or outliers. We compare classical methods based on re-weighting of residuals to our setting in which the wavelet representation of the data computed by our algorithm is used to locate the outliers. As a final application that couples two of the main applications of wavelets (data analysis and operator equations), we propose the use of this least squares data fitting method to evaluate the non-linear term in the wavelet-Galerkin formulation of non-linear PDE problems. At the end of this thesis we discuss efficient implementation issues, with a special interest in the interplay between solution methods and data structures

Nanoelectronic COupled problems solutions - nanoCOPS: modelling, multirate, model order reduction, uncertainty quantification, fast fault simulation

The FP7 project nanoCOPS derives new methods for simulation during development of designs of integrated products. It covers advanced simulation techniques for electromagnetics with feedback couplings to electronic circuits, heat and stress. It is inspired by interest from semiconductor industry and by a simulation tool vendor in electronic design automation. The project is on-going and the paper presents the outcomes achieved after the first half of the project duration