Multigrid methods and stationary subdivisions

Multigridmethods are fast iterative solvers for sparse large ill-conditioned linear systems of equations derived, for instance, via discretization of PDEs in fluid dynamics, electrostatics and continuummechanics problems. Subdivision schemes are simple iterative algorithms for generation of smooth curves and surfaces with applications in 3D computer graphics and animation industry. This thesis presents the first definition and analysis of subdivision based multigrid methods. The main goal is to improve the convergence rate and the computational cost of multigrid taking advantage of the reproduction and regularity properties of underlying subdivision. The analysis focuses on the grid transfer operators appearing at the coarse grid correction step in the multigrid procedure. The convergence of multigrid is expressed in terms of algebraic properties of the trigonometric polynomial associated to the grid transfer operator. We interpreter the coarse-to-fine grid transfer operator as one step of subdivision. We reformulate the algebraic properties ensuring multigrid convergence in terms of regularity and generation properties of subdivision. The theoretical analysis is supported by numerical experiments for both algebraic and geometric multigrid. The numerical tests with the bivariate anisotropic Laplacian ask for subdivision schemes with anisotropic dilation. We construct a family of interpolatory subdivision schemes with such dilation which are optimal in terms of the size of the support versus their polynomial generation properties. The numerical tests confirmthe validity of our theoretical analysis

Beyond B-splines: Exponential pseudo-splines and subdivision schemes reproducing exponential polynomials

The main goal of this paper is to present some generalizations of polynomial B-splines, which include exponential B-splines and the larger family of exponential pseudo-splines. We especially focus on their connections to subdivision schemes. In addition, we generalize a well-known result on the approximation order of exponential pseudo-splines, providing conditions to establish the approximation order of nonstationary subdivision schemes reproducing spaces of exponential polynomial function

Longitudinal analysis of three-dimensional facial shape data

Shape data encompass all the information that is left to describe a shape following removal of location, rotation and scale effects. Much work has been done in the analysis of two-dimensional shapes depicted by anatomical landmarks placed at points of importance. Less has been carried out in the area of three-dimensional shapes, particularly in terms of growth or change over time. This thesis considers the analysis of such longitudinal three-dimensional shape data. In doing so, two well established but normally unrelated areas of Statistics are brought together: those of longitudinal data analysis (specifically, linear mixed effects models) and shape analysis. A recently proposed method of analysing longitudinal high-dimensional data is presented in a novel application within the area of shape analysis, illustrated by a study comparing the facial shapes of cleft-lip and palate children with controls as they grow from three months to two years of age. Both anatomical landmarks and facial curves are considered. Chapter 1 broadly introduces the areas of shape analysis, linear mixed effects models and dimension reduction. Standard methods for measuring shapes are introduced, along with the difficulties inherent in analysing the resulting data. A broad overview of the methods of aligning individual shapes to remove the unwanted effects of location, rotation and scale is given, along with related geometrical issues in terms of the high-dimensional space in which a set of shapes resides. A general introduction to linear mixed effects models compares and contrasts them with simple linear models, explaining the reasons behind using them and presenting the different specifications of the conditional and marginal models. The area of dimension reduction is touched upon, specifically introducing B-splines and principal components analysis, with reference to the analysis of curves consisting of many points at small increments to one another. The data from the cleft-lip and palate study are introduced, along with a discussion of the primary interest of the analysis and the issue of missing data. Chapter 2 presents the statistical definition of a shape and introduces the area of statistical shape analysis in detail, specifically presenting the technicalities of shape space and distances, and methods such as Procrustes alignment of a set of shapes to remove unwanted effects. The concept of tangent coordinates is introduced as a projection of shape data into a Euclidean space, to enable the use of multivariate methods, and an outline given of thin-plate splines and deformations for the analysis of surfaces. Recent literature in the area of shape analysis is presented. Further recent literature addressing the modelling of growth in shapes is presented in Chapter 3, which goes on to discuss the use of linear mixed models on univariate and multivariate longitudinal data. The difficulties of applying mixed models to multivariate data are discussed and a recently proposed alternative method introduced, which involves fitting mixed models to the responses on pairs of outcomes rather than the full set. A description of the R function written as part of this thesis to fit such pairwise models follows, and this is applied to simulated triangles and quadrilaterals as an illustration. The initial application of the pairwise method to the cleft-lip and palate landmark data is presented in Chapter 4. The landmarks are described and the models are fitted to the tangent coordinate responses with different covariance structures for the random effects. The problems that arise and the deficiencies of the fitted models are extensively discussed. Chapter 5 goes on to address the issues raised in Chapter 4. A method of aligning the individual shapes based upon a subset of landmarks is suggested, along with a model that assumes independence of coordinates between dimensions but correlation within, and the benefits of these approaches compared. A simulation study is carried out to investigate the reasons behind and effects of random effects correlations that are estimated as being close to one, concluding that the problem lies in small variances that are poorly estimated, but that this is unlikely to be of severe detriment to the fixed effects estimates. A method of taking the principal components of the tangent coordinates is suggested, where the model responses are the principal components scores, and this proves to be the most appropriate way of applying the pairwise models in terms of model fit and computational efficiency. In Chapter 6, recent literature on the topic of curve analysis is presented, along with the way the facial curves are measured and the need for dimension reduction. Two methods are presented to this end: B-splines and principal components analysis, with the former suffering similar problems to the landmark analyses in terms of poorly estimated random effects variances, and the latter proving more successful. The application of the pairwise models to the principal components scores of the tangent coordinates provides a detailed analysis of the cleft-lip and palate data. Issues surrounding model comparison are addressed in Chapter 7, with several hypothesis tests presented and applied to simulated data. Drawbacks with some of the tests when applied to high dimensional or longitudinal data result in poor performance, but a method suggested by Faraway (1997) and a modification of the likelihood ratio test, both using bootstrapping, show similarly successful results. These are subsequently used to test for any differences in the time trends for the cleft and control groups post-surgery and find that there are significant differences. Condensed forms of this thesis have been presented at invited seminars and international conferences, and may be found in published form in Barry & Bowman (2006), Barry & Bowman (2007) and Barry & Bowman (2008)

New Techniques for the Modeling, Processing and Visualization of Surfaces and Volumes

With the advent of powerful 3D acquisition technology, there is a growing demand for the modeling, processing, and visualization of surfaces and volumes. The proposed methods must be efficient and robust, and they must be able to extract the essential structure of the data and to easily and quickly convey the most significant information to a human observer. Independent of the specific nature of the data, the following fundamental problems can be identified: shape reconstruction from discrete samples, data analysis, and data compression. This thesis presents several novel solutions to these problems for surfaces (Part I) and volumes (Part II). For surfaces, we adopt the well-known triangle mesh representation and develop new algorithms for discrete curvature estimation,detection of feature lines, and line-art rendering (Chapter 3), for connectivity encoding (Chapter 4), and for topology preserving compression of 2D vector fields (Chapter 5). For volumes, that are often given as discrete samples, we base our approach for reconstruction and visualization on the use of new trivariate spline spaces on a certain tetrahedral partition. We study the properties of the new spline spaces (Chapter 7) and present efficient algorithms for reconstruction and visualization by iso-surface rendering for both, regularly (Chapter 8) and irregularly (Chapter 9) distributed data samples