The linear algebra of interpolation with finite applications giving computational methods for multivariate polynomials

Thesis (Ph.D.) University of Alaska Fairbanks, 1988Linear representation and the duality of the biorthonormality relationship express the linear algebra of interpolation by way of the evaluation mapping. In the finite case the standard bases relate the maps to Gramian matrices. Five equivalent conditions on these objects are found which characterize the solution of the interpolation problem. This algebra succinctly describes the solution space of ordinary linear initial value problems. Multivariate polynomial spaces and multidimensional node sets are described by multi-index sets. Geometric considerations of normalization and dimensionality lead to cardinal bases for Lagrange interpolation on regular node sets. More general Hermite functional sets can also be solved by generalized Newton methods using geometry and multi-indices. Extended to countably infinite spaces, the method calls upon theorems of modern analysis

High-order adaptive methods for computing invariant manifolds of maps

The author presents efficient and accurate numerical methods for computing invariant manifolds of maps which arise in the study of dynamical systems. In order to decrease the number of points needed to compute a given curve/surface, he proposes using higher-order interpolation/approximation techniques from geometric modeling. He uses B´ezier curves/triangles, fundamental objects in curve/surface design, to create adaptive methods. The methods are based on tolerance conditions derived from properties of B´ezier curves/triangles. The author develops and tests the methods for an ordinary parametric curve; then he adapts these methods to invariant manifolds of planar maps. Next, he develops and tests the method for parametric surfaces and then he adapts this method to invariant manifolds of three-dimensional maps

Interval simplex splines for scientific databases

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Ocean Engineering, 1995.Includes bibliographical references (p. 130-138).by Jingfang Zhou.Ph.D

Reconstruction of multidimensional signals from multiple level threshold crossings

Also issued as Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1987.Includes bibliographical references.Supported in part by the Advanced Research Projects Agency monitored by ONR. N00014-81-K-0742 Supported in part by the National Science Foundation. ECS 84-07285 Supported in part by a Fanny and John Hertz Foundation Fellowship.Avideh Zakhor

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

Bivariate splines

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