Morten Dæhlen

Approximation of scattered data using smooth grid functions

We present a three-stage scheme for constructing smooth grid functions approximating data defined over scattered point sets in Rs. The scheme is useful for approximating large scattered data sets and is particularly successful when the data points are unevenly distributed. The paper includes several examples of grid surface construction over the plane.

Multiresolution analysis over triangles, based on quadratic Hermite interpolation

Given a triangulation T of R 2 , a recipe to build a spline space S(T) over this triangulation, and a recipe to rene the triangulation T into a triangulation T 0 , the question arises whether S(T)S(T 0 ), i.e., whether any spline surface over the original triangulation T can also be represented as a spline surface over the rened triangulation T 0 . In this paper we will discuss how to construct such a nested sequence of spaces based on Powell{Sabin 6-splits for a regular triangulation. The resulting spline space consists of piecewise C 1 -quadratics, and renement is obtained by subdividing every triangle into four subtriangles at the edge midpoints. We develop explicit formulas for wavelet transformations based on quadratic Hermite interpolation, and give a stability result with respect to a natural norm. c 2000 Elsevier Science B.V. All rights reserved. MSC: 41A63; 65D07; 65D17; 65T60

Box Splines and Applications

We give an elementary introduction to box spline methods for the representation of surfaces. First, we derive basic properties of box splines starting with the univariate cardinal case. Proofs of most of the results are included. We proceed with a detailed presentation of refinement and evaluation methods for box splines. We discuss shape preserving properties, the construction of non-rectangular box spline surfaces, applications of box splines to surface modelling and problems related to an imbedding of box spline surfaces within a tensor product surface.

Simultaneous curve simplification

In this paper we present a method for simultaneous simplification of a collection of piecewise linear curves in the plane. The method is based on triangulations, and the main purpose is to remove line segments from the piecewise linear curves without changing the topological relations between the curves. The method can also be used to construct a multi-level representation of a collection of piecewise linear curves. We illustrate the method by simplifying cartographic contours and a set of piecewise linear curves representing a road network.

Surface Modelling from Biomedical Data

In the present chapter, we present a method for modelling smooth surfaces on the basis of stacks of boundaries taken from serial sections. We also present a method for automatically defining the outer closed boundary of tight clusters of points; the defined boundaries are in turn used for a surface modelling. We exemplify the use of our methods with material of physical sections from the field of experimental brain research. The surfaces modelled represent the exterior of the brain, internal regions, and zones containing specifically labelled tissue elements coded as point clusters.

Box spline interpolation; a computational study

Abstract We present a computational study of box spline interpolation in two space dimensions. We are in particular concerned with the problem of computing the box spline coefficients in the presence of large data sets. The issues of convergence and regularity of the interpolants are also considered from a computational point of view.

Data Reduction of Piecewise Linear Curves

We present and study two new algorithms for data reduction or simplification of piecewise linear plane curves. Given a curve P and a tolerance e ≥ 0, both methods determine a new curve Q, with few vertices, which is at most e in Hausdorff distance from P. The methods differ from most existing methods in that they do not require a vertex in Q to be a vertex in P. Several examples are given where we show that the methods presented here compare favorably to other methods found in the literature. We also show how the vertices of a curve can be reordered so that the first, say n, vertices of the reordered sequence form an approximation to the curve itself.

Iterative polynomial interpolation and data compression

In this paper we look at some iterative interpolation schemes and investigate how they may be used in data compression. In particular, we use the pointwise polynomial interpolation method to decompose discrete data into a sequence of difference vectors. By compressing these differences, one can store an approximation to the data within a specified tolerance using a fraction of the original storage space (the larger the tolerance, the smaller the fraction).We review the iterative interpolation scheme, describe the decomposition algorithm and present some numerical examples. The numerical results are that the best compression rate (ratio of non-zero data in the approximation to the data in the original) is often attained by using cubic polynomials and in some cases polynomials of higher degree.

Grid point interpolation on finite regions using C 1 box splines

Multivariate grid point interpolation on finite regions is considered by translates of

Matrix decomposition and data reduction

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