Methods of Shape-Preserving Spline Approximation

Abstract

in 1995. Dr. Kvasov has worked in various fields of numerical analysis and its ap plications. Originally, he was interested in finite differences schemes of high order accuracy which he combined with the method of fractional steps to solve different problems in mathematical physics. Later he switched to the the ory of splines and their applications to computer aided geometric design. Dr. Kvasov authored or coauthored more than 70 professional papers and numer ous research reports on the subject. He is co-author of the book Methods of Spline Functions published by Nauka in Moscow, which has become a stan dard textbook for students, researchers and engineers in Russia. His current research interests include numerical analysis, approximation theory, mathe matical methods in computer aided geometric design, shape preserving spline approximation and parametrization, scientific visualization. PrefaceSpline functions are a fundamental and prevalent ingredient in the en deavors of scientists and engineers. They constitute the main tool in computer aided geometric design (CAGD for short) which is concerned with the approxi mation and representation of curves and surfaces that arises when these objects have to be processed by a computer. The design of curves and surfaces plays an important role not only in the construction of different products such as car bodies, ship hulls, airplane fuselages and wings, propellers blades, etc., but also in the description of geological, physical and even medical phenomena. New areas of CAGD applications include computer vision and inspection of man ufactured parts, medical research (software for digital diagnostic equipment), image analysis, high resolution TV systems, cartography, the film industry, etc.In the majority of these applications, it is important to construct curves and surfaces which have certain shape properties. For example, we may want the surface to be positive, monotone, or convex in some sense. Standard meth ods of spline functions do not preserve these properties of the data. Therefore, when fitting spline curves and surfaces to functions and data one needs to have more refined methods available which preserve the shape of the data. Based on spline functions, these methods are usually called methods of shape preserving spline approximation. By introducing some parameters into the spline struc ture, one can preserve various characteristics of the data, including positivity, monotonicity, convexity, as well as linear and planar sections. By increasing one or more of these parameters the curve is pulled towards an inherent shape, usually a piecewise linear curve, at the same time keeping the smoothness of the curve.The purpose of this book is to present methods of shape preserving spline approximation using a unified approach based on generalized tension splines. These generalize and unify various of the constructions proven in practice, and include among others rational, exponential, hyperbolic, variable order splines, splines with additional knots. The major idea in ...

Conditions of matrices in discrete tension spline approximations of DMBVP

“…Since the choice of tension parameters comes from practical applications, like shape preserving approximation (see [6] and references therein), it is our hope that such a choice of tension parameters can be made, that both, shapepreserving requirements, and numerical stability can be achieved. The delicate balance between the two is at this moment not completely understood.…”

Length Estimation for Exponential Parameterization and ε-Uniform Samplings

“…5 guaranteeing ψ to be a re-parameterization (see [5]) are fulfilled. The latter is automatically satisfied by [14] for ε > 0 and λ ∈ [0, 1] or by [4] for λ = 1 and samplings (2)…”

“…[1], [2], [10] or [11]). Here the corresponding interpolation knots {t i } m i=0 are not available and as such they need first to be estimated somehow.…”

“…More real-life applications referring to n-dimensional reduced data Q m fitted with piecewise-quadraticsγ 2 or any other interpolation schemes based on exponential parameterization (8) can be found e.g. in [1], [2] or [11].…”

“…The left inequality in (3) excludes samplings with distance between consecutive knots less then βδ. The right inequality of (3) follows from (2). Condition (3), as shown in [7], can be replaced by the equivalent condition (4) holding for each i = 0, 1, .…”