Multidimensional spline integration of scattered data

We introduce a numerical method for reconstructing a multidimensional surface using the gradient of the surface measured at some values of the coordinates. The method consists of defining a multidimensional spline function and minimizing the deviation between its derivatives and the measured gradient. Unlike a multidimensional integration along some path, the present method results in a continuous, smooth surface, furthermore, it also applies to input data that are non-equidistant and not aligned on a rectangular grid. Function values, first and second derivatives and integrals are easy to calculate. The proper estimation of the statistical and systematical errors is also incorporated in the method.     

Image warping with scattered data interpolation

Discusses a new approach to image warping based on scattered data interpolation methods which provides smooth deformations with easily controllable behavior. A new, efficient deformation algorithm underlies the method     

An algorithm for computing shape-preserving cubic spline interpolation to data

We present an algorithm for the construction of shape-preserving cubic splines interpolating a set of data point. The method is based upon some existence properties recently developed. Graphical examples are given.     

Powell-Sabin splines in range restricted interpolation of scattered data

Computing - The construction of range restricted bivariateC 1 interpolants to scattered data is considered. In particular, we deal with quadratic spline interpolation on a Powell-Sabin refinement...     

An iterative algorithm for spline interpolation

One of the fundamental results in spline interpolation theory is the famous Schoenberg-Whitney Theorem, which completely characterizes those distributions of interpolation points which admit unique interpolation by splines. However, until now there exists no iterative algorithm for the explicit computation of the interpolating spline function, and the only practicable method to obtain this function is to solve explicitly the corresponding system of linear equations. In this paper we suggest a method which computes iteratively the coefficients of the interpolating function in its B-spline basis representation; the starting values of our one-step iteration scheme are quotients of two low order determinants in general, and sometimes even just of two real numbers. Furthermore, we present a generalization of Newton's interpolation formula for polynomials to the case of spline interpolation, which corresponds to a result of G. Mühlbach for Haar spaces.     

Convexity preserving scattered data interpolation using Powell–Sabin elements

The problem of strict convexity preserving interpolation by a piecewise quadratic smooth function is solved. A constructive procedure is provided.     

Some physical and mathematical properties of inverse distance weighted methods for scattered data interpolation

A class of inverse distance weighted formulas for scattered data interpolation, such as the well-known Shepard one, is reconsidered from a physical viewpoint and in particular with respect to electrostatic or gravitational fields. In this way, simple properties are explained, that permit the construction of either parallel or recursive algorithms to calculate these formulas. The formulas are then interpreted as means in accordance with the definitions of Cauchy, Chisini and others. As a consequence, some significant properties of the means can be used. Finally, a few comments are made on the numerical evaluation of the formulas considered. this work has been supported by the Italian Ministry of Scientific and Technological Research and the National Research Council.     

Computational complexity of spline interpolation

The procedure for composing the spline function of order m that interpolates n data is roughly divided into two stages: (1) constructing the matrix A_n that transforms the fl-spline coefficient vector c into the sample value vector s; and (2) calculating the vector c. Then the effects of boundary conditions and the locations of the sampling points and the knots on the number of computations for spline interpolation are evaluated. There are no boundary condition effects of the construction of A" for equispacing. Non-periodic boundary conditions reduce the number of computations needed to calculate c from O(m²n) to O(m²n/4).

In the construction of A_n, choosing equispaced sampling points reduces the number of computations from O(m²n) to O(m²) in the case of sampling points located at a constant interval between each pair of adjacent knots. In the calculation of c, equispaced sampling points reduce the number of computations from O(m²n) to O(m²n /2) in the case where m is even and the locations of the knots are identical with those of the sampling points.     

A one-pass algorithm for shape-preserving quadratic spline interpolation

Schumaker (1983) and McAllister and Roulier (1981) have proposed algorithms for shape-preserving interpolation using quadratic splines. The former requires the user to provide and perhaps to adjust estimates of the slope at the data points. Here we show that, for a particular slope estimation technique, the two methods are identical, and that in this case the Schumaker algorithm automatically generates shape-preserving interpolants. Furthermore, in case of convex data the slopes are improved iteratively to produce more visually pleasing curves.     

High accuracy spline interpolation for 5-axis machining

This article presents a new algorithm for simultaneous 5-axis spline interpolation. The algorithm basically merges two concepts: (1) the interpolation of the toolpath with Pythagorean Hodograph (PH) curves and (2) the analytic solution of the inverse kinematic problem using the template equation method. The first method allows one to obtain a analytic relation between the arclength and the parameter of the toolpath curve. This one enables one to control the velocity of the tool on the workpiece. The second method allows one to determine the analytic solution of the parameterized inverse kinematic problem that permits us to introduce arbitrary number of geometric parameters. A natural selection of the possible parameters can be the parameters of tool geometry and the workpiece placement. This way, the off-line generated inverse solutions—that transform the cutter contact curve into axis values—can be online compensated, as soon as the exact parameter values are become known. Based on these two approaches, a robust and fast method for the simultaneous 5-axis spline interpolation is developed. The result of this new algorithm is time-dependent axis splines which represent the given toolpath with high accuracy.     

Error bounds for polynomial tensor product interpolation

We provide estimates for the maximum error of polynomial tensor product interpolation on regular grids in ${\mathbb{R}^d}$ . The set of partial derivatives required to form these bounds depends on the clustering of interpolation nodes. Also bounds on the partial derivatives of the error are derived.     

Approximation by a smooth interpolation spline

The properties of a smooth continuous spline approximation are considered. The existence conditions are established and an algorithm is proposed to determine the parameters of such a spline with segments as the sum of a polynomial and an exponent. The errors of approximating a function and its derivative by such a spline with polynomial segments and segments in the form of the sum of a polynomial and an exponent are estimated.     

C~2 quartic spline surface interpolation

This paper discusses the problem of constructing C2 quartic spline surface interpolation. Decreasing the continuity of the quartic spline to C2 offers additional freedom degrees that can be used to adjust the precision and the shape of the interpolation surface. An approach to determining the freedom degrees is given, the continuity equations for constructing C2 quartic spline curve are discussed, and a new method for constructing C2 quartic spline surface is presented. The advantages of the new method are that the equations that the surface has to satisfy are strictly row diagonally dominant, and the discontinuous points of the surface are at the given data points. The constructed surface has the precision of quartic polynomial. The comparison of the interpolation precision of the new method with cubic and quartic spline methods is included.     

Monotone and convex cubic spline interpolation

Given a set of monotone and convex data, we present a necessary and sufficient condition for the existence of cubic differentiable interpolating splines which are monotone and convex. Further, we discuss their approximation properties when applied to the interpolation of functions having preassigned degree of smoothness.     

Document segmentation using polynomial spline wavelets

Wavelet transforms have been widely used as effective tools in texture segmentation in the past decade. Segmentation of document images, which usually contain three types of texture information: text, picture and background, can be regarded as a special case of texture segmentation. B-spline wavelets possess some desirable properties such as being well localized in time and frequency, and being compactly supported, which make them an effective tool for texture analysis. Based on the observation that text textures provide fast-changed and relatively regular distributed edges in the wavelet transform domain, an efficient document segmentation algorithm is designed via cubic B-spline wavelets. Three-means or two-means classification is applied for classifying pixels with similar characteristics after feature estimation at the outputs of high frequency bands of spline wavelet transforms. We examine and evaluate the contributions of different factors to the segmentation results from the viewpoints of decomposition levels, frequency bands and wavelet functions. Further performance analysis reveals the advantages of the proposed method.