Paul Dierckx

Curve and surface fitting with splines

Part 1 Spline functions: univariate splines bivariate splines. Part 2 Curve fitting: an introduction least-squares spline curve fitting smoothing spline curve fitting more smoothing spline curves fitting with convexity constraints. Part 3 Surface fitting: an introduction scattered data surface fitting mesh data surface fitting more scattered data smoothing more mesh data smoothing Part 4 Fitpack: available software.

On calculating normalized Powell-Sabin B-splines

Abstract An algorithm is presented for calculating a suitable normalized B-spline representation for Powell-Sabin splines in which the basis functions are all positive, have local support and form a partition of unity. Computationally, the problem is reduced to the solution of a number of linear or quadratic programming problems of small size. Geometrically, each of these can be interpreted as a problem of determining a triangle of minimal area, containing a specific subset of Bezier points. We further consider a number of CAGD applications such as the determination of a suitable set of tangent control triangles and the efficient and stable calculation of the Bezier net of the PS-spline surface.

An algorithm for smoothing, differentiation and integration of experimental data using spline functions

Abstract This paper presents an algorithm for fitting a smoothing spline function to a set of experimental or tabulated data. The obtained spline approximation can be used for differentiation and integration of the given discrete function. Because of the ease of computation and the good conditioning properties we use normalised B-splines to represent the smoothing spline. A Fortran implementation of the algorithm is given.

Generating the Be´zier points of a b-spline curve

Abstract We present an efficient algorithm for computing the Bezier points of a generalized cubic β-spline curve and show the connection with multiple knot insertion. We also consider the inverse problem of determining the β-spline vertices of a composite G 2 Bezier curve. Finally, we briefly discuss how to construct the Bezier net of a tensor product β-spline surface.

Algorithm/algorithmus 42 an algorithm for cubic spline fitting with convexity constraints

In this paper an algorithm is presented for fitting a cubic spline satisfying certain local concavity and convexity constraints, to a given set of data points. When using theL2 norm, this problem results in a quadratic programming problem which is solved by means of the Theil-Van de Panne procedure. The algorithm makes use of the well-conditioned B-splines to represent the cubic splines. The knots are located automatically, as a function of a given upper limit for the sum of squared residuals. A Fortran IV implementation is given.ZusammenfassungEs wird ein Algorithmus vorgestellt für den Ausgleich durch kubische Spline-Funktionen, die gewissen Konvexitätsbedingungen genügen müssen. Wenn man dieL2-Norm verwendet, führt dieses Problem auf ein quadratisches Programmierungsproblem, das man mit dem Verfahren von Theil und Van de Panne lösen kann. Für die Darstellung der kubischen Splines verwendet unser Algorithmus die gut konditionierten B-splines. Die Knoten werden automatisch in Abhängigkeit von einer Obergrenze für die Fehlerquadratsumme lokalisiert. Eine Fortran-IV-Version des Algorithmus ist beigefügt.

Algorithms for smoothing data with periodic and parametric splines

Abstract This paper deals with the problem of fitting splines to measured data which either describe a periodic function or some parametric curve, closed, or otherwise. Algorithms are presented which are extensions of an existing semiautomatic curve fitting algorithm. The knots of the splines are chosen automatically but a single parameter is expected to control the tradeoff between closeness of fit and smoothness of fit. The user will interactively change this smoothing factor and examine the corresponding curve graphically, until he can accept the result as satisfactory. Numerical and practical examples illustrate the accuracy of the algorithms and their applicability in image processing and related fields.

Algorithms for smoothing data on the sphere with tensor product splines

Algorithms are presented for fitting data on the sphere by using tensor product splines which satisfy certain boundary constraints. First we consider the least-squares problem when the knots are given. Then we discuss the construction of smoothing splines on the sphere. Here the knots are located automatically. A Fortran IV implementation of these two algorithms is described.ZusammenfassungAlgorithmen werden vorgestellt für den Ausgleich über die Sphäre mit Hilfe von Tensorprodukt-Splines, die gewissen Randbedingungen genügen müssen. Erst untersuchen wir das Problem der kleinsten Quadrate, wenn die Knoten gegeben sind. Dann besprechen wir die Konstruktion von Ausgleichssplines über die Sphäre. Die Knoten werden hier automatisch lokalisiert. Eine Fortran-IV-Version dieser zwei Algorithmen wird beschrieben.

Surface fitting using convex Powell-Sabin splines

Abstract Convexity conditions for Powell—Sabin splines are derived and an algorithm is presented for fitting a convex Powell—Sabin spline to a set of scattered data. Part of this algorithm deals with the elimination of as many redundant conditions as possible. The Powell—Sabin splines are defined in terms of basis functions (B-splines) with local support. Making use of their Bernstein—Bezier representation leads to an easy handling of and an efficient calculating with these B-splines. Numerical examples are supplied to illustrate the usefulness of surface fitting by means of convex Powell—Sabin splines.

Quasi-hierarchical Powell--Sabin B-splines

Hierarchical Powell-Sabin splines are C^1-continuous piecewise quadratic polynomials defined on a hierarchical triangulation. The mesh is obtained by partitioning an initial conforming triangulation locally with a triadic split, so that it is no longer conforming. We propose a normalized quasi-hierarchical basis for this spline space. The basis functions have a local support, they form a convex partition of unity, and they admit local subdivision. We show that the basis is strongly stable on uniform hierarchical triangulations. We consider two applications: data fitting and surface modelling.

Automatic construction of control triangles for subdivided Powell-Sabin splines

In this paper we present an algorithm for calculating the B-spline representation of a Powell-Sabin spline surface on a refinement of the given triangulation. The resulting subdivision scheme is a √3 scheme; a new vertex is added inside every original triangle. Applying the √3 scheme twice yields a triadic scheme, every original edge is split into three new edges, but special care is needed at the boundaries. The scheme is numerically stable and generally applicable, there are no restrictions on the initial triangulation.