Shape-preserving interpolation of irregular data by bivariate curvature-based cubic splines in spherical coordinates

We investigate C¹-smooth bivariate curvature-based cubic L₁ interpolating splines in spherical coordinates. The coefficients of these splines are calculated by minimizing an integral involving the L₁ norm of univariate curvature in four directions at each point on the unit sphere. We compare these curvature-based cubic L₁ splines with analogous cubic L₂ interpolating splines calculated by minimizing an integral involving the square of the L₂ norm of univariate curvature in the same four directions at each point. For two sets of irregular data on an equilateral tetrahedron with protuberances on the faces, we compare these two types of curvature-based splines with each other and with cubic L₁ and L₂ splines calculated by minimizing the L₁ norm and the square of the L₂ norm, respectively, of second derivatives. Curvature-based cubic L₁ splines preserve the shape of irregular data well, better than curvature-based cubic L₂ splines and than second-derivative-based cubic L₁ and L₂ splines. Second-derivative-based cubic L₂ splines preserve shape poorly. Variants of curvature-based L₁ and L₂ splines in spherical and general curvilinear coordinate systems are outlined.     

Smoothing of cubic parametric splines

The most common curve representation in CADCAM systems of today is the cubic parametric spline. Unfortunately this curve will sometimes oscillate and cause unwanted inflexions which are difficult to deal with. This paper has developed from the need to eliminate oscillations and remove inflexions from such splines, a need which may occur for example when interpolating data measured from a model. A method for interactive smoothing is outlined and a smoothing algorithm is described which is mathematically comparable to manual smoothing with a physical spline.     

Spherical splines and orientation interpolation

This paper presents a method for designing spherical curves by two weighted spatial rotations. This approach is for the design of interpolating spherical curves and orientation interpolation. The same approach can be used for smoothing orientations or corners on a sphere. The designed curves have the following features: C¹ continuity, local control, and invariance under orthogonal transformations of coordinate systems.     

Quasi interpolation with Voronoi splines

We present a quasi interpolation framework that attains the optimal approximation-order of Voronoi splines for reconstruction of volumetric data sampled on general lattices. The quasi interpolation framework of Voronoi splines provides an unbiased reconstruction method across various lattices. Therefore this framework allows us to analyze and contrast the sampling-theoretic performance of general lattices, using signal reconstruction, in an unbiased manner. Our quasi interpolation methodology is implemented as an efficient FIR filter that can be applied online or as a preprocessing step. We present visual and numerical experiments that demonstrate the improved accuracy of reconstruction across lattices, using the quasi interpolation framework.     

Calculations of curvature continuous cubic splines

The connection between the recursion formula for B-splines and the de Boor algorithm is well-known. This connection can be transferred to the curvature continuous cubic case where the use of results of Goodman & Unsworth (for a recursion formula) and Boehm (for a de Boor-like algorithm) yields two different pairs of recursion formulas and de Boor-like algorithms. Some properties are discussed.     

Convexity preserving interpolation with exponential splines

Sufficient and necessary conditions are derived under which interpolating splines are convex if the data set is in convex position. In order to select one of the interpolants, by means of a well-known objective function a quadratic optimization problem is stated which can be solved effectively by passing to a dual program.     

Positive interpolation with rational quadratic splines

A necessary and sufficient criterion is presented under which the property of positivity carry over from the data set to rational quadratic spline interpolants. The criterion can always be satisfied if the occuring parameters are properly chosen.     

On-line state estimation using modified cubic splines

A procedure is proposed for rapidly fitting a modified cubic approximation to a moving window of equally spaced measurements.     

On interpolation by generalized planar splines I: The polynomial case

Complex planar splines were introduced by Opfer and Puri [4] and further investigated by several authors, cf. [3, 5, 6, 8]. These papers were mainly concerned with the properties of piecewise linear or quadratic polynomials. In the present paper polynomial planar splines of arbitrary degree are investigated. Our results are mainly concerned with their interpolatory properties on polygonal regions and contain those of [3, 4, 5] for triangular and rectangular regions as special cases.     

Upsilon-quaternion splines for the smooth interpolation of orientations

We present a new method for smoothly interpolating orientation matrices. It is based upon quaternions and a particular construction of /spl nu/-spline curves. The new method has tension parameters and variable knot (time) spacing which both prove to be effective in designing and controlling key frame animations.     

An explicit method ofC 2 interpolation using splines

The interpolation of a discrete set of data, on the interval [a, b], representing the functionf is obtained using explicit splines. Estimations of interpolation accuracy are obtained.     

Numerical integration formulas based on iterated cubic splines

We consider an application of iterated cubic splines to the Euler-MacLaurin integration formula. Some numerical examples are given to illustrate the usefulness of our methods.     

An offset spline approximation for plane cubic splines

A plane cubic spline segment is given. We want to approximate its offset line by another cubic spline segment. Therefore we take the known curvatures and tangents at the end-points of the offset line and calculate the corresponding spline. Finally examples are given.     

Visually C2 cubic splines

Generalized C² conditions are used to define a class of cubic splines. A B spline-like design scheme is provide for these curves.     

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.     

Family of cubic splines with one degree of freedom

A plane cubic spline segment can be specified in many ways. In this paper a specification method that can be divided into two phases is presented. The first phase relates the segment to the geometry of a deining triangle, but leaves one degree of freedom. This degree of freedom is in the polynomial coefficients as a linear parameter and the variation of the geometric relationship of the segment to the defining triangle as this parameter varies is explored. The second phase of the specification requires a choice of value for this parameter in accordance with a criterion of fairness that is specified. To support the method of this phase the variation of the curvature of the segment with changes in the parameter value is demonstrated.To show how the specification method can be used, the paper shows that the interpolation problem for general points in the plane can give rise to a set of defining triangles. An algorithm for generating such triangles for an open or closed curve problem for a given set of points is described. Results of using an implementation of this algorithm together with the specification method are presented.