Tension Spline Research Articles

Energy-minimizing splines in manifolds

Variational interpolation in curved geometries has many applications, so there has always been demand for geometrically meaningful and efficiently computable splines in manifolds. We extend the definition of the familiar cubic spline curves and splines in tension, and we show how to compute these on parametric surfaces, level sets, triangle meshes, and point samples of surfaces. This list is more comprehensive than it looks, because it includes variational motion design for animation, and allows the treatment of obstacles via barrier surfaces. All these instances of the general concept are handled by the same geometric optimization algorithm, which minimizes an energy of curves on surfaces of arbitrary dimension and codimension.

The Wilson–Burg method of spectral factorization with application to helical filtering

ABSTRACTSpectral factorization is a computational procedure for constructing minimum‐phase (stable inverse) filters required for recursive inverse filtering. We present a novel method of spectral factorization. The method iteratively constructs an approximation of the minimum‐phase filter with the given autocorrelation by repeated forward and inverse filtering and rearranging of the terms. This procedure is especially efficient in the multidimensional case, where the inverse recursive filtering is enabled by the helix transform.To exemplify a practical application of the proposed method, we consider the problem of smooth two‐dimensional data regularization. Splines in tension are smooth interpolation surfaces whose behaviour in unconstrained regions is controlled by the tension parameter. We show that such surfaces can be efficiently constructed with recursive filter preconditioning and we introduce a family of corresponding two‐dimensional minimum‐phase filters. The filters are created by spectral factorization on a helix.

Interpolation of geophysical data using continuous global surfaces

A wide class of interpolation methods, including thin‐plate and tension splines, kriging, sinc functions, equivalent‐source, and radial basis functions, can be encompassed in a common mathematical framework involving continuous global surfaces (CGSs). The difficulty in applying these techniques to geophysical data sets has been the computational and memory requirements involved in solving the large, dense matrix equations that arise. We outline a three‐step process for reducing the computational requirements: (1) replace the direct inversion techniques with iterative methods such as conjugate gradients; (2) use preconditioning to cluster the eigenvalues of the interpolation matrix and hence speed convergence; and (3) compute the matrix–vector product required at each iteration with a fast multipole or fast moment method.We apply the new methodology to a regional gravity compilation with a highly heterogeneous sampling density. The industry standard minimum‐curvature algorithms and several scale‐dependent CGS methods are unable to adapt to the varying data density without introducing spurious artifacts. In contrast, the thin‐plate spline is scale independent and produces an excellent fit. When applied to an aeromagnetic data set with relatively uniform sampling, the thin‐plate spline does not significantly improve results over a standard minimum‐curvature algorithm.

Numerical solution of singularly perturbed two-point boundary value problems by spline in tension

Some difference schemes for singularly perturbed two-point boundary value problems are derived using spline in tension. These schemes are second-order accurate. Numerical examples are given in support of the theoretical results.

Spline Techniques for the Numerical Solution of Singular Perturbation Problems

One-dimensional singularly-perturbed two-point boundary-value problems arising in various fields of science and engineering (for instance, fluid mechanics, quantum mechanics, optimal control, chemical reactor theory, aerodynamics, reaction-diffusion processes, geophysics, etc.) are treated. Either these problems exhibits boundary layer(s) at one or both ends of the underlying interval or they possess oscillatory behavior depending on the nature of the coefficient of the first derivative term. Some spline difference schemes are derived for these problems using splines in compression and splines in tension. Second-order uniform convergence is achieved for both kind of schemes. By making use of the continuity of the first-order derivative of the spline function, a tridiagonal system is obtained which can be solved efficiently by well-known algorithms. Numerical examples are given to illustrate the theory.

Assessment of radionuclide contamination in the Black Sea using POSEIDON/RODOS system

The coupled system of the 3D hydrodynamical model THREETOX and the radiological box model POSEIDON-R, developed in the framework of the European system RODOS for emergency response to a nuclear accident, was applied to the Black Sea for reconstruction and outlook of the radionuclide contamination in result of bomb testing and the Chernobyl accident. For the model verification hydrophysical data was obtained from historical records of up to 15180 observational stations in the Black Sea. The data was grouped by months in 1°by1° spatial squares and homogenised in the vertical with application of the method of Reininger and Ross. The radiological data was derived from literature sources mainly based on the expeditions of Typhoon, IBSS, MHI, Hydrometeorological service (all from the former USSR), K. Piri and Knorr (US-Turkish) expeditions in the Black Sea. The vertical radionuclide stratification was obtained as one profile mean for the whole sea due to the scarce data. The spatial reconstruction of the hydrophyscal and radiological fields were done by gridding of the arbitrarily spaced data applying the method of splines in tension and eliminating aliasing by the median method. The THREETOX model was used for re-analysis of radionuclide transport. The 137 Cs concentrations calculated with POSEIDON-R in water and biota are compared with the collected data. A good agreement was found due to the implemented dynamic food chain model.

Tension Spline for the Solution of Self-adjoint Singular Perturbation Problems

A numerical method based on spline in tension is given for the self-adjoint singularly perturbed two point boundary value problems. The schemes derived in this method are second order accurate. One numerical example is given to support the predicted theory.

Tension spline collocation methods for singularly perturbed Volterra integro-differential and Volterra integral equations

We consider the numerical discretization of singularly perturbed Volterra integro-differential equations (VIDE) (∗) εy′(t)=q 1(t)−q 2(t)y(t)+ ∫ 0 t K(t,s)y(s) ds, t∈I ≔ [0,T],y(0)=y 0 and Volterra integral equations (VIE) (∗∗) εy(t)=g(t)− ∫ 0 t K(t,s)y(s) ds, t∈I by tension spline collocation methods in certain tension spline spaces, where ε is a small parameter satisfying 0< ε⪡1, and q 1, q 2, g and K are functions sufficiently smooth on their domains to ensure that Eqs. (∗) and (∗∗) posses a unique solution. We give an analysis of the global convergence properties of a new tension spline collocation solution for 0< ε⪡1 for singularly perturbed VIDE and VIE; thus, extending the existing theory for ε=1 to the singularly perturbed case.

A fourth/second order accurate collocation method for singularly perturbed two-point boundary value problems using tension splines

The collocation tension spline is considered as a numerical solution of a singularly perturbed two-point boundary value problem: \(\varepsilon y'' + b y' + c y = f\). The collocation points are chosen as a generalization of the classical Gaussian points. Unlike the traditional approach, we employ the B-spline representation in the analysis. This leads to global quadratic convergence of the method for small perturbation parameters, and, for large values, the order of convergence is four.

Hilbertian Approach for Univariate Spline with Tension

In this work, a new approach is proposed for constructing splines with tension. The basic idea is in the use of distributions theory, which allows us to define suitable Hilbert spaces in which the tension spline minimizes some energy functional. Classical orthogonal conditions and characterizations of the spline in terms of a fundamental solution of a differential operator are provided. An explicit representation of the tension spline is given. The tension spline can be computed by solving a linear system. Some numerical examples are given to illustrate this approach.

Generalized discrete tension splines

Direct Algorithms for constructing generalized discrete tension splines of arbitrary degree are given. We derive main property of generalized discrete tension splines, i.e., partition of unity. We also give some examples of generalized discrete tension splines.

Elastic Curves as Solutions of Riemannian and Sub-Riemannian Control Problems

We consider the nonlinear dynamic interpolation problem on Riemannian manifolds and, in particular, on connected and compact Lie groups. Basically we force the dynamic variables of a control system to pass through specific points in the configuration space, while minimizing a certain energy function, by a suitable choice of the controls. The energy function we consider depends on the velocity and acceleration along trajectories. The solution curves can be seen as generalizations of the classical splines in tension for the Euclidean space. The relations with sub-Riemannian optimal control problems are explained.

Refinement of Gravimetric Geoid Using GPS and Leveling Data

A gravimetric geoid model does not allow the accurate transformation of Global Positioning System (GPS) ellipsoidal heights to Australian Height Datum (AHD) heights in the Perth region of Western Australia. This is predominantly due to the effect of the geological structures associated with the Darling Fault, the availability and quality of data, and the possibility of distortions in the AHD. Therefore a combined solution is sought, where 99 collocated GPS and AHD heights are used to adjust the gravimetric geoid such that it provides a model of the separation between the AHD and the GRS80 reference ellipsoid. This data combination was investigated using least-squares collocation and continuous curvature splines in tension. The least-squares collocation technique was found to give an improved model of the AHDGRS80 separation, as shown by a standard deviation of fit of the combined model to the control data of 8 mm. In comparison, the gravimetric geoid gives a standard deviation of fit to the control data of 128 mm.

On discrete hyperbolic tension splines

A hyperbolic tension spline is defined as the solution of a differential multipoint boundary value problem. A discrete hyperbolic tension spline is obtained using the difference analogues of differential operators; its computation does not require exponential functions, even if its continuous extension is still a spline of hyperbolic type. We consider the basic computational aspects and show the main features of this approach.

An integrated framework for interpolating airborne geophysical data with special reference to radiometrics

New processing techniques for airborne radiometric data make use of the information contained in all 256 channels of a radiometric spectrum, improving the final quality obtained. However, visualisation and interpretation of the processed data require interpolation to a regular grid and current methods for doing this are generally unsatisfactory. We highlight alternative interpolation techniques (kriging, radial basis functions, tension splines, smoothing splines) that overcome some of the disadvantages of existing methods. These techniques are formulated in a common mathematical framework and can be used for exact or smooth interpolation of the processed data. The resulting grids can be made to inherit certain desirable characteristics, such as smoothness or minimum variance. Further, the framework generates a continuous model of the data that can be updated rapidly when the image is visualised at different scales. Until recently, the main impediments to the application of the technique to large geophysical surveys have been the computer memory and effort needed to solve the resulting matrix equations. We describe some recent advances that reduce the computational requirements to acceptable levels.We describe an extension to the usual multi-channel technique that, during processing, preserves the original signal (as embodied in the 256-channel spectra) for as long as possible. We show that better images can result if the spectral components are gridded and the spectra reconstructed at the grid points. The reduction to standard 4-channel count-rates and conversion to ground concentration occur in the final processing step.

A note on interest rate term structure estimation using tension splines

Tension splines are proposed as a flexible tool for interest rate term structure estimation to circumvent some difficulties arising with the ordinary cubic spline estimators. A few computational experiments on test problems support the merits of the proposed approach.

Interpolation with splines in tension : A Green's function approach

Interpolation and gridding of data are procedures in the physical sciences and are accomplished typically using an averaging or finite difference scheme on an equidistant grid. Cubic splines are popular because of their smooth appearances; however, these functions can have undesirable oscillations between data points. Adding tension to the spline overcomes this deficiency. Here, we derive a technique for interpolation and gridding in one, two, and three dimensions using Green's functions for splines in tension and examine some of the properties of these functions. For moderate amounts of data, the Green's function technique is superior to conventional finite-difference methods because (1) both data values and directional gradients can be used to constrain the model surface, (2) noise can be suppressed easily by seeking a least-squares fit rather than exact interpolation, and (3) the model can be evaluated at arbitrary locations rather than only on a rectangular grid. We also show that the inclusion of tension greatly improves the stability of the method relative to gridding without tension. Moreover, the one-dimensional situation can be extended easily to handle parametric curve fitting in the plane and in space. Finally, we demonstrate the new method on both synthetic and real data and discuss the merits and drawbacks of the Green's function technique.

Monotonicity preserving regression techniques for interest rate term structure estimation: A note

Alternative methodologies are compared for measuring the term structure of interest rates via monotone approximations to the discount function. Some empirical comparisons using coupon bond data show that a simple linear methodology based on rational tension spline parametrizations turns out to yield the most efficient compromise between the conflicting goals of goodness of fit and smoothuess of the interpolaut

A collocation method for singularly perturbed two-point boundary value problems with splines in tension

An error bound for the collocation method by spline in tension is developed for a nonlinear boundary value problemay″+by′+cy=f(·,y),y(0)=y0,y(1)=y1. Sharp error bounds for the interpolating splines in tension are used in conjunction with recently obtained formulae for B-splines, to develop an error bound depending on the tension parameters and net spacing. For singularly perturbed boundary value problems (|a|=e≪1), the representation of the error motivates a choice of tension parameters which makes the convergence of the collocation method problem at least linear. The B-representation of the spline in tension is also used in the actual computations. Some numerical experiments are given to illustrate the theory.

A sinusoidal polynomial spline and its Bezier blended interpolant

Functional polynomials composed of sinusoidal functions are introduced as basis functions to construct an interpolatory spline. An interpolant constructed in this way does not require solving a system of linear equations as many approaches do. However there are vanishing tangent vectors at the interpolating points. By blending with a Bezier curve using the data points as the control points, the blended curve is a proper smooth interpolant. The blending factor has the effect similar to the “tension” control of tension splines. Piecewise interpolants can be constructed in an analogous way as a connection of Bezier curve segments to achieve C 1 continuity at the connecting points. Smooth interpolating surface patches can also be defined by blending sinusoidal polynomial tensor surfaces and Bezier tensor surfaces. The interpolant can very efficiently be evaluated by tabulating the sinusoidal function.