Showing papers on "Monotone cubic interpolation published in 2008"
TL;DR: A numerical technique is presented for the solution of nonlinear system of second-order boundary value problems by expanding the required approximate solution as the elements of cubic B-spline scaling function using the operational matrix of derivative.
Abstract: A numerical technique is presented for the solution of nonlinear system of second-order boundary value problems. This method uses the cubic B-spline scaling functions. The method consists of expanding the required approximate solution as the elements of cubic B-spline scaling function. Using the operational matrix of derivative, we reduce the problem to a set of algebraic equations. Numerical examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces very accurate results.
62 citations
TL;DR: In this article, the convergence of natural cubic spline fractal interpolation functions towards the original function with respect to the data provided by the data is studied and an upper bound of the difference between the natural cubic Spline blended Fractal Interpolation Function and the original Function is derived.
Abstract: Fractal Interpolation functions provide natural deterministic approximation of complex phenomena Cardinal cubic splines are developed through moments (ie second derivative of the original function at mesh points) Using tensor product, bicubic spline fractal interpolants are constructed that successfully generalize classical natural bicubic splines An upper bound of the difference between the natural cubic spline blended fractal interpolant and the original function is deduced In addition, the convergence of natural bicubic fractal interpolation functions towards the original function providing the data is studied
44 citations
TL;DR: Several Hermite-type interpolation methods for rational cubics are presented, in case the input data come from a circular arc, and the rational cubic will reproduce it.
Abstract: We present several Hermite-type interpolation methods for rational cubics. In case the input data come from a circular arc, the rational cubic will reproduce it.
36 citations
TL;DR: By suitably choosing the parameters most of the previous known methods for homogeneous and non-homogeneous cases can be obtained from this method and a new high accuracy scheme is obtained.
Abstract: Second-order parabolic partial differential equations are solved by using a new three level method based on non-polynomial cubic spline in the space direction and finite difference in the time direction. Stability analysis of the method has been carried out and we have shown that our method is unconditionally stable. It has been shown that by suitably choosing the parameters most of the previous known methods for homogeneous and non-homogeneous cases can be obtained from our method. We also obtain a new high accuracy scheme of O(k4+h4). Numerical examples are given to illustrate the applicability and efficiency of the new method.
35 citations
03 Mar 2008
TL;DR: This paper develops two image models that capture the important characteristics of an image and uses the models to derive optimal kernels, which results in linear interpolation and a piece-wise cubic kernel similar to that of cubic spline.
Abstract: Image super-resolution involves interpolating a non-uniformly sampled composite image at uniform locations of a high-resolution image. Interpolation methods used in the literature are generally based on arbitrary functions. Optimal (in least squares sense) interpolation kernels can be derived if the ground-truth high-resolution data is known, this is obviously impractical. An observation that the optimal kernels for very different images are similar suggests that a kernel derived on one image can interpolate another image with good results. This paper extends this idea by developing two image models that capture the important characteristics of an image and uses the models to derive optimal kernels. One of the models results in linear interpolation and the other one results in a piece-wise cubic kernel similar to that of cubic spline. This later model is experimentally shown to be near optimal for three different images. The notion of deriving optimal interpolators from the image model and the model of image capturing process provides a unifying framework that brings together linear and cubic interpolators and gives them a theoretic backing.
33 citations
Posted Content•
TL;DR: The postrcspline package as mentioned in this paper can help with the interpretation of a model that uses a restricted cubic spline as one of the explanatory variables by displaying a graph of the predicted values against the spline variable adjusted for the other covariates.
Abstract: The postrcspline package consists of programs that can help with the interpretation of a model that uses a restricted cubic spline as one of the explanatory variables by displaying a graph of the predicted values against the spline variable adjusted for the other covariates, or the marginal effects of the spline variable.
33 citations
TL;DR: In this paper, a general framework for a higher-order spline level-set (HLS) method was presented and applied to biomolecule surfaces construction. But this method is not suitable for the optimization of biomolecules.
Abstract: We present a general framework for a higher-order spline level-set (HLS) method and apply this to biomolecule surfaces construction. Starting from a first order energy functional, we obtain a general level set formulation of geometric partial differential equation, and provide an efficient approach to solving this partial differential equation using a C2 spline basis. We also present a fast cubic spline interpolation algorithm based on convolution and the Z-transform, which exploits the local relationship of interpolatory cubic spline coefficients with respect to given function data values. One example of our HLS method is demonstrated, which is the construction of biomolecule surfaces (an implicit solvation interface) with their individual atomic coordinates and solvated radii as prerequisites.
31 citations
TL;DR: The extensions of interpolating α-B-spline based on the new B-splines and the singular blending technique are presented and their applications in data interpolation and polygonal shape deformation are investigated.
29 citations
TL;DR: A recently developed cubic L 1 spline model is proposed for term structure analysis that preserves the shape of the data, exhibit no extraneous oscillation and have small fitting errors.
21 citations
TL;DR: In this article, the authors developed and applied three methods, namely, standard cubic spline collocation, perturbation cubic splines collocation and perturbed cubic spliners collocation tau method with exponential fitting, for solving fourth order boundary value problems.
Abstract: Problem Statement: Many boundary value problems that arise in real life situations defy analytical solution; hence numerical techniques are desirable to find the solution of such equations. New numerical methods which are comparatively better than the existing ones in terms of efficiency, accuracy, stability, convergence and computational cost are always needed. Approach: In this study, we developed and applied three methods-standard cubic spline collocation, perturbed cubic spline collocation and perturbed cubic spline collocation tau method with exponential fitting, for solving fourth order boundary value problems. A mathematical software MATLAB was used to solve the systems of equations obtained in the illustrative examples. Results: The results obtained, from numerical examples, show that the methods are efficient and accurate with perturbed cubic spline collocation tau method with exponential fitting been the most efficient and accurate method with little computational effort involved. Conclusion: These methods are preferable to some existing methods because of their simplicity, accuracy and less computational cost involved.
15 citations
TL;DR: A class of non-linear singular boundary value problems is solved by new methods based on non-polynomial cubic spline based on the quasilinearization technique to reduce the given non- linear problem to a sequence of linear problems.
Abstract: A class of non-linear singular boundary value problems is solved by new methods based on non-polynomial cubic spline. We use the quasilinearization technique to reduce the given non-linear problem to a sequence of linear problems. We modify the resulting set of differential equations at the singular point then treat this set of boundary value problems by using a non-polynomial cubic spline approximation. Convergence of the methods is shown through standard convergence analysis. Numerical examples are given to illustrate the applicability and efficiency of our methods.
TL;DR: It is shown that such an interpolatory curve exists provided that the data polygon, formed by the interpolation points, is convex, and satisfies an additional restriction on its angles.
02 Jul 2008
TL;DR: A new kind of Hermite interpolation on arbitrary domains, matching derivative data of arbitrary order on the boundary, and minimizing over derivatives of polynomials of arbitrary odd degree is proposed.
Abstract: In this paper we propose a new kind of Hermite interpolation on arbitrary domains, matching derivative data of arbitrary order on the boundary. The basic idea stems from an interpretation of mean value interpolation as the pointwise minimization of a radial energy function involving first derivatives of linear polynomials. We generalize this and minimize over derivatives of polynomials of arbitrary odd degree. We analyze the cubic case, which assumes first derivative boundary data and show that the minimization has a unique, infinitely smooth solution with cubic precision. We have not been able to prove that the solution satisfies the Hermite interpolation conditions but numerical examples strongly indicate that it does for a wide variety of planar domains and that it behaves nicely.
TL;DR: In this paper, the authors used a minimal energy method to find Hermite interpolation based on bivariate spline spaces over a triangulation of the scattered data locations, and they showed that the Hermite spline interpolation converges to a given sufficiently smooth function f if the data values are obtained from this f.
TL;DR: In this article, a new Bayesian approach for monotone curve fitting based on the isotonic regression model is proposed, where the unknown linear regression function is approximated by a cubic spline and the constraints are represented by the intersection of quadratic cones.
Abstract: This article proposes a new Bayesian approach for monotone curve fitting based on the isotonic regression model. The unknown monotone regression function is approximated by a cubic spline and the constraints are represented by the intersection of quadratic cones. We treat the number and locations of knots as free parameters and use reversible jump Markov chain Monte Carlo to obtain posterior samples of knot configurations. Given the number and locations of the knots, second-order cone programming is used to estimate the remaining parameters. Simulation results suggest the method performs well and we illustrate the approach using the ASA car data.
TL;DR: A bivariate C^1 cubic super spline is constructed on Powell-Sabin type-1 split with the additional smoothness at vertices in the original triangulation being C^2, which permits the Hermite interpolation up to the second order partial derivatives exactly on all the vertices of the original Triangulation.
Abstract: A bivariate C^1 cubic super spline is constructed on Powell-Sabin type-1 split with the additional smoothness at vertices in the original triangulation being C^2, which permits the Hermite interpolation up to the second order partial derivatives exactly on all the vertices in the original triangulation. The locally supported dual basis and computational details using derivatives around each vertex are given.
TL;DR: A new dimension of image compression using random pixels of irregular sampling and image reconstruction using cubic-spline interpolation techniques proposed in this paper would provide a better efficiency both in pixel reconstruction and color reproduction.
Abstract: A new dimension of image compression using random pixels of irregular sampling and image reconstruction using cubic-spline interpolation techniques proposed in this paper. It also covers the wide field of multimedia communication concerned with multimedia messaging (MMS) and image transfer through mobile phones and tried to find a mechanism to transfer images with minimum bandwidth requirement. This method would provide a better efficiency both in pixel reconstruction and color reproduction. The discussion covers theoretical techniques of random pixel selection, transfer and implementation of efficient reconstruction with cubic spline interpolation.
TL;DR: The present paper analyzes linear interpolation, cubic splines and parametric (or "damped") splines for the interpolation task and found Cubic splines to be the most recommendable method.
Abstract: The robust algorithm OPED for the reconstruction of images from Radon data has been recently developed. This reconstructs an image from parallel data within a special scanning geometry that does not need rebinning but only a simple re-ordering, so that the acquired fan data can be used directly for the reconstruction. However, if the number of rays per fan view is increased, there appear empty cells in the sinogram. These cells need to be filled by interpolation before the reconstruction can be carried out. The present paper analyzes linear interpolation, cubic splines and parametric (or “damped”) splines for the interpolation task. The reconstruction accuracy in the resulting images was measured by the Normalized Mean Square Error (NMSE), the Hilbert Angle, and the Mean Relative Error. The spatial resolution was measured by the Modulation Transfer Function (MTF). Cubic splines were confirmed to be the most recommendable method. The reconstructed images resulting from cubic spline interpolation show a significantly lower NMSE than the ones from linear interpolation and have the largest MTF for all frequencies. Parametric splines proved to be advantageous only for small sinograms (below 50 fan views).
TL;DR: This work studies a class of first-derivative-based smooth univariate cubic L1 splines, aimed at generating “shape-preserving” smooth interpolating curves that represent data with abrupt changes in magnitude and/or knot spacing.
Abstract: With the objective of generating "shape-preserving" smooth interpolating curves that represent data with abrupt changes in magnitude and/or knot spacing, we study a class of first-derivative-based $$\mathcal{C}^1$$ -smooth univariate cubic L 1 splines. An L 1 spline minimizes the L 1 norm of the difference between the first-order derivative of the spline and the local divided difference of the data. Calculating the coefficients of an L 1 spline is a nonsmooth non-linear convex program. Via Fenchel's conjugate transformation, the geometric dual program is a smooth convex program with a linear objective function and convex cubic constraints. The dual-to-primal transformation is accomplished by solving a linear program.
TL;DR: In this paper, a fast pattern synthesis method for a broadband array antenna using the particle swarm optimization (PSO) and cubic spline interpolation (CSI) is described. But, the method is not suitable for high-frequency applications.
Abstract: This paper describes a fast pattern synthesis method for a broadband array antenna using the particle swarm optimization (PSO) and cubic spline interpolation (CSI). Being an indispensable part of a high speed space-division communication system, the array antenna operates in a wide frequency band (200–400 MHz) and has stable patterns with 60-degree half power beam width (HPBW) in the whole frequency band. Firstly, by establishing a versatile objective function, the complex excitations of the circular array at the selected seven frequency points are determined via the PSO algorithm. Then, the complex excitations of the circular array at arbitrary frequency points in the whole working frequency band are calculated effectively using the CSI method. A uniform circular array with six broadband dipole elements is examined. The broadband patterns with 60-degree HPBW and the accuracy of the interpolation method are demonstrated.
15 Aug 2008
TL;DR: A novel and efficient local spline interpolation algorithm is presented that performs a constant number of iteration that affects only a small number of control points over time and the cost of the interpolation does not depend on the total control point number.
Abstract: We present a novel and efficient local spline interpolation algorithm and apply it into our application of key frame based 2D animation. Unlike global algorithms which need to solve a linear system every time a vertex is moved, our method performs a constant number of iteration that affects only a small number of control points over time. Therefore the cost of the interpolation does not depend on the total control point number.
01 Jan 2008
TL;DR: In this article, the authors used Hilbert transform (Hilbert-Huang Transform) to analyze harmonic in order to overcome the shortcoming of FFT method and wavelet analysis method.
Abstract: To analyze harmonic accurately is significant to the stability of power system.HHT(Hilbert-Huang Transform) is adopted to analyze harmonic in order to overcome the shortcoming of FFT method and wavelet analysis method.The power quality signals are decomposed by empirical mode decomposition(EMD) method,and then a series of intrinsic mode function(IMF) are obtained.Owing to different IMFs correspond to different harmonic,all harmonic’ amplitudes,frequencies and phases can be achieved via Hilbert transform(HT) plus Least Square fitting,to realize accurate analysis of power system harmonic.During the procession of EMD,piecewise Cubic Hermite Interpolating Polynomial is employed.Furthermore,in order to alleviate edge effects the method of adding extremes is used.Simulation indicates that Hermite interpolation is superior to the cubic spline interpolation in harmonic analysis.The method has high precise and is satisfied to analysis power system harmonic.
TL;DR: In this paper, G1 continuous cubic spline interpolation of data points in \( √ R √ 3 ) was studied, based on a discrete approximation of the strain energy, and simple geometric conditions on data are presented that guarantee the existence of the interpolant.
Abstract: In this paper, G1 continuous cubic spline interpolation of data points in \({\mathbb{R}^3}\) , based on a discrete approximation of the strain energy, is studied. Simple geometric conditions on data are presented that guarantee the existence of the interpolant. The interpolating spline is regular, loop-, cusp- and fold-free.
TL;DR: In this paper, four types of numerical methods namely: Natural Cubic Spline, Special A-D cubic spline, FTCS and Crank-Nicolson are applied to both advection and diffusion terms of the one-dimensional adveection-diffusion equations with constant coefficients.
Abstract: Four types of numerical methods namely: Natural Cubic Spline, Special A-D Cubic Spline, FTCS and Crank–Nicolson are applied to both advection and diffusion terms of the one-dimensional advection-diffusion equations with constant coefficients. The numerical results from two examples are tested with the known analytical solution. The errors are compared when using different Peclet numbers.
Journal Article•
TL;DR: In this paper, the differentiation formulas of the cubic spline interpolation on the three boundary conditions are put forward based on analysis of cubic splines interpolation, and examples are given.
Abstract: Based on analysis of cubic spline interpolation, the differentiation formulas of the cubic spline interpolation on the three boundary conditions are put up forward. At last, examples are given.
TL;DR: The concept of single-side/double-side cubic curves is presented and the necessary and sufficient condition of a cubic curve being a single- side/ double-side curve is obtained and a new method for the problem of cubic polynomial interpolation is presented.
Abstract: ''The NURBS Book'' [L. Piegl, W. Tiller, The NURBS Book, second edn, Springer, 1997] is very popular in the fields of computer aided geometric design (CAGD) and geometric modeling. In Section 9.5.2 of the book, the well-known problem of the local cubic spline approximation is discussed. The key in local cubic spline approximation is cubic polynomial interpolation. In this short paper, we present the concept of single-side/double-side cubic curves and obtain the necessary and sufficient condition of a cubic curve being a single-side/double-side curve. Based on this result, for some cases of two end tangents being nearly parallel we present a new method for the problem of cubic polynomial interpolation. We also point out a flaw in Section 9.5.2 of the book and give the correction result.
01 Jan 2008
TL;DR: In this article, G 1 continuous cubic spline interpolation of data points in [FORMULA], based on a discrete approximation of the strain energy, is studied, and simple geometric conditions on data are presented that guarantee the existence of the interpolant.
Abstract: In this paper, G 1 continuous cubic spline interpolation of data points in [FORMULA] , based on a discrete approximation of the strain energy, is studied. Simple geometric conditions on data are presented that guarantee the existence of the interpolant. The interpolating spline is regular, loop-, cusp- and fold-free.
27 Feb 2008
TL;DR: In this paper, the authors consider piecewise interpolants of higher degree and show how many pieces of information are needed to fit a cubic between two points, where the derivative is not specified but enforced.
Abstract: • While we expect function not to vary, we expect it to also be smooth • So we could consider piecewise interpolants of higher degree • How many pieces of information do we need to fit a cubic between two points? – y=a+bx+cx 2 +dx 3 – 4 coefficients – Need 4 pieces of information – 2 values at end points – Need 2 more pieces of information – Derivatives? • However for Hermite, the derivative needs to be specified • Cubic splines, the derivative is not specified but enforced