Splines Of Degree Research Articles

New Construction and New Error Bounds for (0, 2, 4) Lacunary Interpolation By Six Degree Spline

The object of this paper obtains the existence, uniqueness and upper bounds for errors of six degree splines interpolating the lacunary data (0, 2, 4). We also show that the changes of the boundary conditions and the class of spline functions has a main role in minimizing the upper bounds for error in lacunary interpolation problem. For this reason, in the construction of our spline function which interpolates the lacunary data (0, 2, 4). We changed the boundary condition and the class of spline functions which are given by [4] from first derivative to third derivative and the class of spline function from to .

Real time cardiac MRI: spline-based spatio-temporal reconstruction of spiral data

We propose a novel reconstruction method using a spline-based image model in both spatial and temporal dimensions that takes the advantage of the precise timing of each k-space sample to reconstruct image series at high time frames, independently from the original sampling rate of data and avoiding the temporal blurring that can affect other reconstruction methods like sliding window. While MRI techniques have undergone considerable improvements since the early days, performing real time imaging is still challenging today. Despite the various original methods proposed until very recently, the usual way to reconstruct data at a higher frame rate is the sliding window technique that brings along intrinsic temporal blurring The proposed image model is the following: ρ(x,y,t)=Σ n1,n2,n3cn1,n2,n3β1(x-n1)β1(y-n2)βα(t-n3) where βα is a spline function (α the spline degree). The spline model has the main advantage of allowing image reconstruction at arbitrary time points. In order to recover the Nyquist criteria, more data samples are included in the reconstruction by increasing the spline degree but they are weighted with the temporal information (zeroth degree neglects the temporal information of the samples as in nearest neighbor interpolation). The model was validated on an analytic phantom and then applied to real time data acquired from a healthy volunteer on a 3T scanner using a spiral sequence. Results obtained on the numerical phantom demonstrated that the time resolution of the reconstructed data could be improved by a factor of 3 when using a high enough spline degree, without sacrifying the SNR (11.5 dB for sliding window and 10.3 dB for the proposed method with α=5, whereas only 5.78 dB for α=0) (figure ​(figure1).1). Moreover, the edge strength of the endocardium was recovered for the proposed method (edge strength was 10.4e-2a.u. for sliding window and 13.9e-2a.u. for our method with α=5) (figure ​(figure2).2). The temporal profile obtained on real data with 5th degree splines was comparable to the one obtained with the classical sliding window. Note that the number of spline coefficients (degrees of freedom of the model) remains fixed. Figure 1 Upper, temporal profiles of sliding window (left) and proposed (right) reconstruction, for the numerical phantom; lower, for real-time data acquired on a healthy volunteer. Figure 2 Illustration of blurring on sliding window reconstruction compared to proposed method for the analytic phantom. This novel approach is able to increase up to 3 times the temporal resolution of reconstructed images without introducing temporal blurring.

DIRECT DEGREE ELEVATION OF NURBS CURVES

In this paper we provide the guidelines for the direct degree elevation of NURBS curves. Through the analysis of linear equation systems of quartic and lower degree splines we derive a direct relation between the knot vector of a spline and the degree elevation coefficients. We also present a direct degree elevation scheme and several algorithms based on the discovered relation. Experimental results indicate that the direct degree elevation algorithms are up to twice more time-efficient than Piegl and Tiller’s degree elevation method. This proves the inefficiency of Bspline calculation schemes based on blossoming that involve redundant operations with control points. It also negates Piegl and Tiller’s claim about the inefficiency of linear equation system solving method for quartic and lower degrees.

Robust univariate spline models for interpolating interval data

We consider spline interpolation problems where information about the approximated function is given by means of interval estimates for the function values over ranges of x -values instead of specific knots. We propose two robust univariate spline models formulated as convex semi-infinite optimization problems. We present simplified equivalent formulations of both models as finite explicit convex optimization problems for splines of degrees up to 3. This makes it possible to use existing convex optimization algorithms and software.

О применении специальных многомерных сплайнов произвольной степени в численном анализе

О применении специальных многомерных сплайнов произвольной степени в численном анализе

The Existence, Uniqueness And Upper Bounds For Errors Of Six Degree Spline Interpolating The Lacunary Data (0,2,5)

The object of this paper is to obtain the existence, uniqueness and upper bounds for errors of six degree spline interpolating the lacunary data (0,2,5). We also showed that the changes of the boundary conditions and the class of spline functions has a main role in minimizing the upper bounds for error in lacunary interpolation problem. For this reason, in the construction of our spline function which interpolates the lacunary data (0,2,5), we changed the boundary conditions and the class of spline functions which are given by [1] from first derivative to third derivative and the class of spline function from to .

High accuracy asymptotics for a least squares spline error

The error of a least squares spline of an arbitrary degree p and all its derivatives is considered. Highly accurate asymptotic approximation is constructed for the dependence of the error coefficient on p.

Inverses of cyclic band matrices and the convergence of interpolation processes for derivatives of periodic interpolation splines

Entries of matrices inverse to cyclic band ones, as well as norms of the inverse matrices, are estimated. The estimates obtained are used to specify conditions under which interpolation processes with periodic odd degree splines converge for different derivatives. In particular, we give a positive solution to the C. de Boor problem on unconditional convergence of one of two middle derivatives without imposing any restrictions on a grid in the periodic case.

Changeable degree spline basis functions

A B-spline basis function is a piecewise function of polynomials of equal degree on its support interval. This paper extends B-spline basis functions to changeable degree spline (CD-spline for short) basis functions, each of which may consist of polynomials of different degrees on its support interval. The CD-spline basis functions possess many B-spline-like properties and include the B-spline basis functions as subcases. Their corresponding parametric curves, called CD-spline curves, are like B-spline curves and also have many good properties. If we use the CD-spline basis functions to design a curve made up of polynomial segments of different degrees, the number of control points may be decreased.

MODAL METHOD BASED ON SPLINE EXPANSION FOR THE ELECTROMAGNETIC ANALYSIS OF THE LAMELLAR GRATING

This paper reports an exact and explicit representation of the difierential operators from Maxwell's equations. In order to solve these equations, the spline basis functions with compact support are used. We describe the electromagnetic analysis of the lamellar grating as an eigenvalues problem. We choose the second degree spline as basis functions. The basis functions are projected onto a set of test functions. We use and compare several test functions namely: Dirac, Pulse and Spline. We show that the choice of the basis and test functions has a great in∞uence on the convergence speed. The outcomes are compared with those obtained by implementing the Finite-Difierence Modal Method which is used as a reference. In order to improve the numerical results an adaptive spatial resolution is used. Compared to the reference method, we show a signiflcantly improved convergence when using the spline expansion projected onto spline test functions.

10.1007/s11470-008-2003-5

A Hermite spline of third degree is constructed on a three-dimensional simplex. The deviation of its directional derivatives up to the third order is estimated in angle-free terms. The resulting estimates hold for any tetrahedron irrespective of its geometry.

Estimation of a k-monotone density: characterizations, consistency and minimax lower bounds.

The classes of monotone or convex (and necessarily monotone) densities on ℝ(+) can be viewed as special cases of the classes of k-monotone densities on ℝ(+). These classes bridge the gap between the classes of monotone (1-monotone) and convex decreasing (2-monotone) densities for which asymptotic results are known, and the class of completely monotone (∞-monotone) densities on ℝ(+). In this paper we consider non-parametric maximum likelihood and least squares estimators of a k-monotone density g(0).We prove existence of the estimators and give characterizations. We also establish consistency properties, and show that the estimators are splines of degree k - 1 with simple knots. We further provide asymptotic minimax risk lower bounds for estimating the derivatives[Formula: see text], at a fixed point x(0) under the assumption that [Formula: see text].

Expansion in the system of shifts of a B-spline

A method of approximating functions by linear combinations of shifts of a fixed function, a B-spline of the first, second, or third degree, is proposed. This method is exact for the classes of splines of the corresponding degree with deficiency 1. We obtain estimates of strong type for the norms of approximating operators in certain spaces, from which it follows immediately that the approximation in these spaces coincides in order with the best approximation by splines.

Introduction to the Contents of Issue 49:3

In this issue, we have happened to get several contributions in the Approximation and Numerical Quadrature area. In the first paper Rafael Campos and Fransisco Meijo Diaz develop quadrature formulas for discrete two sided Laplace transforms. These formulas give convergent approximations for functions that fall off to zero fast enough. An asymptotic formula yields an algorithm that computes a discrete Laplace transform using only exponentials. In another paper, Frances Kuo, Gregorz Wasilkowski and Henryk Woźniakowski investigate the power of algorithms to compute approximate values of a function in a reproducing Hilbert kernel space. They show that standard information, using n function values at randomly distributed points, is nearly as good as generalized information that uses n values of arbitrary linear functionals. In another paper, Paula Lamberti develops cubature formulas based on two dimensional B splines over an irregular triangulation. Two papers deal with how to find an approximation to a function given at irregularly distributed data points. A Lamni, Hamid Mraoui, Dress Sbibih and Ahmed Zidna approximate a function over a sphere by tensor product trigonometric splines or periodic algebraic trigonometric wavelets. The intention is to use such approximations for data compression. Francesca Mazzia and Alessandra Sestini develop a quasi interpolation scheme based on Hermite B splines of arbitrary degree. A small linear system is solved for each knot. For reasonably distributed knots, it can be used as a collocation method for solving a system of ordinary differential equations. In two related papers, it is shown how to develop higher order splitting methods for dissipative evolution equations. Complex time steps with positive real parts are used independently by Francois Castella, Philippe Chartier, Stephane Descombes

Estimation and Classification of BOLD Responses Over Multiple Trials

In this article, we model functional magnetic resonance imaging (fMRI) data for event-related experiment data using a fourth degree spline to fit voxel specific blood oxygenation level-dependent (BOLD) responses. The data are preprocessed for removing long term temporal components such as drifts using wavelet approximations. The spatial dependence is incorporated in the data by the application of 3D Gaussian spatial filter. The methodology assigns an activation score to each trial based on the voxel specific characteristics of the response curve. The proposed procedure has a capability of being fully automated and it produces activation images based on overall scores assigned to each voxel. The methodology is illustrated on real data from an event-related design experiment of visually guided saccades (VGS).

Construction functions of spline of fifth degree

Construction functions of spline of fifth degree

2D SCHEME FOR COMPUTING WATER SOURCE COMPONENTS IN COASTAL AREA - A TOOL FOR CREATING BASIC ENVIRONMENT MAP

2D scheme for hydrodynamic and water source components problems is created on improved finite difference scheme KOD-02 with rectangular network that may be added a number of curved triangles at some boundaries. Water resource components problem is solved via so-called second degree spline functions. In all computational process the alternative direction is used. The computation results lead to the formation of the water source components map, which is the basic environment map in coastal area.

Gauss–Green cubature and moment computation over arbitrary geometries

We have implemented in Matlab a Gauss-like cubature formula over arbitrary bivariate domains with a piecewise regular boundary, which is tracked by splines of maximum degree p (spline curvilinear polygons). The formula is exact for polynomials of degree at most 2 n − 1 using N ∼ c m n 2 nodes, 1 ≤ c ≤ p , m being the total number of points given on the boundary. It does not need any decomposition of the domain, but relies directly on univariate Gauss–Legendre quadrature via Green’s integral formula. Several numerical tests are presented, including computation of standard as well as orthogonal moments over a nonstandard planar region.

A multiresolution analysis for tensor-product splines using weighted spline wavelets

We construct biorthogonal spline wavelets for periodic splines which extend the notion of “lazy” wavelets for linear functions (where the wavelets are simply a subset of the scaling functions) to splines of higher degree. We then use the lifting scheme in order to improve the approximation properties with respect to a norm induced by a weighted inner product with a piecewise constant weight function. Using the lifted wavelets we define a multiresolution analysis of tensor-product spline functions and apply it to image compression of black-and-white images. By performing–as a model problem–image compression with black-and-white images, we demonstrate that the use of a weight function allows to adapt the norm to the specific problem.

The Existence, Uniqueness and Error Bounds of Approximation Splines Interpolation for Solving Second-Order Initial Value Problems

Problem statement: The lacunary interpolation problem, which we had investigated in this study, consisted in finding the six degree spline S (x) of deficiency four, interpolating data given on the function value and third and fifth order in the int erval (0,1). Also, an extra initial condition was prescribed on the first derivative. Other purpose o f this construction was to solve the second order differential equations by two examples showed that the spline function being interpolated very well. The convergence analysis and the stability of the a pproximation solution were investigated and compared with the exact solution to demonstrate the prescribed lacunary spline (0, 3, 5) function interpolation. Approach: An approximation solution with spline interpolatio n functions of degree six and deficiency four was derived for solving initial value problems, with prescribed nonlinear endpoint conditions. Under suitable assumptions, the existen ces; uniqueness and the error bounds of the spline (0, 3, 5) function had been investigated; also the uppe r bounds of errors were obtained. Results: Numerical examples, showed that the presented spline function proved their effectiveness in solving the second order initial value problems. Also, we noted that, the better error bounds were obtained for a small s tep size h. Conclusion: In this study we treated for a first time a lacuna ry data (0,3,5) by constructing spline function of degree six which interpolated th e lacunary data (0,3,5) and the constructed spline function applied to solve the second order initial value problems.