Ideal Interpolation Research Articles

Research on NURBS Curve Interpolation Algorithm Based on Mline-Gear

Aiming at the problem of speed fluctuation caused by the large error between the ideal interpolation step and the actual interpolation step in the traditional non-uniform rational B-spline ( NURBS ) curve interpolation, a NURBS curve interpolation algorithm based on Mline-Gear is proposed. The differential and derivative definitions are derived to transform the Mline and Gear formulas respectively, and the prediction and correction formulas of the interpolation parameters are obtained. Then, it is judged whether the interpolation parameter values obtained by the correction formula meet the speed fluctuation rate requirements, and the interpolation can be continued. Otherwise, the Steffensen descent formula is used to iteratively correct the interpolation parameter values. In order to ensure the calculation accuracy and avoid solving high-order derivatives, the fourth-order Runge-Kutta method is used to solve the initial two interpolation parameters. Finally, the algorithm is simulated. The simulation results show that the algorithm reduces the feed speed fluctuation rate while ensuring a small number of iterations.

Algorithms for computing Gröbner bases of ideal interpolation

Algorithms for computing Gröbner bases of ideal interpolation

Symmetry in multivariate ideal interpolation

An interpolation problem is defined by a set of linear forms on the (multivariate) polynomial ring and values to be achieved by an interpolant. For Lagrange interpolation the linear forms consist of evaluations at some nodes, while Hermite interpolation also considers the values of successive derivatives. Both are examples of ideal interpolation in that the kernels of the linear forms intersect into an ideal. For an ideal interpolation problem with symmetry, we address the simultaneous computation of a symmetry adapted basis of the least interpolation space and the symmetry adapted H-basis of the ideal. Beside its manifest presence in the output, symmetry is exploited computationally at all stages of the algorithm. For an ideal invariant, under a group action, defined by a Gröbner basis, the algorithm allows to obtain a symmetry adapted basis of the quotient and of the generators. We shall also note how it applies surprisingly but straightforwardly to compute fundamental invariants and equivariants of a reflection group.

Cyclic matrices and polynomial interpolation over division rings

As is well known, any complex cyclic matrix A is similar to the unique companion matrix associated with the minimal polynomial of A. On the other hand, a cyclic matrix over a division ring F is similar to a companion matrix of a polynomial which is defined up to polynomial similarity. In this paper we study more rigid canonical forms by embedding a given cyclic matrix over a division ring F into a controllable or an observable pair. Using the characterization of ideals in F[z] in terms of controllable and observable pairs we consider ideal interpolation schemes in F[z] which merge into polynomial interpolation problems containing both left and right interpolation conditions.

Optimization of the 3p keys kernel parameters by minimizing the ripple of the spectral characteristic

The ideal interpolation kernel is described by the sinc function, and its spectral characteristic is the box function. Due to the infinite length of the ideal kernel, it is not achievable. Therefore, convolutional interpolation kernels of finite length, which should better approximate the ideal kernel in a specified interval, are formed. The approximation function should have a small numerical complexity, so as to reduce the interpolation execution time. In the scientific literature, great attention is paid to the polynomial kernel of the third order. However, the time and spectral characteristic of the third-order polynomial kernels differs significantly from the shape of the ideal kernel. Therefore, the accuracy of cubic interpolation is lower. By optimizing the kernel parameters, it is possible to better approximate the ideal kernel. This will increase the accuracy of the interpolation. The first part of the paper describes a three-parameter (3P) Keys interpolation kernel, r. After that, the algorithm for optimizing the parameters of the 3P Keys kernel, is shown. First, the kernel is disassembled into components, and then, over each kernel component, Fourier transform is applied. In this way the spectral characteristic of the 3P Keys kernel, H, was determined. Then the spectral characteristic was developed in the Taylor series, HT. With the condition for the elimination of the members of the Taylor series, which greatly affect the ripple of the spectral characteristic, the optimal kernel parameters (?opt, ?opt, gopt) were determined. The second part of the paper describes an experiment, in which the interpolation accuracy of the 3P Keys kernel, was tested. Parametric cubic convolution (PCC) interpolation, with the 3P kernel, was performed over the images from the Test database. The Test database is created with standard Test images, which are intensively used in Digital Image Processing. By analyzing the interpolation error, which is represented by the Mean Square Error, MSE, the accuracy of the interpolation was determined. The results (?opt, ?opt, gopt, MSEmin) are presented on tables and graphs. Detailed comparative analysis showed higher interpolation accuracy with the proposed 3P Keys interpolation kernel, compared to the interpolation accuracy with, 1P Keys and 2P Keys interpolation kernels. Finally, the numerical values of the optimal kernel parameters, which are determined by the optimization algorithm proposed in this paper, were experimentally verified.

Comparative suitability of ordinary kriging and Inverse Distance Weighted interpolation for indicating intactness gradients on threatened savannah woodland and forest stands

Mapping spatial variations in tree density and woody species diversity, as indicators of intactness of tropical forests and woodlands, is potentially useful to nature conservation managers. This paper examines the comparative suitability of ordinary kriging and Inverse Distance Weighted (IDW) interpolation for mapping these variables. Two study sites representing the dense tree cover of tropical forest and the scattered tree cover in savannah woodland were used. As an interpolation variable, savannah tree density presents the challenge that tree cover is characteristically patchy. Field sampling at unevenly distributed sampling sites yielded spatial gradients in tree and shrub density and number of woody species per hectare, in relation to human harvesting. Autocorrelation indicated suitability of the data for kriging. Based on least error cross-validation statistics, the exponential semivariogram model was the most suitable for the data. Low neighborhood search radius values were more appropriate for IDW interpolation. The accuracy of the resulting tree density interpolations was judged against Normalized Difference Vegetation Index (NDVI) values from near-concurrent SPOT images. For the woodland, IDW interpolation had higher correlation with the NDVI (r = 0.782, P < 0.05) than ordinary kriging (r = 0.731, P < 0.05). For the forest, ordinary kriging had higher correlation (r = 0.921, P < 0.01) than IDW interplation (r = 0.875, P < 0.01). The results indicated that ordinary kriging is suitable for mapping tree density in dense forests, while IDW interpolation is more appropriate for scattered tree savannah woodland. The ideal interpolation technique can inform environmental management interventions.

The Representation of D-Invariant Polynomial Subspaces Based on Symmetric Cartesian Tensors

Multivariate polynomial interpolation plays a crucial role both in scientific computation and engineering application. Exploring the structure of the D-invariant (closed under differentiation) polynomial subspaces has significant meaning for multivariate Hermite-type interpolation (especially ideal interpolation). We analyze the structure of a D-invariant polynomial subspace Pn in terms of Cartesian tensors, where Pn is a subspace with a maximal total degree equal to n,n≥1. For an arbitrary homogeneous polynomial p(k) of total degree k in Pn, p(k) can be rewritten as the inner products of a kth order symmetric Cartesian tensor and k column vectors of indeterminates. We show that p(k) can be determined by all polynomials of a total degree one in Pn. Namely, if we treat all linear polynomials on the basis of Pn as a column vector, then this vector can be written as a product of a coefficient matrix A(1) and a column vector of indeterminates; our main result shows that the kth order symmetric Cartesian tensor corresponds to p(k) is a product of some so-called relational matrices and A(1).

Interpolation of Discrete-Time Series

AbstractInterpolation can be used to up-sample a sparsely recorded or simulated data set. Ideal interpolation models such as a Nyquist-bandlimited filter in the frequency domain and sinc function i...

Optimal Interpolation and Compatible Relaxation in Classical Algebraic Multigrid

In this paper, we consider a classical algebraic multigrid (AMG) form of optimal interpolation that directly minimizes the two-grid convergence rate and compare it with a so-called ideal interpolation that minimizes a weak approximation property of the coarse space. We study compatible relaxation type estimates for the quality of the coarse grid and derive a new sharp measure using optimal interpolation that provides a guaranteed lower bound on the convergence rate of the resulting two-grid method for a given grid. In addition, we design a generalized bootstrap AMG setup algorithm that computes a sparse approximation to the optimal interpolation matrix. We demonstrate numerically that the bootstrap AMG method with sparse interpolation matrix (and spanning multiple levels) converges faster than the two-grid method with the standard ideal interpolation (a dense matrix) for various scalar diffusion problems with highly varying diffusion coefficient.

On the Ideal Interpolation Operator in Algebraic Multigrid Methods

Various algebraic multigrid algorithms have been developed for solving problems in scientific and engineering computation over the past decades. They have been shown to be well-suited for solving discretized partial differential equations on unstructured girds in practice. One key ingredient of algebraic multigrid algorithms is a strategy for constructing an effective prolongation operator. Among many questions on constructing a prolongation, an important question is how to evaluate its quality. In this paper, we establish new characterizations (including sufficient condition, necessary condition, and equivalent condition) of the so-called ideal interpolation operator. Our result suggests that, compared with common wisdom, one has more room to construct an ideal interpolation, which can provide new insights for designing algebraic multigrid algorithms. Moreover, we derive a new expression for a class of ideal interpolation operators.

Hierarchical MK Splines: Algorithm and Applications to Data Fitting

In the era of big data, it is very important to study large-scale data fitting methods. In order to ensure the calculation speed and accuracy, we propose a new kind of hierarchical many-knot splines (hereinafter called “hierarchical MK splines,” generally abbreviated as HMK splines) in this paper. The HMK splines method produces a sequence of MK spline functions. These MK spline functions are constructed into one ideal interpolation function by the MK spline refinement. In the case of regular sampling data, HMK splines can achieve the purpose of accurate approximation for the given data points without solving systems of equations. Further, in order to deal with the issues of scattered data fitting, the use of least-squares method will lead to the necessary of solving a linear system of equations. Since the ill-conditioned systems of equations often lead to unacceptable deviation of calculation results, one tries to avoid it as much as possible. The HMK splines algorithm can meet this requirement; it can avoid the intolerable deviation caused by solving systems of equations. Experimental results show that large-scale scattered data fitting can be easily achieved by the HMK splines algorithm and the reconstruction of nonuniform samples has a high accuracy.

An algorithm for the discretization of an ideal projector

Ideal interpolation is a generalization of the univariate Hermite interpolation. It is well known that every univariate Hermite interpolant is a pointwise limit of some Lagrange interpolants. However, a counterexample provided by Shekhtman Boris shows that, for more than two variables, there exist ideal interpolants that are not the limit of any Lagrange interpolants. So it is natural to consider: Given an ideal interpolant, how to find a sequence of Lagrange interpolants (if any) that converge to it. The authors call this problem the discretization for ideal interpolation. This paper presents an algorithm to solve the discretization problem. If the algorithm returns “True”, the authors get a set of pairwise distinct points such that the corresponding Lagrange interpolants converge to the given ideal interpolant.

The discretization for bivariate ideal interpolation

Carl de Boor conjectured that every ideal interpolant over complex field is the pointwise limit of Lagrange interpolants. Boris Shekhtman proved that the conjecture is true in two variables, and he also provided a counterexample for more than three variables. However, Shekhtman only mentioned the existence of some mathematical objects (without giving a method to compute them) in his proof of the bivariate case. For general interpolation condition functionals on the interpolation sites, we improve Shekhtman’s method to find a sequence of interpolation sites (also called discrete sites), such that the corresponding Lagrange interpolants converge to the given bivariate ideal interpolant. We discuss a special case where the multiplicity space is of breadth one. The results in this paper give a completely algorithmic way to realize Shekhtman’s method.

Error formulas for ideal interpolation

We study the algebraic structure of error formulas for ideal interpolation. We introduce the so-called “normal” error formulas and prove that the lexicographic order reduced Grobner basis admits such a formula for all ideal interpolations. This formula is a generalization of the “good” error formula proposed by Carl deBoor. Finally, we discuss a Shekhtman’s example and give an explicit form of “normal” error formula for this example.

The structure of a second-degree [formula omitted]-invariant subspace and its application in ideal interpolation

The D-invariant polynomial subspaces play a crucial role in ideal interpolation. In this paper, we analyze the structure of a second-degree D-invariant polynomial subspace P2. As an application for ideal interpolation, we solve the discrete approximation problem for δzP2(D) under certain conditions, i.e., we compute pairwise distinct points, such that the limiting space of the evaluation functionals at these points is the given space δzP2(D), as the evaluation sites all coalesce at one site z.

The equivalent representation of the breadth-one D-invariant polynomial subspace and its discretization

This paper demonstrates the equivalence of two classes of D-invariant polynomial subspaces, i.e., these two classes of subspaces are different representations of the breadth-one D-invariant subspace. Moreover, the authors solve the discrete approximation problem in ideal interpolation for the breadth-one D-invariant subspace. Namely, the authors find the points, such that the limiting space of the evaluation functionals at these points is the functional space induced by the given D-invariant subspace, as the evaluation points all coalesce at one point.

Experimental Analysis of the Errors due to Polynomial Interpolation in Digital Image Correlation

AbstractDigital image correlation attempts to estimate displacement fields by digitally correlating two images acquired before and after motion. To do so, pixel intensity has to be interpolated at non‐integer locations. The ideal interpolator is the sinc, but as it requires infinite support, it is not normally used and is replaced by polynomials. Polynomial interpolation produces visually appealing results but introduces positional errors in the signal, thus causing the digital image correlation algorithms to converge to incorrect results. In this work, an experimental campaign is described, that aims to characterise the errors introduced by interpolation, focusing in particular on the systematic error and the standard deviation of displacements.

On non-existence of certain error formulas for ideal interpolation

We answer a question raised by Carl de Boor regarding the existence of certain “good” error formulas for ideal interpolation. We will show that, for a specific form of ideal interpolation by linear polynomials in two variables, such formulas do not exist.

Continuous time domain properties of causal cubic splines

The paper studies the continuous time domain properties of causal cubic splines with equidistant knots. Causality is achieved by truncating the impulse response of the ideal interpolator to a chosen length. The introduced error affects the final interpolation differently depending on the considered output filter formulation. For the first formulation it is the regularity that is sacrificed while for the second it is the interpolation and splitting of unit delay properties. The analysis of the casual cardinal splines’ spectra also demonstrates different manifestation of the causality induced error that affects either the pass-band region only or both the pass-band and the stop-band region. The frequency kernel is used for objective comparison of the causal and the non-causal interpolators including the causal piecewise cubic MOMS interpolator. Their relative interpolation performance was determined both for wide-band as well as for oversampled narrow-band input signals. The transparent delay is identified for which the causal and the non-causal interpolators achieve identical band-limited reconstruction accuracy.

Equivalence of the Lagrange interpolator for uniformly sampled signals and the scaled binomially windowed shifted sinc function

The maximally flat (MF) fractional delay (FD) filter which is in fact a Lagrange interpolator for uniformly sampled signals, has previously been shown to be equal to the scaled binomially windowed shifted version of the sinc function; the ideal interpolation kernel for band-limited signals. In this paper, another proof for this equivalence is presented. Unlike its counterparts available in the literature, the proof given here is neither strictly algebraic, nor deploys the explicit coefficient formulas of the MFFD filter. It follows a frequency domain approach based on the definition of this filter instead, and aims to provide more insight into the corresponding equivalence.