Scattered Data Interpolation Research Articles

Bivariate Polynomial Interpolation over Nonrectangular Meshes

AbstractIn this paper, bymeans of a new recursive algorithm of non-tensor-product-typed divided differences, bivariate polynomial interpolation schemes are constructed over nonrectangular meshes firstly, which is converted into the study of scattered data interpolation. And the schemes are different as the number of scattered data is odd and even, respectively. Secondly, the corresponding error estimation is worked out, and an equivalence is obtained between high-order non-tensor-product-typed divided differences and high-order partial derivatives in the case of odd and even interpolating nodes, respectively. Thirdly, several numerical examples illustrate the recursive algorithms valid for the non-tensor-product-typed interpolating polynomials, and disclose that these polynomials change as the order of the interpolating nodes, although the node collection is invariant. Finally, from the aspect of computational complexity, the operation count with the bivariate polynomials presented is smaller than that with radial basis functions.

Interpolating sparse scattered data using flow information

Scattered data interpolation and approximation techniques allow for the reconstruction of a scalar field based upon a finite number of scattered samples of the field. In general, the fidelity of the reconstruction with respect to the original scalar field tends to deteriorate as the number of samples decreases. For the situation of very sparse sampling, the results may not be acceptable at all. However, if it is known that the scalar field of interest is correlated with a known flow field – as is the case when the scalar field represents the value of an oceanographic tracer that propagates under the influence of the ocean's flow – then this knowledge can be exploited to enhance the scattered data reconstruction method. One way to exploit flow field information is to use it to construct a modified notion of distance between points. Replacing the standard Euclidean distance metric with a flow-field-aware notion of distance provides a method for extending standard scattered data interpolation methods into flow-based methods that produce superior results for very sparse data. The resulting reconstructions typically have lower root-mean-square errors than reconstructions that do not use the flow information, and qualitatively they often appear physically more realistic.

Comparison of nonparametric methods for static visual field interpolation

Visual field testing with standard automated perimetry produces a sparse representation of a sensitivity map, sometimes called the hill of vision (HOV), for the retina. Interpolation or resampling of these data is important for visual display, clinical interpretation, and quantitative analysis. Our objective was to compare several popular interpolation methods in terms of their utility to visual field testing. We evaluated nine nonparametric scattered data interpolation algorithms and compared their performances in normal subjects and patients with retinal degeneration. Interpolator performance was assessed by leave-one-out cross-validation accuracy and high-density interpolated HOV surface smoothness. Radial basis function (RBF) interpolation with a linear kernel yielded the best accuracy, with an overall mean absolute error (MAE) of 2.01 dB and root-mean-square error (RMSE) of 3.20 dB that were significantly better than all other methods (p ≤ 0.003). Thin-plate spline RBF interpolation yielded the best smoothness results (p < 0.001) and scored well for accuracy with overall MAE and RMSE values of 2.08 and 3.28 dB, respectively. Natural neighbor interpolation, which may be a more readily accessible method to some practitioners, also performed well. While no interpolator will be universally optimal, these interpolators are good choices among nonparametric methods.

Pseudo-inverses of difference matrices and their application to sparse signal approximation

We derive new explicit expressions for the components of Moore–Penrose inverses of symmetric difference matrices. These generalized inverses are applied in a new regularization approach for scattered data interpolation based on partial differential equations. The columns of the Moore–Penrose inverse then serve as elements of a dictionary that allow a sparse signal approximation. In order to find a set of suitable data points for signal representation we apply the orthogonal matching pursuit (OMP) method.

Tool path optimisation for turning of non-rotational symmetrical surfaces based on scattered data interpolation

Non-rotational symmetric surfaces have been widely applied in the fields of imaging, illumination, precision measurement and laser beam reshaping. Commonly used tool nose compensation method would introduce high frequency compensation components both in X and Z direction. This high frequency motion especially that of X slide is disadvantageous for the fabrication. To solve the above problem, an algorithm based on scattered data interpolation is brought out which can achieve the tool nose compensation only in Z direction. The moving least squares (MLS) method is applied for the surface fitting to calculate the tool path, tangent vector and normal vector. Then the factors influencing the accuracy of the data interpolation are discussed. Finally, a facial 3D surface was successfully fabricated and the operation time shows a high efficiency of the path optimisation algorithm. The machined results further verified the proposed tool path generation and optimisation algorithm.

Differentiable positive definite functions on two-point homogeneous spaces

In this paper we study continuous kernels on compact two point homogeneous spaces which are positive definite and zonal (isotropic). Such kernels were characterized by R. Gangolli some forty years ago and are very useful for solving scattered data interpolation problems on the spaces. In the case the space is the d-dimensional unit sphere, J. Ziegel showed in 2013 that the radial part of a continuous positive definite and zonal kernel is continuously differentiable up to order ⌊(d−1)/2⌋ in the interior of its domain. The main issue here is to obtain a similar result for all the other compact two point homogeneous spaces.

Robust, Error-Tolerant Photometric Projector Compensation.

We propose a novel error tolerant optimization approach to generate a high-quality photometric compensated projection. The application of a non-linear color mapping function does not require radiometric pre-calibration of cameras or projectors. This characteristic improves the compensation quality compared with related linear methods if this approach is used with devices that apply complex color processing, such as single-chip digital light processing projectors. Our approach consists of a sparse sampling of the projector's color gamut and non-linear scattered data interpolation to generate the per-pixel mapping from the projector to camera colors in real time. To avoid out-of-gamut artifacts, the input image's luminance is automatically adjusted locally in an optional offline optimization step that maximizes the achievable contrast while preserving smooth input gradients without significant clipping errors. To minimize the appearance of color artifacts at high-frequency reflectance changes of the surface due to usually unavoidable slight projector vibrations and movement (drift), we show that a drift measurement and analysis step, when combined with per-pixel compensation image optimization, significantly decreases the visibility of such artifacts.

Implicit local radial basis function interpolations based on function values

In this paper we propose two fast localized radial basis function fitting algorithms for solving large-scale scattered data interpolation problems. For each given point in the given data set, a local influence domain containing a small number of nearest neighboring points is established and a global interpolation is performed within this restricted domain. A sparse matrix is formulated based on the global interpolation in these local influence domains. The proposed methods have achieved both low computational cost and minimal memory storage. In comparison with the compactly supported radial basis functions, the proposed fitting algorithms are highly accurate. The numerical examples have provided strong evidence that the two proposed algorithms are indeed highly efficient and accurate. In the two proposed algorithms, we have successfully solved a large-scale interpolation problem with 225,000 interpolation points in two dimensional space.

Scattered data interpolation with nonnegative preservation using bivariate splines and its application

We study how to use bivariate splines for scattered data interpolation with nonnegativity preservation of the given data values. That is, we propose a constrained minimal energy method to find a C1 smooth interpolation of nonnegative data values over scattered locations using bivariate splines. We establish the existence and uniqueness of the minimizer under mild assumptions on the data locations and triangulations. Next we show some approximation properties of the minimizer. We then use the classic projected gradient algorithm to find the minimizer using a simplified nonnegative constraint. Some synthetic as well as a real life example demonstrate the performance, producing nonnegative interpolatory spline surfaces from nonnegative data values. Experimental results also show that the approximation of the nonnegative interpolatory spline surfaces is slightly better than the approximation of the classic minimal energy interpolatory splines.

Resting State fMRI in the moving fetus: A robust framework for motion, bias field and spin history correction

There is growing interest in exploring fetal functional brain development, particularly with Resting State fMRI. However, during a typical fMRI acquisition, the womb moves due to maternal respiration and the fetus may perform large-scale and unpredictable movements. Conventional fMRI processing pipelines, which assume that brain movements are infrequent or at least small, are not suitable. Previous published studies have tackled this problem by adopting conventional methods and discarding as much as 40% or more of the acquired data.In this work, we developed and tested a processing framework for fetal Resting State fMRI, capable of correcting gross motion. The method comprises bias field and spin history corrections in the scanner frame of reference, combined with slice to volume registration and scattered data interpolation to place all data into a consistent anatomical space. The aim is to recover an ordered set of samples suitable for further analysis using standard tools such as Group Independent Component Analysis (Group ICA).We have tested the approach using simulations and in vivo data acquired at 1.5T. After full motion correction, Group ICA performed on a population of 8 fetuses extracted 20 networks, 6 of which were identified as matching those previously observed in preterm babies.

Range restricted positivity-preserving scattered data interpolation

The construction of a range restricted bivariate C1 ( or G1 ) interpolant to scattered data is considered in which the interpolant is positive everywhere if the original data are positive. This study is motivated by earlier work in which sufficient conditions are derived on Bézier points in order to ensure that surfaces comprising cubic Bézier triangular patches are always positive and satisfy C1 continuity conditions. In the current work, simpler and more relaxed conditions are derived on the Bézier points. The gradients at the data sites are then calculated (and modified if necessary) to ensure that these conditions are satisfied. Each triangular patch of the interpolating surface is formed as a convex combination of three quartic Bézier triangular patches. Its construction is local and easily extended to include as upper and lower constraints to the interpolant surfaces of the form z = P(x,y) where P is a polynomial of degree less or equal to 4. Moreover, C1 ( or G1 ) piecewise polynomial surfaces consisting of polynomial pieces of the form z = P(x,y) on the triangulation of the data sites are also admissible constraints. A number of examples are presented.

Regularized symmetric positive definite matrix factorizations for linear systems arising from RBF interpolation and differentiation

Scattered data interpolation using Radial Basis Functions involves solving an ill-conditioned symmetric positive definite (SPD) linear system (with appropriate selection of basis function) when the direct method is used to evaluate the problem. The standard algorithm for solving a SPD system is a Cholesky factorization. Severely ill-conditioned theoretically SPD matrices may not be numerically SPD (NSPD) in which case a Cholesky factorization fails. An alternative symmetric matrix factorization, the square root free Cholesky factorization, has the same flop count as a Cholesky factorization and is successful even when a matrix ceases to be NSPD. A regularization method can be used to prevent the failure of the Cholesky factorization and to improve the accuracy of both SPD matrix factorizations when the matrices are severely ill-conditioned. The specification of the regularization parameter is discussed as well as convergence/stopping criteria for the algorithm. The formation of differentiation matrices with the regularized SPD factorizations is demonstrated to improve eigenvalue stability properties of RBF methods for hyperbolic PDEs.

Error Estimate for Spherical Neural Networks Interpolation

We present a type of spherical neural network (SNN) with bounded sigmoidal activation function and study its interpolation capability. We find that the provided SNN can exactly interpolate the training samples. Furthermore, based on the special structure of the presented SNN, we can bound the interpolation error by the modulus of smoothness of the target function, which is different from the previous results on the spherical scattered data interpolation problem.

Foundation of 3D Coal Seam Model Based on the Incremental Insertion and the Shortest Diagonal Methods

As the borehole data is sparse , the particle swarm neural network (PSO-BP) spatial data interpolation method is used for scattered data interpolation, which enhances the availability of the data and improves the accuracy of data interpolation. Furthermore, the incremental insertion and the shortest diagonal methods are used in modeling because the modeling speed is high and dynamically updating is convenient, finally, 3D visualization model of the coal seam on the base of Matlab language is realized.

The Fourier approximation of smooth but non-periodic functions from unevenly spaced data

We develop an algorithm to extend, to the nonequispaced case, a recently-introduced fast algorithm for constructing spectrally-accurate Fourier approximations of smooth, but nonperiodic, data. Fast Fourier continuation algorithms, which allow for the Fourier approximation to be periodic in an extended domain, are combined with the underlying ideas behind nonequispaced fast Fourier transform (NFFT) algorithms. The result is a method which allows for the fast and accurate approximation of unevenly sampled nonperiodic multivariate data by Fourier series. A particular contribution of the proposed method is that its formulation avoids the difficulties related to the conditioning of the linear systems that must be solved in order to construct a Fourier continuation. The efficiency, essentially equivalent to that of an NFFT, and accuracy of the algorithm is shown through a number of numerical examples. Numerical results demonstrate the spectral rate of convergence of this method for sufficiently smooth functions. The accuracy, for sufficiently large data sets, is shown to be improved by several orders of magnitudes over previously published techniques for scattered data interpolation.

Comparison study of Gauss, MQ and TPS for interpolation application

There are mainly two parts in this paper. In the first part, we briefly introduce the basic theory of the radial basis function (RBF) and its application in scattered data interpolation, and investigate the difference among three kernel functions: multi-quadric, Gauss and thin plate spline, in detail. According to the multiple examples, we make a lot of experiments, and develop a system in the C++ programme language and the OpenGL open graphics library to make simulation experiments. Finally, we analyse and summarise the different characteristics of the each kernel function in the process of interpolation. In addition, on the basis of the well-known interpolation theory and the analysis of the experimental data, we list several examples for interpolation comparisons. In the second part, the distance of the current marked points down to its most neighbouring marked points is used as shape parameter. Combining with other shape parameter for analysis and comparisons, the results represent the different influences of RBF shape parameter on the curve and surface construction. Our experiment results obtained in this article have theoretical guide values in graphic interpolation modelling.

Regularized Minimum Length Method in Scattered Data Interpolation

In an attempt of accumulating more experiences of interpolating scattered data using the minimum length method, this study chooses new kernel functions from the machine learning technique to implementing this minimum length method. But, consulting with the regularization theory, a regularized minimum length method is created by solving coefficient of it in a penalized least squares approximation problem. The purpose of creating this regularized minimum length method is responding to a pilot observation finding the instability of original minimum length method under dense interpolation points. Testing the regularized minimum length method finds that applying it is time-saving but its performance is comparable to the radial point interpolation with polynomial reproduction. Inverse multiquadric and rational quadric kernel functions are two preferred kernel function to perform the regularized minimum length method. In conclusion, the proposed regularized minimum length method can be a useful scattered data interpolation method.

Anisotropic scattered data interpolation for pushbroom image rectification.

This paper deals with fast and accurate visualization of pushbroom image data from airborne and spaceborne platforms. A pushbroom sensor acquires images in a line-scanning fashion, and this results in scattered input data that need to be resampled onto a uniform grid for geometrically correct visualization. To this end, we model the anisotropic spatial dependence structure caused by the acquisition process. Several methods for scattered data interpolation are then adapted to handle the induced anisotropic metric and compared for the pushbroom image rectification problem. A trick that exploits the semiordered line structure of pushbroom data to improve the computational complexity several orders of magnitude is also presented.

A novel triangle-based method for scattered data interpolation

Local numerical methods for scattered data interpolation often require a smart subdivision of the domain in geometrical polyhedral structures. In particular triangulations in the plane (2D) and tetrahedrizations in the space (3D) are widely used to dene interpolation models. In this paper we give a short survey on the main methods for the scattered data problem and we recall preliminaries on triangulations and their related properties. Finally, combining two well-known ideas we present a new triangle-based interpolation method and show its application to a case study.

Adaptive Refinement of the Flow Map Using Sparse Samples

We present a new efficient and scalable method for the high quality reconstruction of the flow map from sparse samples. The flow map describes the transport of massless particles along the flow. As such, it is a fundamental concept in the analysis of transient flow phenomena and all so-called Lagrangian flow visualization techniques require its approximation. The flow map is generally obtained by integrating a dense 1D, 2D, or 3D set of particles across the domain of definition of the flow. Despite its embarrassingly parallel nature, this computation creates a performance bottleneck in the analysis of large-scale datasets that existing adaptive techniques alleviate only partially. Our iterative approximation method significantly improves upon the state of the art by precisely modeling the flow behavior around automatically detected geometric structures embedded in the flow, thus effectively restricting the sampling effort to interesting regions. Our data reconstruction is based on a modified version of Sibson's scattered data interpolation and allows us at each step to offer an intermediate dense approximation of the flow map and to seamlessly integrate regions that will be further refined in subsequent steps. We present a quantitative and qualitative evaluation of our method on different types of flow datasets and offer a detailed comparison with existing techniques.