Quasi Interpolation Research Articles

Local spline refinement driven by fault jump estimates for scattered data approximation

We present new fault jump estimates to guide local refinement in surface approximation schemes with adaptive spline constructions. The proposed approach is based on the idea that, since discontinuities in the data should naturally correspond to sharp variations in the reconstructed surface, the location and size of jumps detected in the input point cloud should drive the mesh refinement algorithm. To exploit the possibility of inserting local meshlines in one or the other coordinate direction, as suggested by the jump estimates, we propose a quasi-interpolation (QI) scheme based on locally refined B-splines (LR B-splines). Particular attention is devoted to the construction of the local operator of the LR B-spline QI scheme, which properly adapts the spline approximation according to the nature and density of the scattered data configuration. A selection of numerical examples outlines the performance of the method on synthetic and real datasets characterized by different geographical features.

Cutting-Edge Computational Approaches for Approximating Nonlocal Variable-Order Operators

This study presents an algorithmically efficient approach to address the complexities associated with nonlocal variable-order operators characterized by diverse definitions. The proposed method employs integro spline quasi interpolation to approximate these operators, aiming for enhanced accuracy and computational efficiency. We conduct a thorough comparison of the outcomes obtained through this approach with other established techniques, including finite difference, IQS, and B-spline methods, documented in the applied mathematics literature for handling nonlocal variable-order derivatives and integrals. The numerical results, showcased in this paper, serve as a compelling validation of the notable advantages offered by our innovative approach. Furthermore, this study delves into the impact of selecting different variable-order values, contributing to a deeper understanding of the algorithm’s behavior across a spectrum of scenarios. In summary, this research seeks to provide a practical and effective solution to the challenges associated with nonlocal variable-order operators, contributing to the applied mathematics literature.

Spline manipulations for empirical mode decomposition (EMD) on bounded intervals and beyond

Empirical mode decomposition (EMD), introduced by N.E. Huang et al. in 1998, is perhaps the most popular data-driven computational scheme for the decomposition of a non-stationary signal or time series f(t), with time-domain R:=(−∞,∞), into finitely many oscillatory components {f1(t),⋯,fK(t)}, called intrinsic mode functions (IMFs), and some “almost monotone” remainder r(t), called the trend of f(t). The core of EMD is the iterative “sifting process” applied to each function mk−1(t) to compute fk(t), for k=1,⋯,K, where m0(t):=f(t) and mk(t):=mk−1(t)−fk(t), with trend r(t):=mK(t). For the computation of each IMF, the sifting process depends on cubic spline interpolation of the local maxima and local minima for computing the upper and lower envelopes, respectively, and on subtracting the mean of the two envelopes from the result of the previous iterative step. Since it is not feasible to search for all local extrema in the entire time-domain (−∞,∞), implementation of the sifting process is commonly performed on some desired truncated bounded interval [a,b]. The main objective of this paper is to introduce and develop four “cubic spline manipulation engines”, called “quasi-interpolation (QI)”, “enhanced quasi-interpolation (EQI)”, “local interpolation (LI)”, and “improved global interpolation (IGI)” cubic spline manipulation engines, in order to significantly improve the performance of EMD on the truncated time-domains with minimal boundary artifacts, computational efficiency, accuracy, and consistency. Introduction and construction of the “fundamental quasi-interpolation” (FQI) splines as basis functions of the QI manipulation engine eliminates the need of matrix inversion for computing (global) cubic spline interpolation, since the local maximum values and local minimum values are used as coefficients of their FQI spline series representations, respectively. For the EQI spline manipulation engine, the FQI functions are formulated in terms of the same cubic B-spline basis for both the upper and lower envelopes; and for the LI spline manipulation engine, the “cubic spline blending” operation is applied to further modify the FQI splines to enable true cubic spline interpolation by “correcting the approximate interpolation error” of the EQI engine. As a consequence, the EQI and LI manipulation engines have the common property that in computing the means of the upper and lower envelopes, the only computation is averaging the B-spline coefficients, instead of computing the upper and lower envelopes separately. Furthermore, fast cubic spline pre-processing of the given f(t) is also introduced to assure numerical stability in the computation of the Hilbert transform of the first IMF f1(t) on the truncated time-domain. The theory, along with methods and explicit formulas, developed in this paper are intended for other applications beyond EMD.

Hermite–Birkhoff spline Quasi-Interpolation with application as dense output for Gauss–Legendre and Gauss–Lobatto Runge–Kutta schemes

Spline Quasi-Interpolation (QI) of even degree 2R on general partitions is introduced, where derivative information up to order R≥1 at the spline breakpoints is required and maximal convergence order can be proved. Relying on the B-spline basis with possible multiple inner knots, a family of quasi-interpolating splines with smoothness of order R is associated with each R≥1, since there is the possibility of using different local sequences of breakpoints to define each QI spline coefficient. By using suitable finite differences approximations of the necessary discrete derivative information, each QI spline in this family can be associated also with a twin approximant belonging to the same spline space, but requiring just function information at the breakpoints.Among possible different applications of the introduced QI scheme, a smooth continuous extension of the numerical solution of Gauss-Lobatto and Gauss-Legendre Runge-Kutta methods is here considered. When R>1, such extension is based on the use of the variant of the QI scheme with derivative approximation which preserves the approximation power of the original Runge-Kutta scheme.

Spline quasi-interpolation in the Bernstein basis on the Powell–Sabin 6-split of a type-1 triangulation

In this paper, we provide quasi-interpolation schemes defined on a uniform triangulation of type-1 endowed with a Powell–Sabin refinement. In contrast to the usual construction of quasi interpolation splines on the 6-split, the approach described in this work does not require a set of appropriate basis functions. The approximating splines are directly defined by setting their Bézier ordinates to suitable combinations of the given data values. The resulting quasi-interpolants are C1 continuous and reproduce quadratic polynomials. Some numerical tests are given to confirm the theoretical results.

Weighted Quasi-Interpolant Spline Approximations of Planar Curvilinear Profiles in Digital Images

The approximation of curvilinear profiles is very popular for processing digital images and leads to numerous applications such as image segmentation, compression and recognition. In this paper, we develop a novel semi-automatic method based on quasi-interpolation. The method consists of three steps: a preprocessing step exploiting an edge detection algorithm; a splitting procedure to break the just-obtained set of edge points into smaller subsets; and a final step involving the use of a local curve approximation, the Weighted Quasi Interpolant Spline Approximation (wQISA), chosen for its robustness to data perturbation. The proposed method builds a sequence of polynomial spline curves, connected C0 in correspondence of cusps, G1 otherwise. To curb underfitting and overfitting, the computation of local approximations exploits the supervised learning paradigm. The effectiveness of the method is shown with simulation on real images from various application domains.

Mathematical models for the improvement of detection techniques of industrial noise sources from acoustic images

In this paper, a procedure for the detection of the sources of industrial noise and the evaluation of their distances is introduced. The above method is based on the analysis of acoustic and optical data recorded by an acoustic camera. In order to improve the resolution of the data, interpolation and quasi interpolation algorithms for digital data processing have been used, such as the bilinear, bicubic, and sampling Kantorovich (SK). The experimental tests show that the SK algorithm allows to perform the above task more accurately than the other considered methods.

A B-Spline Quasi Interpolation Crank–Nicolson Scheme for Solving the Coupled Burgers Equations with the Caputo–Fabrizio Derivative

In this paper, a Crank–Nicolson finite difference scheme based on cubic B-spline quasi-interpolation has been derived for the solution of the coupled Burgers equations with the Caputo–Fabrizio derivative. The first- and second-order spatial derivatives have been approximated by first and second derivatives of the cubic B-spline quasi-interpolation. The discrete scheme obtained in this way constitutes a system of algebraic equations associated with a bi-pentadiagonal matrix. We show that the proposed scheme is unconditionally stable. Numerical examples are provided to verify the efficiency of the method.

Weighted quasi-interpolant spline approximations: Properties and applications

Continuous representations are fundamental for modeling sampled data and performing computations and numerical simulations directly on the model or its elements. To effectively and efficiently address the approximation of point clouds we propose the Weighted Quasi Interpolant Spline Approximation method (wQISA). We provide global and local bounds of the method and discuss how it still preserves the shape properties of the classical quasi-interpolation scheme. This approach is particularly useful when the data noise can be represented as a probabilistic distribution: from the point of view of nonparametric regression, the wQISA estimator is robust to random perturbations, such as noise and outliers. Finally, we show the effectiveness of the method with several numerical simulations on real data, including curve fitting on images, surface approximation and simulation of rainfall precipitations.

Estimation of effluent parameters of slaughterhouse wastewater treatment with artificial neural network and B-spline quasi interpolation

Effluent of slaughterhouse wastewater treatment by combined up-flow anaerobic sludge blanket (UASB) reactor and extended aeration reactor was estimated through artificial neural networks (ANN), ANN-genetic algorithm (GA) and B-spline quasi interpolation. The overall system operated at two runs with average total chemical oxygen demand (TCOD) of 1514.65 and 3160.2 mg/L for the first and second run, respectively; with two overall hydraulic retention times of 73 and 104 h for each run. The overall system could remove TCOD, ammonia, phosphate and turbidity to a high extent. The multilayer perceptron artificial neural network (MLPANN) trained by Levenberge–Marquardt algorithm was employed to predict the TCOD, ammonia, phosphate and turbidity of the effluent which resulted in R of 0.8257, 0.6274, 0.7961 and 0.6884, respectively. The optimization of MLPANN by GA performed better than MLPANN with R of 0.8390, 0.7650, 0.8107 and 0.7365 for TCOD, ammonia, phosphate and turbidity, respectively. The B-spline quasi interpolation indicates a more accurate prediction due to its R of 0.9619, 0.8806, 0.8307 and 0.7856 for TCOD, ammonia, phosphate and turbidity, respectively. The B-spline quasi interpolation operation time is notably lower than ANN and ANN-GA. In addition, it has a simple algorithm and is implemented easier than the artificial neural network model.

Data-driven quasi-interpolant spline surfaces for point cloud approximation

In this paper we investigate a local surface approximation, the Weighted Quasi Interpolant Spline Approximation (wQISA), specifically designed for large and noisy point clouds. We briefly describe the properties of the wQISA representation and introduce a novel data-driven implementation, which combines prediction capability and complexity efficiency. We provide an extended comparative analysis with other continuous approximations on real data, including different types of surfaces and levels of noise, such as 3D models, terrain data and digital environmental data.

Ambient Approximation on Embedded Submanifolds

In this paper, we present a generalization of Lehmann’s approach for solving approximation problems on hypersurfaces to situations with arbitrary codimension. We show that as in the case of hypersurfaces, the method is able to transfer approximation orders from the ambient space to the submanifold. In particular, the resulting approximant is $${\mathrm {C}}^{m-2}$$ and the error decays at an optimal $$h^m$$ for tensor product B-splines of order m. Additionally, the method is easily implemented and comes with an optimal computational expense when applying quasi interpolation techniques. Applications include, in particular, surfaces embedded into $${{\mathbb {R}}}^4$$ but not into $${{\mathbb {R}}}^3$$ .

Biological treatment of slaughterhouse wastewater: kinetic modeling and prediction of effluent.

In this study three modeling approaches consisting Modified Stover-Kincannon, multilayer perceptron neural network (MLPANN) and B-Spline quasi interpolation were applied in order to predict effluent of up-flow anaerobic sludge blanket (UASB) reactor and also to find the reaction kinetics. At first run, the average total chemical oxygen demand (TCOD) removal efficiency was 48.3% with hydraulic retention time (HRT) of 26h and 63.8% with HRT of 37h, at OLR of 0.77-1.66kg TCOD/m3 d. At the second run, UASB reactor operated with OLR of 1.94-3.1kg TCOD/m3 d and achieved the average TCOD removal efficiency of 64.74 and 72.48% with HRT of 26 and 37h, respectively. The Modified Stover-Kincannon performed well in terms of kinetic determination with a high value of regression coefficient over 0.98. The B-Spline quasi interpolation and MLPANN indicated a great fit for effluent prediction with average R of 0.9984 and 0.9986, and MSE of 157.6050 and 129.7796, respectively; however, they gave no information about reactions occurred in the system.

An adaptive IgA‐BEM with hierarchical B‐splines based on quasi‐interpolation quadrature schemes

SummaryThe isogeometric formulation of the boundary element method (IgA‐BEM) is investigated within the adaptivity framework. Suitable weighted quadrature rules to evaluate integrals appearing in the Galerkin BEM formulation of 2D Laplace model problems are introduced. The proposed quadrature schemes are based on a spline quasi‐interpolation (QI) operator and properly framed in the hierarchical setting. The local nature of the QI perfectly fits with hierarchical spline constructions and leads to an efficient and accurate numerical scheme. An automatic adaptive refinement strategy is driven by a residual‐based error estimator. Numerical examples show that the optimal convergence rate of the Galerkin solution is recovered by the proposed adaptive method.

Ambient Approximation on Hypersurfaces

We present a novel general approach for solving approximation problems on hypersurfaces drawing on standard approximation techniques in the ambient space. The method is capable of transferring approximation orders from the ambient space to the hypersurface under mild conditions. When based on tensor product B-splines of order n and knot spacing h, the resulting approximant is $$C^{n-2}$$ , while the error decays at the optimal rate $$h^n$$ . Further, the method is easy to implement and, when applying quasi interpolation techniques, optimally efficient as the computational expense is linear in the number of spline coefficients. Applications include the representation of smooth surfaces of arbitrary genus.

Quasi-interpolation scheme for arbitrary dimensional scattered data approximation based on natural neighbors and RBF interpolation

A local quasi interpolation scheme is presented in the paper, which can be applied to arbitrary dimensional scattered data approximation with high accuracy, high efficiency and high stability. Under our new quasi-interpolation scheme, we treat the natural neighbor set of the point to be estimated in a given scattered node set as a local interpolation node set. Based on the local node set, a local radial basis function interpolation with algebraic precision of any degree is constructed through a modified Taylor expansion of sampled function at each natural neighbor to compute approximate values. Given the exclusion of the point to be estimated from its natural neighbor set, i.e. interpolation node set, the interpolation scheme we construct is a local quasi-interpolation scheme, and its approximation error is given. The numerical experimental results show that, when the derivative information is unavailable, our method outperforms the famous natural neighbor interpolation method in both approximation accuracy and efficiency. As algebraic precision increases, the approximation accuracy can be further improved.

Max-product neural network and quasi-interpolation operators activated by sigmoidal functions

The max-product neural network (NN) and quasi-interpolation (QI) operators are here introduced and studied. The density functions considered as kernels for the above operators are generated by certain finite linear combination of sigmoidal functions, and from them inherit very useful approximation properties. The convergence and the rate of approximation for the max-product NN and QI operators are studied. Estimates involving the modulus of continuity of the functions being approximated have been derived. Several examples are provided together with some applications and graphical representations. The relations with the general theory of neural networks and sampling operators are discussed in detail.

Bivariate hierarchical Hermite spline quasi-interpolation

Spline quasi-interpolation (QI) is a general and powerful approach for the construction of low cost and accurate approximations of a given function. In order to provide an efficient adaptive approximation scheme in the bivariate setting, we consider quasi-interpolation in hierarchical spline spaces. In particular, we study and experiment the features of the hierarchical extension of the tensor-product formulation of the Hermite BS quasi-interpolation scheme. The convergence properties of this hierarchical operator, suitably defined in terms of truncated hierarchical B-spline bases, are analyzed. A selection of numerical examples is presented to compare the performances of the hierarchical and tensor-product versions of the scheme.

Optimal homotopy asymptotic method for solving nonlinear Fredholm integral equations of second kind

The aim of this communication is to present the effectiveness of optimal homotopy asymptotic method (OHAM) for solving nonlinear Fredholm integral equations of second kind. Here, we consider the first, second and third order Fredholm integral equations of second kind. Comparisons are made with Adomian Decomposition Method, Discrete Adomian Decomposition Method, Homotopy Perturbation Method, Quasi Interpolation Method, Triangular Functions Method, Trapezoidal rule and Simpson rule. The results reveal that the OHAM is very effective, simple and has close agreement with exact solutions.

Quasi Interpolation With Voronoi Splines

We present a quasi interpolation framework that attains the optimal approximation-order of Voronoi splines for reconstruction of volumetric data sampled on general lattices. The quasi interpolation framework of Voronoi splines provides an unbiased reconstruction method across various lattices. Therefore this framework allows us to analyze and contrast the sampling-theoretic performance of general lattices, using signal reconstruction, in an unbiased manner. Our quasi interpolation methodology is implemented as an efficient FIR filter that can be applied online or as a preprocessing step. We present visual and numerical experiments that demonstrate the improved accuracy of reconstruction across lattices, using the quasi interpolation framework.