Approximation Of Piecewise Smooth Functions Research Articles

Approximation of piecewise smooth functions by nonlinear bivariate C2 quartic spline quasi-interpolants on criss-cross triangulations

In this paper we focus on the space of C2 quartic splines on uniform criss-cross triangulations and we propose a method based on weighted essentially non-oscillatory techniques and obtained by modifying classical spline quasi-interpolants in order to approximate piecewise smooth functions avoiding Gibbs phenomenon near discontinuities and, at the same time, maintaining the high-order accuracy in smooth regions. We analyse the convergence properties of the proposed quasi-interpolants and we provide some numerical and graphical tests confirming the theoretical results.

Fully accurate approximation of piecewise smooth functions using corrected B-spline quasi-interpolants

This article is devoted to introducing new spline quasi-interpolants for the sharp approximation of data in one and several dimensions. The new construction aims to obtain accurate approximations close to singularities of the function from which the data are obtained. The technique relies in an accurate knowledge of the position of the singularity, which can be known or approximated, and that allows for the obtention of accurate approximations of the jumps of the function and its derivatives at this point. With this information, it is possible to compute correction terms for the B-spline bases of the spline that are affected by the presence of singularities. The result is a piecewise smooth reconstruction with full accuracy (meaning by full accuracy, the accuracy of the spline at smooth zones). The numerical experiments presented support the theoretical results obtained.

Nonlinear 2D C1 Quadratic Spline Quasi-Interpolants on Triangulations for the Approximation of Piecewise Smooth Functions

The aim of this paper is to present and study nonlinear bivariate C1 quadratic spline quasi-interpolants on uniform criss-cross triangulations for the approximation of piecewise smooth functions. Indeed, by using classical spline quasi-interpolants, the Gibbs phenomenon appears when approximating near discontinuities. Here, we use weighted essentially non-oscillatory techniques to modify classical quasi-interpolants in order to avoid oscillations near discontinuities and maintain high-order accuracy in smooth regions. We study the convergence properties of the proposed quasi-interpolants and we provide some numerical and graphical tests confirming the theoretical results.

Optimal approximation of piecewise smooth functions using deep ReLU neural networks

We study the necessary and sufficient complexity of ReLU neural networks – in terms of depth and number of weights – which is required for approximating classifier functions in an Lp-sense.As a model class, we consider the set Eβ(Rd) of possibly discontinuous piecewise Cβ functions f:[−12,12]d→R, where the different “smooth regions” of f are separated by Cβ hypersurfaces. For given dimension d≥2, regularity β>0, and accuracy ε>0, we construct artificial neural networks with ReLU activation function that approximate functions from Eβ(Rd) up to an L2 error of ε. The constructed networks have a fixed number of layers, depending only on d and β, and they have O(ε−2(d−1)∕β) many nonzero weights, which we prove to be optimal. For the proof of optimality, we establish a lower bound on the description complexity of the class Eβ(Rd). By showing that a family of approximating neural networks gives rise to an encoder for Eβ(Rd), we then prove that one cannot approximate a general function f∈Eβ(Rd) using neural networks that are less complex than those produced by our construction.In addition to the optimality in terms of the number of weights, we show that in order to achieve this optimal approximation rate, one needs ReLU networks of a certain minimal depth. Precisely, for piecewise Cβ(Rd) functions, this minimal depth is given – up to a multiplicative constant – by β∕d. Up to a log factor, our constructed networks match this bound. This partly explains the benefits of depth for ReLU networks by showing that deep networks are necessary to achieve efficient approximation of (piecewise) smooth functions.Finally, we analyze approximation in high-dimensional spaces where the function f to be approximated can be factorized into a smooth dimension reducing feature map τ and classifier function g – defined on a low-dimensional feature space – as f=g∘τ. We show that in this case the approximation rate depends only on the dimension of the feature space and not the input dimension.

High-accuracy approximation of piecewise smooth functions using the Truncation and Encode approach

Abstract In the present work, we analyze a technique designed by Geraci et al. in [1,11] named the Truncate and Encode (TE) strategy. It was presented as a non-intrusive method for steady and non-steady Partial Differential Equations (PDEs) in Uncertainty Quantification (UQ), and as a weakly intrusive method in the unsteady case. We analyze the TE algorithm applied to the approximation of functions, and in particular its performance for piecewise smooth functions. We carry out some numerical experiments, comparing the performance of the algorithm when using different linear and non-linear interpolation techniques and provide some recommendations that we find useful in order to achieve a high performance of the algorithm.

Wavelet frame based scene reconstruction from range data

Abstract How to reconstruct the scene (a visible surface) from a set of scattered, noisy and possibly sparse range data is a challenging problem in robotic navigation and computer graphics. As most real scenes can be modeled by piecewise smooth surfaces, traditional surface fitting techniques (e.g. smoothing spline) generally can not preserve sharp discontinuities of surfaces. Based on sparse approximation of piecewise smooth functions in frame domain, we propose a new tight frame based formulation for reconstructing a piecewise smooth surface from a sparse range data set, which is robust to both additive noise and outliers. Furthermore, the resulting minimization problem from our formulation can be efficiently solved by the split Bregman method [1] , [2] . The numerical experiments show that the proposed approach is capable of reconstructing a piecewise smooth surface with sharp edges from sparse range data corrupted with noise and outliers.

Edge detection insensitive to changes of illumination in the image

In this paper we present new edge detection algorithms which are motivated by recent developments on edge-adapted reconstruction techniques [F. Aràndiga, A. Cohen, R. Donat, N. Dyn, B. Matei, Approximation of piecewise smooth functions and images by edge-adapted (ENO-EA) nonlinear multiresolution techniques, Appl. Comput. Harmon. Anal. 24 (2) (2008) 225–250]. They are based on comparing local quantities rather than on filtering and thresholding. This comparison process is invariant under certain transformations that model light changes in the image, hence we obtain edge detection algorithms which are insensitive to changes in illumination.

Smoothing transforms for wavelet approximation of piecewise smooth functions

Multi-resolution analysis with high vanishing moment wavelets provides a framework to efficiently approximate smooth functions. However, it is a well-known fact that wavelet approximation usually cannot achieve the same order of approximation in the vicinity of discontinuous points of functions as that in the smooth regions. Ringing artefacts in the reconstructed functions inevitably appear around discontinuous points. To reduce these artefacts, the authors propose to locally smooth piecewise smooth functions at the discontinuous points, prior to applying the wavelet transform, via a smoothing transform. The numerical experiments for one- and two-dimensional signals show the effectiveness of the proposed strategy.

Approximation of piecewise smooth functions and images by edge-adapted (ENO-EA) nonlinear multiresolution techniques

This paper introduces and analyzes new approximation procedures for bivariate functions. These procedures are based on an edge-adapted nonlinear reconstruction technique which is an intrinsically two-dimensional extension of the essentially non-oscillatory and subcell resolution techniques introduced in the one-dimensional setting by Harten and Osher. Edge-adapted reconstructions are tailored to piecewise smooth functions with geometrically smooth edge discontinuities, and are therefore attractive for applications such as image compression and shock computations. The local approximation order is investigated both in L p and in the Hausdorff distance between graphs. In particular, it is shown that for general classes of piecewise smooth functions, edge-adapted reconstructions yield multiscale representations which are optimally sparse and adaptive approximations with optimal rate of convergence, similar to curvelets decompositions for the L 2 error.

Determining Analyticity for Parameter Optimization of the Gegenbauer Reconstruction Method

The Gegenbauer reconstruction method effectively eliminates the Gibbs phenomenon and restores exponential accuracy to the approximations of piecewise smooth functions. Recent investigations show that its success depends upon choosing parameters in such a way that the regularization and the truncation error estimates are equally considered. This paper shows that the underlying analyticity of the function in smooth regions plays a critical role in the regularization error estimate. Hence we develop a technique that first analyzes the behavior of the function in its regions of smoothness and then applies this knowledge to refine the regularization error estimate. Such refinement yields better parameter choices for the Gegenbauer reconstruction method, and is confirmed both by better accuracy and more robustness in the approximation of piecewise smooth functions.

Interpolation and Approximation of Piecewise Smooth Functions

This paper provides approximation orders for a class of nonlinear interpolation procedures for uniformly sampled univariate data. The interpolation is based on essentially nonoscillatory (ENO) and subcell resolution (SR) reconstruction techniques. These nonlinear techniques aim at reducing significantly the approximation error for functions with isolated singularities and are therefore attractive for applications such as shock computations or image compression. We prove that in the presence of isolated singularities, the approximation order provided by the interpolation procedure is improved by a factor of h relative to the linear methods, where h is the sampling rate. Moreover, for h below a critical value, we recover the optimal approximation order as for uniformly smooth functions.

Effective approximation of piecewise smooth functions by their expansion into fast convergent series in terms of functions formed by eigenfunctions of Sturm–Liouville problems

Using the eigenfunctions of two Sturm–Liouville problems (with the same operator of a general form but with two different versions of boundary conditions), a method for constructing these specific basis functions is developed. The corresponding expansions of smooth and piecewise smooth functions in terms of such basis lead to fast convergent series. This result makes it possible to approximate the functions of the above class by a small number of terms. We also give brief information on a method for constructing multidimensional (in particular, two-dimensional) specific basis functions with the above properties. The proposed method is based on the ideas of the author's earlier works. However, it is, in essence, a new method that has substantially improved characteristics and is mainly oriented to the approximation of piecewise smooth functions. We also consider several important special cases.

Effective approximation of piecewise smooth functions by their expansion into fast convergent series in terms of functions formed by eigenfunctions of Sturm–Liouville problems

Using the eigenfunctions of two Sturm–Liouville problems (with the same operator of a general form but with two different versions of boundary conditions), a method for constructing these specific basis functions is developed. The corresponding expansions of smooth and piecewise smooth functions in terms of such basis lead to fast convergent series. This result makes it possible to approximate the functions of the above class by a small number of terms. We also give brief information on a method for constructing multidimensional (in particular, two-dimensional) specific basis functions with the above properties. The proposed method is based on the ideas of the author's earlier works. However, it is, in essence, a new method that has substantially improved characteristics and is mainly oriented to the approximation of piecewise smooth functions. We also consider several important special cases.

Wavelet footprints: theory, algorithms, and applications

Wavelet-based algorithms have been successful in different signal processing tasks. The wavelet transform is a powerful tool because it manages to represent both transient and stationary behaviors of a signal with few transform coefficients. Discontinuities often carry relevant signal information, and therefore, they represent a critical part to analyze. We study the dependency across scales of the wavelet coefficients generated by discontinuities. We start by showing that any piecewise smooth signal can be expressed as a sum of a piecewise polynomial signal and a uniformly smooth residual (Theorem 1). We then introduce the notion of footprints, which are scale space vectors that model discontinuities in piecewise polynomial signals exactly. We show that footprints form an overcomplete dictionary and develop efficient and robust algorithms to find the exact representation of a piecewise polynomial function in terms of footprints. This also leads to efficient approximation of piecewise smooth functions. Finally, we focus on applications and show that algorithms based on footprints outperform standard wavelet methods in different applications such as denoising, compression, and (nonblind) deconvolution. In the case of compression, we also prove that at high rates, footprint-based algorithms attain optimal performance (Theorem 3).

Approximation and compression of piecewise smooth functions

Wavelet or sub–band coding has been quite successful in compression applications, and this success can be attributed in part to the good approximation properties of wavelets. In this paper, we revisit rate–distortion (RD) bounds for the wavelet approximation of piecewise smooth functions, and piecewise polynomial functions in particular. We contrast these results with RD bounds achievable using an oracle–based method. We then introduce a practical dynamic programming algorithm, which achieves performance similar to the oracle method, and present experimental results.

Characterization of weights in best rational weighted approximation of piecewise smooth functions, I

Caracterisation des poids dans l'approximation rationnelle ponderee des fonctions regulieres par morceaux

Characterization of weights in best rational weighted approximation of piecewise smooth functions, II

Characterization of weights in best rational weighted approximation of piecewise smooth functions, II

On solving optimal control problems by means of splines

We consider spline-methods of solving problems, in which piecewise-smooth functions can be the sought solution. We provide and ground the algorithm of approximation of piecewise-smooth functions by splines with automatic search for points of discontinuity of the function and its derivatives. On the base of this algorithm, we describe methods to determine the spline-solution for some optimal control problems.