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A new approach to numerical integration based on Coifman wavelets

Dehda Bachir

Journal of Applied Mathematics and Computational Mechanics

2019

Vol. 18, nr 3

31--44

In this paper, we present a new approach based on Coifman wavelets to find approximate values of definite integrals. This approach overcomes both CAS and Haar wavelets and hybrid functions in terms of absolute errors. The algorithm based on Coifman wavelets can be easily extended to find numerical approximations for double and triple integrals. Illustrative examples implemented using Matlab show the efficiency and effectiveness of this new method.

Ocena wpływu metodyki doboru falki bazowej na analizę falkową zarysów nierówności powierzchni

Makieła W., Stępień K.

Pomiary Automatyka Kontrola

2010

R. 56, nr 1

32-34

W referacie przedstawiono jedno z ważniejszych zagadnień analizy falkowej, a mianowicie metodykę doboru postaci falki bazowej. W treści zaprezentowano metodykę i wyniki obliczeń dla sygnału reprezentującego zarys nierówności powierzchni z wykorzystaniem trzech kryteriów doboru falki: testu autokorelacji, funkcji korelacji wzajemnej i procedury z wykorzystaniem wskaźników entropii. Obliczenia i wykresy wykonano z wykorzystaniem pakietu Wavelet toolbox środowiska MATLAB dla wybranego profilu chropowatości powierzchni.

The paper deals with one of the most important problems of the wavelet analysis - the methodology of selection of basic wavelets. Correct selection of the basic wavelet (so-called the mother wavelet) allows distinguishing the main signal and removal of signal disturbances. This problem is fundamental because the basic function form influences significantly the spectrum components. In metrological practice, there are no criteria that could be used for selection of the mother wavelet. Such decision is usually taken on a basis of the fundamental of visual evaluation of an analyzed signal. In the paper there are described three criteria that can be used for selection of the basic wavelets: the statistical test of autocorrelation of the coefficients of the decomposition details [3, 4], values of the normalized cross-correlation function [5, 6] and minimisation of the Shannon entropy [1, 7]. The calculations were performed with use of Wavelet toolbox - the component of the Matlab environment, for a selected roughness profile. The investigations were carried out for the roughness profile consisting of N=1024 sampling points using following wavelet types: Haar, Daubechies, Coiflets, Symlet and biorthogonal ones. The calculation results are given in Table 1. The authors draw a conclusion from them that the most promising criterion is investigation of the entropy values (however, other methods should not be neglected). The reason is that the method of entropy minimisation is not limited to one decomposition only, but it allows searching an optimum decomposition tree, applying multilevel analysis by the wavelet packets.