Mollification Based on Wavelets

Abstract

Abstract:The mollification obtained by truncating the expansion in wavelets is studied, where the wavelets are so chosen that noise is reduced and the Gibbs phenomenon does not occur. The estimations of the error of approximation of the mollification are given for the case when the fractional derivative of a function is calculated. Noting that the estimations are applicable even when the orthogonality of the wavelets is not satisfied, we study mollifications using unorthogonalized wavelets, as well as those using orthogonal wavelets.

Detection of Edge with the Aid of Mollification Based on Wavelets

“…Following [4], we consider the following requirements in evaluating the mollifiers. The first two were mentioned in [3] takes nonzero values is narrow.…”

“…m − -th-order B-spline [7]. In [4], Mollifier 3 is called the mollifier based on the scaled unorthogonalized Franklin wavelet, since the scaling functions of the Franklin wavelet is constructed by orthogonalizing the scaling functions of the first-order B-spline wavelet.…”

“…When a function involving noise, ( ) g x , is given, Murio [1] proposed to use is a Gaussian probability density function. In our preceding papers [2]- [4], the mollification based on wavelets is studied for the problem of calculating the derivative or the fractional derivative (fD) of a function involving noise, and an estimation of the error of approximation is given in terms of fD. In [4], we chose three mollifiers based on wavelets, by which the noise in a noisy data is removed and the Gibbs phenomenon is not observed.…”

“…The results are compared with the derivative calculated by the method of mollification given in [3]. The calculation is done by using the mollifiers proposed in [4].…”

“…In Section 2, we review the preceding papers [2]- [4]. In Section 3, we numerically study the edge detection by applying the our method of mollification to the calculation of a function involving noise.…”