In the present paper, we take up the problem of detecting an edge for a function involving noise. For a function, an edge is a point where the derivative is maximum or minimum.

Calculation of the derivative of a function is an ill-posed problem, in the sense that, when a function involes noise, the derivative emphasizes the noise. In the method of mollification [1] to cope with the problem, the data involving noise is mollified before the derivative is taken. When a function involving noise, , is given, Murio [1] proposed to use

as the mollified function where the mollifier 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.

In the problem of detecting an edge of an image, Mathieu et al. [5] [6] proposed the use of the CRONE detector. For a function, an edge is a point where the derivative is maximum or minimum. In order to make the point clearer, they propose to use the difference of an fD in increasing variable and an fD in decreasing variable, when there exists no noise. We note that the difference is equal to the Riesz fD of the derivative. We shall call that detector the primitive CRONE fD detector. The calculation of fD is an ill-posed problem, and this is powerless when there exists noise. When there exists noise, they propose to use the fractional integral (fI), to reduce noise. If we use fI, the peak of the derivative is made broad, compared with the simple derivative of the mollification. In practice, they truncate the function to be convoluted in the calculation of fI, and it is not seen to be a direct application of fI. They call this detector also as the CRONE detector. We shall not discuss that method in the present paper.

In the present paper, we study the application of mollification to the Riesz fD of the derivative, for the case when there exists noise. 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. In Section 4, we recall the definitions of fDs and the primitive CRONE fD detector. In Section 5, we study the application of the primitive CRONE fD detector to a function without noise. In Section 6, we numerically study the mollification of a function involving noise, and the application of the primitive CRONE fD detector to it. Section 7 is for conclusion.

We use notations and to represent the sets of all real numbers and of all integers, respectively. We

also use, and for. For a function, that is

integrable on in the sense of Lebesgue, and its Fourier transform is denoted by or, so that

We denote the Heaviside step function by, so that for and for.

In the present study of mollification, we choose a mollifier in unit scale, and a scale.

The mollification of a function by the mollifier in the scale is given by

where the mollifier is assumed to be given by, so that. Now the

Fourier transform of is given by

Following Mathieu et al. [5] [6] , we take up the function given by

This function and its derivative are shown in Figure 4. We note from Figure 4(b), that

At the point, takes the maximum value. We take this as the place of the edge.

We now consider a noisy data given by

for for, where, and for each is a random number chosen from the uniform

distribution in the interval. In Figure 5, we show the graphs of and for, 0.01 and 0.1, where

From Figure 5(b) for very small, we can detect the point of the edge, but from Figure 5(f) for, we cannot see the existence of an edge.

We are interested in the place of an edge where the derivative of the function is maximum, but we assume that we only know a noisy function in place of. Then in the method of mollification, we calculate the derivative of the mollified function. If is locally integrable, is given by (2.1). We now know only discrete values for, and we use

for. Since this is a differentiable function, its derivative is denoted by

.

In Figure 6, we show the curves of and for Mollifier 1. The values of and are found in the respective figures. For each, the noise is reduced as decreases. The chosen values of are the highest values for which the noise in is removed fairly well. We can now point out the place at which the derivative is maximum even for the case of. In Figure 7 and Figure 8, the corresponding curves of are shown for Mollifiers 2 and 3, respectively. The curves in Figure 8 for Mollifier 3 resemble very closely to the corresponding curves in Figure 6. The curves for in Figure 7 for Mollifier 2 are noisier than the other figures.

In Figure 7 and Figure 8, the mollification of, that is is drawn in place of

on the leftmost column. They are obtained by applying the mollification to the on the second column. We note that the additional application of mollification improves the result. In fact, the following fact follows from construction of Mollifiers 2 and 3.

Remark 2 for in Figure 7 must be equal to for in Figure 8.

Since the calculation of mollification is simple for Mollifier 2, the use of for Mollifier 2 is recom-

mended. If is to be used, we have to use it for Mollifier 1 or 3.

In formulating primitive CRONE fD detector, fDs are used. These are usually defined in terms of fIs.

In the present section, we are concerned with the function given by (3.1) without noise. This function and its derivative are shown in Figure 4.

The function given by (3.1) is expressed as

Its Liouville fD of order satisfying is given by

When, this takes only finite values.

By using (4.2), Lemma 1 and (3.2), we obtain

For without noise, the primitive CRONE fD detector applied to it is calculated by using (4.14), (5.2)

and (5.3). In Figure 9, we compare for and 0.75, with

. Here and 0.75 are chosen as typical values between 1 and 2 and between 0 and 1.

is an even function around the point of a peak, and hence is also an even function around

the point, as seen in Figure 9. We note that the latter has a sharper peak, for. Based on this fact,

Mathieu el al. [5] [6] claim that for is more favorable than as a detector of edge.

In the present section, we are concerned with noisy data of the function given in Section 3.

We now investigate the primitive CRONE fD detector applied to, and hence calculate

given by

for and 0.75. This is compared with.

Numerical calculation of the righthand side of (6.1) is made by using

for and, and satisfying. Here for. Note that (6.2) is applicable for. This equation is obtained by applying the trapezoidal rule of integration to the righthand side of the first equation of (4.12), with the aid of (4.9). In Figure 10 and Figure 11, we show the curves of

for Mollifiers 2 and 3, respectively. The curves for are the same as in Figure 7 and Figure 8. Here we do not give the figures for Mollifier 1, since they are so close to those for Mollifier 3, shown in Figure 11. The values of and are found in the respective figures. In some of the figures, the curve of is drawn but the curve of is not drawn, that is the case when the is too noisy to draw. On the leftmost column, are drawn. As compared with other figures for the same in Figures 6-8, Figure 10 and Figure 11, they are smeared and poor. On the second column, are shown, which are obtained by applying the mollification to given on the third column. We note that it is well mollified compared with.

The curves of in Figure 10 for Mollifier 2 are noisier than the corresponding curves in Figure 11 for Mollifier 3. Corresponding to Remark 2, we note here the following fact.

Remark 4 for in Figure 10 must be equal to for in Figure 11.

Hence the best choice in this case is to use for Mollifier 2, for which the mollification is very simple. If is to be used, then we have to use it for Mollifiers 1 or 3.

The method of mollification based on wavelets is applied to the detection of the edge of a function, when the given data involve noise. Here an edge of a function is the place where the derivative of the function is maximum or minimum. In Section 3, noisy data are given for, 0.01 and 0.1. The data and its difference are shown in Figure 5. The primitive CRONE fD detector is given in Section 4.3.

In detecting the edge of a function, we calculate, which is the derivative of mollified

data function, and its mollification in Section 3. In Section 6, we calculate

, which is the result of the application of the primitive CRONE fD detector to the molli-

fied data function, and its mollification. Calculations are made for three mollifiers. The

results for Mollifiers 1 and 3 are very close, and the results for Mollifier 1 are not given in Section 6. In these calculations, the results for Mollifier 2 are noisier than the others.

In Section 3. are found to improve the results of. The curves of for Mollifier 2 are so improved that they are very close to those for Mollifiers 1 and 3. This section is concluded as follows. Since the

calculation of mollification is simple for Mollifier 2, the use of for Mollifier 2 is most recommended. If is to be used, we have to use it for Mollifiers 1 or 3.

In Section 6, is also calculated, but it is too smeared and is not useful. In Section 6, the curves of are found to improve those of. This section is concluded as follows. The best choice in this case is to use for Mollifier 2, for which the mollification is very simple. If is to be used, then we have to use it for Mollifiers 1 or 3.

We finally compare the curves of and, which are given in Sections 3 and 6, respectively. The curves of are calculated for a larger value of than the curves of for the same, and hence the former have a sharper top. The general form of the curves of is slender than that of. The calculation is simpler for for Mollifier 2.

The authors are grateful to Professor Hiroaki Hara, who showed the recent book of Ortigueira. A preliminary report of the content of this paper was done orally by T. Morita, in a semi-plenary lecture in the 5th Symposium on Fractional Differentiation and Its Applications, held in Nanjing, China, on May 14-17, 2012. The authors are indebted to Professor Nobuyuki Shimizu, for giving the authors this opportunity.