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In preceding papers, the present authors proposed the application of the mollification based on wavelets to the calculation of the fractional derivative (fD) or the derivative of a function involving noise. We study here the application of that method to the detection of edge of a function. Mathieu et al. proposed the CRONE detector for a detection of an edge of an image. For a function without noise, we note that the CRONE detector is expressed as the Riesz fractional derivative (fD) of the derivative. We study here the application of the mollification to the calculation of the Riesz fD of the derivative for a data involving noise, and compare the results with the results obtained by our method of applying simple derivative to mollified data.

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 [

as the mollified function where the mollifier

In our preceding papers [

In the problem of detecting an edge of an image, Mathieu et al. [

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 [

In Section 2, we review the preceding papers [

We use notations

also use

integrable on

We denote the Heaviside step function by

In the present study of mollification, we choose a mollifier

The mollification

where the mollifier

Fourier transform of

Following [

Requirement 1

If this is satisfied, noise reduction is expected, since high frequency contribution is important in noise. This is concluded from (2.2).

Requirement 2

If this is satisfied, the Gibbs phenomenon does not appear.

Requirement 3 The region where

If this is satisfied, the mollified function is less smeared.

We proposed three mollifiers based on wavelets in [

Mollifier 1 This mollifier is based on a special one of rapidly decaying harmonic wavelet. It is given by

Mollifier 2 This mollifier is based on the Haar wavelet, and is given by

Mollifier 3 This mollifier is based on the first-order-spline wavelet, which is given by

where

Here

fier 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.

Remark 1 In the method of

In Figures 1-3,

In discussing the Gibbs phenomenon, we use function

and is shown in Figures 1(c)-3(c) by thin line. In Figures 1(c)-3(c),

Mollifier 3 is so scaled that the variance of

standard deviation is then

By Requirement 3, Mollifier 1 is little less smeared.

The evaluations are summarized in

Following Mathieu et al. [

This function

At the point

Requirement 1 | Requirement 2 | Requirement 3 | |
---|---|---|---|

Mollifier 1 | |||

Mollifier 2 | |||

Mollifier 3 |

We now consider a noisy data given by

for

distribution in the interval

From

We are interested in the place of an edge where the derivative of the function

In

In

Remark 2

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

mended. If

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

In this section, we use notations

not less than

Definition 1 We define the Liouville fI and the Weyl fI of order

We define their fDs of order

where

also call

In [

where

When

where

The righthand sides are seen to be equal to the righthand sides of the corresponding equations in (4.2).

Lemma 1 Let

if the righthand side exists.

In [

for

Definition 2 We define the Riesz fD by (4.8) for

Definition 3 We define a related fD by

for

We note that

and the fDs defined by Definitions 2 and 3 are related by

for

Remark 3 In [

In [

conjugate, respectively. In [

By using Lemma 1 and Definitions 2 and 3, we confirm the following lemma.

Lemma 2 Let

Mathieu et al. [

By using (4.2) and (4.8), we can express it also as

If

Lemma 3 If

Proof This follows from Lemma 2 by using (4.15).

In the present section, we are concerned with the function

The function

Its Liouville fD of order

When

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

For

and (5.3). In

the point, as seen in

Mathieu el al. [

In the present section, we are concerned with noisy data of the function

We now investigate the primitive CRONE fD detector applied to

for

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

for

for Mollifiers 2 and 3, respectively. The curves for

The curves of

Remark 4

Hence the best choice in this case is to use

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

In detecting the edge of a function, we calculate

data function, and its mollification

fied data function, and its mollification

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.

calculation of mollification is simple for Mollifier 2, the use of

In Section 6,

We finally compare the curves of

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.