Detection of Edge with the Aid of Mollification Based on Wavelets

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.


Introduction
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, ( ) g x , is given, Murio [1] proposed to use ∫ as the mollified function where the mollifier ( ) t µ 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 m ∈  .For a function ( ) ( ) , that is integrable on  in the sense of Lebesgue, and its Fourier transform is denoted by ( ) ĝ w or ( ) We denote the Heaviside step function by ( )

Mollification Depending on a Scale
In the present study of mollification, we choose a mollifier 1 µ in unit scale, and a scale >0 ν ∈  .The mollification M ν of a function ( ) g x by the mollifier ν µ in the scale ν is given by where the mollifier ν µ is assumed to be given by

Evaluation of Mollifiers
Following [4], we consider the following requirements in evaluating the mollifiers.The first two were mentioned in [3], as Criteria 1 and 2. Requirement 1 ( ) µ is essentially zero for w higher than a threshold frequency.
If this is satisfied, noise reduction is expected, since high frequency contribution is important in noise.This is concluded from (2.2).
Requirement 2 ( ) µ is nonnegative for all x ∈  .If this is satisfied, the Gibbs phenomenon does not appear.

Requirement 3
The region where ( ) takes nonzero values is narrow.If this is satisfied, the mollified function is less smeared.

Mollifiers Based on Wavelets
We proposed three mollifiers based on wavelets in [4].
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 ( ) 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.

Remark 1
In the method of σ -factor of Lanczos [8], ( ) ( ) , and in its extension, ( ) ( ) In Figures 1-3, ( )  does not decay rapidly as w increases for Mollifier 2, and hence Requirement 1 is not well satisfied for this mollifier.
In discussing the Gibbs phenomenon, we use function takes small negative values, but the Gibbs phenomenon is hardly observed for Mollifier 1.We note that Requirement 2 is well satisfied for Mollifiers 2 and 3. Mollifier 3 is so scaled that the variance of ( )    By Requirement 3, Mollifier 1 is little less smeared.
The evaluations are summarized in Table 1.

Detection of Edge of a Function
Following Mathieu et al. [5] [6], we take up the function ( ) x ( ) ( ) ′ takes the maximum value.We take this as the place of the edge.( ) We now consider a noisy data given by ( ) . 2 , we cannot see the existence of an edge.We are interested in the place of an edge where the derivative of the function ( ) 1 f x is maximum, but we assume that we only know a noisy function ( ) Then in the method of mollification, we calculate the derivative of the mollified function

M f x
ν  is given by (2.1).We now know only discrete values ( ) for k ∈  , and we use ( ) ( ) . Since this is a differentiable function, its derivative is denoted by In Figure 6, we show the curves of ( ) ( ) : for Mollifier 1.The values of  and ν are found in the respective figures.For each  , the noise is reduced as ν decreases.The chosen values of ν are the highest values for which the noise in ( ) 1 y x is removed fairly well.We can now point out the place at which the derivative is maximum even for the case of 0.1 =  . In Figure 7 and In Figure 7 and Figure 8, the mollification of ( ) : is drawn in place of ( ) y x on the leftmost column.They are obtained by applying the mollification to the ( ) 1 y x 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.

Fractional Derivatives and Primitive CRONE fD Detector
In formulating primitive CRONE fD detector, fDs are used.These are usually defined in terms of fIs.

Liouville fD and Weyl fD
In this section, we use notations ( ) ( ) ( )   is used to represent the least integer that is not less than x .

Definition We define the Liouville fI and the Weyl fI of order
We define their fDs of order where : . Even when , if the righthand side exists [9].We also call ( ) 2) for λ ∈  , simply the fD as a whole.In [5] [6], the fDs defined by (4.1)-(4.2) for λ ∈  are denoted by where m λ =     , and The righthand sides are seen to be equal to the righthand sides of the corresponding equations in (4.2).
Lemma 1 Let a ∈  be such that ( ) ( ) if the righthand side exists.

Riesz fD
In [10], the Riesz fI is defined by for n ∈  .We note that ( ) ( ) ( ) ( ) and the fDs defined by Definitions 2 and 3 are related by , , for λ ∈  .By using Lemma 1 and Definitions 2 and 3, we confirm the following lemma.

Primitive CRONE fD Detector Applied to a Function without Noise
In the present section, we are concerned with the function ( ) are shown in (a) and (b), respectively, for the three mollifiers.

Figure 1 (Figure 2
Figure 1(a) and Figure 3(a) show that Requirement 1 is well satisfied for Mollifiers 1 and 3.Figure 2(a) shows that ( ) 1 ˆw µdoes not decay rapidly as w increases for Mollifier 2, and hence Requirement 1 is not well satisfied for this mollifier.In discussing the Gibbs phenomenon, we use function 8) and is shown in Figures 1(c)-3(c) by thin line.In Figures 1(c)-3(c), shown by thick lines for the three mollifiers.Figure 1(b) and Figure 1(c) show that ( ) 1 x µ that is the value for Mollifier 2. The standard deviation is then 0.408 σ ≒ .The corresponding values for Mollifier 1 are 2 1 8 σ = and 0.354 σ ≒ .

Figure 4 .
We note from Figure 4(b), that

Figure 4 .
Figure 4.The curves of ( ) 1 f x and r for each k is a random number chosen from the uniform distribution in the interval ( )

4 )From
Figure 5(b) for very small 0.001 =  , we can detect the point of the edge, but from Figure 5(f) for 0.1 = 

Figure 8 , 1 y 6 .in Figure 7
the corresponding curves of ( ) x 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 The curves for for Mollifier 2 are noisier than the other figures.

Figure 7 in Figure 8 . 1 y
must be equal to Since the calculation of mollification is simple for Mollifier 2, the use of ( ) x is to be used, we have to use it for Mollifier 1 or 3.

3 )
should be regarded as expressions of "distributions", and be read as

Remark 3
In[10], are called the modified Riesz potential and its inverse, respectively.

4 . 3 .
Primitive CRONE fD Detector in Terms of Riesz fDMathieu et al.[5] [6]  proposed a detector of an edge which they called the CRONE detector.We call the one proposed for a function without noise as the primitive CRONE fD detector.By using (4.3), we can express it as using (4.2) and (4.8), we can express it also as even function.ProofThis follows from Lemma 2 by using (4.15).

1 f
x given by (3.1) is expressed as

Table 1 .
Summary of the evaluations of the three mollifiers.○ : satisfies very well, and ∆ : satisfies fairly well.