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We study fractional-order derivatives of left-handed Grünwald-Letnikov formula with

How to preserve singularities in image and signal processing is a very important problem [

However, we think fractional derivatives should be studied differently. It is well known that one powerful method to preserve singularities in signal processing is to detect and locate singularities correctly and then to protect them in signal processing. Thus, we think the most important problem in theory analysis should be if the fractional derivatives can detect and locate singularities well.

In this paper, we study fractional-order derivatives of left-handed Grünwald-Letnikov formula with

The rest of this paper is as follows. Section

In contrast to integer-order differentials

Another way to represent the fractional derivatives is by the Grünwald-Letnikov (G-L) formula, which is a generalization of the ordinary discretization formulas for integer-order derivatives. For

The above two definitions have different forms. However, by requiring a reasonable behavior of the function

Let us assume that the function

Generally, the analytic definition given by (

For

Therefore, in order to handle fractional derivative numerically, it is necessary to compute the coefficients

The recurrence relationship of the coefficients of G-L formula

For

Assume that when

The nonlocal operator defined in (

Consider

Since

For

For

Assume that

According to Lemma

For

According to Lemma

For

From Lemma

Thus,

Singularity detection is the name for a set of mathematical methods which aim at identifying points in a digital signal at which the signal value changes sharply or, more formally, has discontinuities.

We can categorize singularities as step, roof, jump, and ramp. They can be represented as

ideal step

ideal roof

ideal impulse

ideal ramp

Singularity detection is the name for a set of mathematical methods which aim at identifying points in a digital signal at which the signal value changes sharply or, more formally, has discontinuities.

The singularity detection by 1-order derivatives detects singularities by first computing a measure of singularity strength, usually a first-order derivative expression, and then searching for local absolute maxima as the locations of singularities. The simplest approach to compute first-order derivatives is to use left-handed differences

ideal step

ideal roof

ideal impulse

ideal ramp

The main steps to locate singularities are as follows: (1) find all points with

For

For

For

For

In summary, for

For

Summarizing the above conclusion, we have the following.

The detection and location of four types of ideal singularities using 1-order derivatives are

ideal step: 1-order derivative can detect and locate ideal step singularities correctly;

ideal roof: 1-order derivative can detect ideal roof singularities when

ideal impulse: the singularity can not be detected;

ideal ramp: 1-order derivative can detect and locate ideal ramp singularities correctly when

The singularity detection by fractional-order derivatives detects singularities by computing fractional-order derivative expression firstly and then searching for local extrema as the locations of singularities.

Fractional-order derivatives of the four types of singularities are as follows.

The main steps to find extrema of fractional-order derivatives

Based on the above discussion, we can detect and locate four types of singularities as follows.

From the summary above, there is only one singularity on

According to Lemma

Thus,

Summarizing the above conclusion, we have the following.

The fractional derivatives can detect and locate four types of ideal singularities correctly.

In this paper, we study fractional-order derivatives of left-handed Grünwald-Letnikov formula with

The authors declare that there is no conflict of interests regarding the publication of this paper.

This paper is supported by the National Natural Science Foundation of China (nos. 60873102 and 60873264), Major State Basic Research Development Program (no. 2010CB732501), and Open Foundation of Visual Computing and Virtual Reality Key Laboratory of Sichuan Province (no. J2010N03). This work was supported by a Grant from the National High Technology Research and Development Program of China (no. 2009AA12Z140).