On Local Fractional Continuous Wavelet Transform

Abstract

We introduce a new wavelet transform within the framework of the local fractional calculus. An illustrative example of local fractional wavelet transform is also presented.

1. Introduction

Wavelet transforms have been applied successfully in the areas of signals analysis, data compression, and sound processing (see, for details, [1–6] and the references cited therein). Although there is scaled and shifted versions of a mother wavelet, the daughter wavelets are structured as follows (see [3–5]): 𝜑_𝑎,𝑏(𝑡)=1𝑎^1/2𝜑𝑡−𝑏𝑎, (1) where 𝑎 is the dyadic dilation, 𝑏 is the dyadic position, and 𝑎^−1/2 is the normalization factor. The expression of a one-dimensional wavelet transform for a given continuous signal 𝑓(𝑡) is given by 𝑊_𝜑𝑓(𝑎,𝑏)=^∞_−∞𝑓(𝑡)𝜑_𝑎,𝑏(𝑡)𝑑𝑡 (2) and the reconstruction formula becomes 𝑓(𝑥)=𝐶_𝜑^∞_−∞1𝑎²𝑊_𝜑𝑓(𝑎,𝑏)𝜑_𝑎,𝑏(𝑡)𝑑𝑎𝑑𝑏, (3) where 𝐶_𝜑=^∞_−∞𝑓(𝑥)²|𝑥|𝑑𝑥. (4)

Recently, fractional wavelet transform, as a generalization of the classical wavelet transform, was proposed in [7]. The one-dimensional fractal wavelet transform of a continuous signal 𝑓(𝑡) has the following form: 𝑊_𝜑𝑓(𝑎,𝑏)=^∞_−∞𝐵(𝑥,𝑡)𝑓(𝑡)𝜑_𝑎,𝑏(𝑥)𝑑𝑡𝑑𝑥, (5) where 𝐵(𝑥,𝑡) denotes a bulk optics kernel.

The reconstructing formula of the input is defined as given by the following expression: 𝑓(𝑥)=1𝐶_𝜑^∞_−∞1𝑎³𝑊_𝜑𝑓(𝑎,𝑏)𝐵(𝑥,𝑡)𝜑_𝑎,𝑏(𝑡)𝑑𝑎𝑑𝑏𝑑𝑡𝑑𝑡. (6)

We notice that the fractional wavelet transforms was applied to image encryption [8], to the simultaneous spectral analysis in [9], and to the composite signals in [10, 11]. For other definition of fractional wavelet transform, see [12] and the references cited therein.

Keeping in mind the study of the fractal signals (local fractional continuous signals), a new local fractional wavelet transform was developed in [13] based upon the local fractional Fourier transform [14] via local fractional calculus [15–18]. In this paper, we investigate the local fractional Fourier transform to deal with the local fractional wavelet transforms by implementing the local fractional calculus.

The organization of the paper is as follows. Section 2 presents the concept of local fractional Fourier transform and wavelet. Section 3 discusses the derivation of the local fractional continuous wavelet transform. Section 4 studies the wave space and Section 5 present an illustrative example. Finally, Section 6 outlines the main conclusions of our present investigation.

2. Local Fractional Fourier Transform and Wavelet

Let 𝑓(𝑥) be local fractional continuous function, which is denoted as follows (see [18]): 𝑓(𝑥)∈𝐶_𝛼(−∞,∞). (7) The space of local fractional continuous functions 𝐶_𝑝,𝛼[𝑎,𝑏], under 𝑝-norm, is given by (see [13]) 𝑓_𝑝,𝛼=1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)^𝑝(𝑑𝑥)^𝛼1/𝑝,for1≤𝑝<∞, (8) where the operator is local fractional operator.

The space 𝐿_𝑝,𝛼[ℝ] norm on 𝐶_𝑝,𝛼[ℝ] is defined by 𝑓_𝑝,𝛼1=∶Γ(1+𝛼)^∞_−∞𝑓(𝑥)^𝑝(𝑑𝑥)^𝛼1/𝑝<∞ (9) for 1≤𝑝<∞. This is infinite for 𝑎 and 𝑏.

The local fractional Fourier transforms in fractal space is defined as follows (see [13, 14]): 𝐹_𝛼𝑓(𝑥)=𝑓_𝜔^𝐹,𝛼(𝜔)1∶=Γ(1+𝛼)^∞_−∞𝐸_𝛼−𝑖^𝛼𝜔^𝛼𝑥^𝛼𝑓(𝑥)(𝑑𝑥)^𝛼. (10) Its inverse is formulated as follows (see [13, 14]): 𝑓(𝑥)=𝐹_𝛼⁻¹𝑓_𝜔^𝐹,𝛼(𝜔1)∶=(2𝜋)^𝛼∞_−∞𝐸_𝛼𝑖^𝛼𝜔^𝛼𝑥^𝛼𝑓_𝜔^𝐹,𝛼(𝜔)(𝑑𝜔)^𝛼,𝑥>0. (11) Let 𝜑(𝑥)∈𝐿_2,𝛼[ℝ] and let 𝜑_𝜔^𝐹,𝛼(𝜔)=1Γ(1+𝛼)^∞_−∞𝐸_𝛼−𝑖^𝛼𝜔^𝛼𝑥^𝛼𝜑(𝑥)(𝑑𝑥)^𝛼,0<𝛼≤1. (12) When 𝜑_𝜔^𝐹,𝛼(𝜔)=1Γ(1+𝛼)^∞_−∞𝜑(𝑥)(𝑑𝑥)^𝛼=0,0<𝛼≤1, (13) the function 𝜑(𝑥) is called a local fractional wavelet [13].

Let 𝜑(𝑥)∈𝐿_2,𝛼[ℝ]. Then, we have 𝜑_{𝑎,𝑏,𝛼}(𝑡)²_1,𝛼=1𝑎^𝛼Γ(1+𝛼)^∞_−∞𝜑_{𝑎,𝑏,𝛼}(𝑡)²(𝑑𝑥)^𝛼=1Γ(1+𝛼)^∞_−∞𝜑(𝑡)²(𝑑𝑥)^𝛼=𝜑²_1,𝛼, (14) so that 𝜑_{𝑎,𝑏,𝛼}(𝑡)=1𝑎^𝛼/2𝜑𝑡−𝑏𝑎, (15) where 𝑎,𝑏∈ℝ and 𝑎≠0.

3. Local Fractional Continuous Wavelet Transform

Let 𝜑∈𝐿_2,𝛼[ℝ]. Then, we arrive at the following relation: 𝜑_{𝑎,𝑏,𝛼}(𝑡)¹_2,𝛼=1𝑎^𝛼Γ(1+𝛼)^∞_−∞𝜑_{𝑎,𝑏,𝛼}(𝑡)²(𝑑𝑡)^𝛼=1Γ(1+𝛼)^∞_−∞𝜑(𝑡)²(𝑑𝑡)^𝛼=𝜑¹_2,𝛼, (16) where 𝜑_{𝑎,𝑏,𝛼}(𝑡)=(1/𝑎^𝛼/2)𝜑((𝑡−𝑏)/𝑎), 𝑎,𝑏∈ℝ, and 𝑎≠0.

Similarly, we get 𝜑_𝑏,𝛼(𝑡)¹_2,𝛼=𝜑¹_2,𝛼. (17)

Taking 𝜑_𝑏,𝛼(𝑡) in place of 𝜑_{𝑎,𝑏,𝛼}(𝑡)𝐸_𝛼(−𝑖^𝛼𝜔^𝛼𝑡^𝛼), we obtain Θ_{𝜑𝑎,𝑏,𝛼}𝑓(𝑎,𝑏)=𝑓(𝑡),𝜑_{𝑎,𝑏,𝛼}(𝑡=1)Γ(1+𝛼)^∞_−∞𝑓(𝑡)𝜑_{𝑎,𝑏,𝛼}(𝑡)(𝑑𝑡)^𝛼. (18) In the special case when 𝑓(𝑡)=1, we have the following relation: Θ_{𝜑𝑎,𝑏,𝛼}()=1𝑎,𝑏Γ(1+𝛼)^∞_−∞𝜑_{𝑎,𝑏,𝛼}(𝑡)(𝑑𝑡)^𝛼 (19) such that 1Γ(1+𝛼)^∞_−∞Θ_{𝜑𝑎,𝑏,𝛼}(𝑎,𝑏)𝜑_{𝑎,𝑏,𝛼}(𝑡)(𝑑𝑏)^𝛼=|𝑎|^𝛼1Γ(1+𝛼)^∞_−∞𝜑(𝑡)(𝑑𝑡)^𝛼2. (20) Hence, there exists the following relation: 1Γ²(1+𝛼)^∞_−∞𝑎^−2𝛼Θ_{𝜑𝑎,𝑏,𝛼}(𝑎,𝑏)𝜑_{𝑎,𝑏,𝛼}(𝑡)(𝑑𝑎)^𝛼(𝑑𝑏)^𝛼=1Γ(1+𝛼)^∞_−∞|𝑥|^−𝛼(𝑑𝑥)^𝛼. (21) In general, we also deduce the following identities: 𝑓(𝑥)=∫^∞_−∞𝑓(𝑥)²/|𝑥|^𝛼(𝑑𝑥)^𝛼Γ³()×1+𝛼_∞−∞𝑎^−2𝛼Θ_{𝜑𝑎,𝑏,𝛼}𝑓(𝑎,𝑏)𝜑_{𝑎,𝑏,𝛼}(𝑡)(𝑑𝑏)^𝛼(𝑑𝑎)^𝛼,Θ_{𝜑𝑎,𝑏,𝛼}𝑓()=1𝑎,𝑏Γ(1+𝛼)^∞_−∞𝑓(𝑡)𝜑_{𝑎,𝑏,𝛼}(𝑡)(𝑑𝑡)^𝛼. (22) Now, we establish the following relations: Θ_{𝜑𝑎,𝑏,𝛼}𝑓()=𝑎𝑎,𝑏^−𝛼/2Γ(1+𝛼)^∞_−∞𝑓(𝑡)𝜑_{𝑎,𝑏,𝛼}(𝑡)(𝑑𝑡)^𝛼,𝑓(𝑥)=∫0<𝛼≤1,_∞−∞𝑓(𝑥)²/|𝑥|^𝛼(𝑑𝑥)^𝛼Γ³()×1+𝛼_∞−∞𝑎^−2𝛼Θ_{𝜑𝑎,𝑏,𝛼}𝑓(𝑎,𝑏)𝜑_{𝑎,𝑏,𝛼}×(𝑡)(𝑑𝑎)^𝛼(𝑑𝑏)^𝛼. (23) Hence, the local fractional continuous wavelet transform takes the following form (see [13]): 𝑊_𝜑,𝛼𝑓()=𝑎𝑎,𝑏^−𝛼/2Γ(1+𝛼)^∞_−∞𝑓(𝑡)𝜑_{𝑎,𝑏,𝛼}(𝑡)(𝑑𝑡)^𝛼,0<𝛼≤1. (24) And the inversion formula of local fractional continuous wavelet transform is derived as follows (see [14]): 𝑓(𝑥)=𝐶_𝜑,𝛼Γ²()×1+𝛼_∞−∞𝑎^−2𝛼𝑊_𝜑,𝛼𝑓(𝑎,𝑏)𝜑_{𝑎,𝑏,𝛼}(𝑡)(𝑑𝑎)^𝛼(𝑑𝑏)^𝛼,0<𝛼≤1, (25) where 𝐶_𝜑,𝛼=1Γ(1+𝛼)^∞_−∞𝑓(𝑥)²|𝑥|^𝛼(𝑑𝑥)^𝛼,0<𝛼≤1. (26)

4. The Wavelet Space

In order to differ the classical wavelets from fractional wavelets, here we formulate a wavelet space as follows. In fact, a wavelet space is defined by 𝑊_𝜑,𝛼[ℝ]=𝜑,𝛼∶𝑊_𝜑,𝛼𝑓()=𝑎𝑎,𝑏_−𝛼/2Γ(1+𝛼)^∞_−∞𝑓(𝑡)𝜑_{𝑎,𝑏,𝛼}(𝑡)(𝑑𝑡)^𝛼,0<𝛼≤1. (27) When the fractal dimension 𝛼 is equal to 1, from (27), we deduce (see [3–5]) 𝑊_𝜑,1[ℝ]=𝜑,1∶𝑊_𝜑,1𝑓(𝑎,𝑏)=𝑎^−1/2∞_−∞𝑓(𝑡)𝜑_𝑎,𝑏,1(𝑡)𝑑𝑡,𝛼=1, (28) where 𝑓(𝑡) is continuous and 𝑊_𝜑,1𝑓(𝑎,𝑏)∈𝑊_𝜑,1[ℝ].

Taking the fractal dimension 0<𝛼<1, we derive a formula given by 𝑊_𝜑,𝛼𝑓()=𝑎𝑎,𝑏^−𝛼/2Γ(1+𝛼)^∞_−∞𝑓(𝑡)𝜑_{𝑎,𝑏,𝛼}(𝑡)(𝑑𝑡)^𝛼 (29) with 𝑊_𝜑,𝛼𝑓(𝑎,𝑏)∈𝑊_𝜑,𝛼[ℝ], where 𝑓(𝑡) is a local fractional continuous function.

5. An Illustrative Example

In order to construct the local fractional continuous wavelet, we suppose that 𝜙(𝑡) is 𝑚𝛼 times the local fractional differentiable function.

We define the local fractional wavelet 𝜑(𝑡) by means of the following expression: 𝜑(𝑡)=𝑑^𝑚𝛼𝑦𝑑𝑥^𝑚𝛼, (30) where the differential operator is the local fractional operator proposed by Yang [18] (for other definition, see [19] and the references cited therein).

Then, we get 1Γ(1+𝛼)^∞_−∞𝜑(𝑡)𝑡^𝑚𝛼Γ()()1+𝑚𝛼𝑑𝑡^𝛼=0(𝑚=0,1,2,…,𝑚). (31) Let us consider the following nondifferentiable signal, namely, 𝜙_𝐻(𝛼)(𝑡)=𝑡^𝛼Γ()11+𝛼,0≤𝑡<2,(1−𝑡)^𝛼Γ(),11+𝛼2≤𝑡<1,0,else. (32) For 0≤𝑡<1/2, we obtain 𝑑^𝛼𝜙_𝐻(𝛼)(𝑡)𝑑𝑡^𝛼=𝑑^𝛼𝑑𝑡^𝛼𝑡^𝛼Γ(1+𝛼)=1. (33) For 1/2≤𝑡<1, we obtain 𝑑^𝛼𝜙_𝐻(𝛼)(𝑡)𝑑𝑡^𝛼=𝑑^𝛼𝑑𝑡^𝛼(1−𝑡)^𝛼Γ(1+𝛼)=−1. (34) In view of (33)-(34), we get a local fractional wavelet given by 𝜑_𝐻(𝛼)(𝑡)=11,0≤𝑡<2,1−1,2≤𝑡<1,0,else. (35) Following (35), we obtain 1Γ(1+𝛼)^∞_−∞𝜑_𝐻(𝛼)(𝑡)(𝑑𝑡)^𝛼1=0,Γ(1+𝛼)^∞_−∞𝜑²_𝐻(𝛼)(𝑡)(𝑑𝑡)^𝛼=1. (36) In view of (15), taking 𝑎=2^−𝑗 and 𝑏=𝑘2^−𝑗, we have 𝜑_{𝑎,𝑏,𝛼}(𝑡)=1𝑎^𝛼/2𝜑𝑡−𝑏𝑎=𝜑_{𝑗,𝑘,𝛼}(𝑡)=𝜑_{2−𝑗,𝑘2−𝑗,𝛼}(𝑡)=2^𝑗𝛼/2𝜑2^𝑗𝑡−𝑘 (37) for integers 𝑗,𝑘∈ℤ.

Hence, we get the following equation: 𝜑_𝐻(𝛼)^𝑗,𝑘(𝑡)=2^𝑗𝛼/2𝜑_𝐻(𝛼)2^𝑗𝑡−𝑘. (38) We thus conclude that 1Γ(1+𝛼)^∞_−∞𝜑_𝐻(𝛼)^𝑗,𝑘(𝑡)²(𝑑𝑡)^𝛼=1Γ(1+𝛼)^∞_−∞2^𝑗𝛼/2𝜑_𝐻(𝛼)2^𝑗𝑡−𝑘²(𝑑𝑡)^𝛼=2^𝑗𝛼1Γ(1+𝛼)^∞_−∞𝜑_𝐻(𝛼)2^𝑗𝑡−𝑘²(𝑑𝑡)^𝛼=1Γ(1+𝛼)^∞_−∞𝜑_𝐻(𝛼)2^𝑗𝑡−𝑘²𝑑2^𝑗𝑡−𝑘^𝛼=1Γ(1+𝛼)^∞_−∞𝜑_𝐻(𝛼)(𝑡)²(𝑑𝑡)^𝛼1=1,Γ(1+𝛼)^∞_−∞𝜑_𝐻(𝛼)^𝑗,𝑘(𝑡)(𝑑𝑡)^𝛼=1Γ(1+𝛼)₀^1/22^𝑗𝛼/2𝜑_𝐻(𝛼)2^𝑗𝑡−𝑘(𝑑𝑡)^𝛼−1Γ(1+𝛼)¹_1/22^𝑗𝛼/2𝜑_𝐻(𝛼)2^𝑗𝑡−𝑘(𝑑𝑡)^𝛼=0. (39)

6. Concluding Remarks and Observations

A novel local fractional wavelet transformation was investigated by using Fourier transform based upon local fractional calculus. This transform has been found to be advantageous in dealing with the functions in fractal space. The wave space is considered and an illustrative example is shown.

Conflict of Interests

The authors declare that they have no conflict of interests regarding this paper.