A discrete fractional angular transform

Available online at www.sciencedirect.com

Optics Communications 281 (2008) 1424–1429 www.elsevier.com/locate/optcom

A discrete fractional angular transform Zhengjun Liu b

a,b

, Muhammad Ashfaq Ahmad

a,c

, Shutian Liu

a,*

a Harbin Institute of Technology, Department of Physics, Harbin 150001, PR China Harbin Institute of Technology, Department of Automation Measurement and Control Engineering, Harbin 150001, PR China c COMSATS Institute of Information Technology, Department of Physics, Lahore 54000, Pakistan

Received 29 March 2007; received in revised form 30 October 2007; accepted 1 November 2007

Abstract A new discrete fractional transform defined by two parameters (angle and fractional order) is presented. All eigenvectors of the transform are obtained by an angle using recursion method. This transform is named as discrete fractional angular transform (DFAT). The computational load of kernel matrix of the DFAT is minimum than all other transforms with fractional order. This characteristics has very important practical applications in signal and image processing. Numerical results and the mathematical properties of this transform are also given. As fractional Fourier transform, this transform can be applied in one and two dimensional signal processing.  2007 Elsevier B.V. All rights reserved. PACS: 42.30.Kq; 42.30.d; 42.30.Va Keywords: Fourier transform; Fractional transform; Fractional Fourier transform; Random transform

1. Introduction It is well known that the mathematical transforms are significant in digital signal and image processing as well as in science and engineering. In the past decades, several useful transforms have been extensively investigated and have been applied to various digital signal processing fields. Among them the fractional Fourier transform has been regarded as a powerful tool in optics and signal processing [1–6]. Pei and Yeh have proposed an effective and fast algorithm of discrete fractional Fourier transform (DFrFT) [7] with high accuracy. This fast fractional Fourier transform can be a useful tool in digital signal processing. Some extensions from DFrFT, such as discrete fractional sine and cosine transforms, multiple parameter DFrFTs [8,9], discrete fractional random transform (DFRNT) [10,11], random Fourier transform (RFT) [12] and random frac-

*

Corresponding author. Tel.: +86 451 86418042; fax: +86 451 86414335. E-mail address: [email protected] (S. Liu). 0030-4018/$ - see front matter  2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.11.012

tional Fourier transform (RFrFT) [13] have also been proposed. The kernel matrices of the discrete transforms can be expressed as the product of the eigenvectors and eigenvalues matrices. We find that the matrices of the eigenvectors and eigenvalues can be changeable. The DFRNT has the same eigenvalues as the DFrFT, however, its eigenvectors are random matrix [10]. The eigenvalues of the RFT we proposed was changed to a random matrix [12]. This feature of the discrete transform motivated us to construct new transforms on purpose to acquire new features. In this paper, we propose a new discrete fractional transform base on the DFrFT and the DFRNT, in which the eigenvectors have been re-defined by a series of vectors generated from an initial angle by recurrence method. The new discrete fractional transform can be defined with only two parameters, a fractional order a and an angle b. Unlike the DFrFT in which the complicated Hermite polynomials should be calculated for eigenvectors, this transform can generate its eigenvectors by simple recurrences and hence the computation speed can be greatly enhanced. Furthermore, the new discrete transform inherits all the mathematical

Z. Liu et al. / Optics Communications 281 (2008) 1424–1429

properties as DFrFT and DFRNT have. For the reasons that the new discrete fractional order transform is defined by an initial angle and a fractional order, we can therefore refer to it as a discrete fractional angular transform (DFAT). The rest of the paper is organized in the following sequence. In Section 2, the new transform is defined and its properties are described. In Section 3, numerical results and comparison of computational speed with other transforms are presented. Some concluding remarks are summarized in the final section.

Mathematically a discrete transform T can be expressed by matrix multiplications as follows: T ¼ VDV t ; where V and D are eigenvector matrix and eigenvalue matrix of the transform, respectively. V t indicates the transpose of the matrix V. Similarly, the kernel matrix Aa;b N of DFAT can also be expressed as follows: b a Aa;b N ¼ V N DN ðV

b t NÞ ;

where the angle b is the main variable in Aa;b N , a is the fractional order. The eigenvectors of DFAT can only be determined by one variable (angle b). This characteristics makes DFAT different from the DFrFT and DFRNT. The DFAT of N-points signal x and image y with M · N size can be expressed as: X a;b ¼ Aa;b N x;

a;b Y a;b ¼ Aa;b M yAN :

2.1. Eigenvectors and eigenvalues of DFAT In order to introduce eigenvectors of new transform DFAT, we give a proposition about the eigenvectors matrix of new discrete transform. Proposition 1. If V N is an orthonormal matrix (i.e. V N V tN ¼ I), then the matrices   V N; V N 1 p ffiffi ffi V 2N ¼ ð1Þ z z 2 V N ; V N and

2

V N;

V N;

1 6 V 2N þ1 ¼ pffiffiffi 4 V 0 ; V 0; 2 z V N ; V zN ;

3 V t0 pffiffiffi 7 25 V

to construct the eigenvectors matrix of the DFRNTs RaN and Ra2N (or Ra2N þ1 ), respectively. For all 2 · 2 matrices, the matrix   cos b; sin b b V2 ¼ ð3Þ  sin b; cos b is an orthonormal matrix. V b2 is the matrix of coordinate transformation for rotation as well. And the matrix 2 3 cos b; sin b; 0 6 7 V b3 ¼ 4 0; 0; 15 ð4Þ  sin b;

2. The definition and properties of the DFAT

ð2Þ

t 0

are also orthonormal matrices. Here the matrix V zN is obtained by flipping the matrix V N in up-down direction, and V 0 is an 1 · N zero vector. This proposition can be proved according to the relation V N V tN ¼ I. An orthonormal matrix can be regarded as the matrix that is made of the eigenvectors of certain DFRNT [10]. Therefore, matrices V N and V 2N (or V 2N þ1 ) can be used

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cos b;

0

is a special kind of orthonormal matrix. The construction of V b2 and V b3 is similar to V b2N and V b2N þ1 , respectively. With the help of Eqs. (1)–(4) we can compute another orthonormal matrices V bN ðN > 3Þ using recurrence algorithm. For example, we can compute V b42 and V b53 through the processes as follows: Eq: ð2Þ

Eq:ð1Þ

Eq:ð2Þ

Eq:ð1Þ

V b2 ! V b5 ! V b10 ! V b21 ! V b42 ; Eq:ð1Þ Eq: ð2Þ Eq: ð1Þ Eq: ð2Þ V b3 ! V b6 ! V b13 ! V b26 ! V b53 : Thus, all V bN ðN > 3Þ can be computed by

ð5Þ ð6Þ

following the processes elaborated in Eqs. (5) and (6). This process is very similar to that of making a fractal. As a matter of fact, the eigenvector matrices can be regarded as fractal patterns (see Fig. 1). It is noted that there is only one variable b exists in the expressions of these matrices V bN . As stated above the matrix V bN is an orthonormal matrix, therefore, the column vectors of matrix V bN can be regarded as the eigenvectors of new transform. All elements of the matrix V bN can be collected from the set    r 1 pffiffiffi ½0; 1;  sin b;  cos b ; SN ¼ ð7Þ 2 where r represents the number of recurrence process. For N ¼ 2n , the set in Eq. (7) takes the following form (  ) n1 1 pffiffiffi S 2n ¼ ½ sin b;  cos b ; ð8Þ 2 which can be regarded as a subset of S N . Because the eigenvectors are obtained from simple matrix using recurrence method. It means that the computational load is very small. Thereby, the eigenvectors of the new transform DFAT can be obtained much faster than the eigenvectors of DFrFT and DFRNT. Similar to DFrFT and DFRNT, the eigenvalues of the DFAT can be expressed as: kaN ¼ ½1; expði2paÞ; expð4ipaÞ; . . . ; expð2ðN  1ÞipaÞ ð9Þ and can be calculated easily as discussed by Liu and coworkers [10]. The diagonal matrix DaN are defined as: DaN ¼ diagðkaN Þ:

ð10Þ

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Z. Liu et al. / Optics Communications 281 (2008) 1424–1429

Fig. 1. The eigenvectors matrix V bN with N = 40 for different values of angle b.

For the special case of N = 1, the absolute value of V b1 should be equal to 1 according to the energy conservation law. Therefore, V b1 can take a simple form of V b1 ¼ expðibÞ and the eigenvalue equals 1 in Eq. (9). The kernel matrix will degenerate to Aa;b 1 ¼ expðibÞ under this condition. 2.2. The properties of DFAT The mathematical properties of DFAT are inherited from the FrFT and DFRNT, which can be outlined as follows: • Linearity. The DFAT is a linear transform, a;b a;b Aa;b N ðax1 þ bx2 Þ ¼ aAN x1 þ bAN x2 , where a and b are constants. • Unitarity. The matrix of DFAT is a unitary matrix,  a;b 1 ðAa;b ¼ ANa;b . Thereby, the inverse DFAT N Þ ¼ ðAN Þ a;b a;b AN is AN . • Index Additivity. This property is universal for all fractional transforms, such as FrFT, DFrFT and DFRNT. ANa1 ;b ANa2 ;b ¼ ANa2 ;b ANa1 ;b ¼ ANa1 þa2 ;b . • Double Multiplicity. The property is tenable for both of a and b, ANaþ1;b ¼ ANa;bþ2p ¼ Aa;b N , for the eigenvectors and eigenvalues are periodic function for a and b, respectively.

• Parseval. The input and output of DFAT P satisfy 2the Paesval energy conservation theorem. k jX a;b ðkÞj ¼ P 2 jxðmÞj . m In addition, the output of DFAT keeps the symmetry of input for an even and odd signals as shown in the Figs. 2 and 3, which is also discussed in [8,11] for the cases of DFrCT(DFrST) and DFRNCT(DFRNST), respectively. The amplitude of DFAT of even and odd signals keeps the symmetry similar to FrFT. Because the column vectors are even vectors or odd vectors. This symmetry can reduce further the computational load of transforms. 3. Numerical results In this section, we shall use the rectangle function to simulate the numerical results. One dimensional rectangle function is given as follows:  1 if jxj < 1; f ðxÞ ¼ ð11Þ 0 otherwise: By applying the process from Eqs. (1) to (6), DFAT of one dimensional rectangle function is calculated and the numerical results are depicted in Fig. 2 for different values of a and b. Here we have taken the number of sample

Z. Liu et al. / Optics Communications 281 (2008) 1424–1429

α = 0.3, β = 0.3

α = 0.3, β = 0.7

α = 0.3, β = 0.5

4

4

2

2

2

0

0

0

-2

-2

-2

-4

-2

0

2

-4

-2

0

4

2

-4

-2

α = 0.5, β = 0.5

α = 0.5, β = 0.3 2

4

0

2

α = 0.5, β = 0.7

2

2

0

1427

0

0 -2 -4

-2

-2 -2

0

2

-4

-2

α = 0.7, β = 0.3

0

2

-4

α = 0.7, β = 0.5 4

4

2

2

2

0

0

0

-2

-2

-2

-2

0

2

-4

-2

0

0

2

α = 0.7, β = 0.7

4

-4

-2

2

-4

-2

0

2

Fig. 2. The results of DFAT of one dimensional rectangle function with different values of parameters a and b. Here (a, b) 2 {0.3, 0.5, 0.7}, N = 200.

points equal to 200 and the sample range is [3, 3]. The values of fractional order a and the angle b are taken from the set {0.3, 0.5, 0.7}. From Fig. 2 it is clear that the symmetry exists in amplitude of the even signal which is also evident in Figs. 3 and 4. It is also evident that the amplitude remains the same for fractional order equal to 0.3 and 0.7 with the same value of b, because ðA0:3;b Þ ¼ A0:3;b ¼ N N 0:7;b AN . Real output is obtained for a = 0.5, which is similar to the case of DFRNT. DFAT of four special functions expðt2 Þ; sinðtÞ; KðtÞ and sincðtÞ is illustrated in Fig. 3, where KðtÞ is the triangular function given as:  KðtÞ ¼

1  jxj if jxj 6 1; 0 otherwise:

ð12Þ

Numerical results of DFAT of three simple images for the two-dimensional case are shown in Fig. 4. This figure also shows the evidence of symmetry. Rectangle function is employed in calculations to make a comparison of calculation speed among DFrFT, DFRNT and DFAT. Five different values of sample numbers (N) are used in this process. Here we have fixed the value of fractional order a = 0.3, angle b = 1 as the parameters of DFAT, and the sample range is taken as [3, 3]. In

our numerical operations, we used a computer with Pentium-IV, 1.3-GHz CPU and 256 Mbytes memory under Windows 2000 system. The simulation is implemented in the environment with MATLAB 6.5. The calculating time t of all the transforms for rectangle function and the eigenvectors of transforms are recorded in Tables 1 and 2, respectively. Numerical values in the Table 1 reveal that the calculating time is much shorter which means that DFAT is fastest as compared to other transforms. From Table 2, the time to obtain the eigenvectors of DFAT is as small as negligible for each value of sample number (N) contrary with other transforms. A necessary step for introducing this transform into the field of optics information processing is to carry out optical implementation of the transform. To our best knowledge, the exact optical implementation of DFAT may be difficult. Recently, however, we proposed an optical system achieved random fractional Fourier transform [13]. Theoretically, the eigenvectors of DFAT can be obtained by designing phase mask in optical system of random fractional Fourier transform. We will consider the corresponding applications of DFAT in optical information processing in our further investigations. The special feature of high speed operation of DFAT and the randomness of

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Z. Liu et al. / Optics Communications 281 (2008) 1424–1429

a

b

4

2

sin (t)

exp (—t 2)

2 0 —2 —4

—5

0

5

4

—4 Input Amplitude Phase 4

0 t

5

—5

0 t

5

2

sinc (t)

Λ (t)

—5

d

2 0 —2 —4

0 —2

t

c

4

0 —2

—5

0 t

5

—4

Fig. 3. The results of DFAT of four functions. (a) expðt2 Þ, (b) sinðtÞ, (c) KðtÞ, (d) sincðtÞ. Here we have fixed a = 0.6, and b = 1 and N = 200.

Fig. 4. The results for three images with a = 0.6 and b = 1. (a) The original image I 1 , (b) the original image I 2 , (c) the original image I 3 , (d) the amplitude of the DFAT for image I 1 , (e) the amplitude of the DFAT for image I 2 , (f) the amplitude of the DFAT for image I 3 . Here the sample numbers are M = 128 and N = 128, respectively.

Table 1 The calculating time t(s) of transforms for rectangle function

Table 2 The calculating time t(s) of the eigenvectors of transforms

Transform

N = 200

N = 400

N = 600

N = 800

N = 1000

Transform

N = 200

N = 400

N = 600

N = 800

N = 1000

DFrFT DFRNT DFAT

0.491 0.511 0.140

2.644 3.535 0.811

8.882 17.005 4.366

30.044 48.629 12.879

69.980 103.199 25.697

DFrFT DFRNT DFAT

0.321 0.411 0.020

1.802 2.814 0.040

5.608 12.719 0.090

18.517 36.192 0.171

44.644 76.430 0.270

Z. Liu et al. / Optics Communications 281 (2008) 1424–1429

our previous proposed transforms [10–13] can directly find various applications in the fields of digital signal image processing. 4. Conclusion A new transform DFAT with two parameters angle and factional order, which determine the eigenvectors and eigenvalues, respectively, has been proposed. Some mathematical properties have been discussed which are similar to general fractional transform. Moreover, it has been proved that DFAT is very fast than DFrFT and DFRNT in calculations and keeps the symmetry for even and odd signals. We expect that the DFAT can be useful in the fields of fast signal analysis and image processing. Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant No. 10674038 and 10604042 and National Basic Research Program of China under Grant 2006CB302901. Mr. M. A. Ahmad

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thanks to COMSTAS Institute of Information Technology (CIIT), Lahore, Pakistan for supporting his PhD fellowship in Harbin Institute of Technology, Harbin, China. His permanent address is COMSATS Institute of Information Technology, Defence Road, off Raiwind Road, Lahore, 54000, Pakistan. References [1] H.M. Ozaktas, Z. Zalevsky, M.A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing, John Wiley & Sons, New York, 2000. [2] B. Zhu, S. Liu, Q. Ran, Opt. Lett. 25 (2000) 1159. [3] S. Liu, Q. Mi, B. Zhu, Opt. Lett. 26 (2001) 1242. [4] B. Hennelly, J.T. Sheridan, Opt. Commun. 226 (2003) 61. [5] S. Jin, S.Y. Lee, Opt. Commun. 207 (2002) 161. [6] Y. Zhang, C.H. Zheng, N. Tanno, Opt. Commun. 202 (2002) 277. [7] S.C. Pei, M.H. Yeh, Opt. Lett. 22 (1997) 1047. [8] S.C. Pei, M.H. Yeh, IEEE Trans. Signal Process. 49 (2001) 1198. [9] S.C. Pei, W.L. Hsue, IEEE Signal Process. Lett. 13 (2006) 329. [10] Z. Liu, H. Zhao, S. Liu, Opt. Commun. 255 (2005) 357. [11] Z. Liu, Q. Guo, S. Liu, Opt. Commun. 265 (2006) 100. [12] Z. Liu, S. Liu, Opt. Lett. 32 (2007) 478. [13] Z. Liu, S. Liu, Opt. Lett. 32 (2007) 2088.