CONVOLUTION THEOREM FOR TWO DIMENSIONAL FRACTIONAL MELLIN TRANSFORM

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Available Online at www.ijpret.com 103

INTERNATIONAL JOURNAL OF PURE AND

APPLIED RESEARCH IN ENGINEERING AND

TECHNOLOGY

A PATH FOR HORIZING YOUR INNOVATIVE WORK

CONVOLUTION THEOREM FOR TWO DIMENSIONAL FRACTIONAL MELLIN

TRANSFORM

V. D. SHARMA1_{, P. B. DESHMUKH}2

1. Department of Mathematics, Arts, Commerce and Science College, Kiran Nagar, Amravati, India 2. Department of Engineering Mathematics, IBSS College of Engineering, Amravati, India

Accepted Date: 05/03/2015; Published Date: 01/05/2015

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Abstract: The fractional Mellin transform is a generalization of the Mellin transform. It has many applications in the electronic world, such as computation of algorithm, cryptographic scheme, vowel recognition, pattern recognition. Signal processing and pattern recognition algorithms make extensive use of convolution. Convolution is an important tool because of its translation invariance property. Also convolution is a powerful way of characterizing the input-output relationship of time invariance system. In this paper we discussed the definition of two dimensional fractional Mellin transform (FRMT) which is extended to the distribution of compact support. The convolution theorem for FRMT is also proved.

Keywords: Fractional Mellin Transform, Mellin Transform, Testing Function space, Generalize function

Corresponding Author: MR.V. D. SHARMA

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How to Cite This Article:

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INTRODUCTION

The fractional integral transform is a backbone of today’s technology, because fractional integral transform has many applications which are very useful in electronic world.

Also Mellin transform has many applications such as navigation, radar system [1], in finding the stress distribution in an infinite wedge [1], also used in digital audio effects [1].

Mellin transform is also useful for segregating the information about the size and shape of vocal track by stabilizing the Mellin image [5]. For the past decades, the Mellin transform has received considerable attention from the optical image processing, radar and sonar signal processing and target classification [6]. Because of scale invariance property of Mellin transform the independent speech recognition of speaker is possible [6].

By using convolution theorem we can tabulate atomic scattering factors by working out the diffraction pattern of atoms place at origin. In pattern recognition convolution is an important tool because of its translation invariance property. It is useful for extraction of information in communication engineering [2].

In this paper we proposed a convolution theorem for two dimensional fractional Mellin transform. Also defined the testing function space and definition of two dimensional fractional Mellin transform.

The two dimensional fractional Mellin transform with parameter θ of f(x,y)denoted by

)} , ( {

FRMT f x y performs a linear operation, given by the integral transform

dxdy v u y x K y x f v u F y x f

FRMT{ ( , )} ( , ) ( , ) ( , , , )

0 0





  

 (1)

where the kernel

] log log [ tan 1 sin 2 1 sin

2 2 2 2 2

) , , ,

( u v x y

i iv iu e y x v u y x

K       

      . 2

0  (2)

where, 𝐶1𝜃 = 2𝜋𝑖

𝑠𝑖𝑛𝜃, 𝐶2𝜃 = 𝜋

𝑡𝑎𝑛𝜃 (3)

For the convolution of two dimensional fractional Mellin transform.

Let, f, g be in W and denote their convolution h is as

ℎ(𝑥, 𝑦) = ∫ ∫ 𝑓̃(𝑟, 𝑠)₀∞ ₀∞ 1_𝑟1_𝑠𝑔̃(𝑥

𝑟, 𝑦

𝑠)𝑑𝑟𝑑𝑠

(3)

For any function 𝑓(𝑥, 𝑦)

𝑓̃(𝑥, 𝑦) = 𝑓(𝑥, 𝑦)𝑒𝑖𝐶2𝜃(𝑙𝑜𝑔2𝑥+𝑙𝑜𝑔2𝑦)₍₅₎

Then for any two functions f & g the convolution operation * is defined as

ℎ(𝑥, 𝑦) = (𝑓 ∗ 𝑔)(𝑥, 𝑦)

= 𝑒−𝑖𝐶2𝜃(𝑙𝑜𝑔2𝑥+𝑙𝑜𝑔2𝑦)_{(𝑓̃ ∗ 𝑔̃)(𝑥, 𝑦)}₍₆₎

where * is the convolution operation for the FRMT as defined in (6).

1.1The test function space E

An infinitely differentiable complex valued function 

 

x,y on n

R belongs to

 

n

R

E , if for each

compact set IS, KSbwhere

} 0 , , : {   

 x x R x a a

S_a S_b {y:yR, y b,b0}, n

R K

I, 

     ) , ( )] , ( [ , , ,

, x y

sup

D x y

q y x K y I x q

E  

 



Thus E(Rn)will denote the space of all (x,y)E(Rn) with compact support contained in S_&

b S .

Note that the space E is complete and therefore a Frechet space. Moreover, we say that f(x,y)

is a fractional Mellin transformable if it is a member of *

E , the dual of E.

1.2Two dimensional fractional Mellin transform (FRMT)

The two dimensional fractional Mellin transform of f(x,y)E*(Rn)can be defined by ) , , , ( ), , ( ) , ( )} , (

{f x y F u v f x y K x y u v

FRMT  _  _

] log log [ tan 1 sin 2 1 sin

2 2 2 2 2

) , , ,

( u v x y

i iv iu e y x v u y x

K       

    

 Right hand side of equation (4) has a

meaning as the application of *

) ,

(x y E

f  to K_(x,y,u,v)E. It

can be extended to the complex space as an entire function given by ) , , , ( ), , ( ) , ( )} , (

{f x y F p k f x y K x y p k

FRMT  _  _ The right hand side is meaningful because for each

n C k

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II. Convolution Theorem of two dimensional fractional Mellin transform:

Let ℎ(𝑥, 𝑦) = (𝑓 ∗ 𝑔)(𝑥, 𝑦) & 𝐹𝜃, 𝐺𝜃, 𝐻𝜃 denotes the two dimensional fractional Mellin transform

of 𝑓, 𝑔, ℎ respectively then 𝐻_𝜃(𝑢, 𝑣) = 𝑒−𝑖𝐶2𝜃(𝑢2+𝑣2)𝐹

𝜃(𝑢, 𝑣)𝐺𝜃(𝑢, 𝑣).

Proof- By using the definition of the two dimensional fractional Mellin transform

𝐻_𝜃(𝑢, 𝑣) = ∫ ∫ ℎ(𝑥, 𝑦)𝑥𝐶1𝜃𝑢−1

∞

0

𝑦𝐶1𝜃𝑣−1

∞

0

𝑒𝑖𝐶2𝜃(𝑢2+𝑣2+𝑙𝑜𝑔2𝑥+𝑙𝑜𝑔2𝑦)_{𝑑𝑥𝑑𝑦}

By using (6)

= ∫ ∫ (𝑓 ∗ 𝑔)(𝑥, 𝑦)𝑥𝐶1𝜃𝑢−1

∞

0

∞

0

= ∫ ∫ 𝑒−𝑖𝐶2𝜃(𝑙𝑜𝑔2𝑥+𝑙𝑜𝑔2𝑦)

∞

0 ∞

0

(𝑓̃ ∗ 𝑔̃)(𝑥, 𝑦)𝑥𝐶1𝜃𝑢−1

𝑦𝐶1𝜃𝑣−1𝑒𝑖𝐶2𝜃(𝑢2+𝑣2+𝑙𝑜𝑔2𝑥+𝑙𝑜𝑔2𝑦)𝑑𝑥𝑑𝑦

By using (4)

= ∫ ∫ 𝑒−𝑖𝐶2𝜃(𝑙𝑜𝑔2𝑥+𝑙𝑜𝑔2𝑦)𝑥𝐶1𝜃𝑢−1

∞

0

∞

0

∫ ∫ 𝑓̃(𝑟, 𝑠)₀∞ ₀∞ 1_𝑟1_𝑠𝑔̃(𝑥

𝑟, 𝑦

𝑠)𝑑𝑟𝑑𝑠

Putting 𝑥

𝑟= 𝑚, 𝑦 𝑠 = 𝑛

⇒ 𝑥

𝑚= 𝑟, 𝑦 𝑛 = 𝑠

Differentiate with respect to m & n respectively

−𝑥

𝑚2𝑑𝑚 = 𝑑𝑟,

−𝑦

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𝐻𝜃(𝑢, 𝑣) = ∫ ∫ 𝑒−𝑖𝐶2𝜃(𝑙𝑜𝑔

2_{𝑥+𝑙𝑜𝑔}2_𝑦)

𝑥𝐶1𝜃𝑢−1

∞

0 ∞

0

𝑦𝐶1𝜃𝑣−1_𝑒𝑖𝐶2𝜃(𝑢2+𝑣2+𝑙𝑜𝑔2𝑥+𝑙𝑜𝑔2𝑦)_{𝑑𝑥𝑑𝑦}

∫ ∫ 𝑓̃ (_𝑚𝑥,𝑦

𝑛) ∞

0 ∞ 0

𝑚 𝑥 𝑛

𝑦𝑔̃(𝑚, 𝑛)

(−𝑥

𝑚2) (

−𝑦

𝑛2) 𝑑𝑚𝑑𝑛

= ∫ ∫ 𝑥𝐶1𝜃𝑢−1

∞

0

𝑦𝐶1𝜃𝑣−1_𝑒𝑖𝐶2𝜃(𝑢2+𝑣2)_{𝑑𝑥𝑑𝑦}

∞

0

∫ ∫ _𝑚𝑛1 𝑓̃ (𝑥

𝑚, 𝑦 𝑛) ∞

0 ∞

0 𝑔̃(𝑚, 𝑛)𝑑𝑚𝑑𝑛

By using (5) = ∫ ∫ 𝑥∞ 𝐶1𝜃𝑢−1

0 𝑦

𝐶1𝜃𝑣−1

∞

0 𝑒

𝑖𝐶2𝜃(𝑢2+𝑣2)𝑑𝑥𝑑𝑦

∫ ∫ _𝑚𝑛1 𝑓 (𝑥

𝑚, 𝑦 𝑛) 𝑒

𝑖𝐶2𝜃(𝑙𝑜𝑔2 𝑥_𝑚+𝑙𝑜𝑔2𝑦_𝑛)

∞ 0 ∞ 0

𝑔(𝑚, 𝑛)𝑒𝑖𝐶2𝜃(𝑙𝑜𝑔2𝑚+𝑙𝑜𝑔2𝑛)_{𝑑𝑚𝑑𝑛}

Putting _𝑚𝑥 = 𝑝 , 𝑦_𝑛= 𝑞

⇒ 𝑥 = 𝑚𝑝 , 𝑦 = 𝑛𝑞

⇒ 𝑑𝑥 = 𝑚𝑑𝑝 , 𝑑𝑦 = 𝑛𝑑𝑞

𝐻𝜃(𝑢, 𝑣) = ∫ ∫ (𝑚𝑝)𝐶1𝜃𝑢−1

∞

0 (𝑛𝑞)

𝐶1𝜃𝑣−1

∞

0

𝑒𝑖𝐶2𝜃(𝑢2+𝑣2)_{∫ ∫} 1

𝑚𝑛𝑓(𝑝, 𝑞)𝑒

𝑖𝐶2𝜃(𝑙𝑜𝑔2𝑝+𝑙𝑜𝑔2𝑞)

∞ 0 ∞ 0

𝑔(𝑚, 𝑛)𝑒𝑖𝐶2𝜃(𝑙𝑜𝑔2𝑚+𝑙𝑜𝑔2𝑛)_{𝑚𝑑𝑝𝑛𝑑𝑞𝑑𝑚𝑑𝑛}

= ∫ ∫ (𝑚)∞ 𝐶1𝜃𝑢−1

0 (𝑛)

𝐶1𝜃𝑣−1

∞

0 𝑔(𝑚, 𝑛)

𝑒𝑖𝐶2𝜃(𝑢2+𝑣2+𝑙𝑜𝑔2𝑚+𝑙𝑜𝑔2𝑛)_{𝑑𝑚𝑑𝑛}_𝑒−𝑖𝐶2𝜃(𝑢2+𝑣2)_{∫ ∫ (𝑝)}∞ 𝐶1𝜃𝑢−1_(𝑞)𝐶1𝜃𝑣−1

0 ∞ 0

𝑓(𝑝, 𝑞)𝑒𝑖𝐶2𝜃(𝑢2+𝑣2+𝑙𝑜𝑔2𝑝+𝑙𝑜𝑔2𝑞)_{𝑑𝑝𝑑𝑞}_{= 𝑒}−𝑖𝐶2𝜃(𝑢2+𝑣2)_𝐺

𝜃(𝑢, 𝑣)𝐹𝜃(𝑢, 𝑣)

= 𝑒−𝑖𝐶2𝜃(𝑢2+𝑣2)_𝐹

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III. Diagram:

CONCLUSION

In this paper we have defined distributional two dimensional fractional Mellin transform with compact support. Convolution theorem for two dimensional fractional Mellin transform is proved.

REFERENCES

1. V.D. Sharma, P.B.Deshmukh: “Inversion theorem of two dimensional fractional Mellin transform”, International Journal of Applied Mathematics & Mechanics, Vol.-3, No.-1 (2014), pp 33-39.

2. V.D. Sharma, P.B.Deshmukh: “A convolution theorem for the two dimensional fractional Fourier Transform in generalized sense”, 3rd_{International Conference of Emerging Trends in}

Engineering & Technology, 2010, IEEE Computer Society.

3. Ahmed I Zayed, “A convolution & product theorem for the fractional Fourier transform”, IEEE Signal Processing letters, vol-5, No.-4, April 1998.

𝑪

_𝟐𝜽

=

𝝅

𝒕𝒂𝒏𝜽

Convolution for Two-Dimensional

Fractional Mellin Transform

𝒆𝒊𝑪𝟐𝜽(𝒍𝒐𝒈𝟐𝒙+𝒍𝒐𝒈𝟐𝒚)

f(x,y)

g(x,y)

C O N V O L U T I O N

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4. V.N.Mahalle, A.S. Gudadhe, R.D.Taywade: “Generalization of linear Scale Invariant System in he fractional domain and some properties of Fractional Complex Mellin Transform”, International Journal of Contemp. Math Sciences, vol-6,2011, no.-2, 577-584.

5. Toshio Irino, Roy D. Patterson: “Segregating information about the size & shape of the vocal tract using a time domain auditory Model, The stbilised wavelet Mellin Transform”, Speech communication 36 (2002), pp 181-203.