Analytical Structure of Distributional Two Dimensional Fourier-Mellin Transform
V. D. Sharma P. D. Dolas
HOD, Department of Mathematics, Arts, Department of Mathematics,
Commerce and Science College, Kiran Nagar, Dr. Rajendra Gode Institute of Technology &
Amravati, India, 444606. Research, Amravati, India, 444602
Abstract
The Fourier-Mellin transform is that it is invariant in rotation, translation, scale and they have numerous applications in engineering such as new paper currency recognition system, image recognition, signal processing. Both transforms are mathematically related with each other. In the present work, we have generalized two dimensional Fourier-Mellin transform in distributional sense and proved the analyticity theorem for distributional Fourier-Mellin transform.
Keywords
Fourier transform, Mellin transform, two dimensional Fourier-Mellin transform, generalized function.
Introduction
Mathematics is everywhere in every phenomenon, technology, observation, experiment etc. Fourier Transform is a mathematical method using the trigonometric functions to transform a time domain spectrum into a frequency domain spectrum. Fourier transform applicable in signal processing including audio, speech, images, videos, seismic data, radio transmissions and also applicable in computer graphics, image processing, and fingerprint analysis, in many modern technologies advances etc.[6]. The Fourier Transform itself is translation invariant and its conversion to log-polar coordinates converts the scale and rotation differences to vertical and horizontal offsets that can be measured [3]. The use of Fourier transforms for deriving probability densities of sums and differences of random variables is well known [5,7]. The Fourier Transform is a tool for solving physical problems and applied to optics, crystallography, solving science problems, acoustics, occurs naturally all throughout physics. The Mellin transform used in place of Fourier transform when scale invariance is more relevant than shift invariance [4].The Mellin transform is used in signal processing as a tool to investigate scale invariance and it gives a transform-space image that is invariant to translation, rotation and scale [3]. Besides its use in mathematics, Mellin transformation has been applied in many different areas of physics and engineering. It is related to the Fourier transform by a logarithmic coordinate transformation. Its modulus is scale invariant [1]. We use the Mellin integral transforms to derive different properties in statistics and probability densities of single continuous random variable [2,7]. The key feature of Fourier-Mellin transform is that it is invariant in rotation, translation and scale. The Fourier-Fourier-Mellin transform is a powerful tool for image recognition because its resulting spectrum is invariant in rotation, translation and scale. New paper currency recognition system based on Fourier-Mellin transform [3]. Motivated from the work of Fourier and Mellin transform, we have generalized two dimensional Fourier-Mellin transform in distributional sense and defined in [9],
* ( )+ ( ) 〈 ( ) ( ) 〉
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1.
Analyticity TheoremIf ( ) 〈 ( ) ( )〉 that is ( )
〈 ( ) ( ) 〉. Then ( ) is analytic for some fixed
and ( ) 〈 ( ) ( )〉 ( )
and ( ) 〈 ( ) ( )〉 ( ) where, ( ) ( ) .
Proof: Let and be an arbitrary but fixed. Choose the real positive number and such that
. Also, let be a complex increment such that . For , we write
( ) ( )
〈 ( ) ( ) 〉 〈 ( )
[ ( ) ]〉 〈 ( )
( ) 〉 〈 ( )
[ ( ) ] ( ) 〉
〈 ( ) ( )〉, (1.3)
where ( )
[ ( ) ] ( )
To prove ( ) , we shall show that as | | , ( ) converges in to zero.
To proceed, let denotes the circle with centre at and radius , where (
). We may interchange differentiation on with differentiation on .
( ) ( )
[( )( ) ( ) ( )( ) ]
( ) ( )
Now applying Cauchy’s integral formula,
( ) ( )
{ ∫
( )( )
∫
( )( )
∫
( )
( ) }
∫ [ ] ( )( )
∫
( )
( )
∫ [
( )( )] ( )( )
∫
( )
( )
∫ [
( )( ) ] ( ) ( )
( )
∫ [( )( ) ] ( )
Now for all and ,
|
( ) ( ) |
where is a constant independent of and . Moreover, | | and | | ,
*| | +, consequently
|
( ) ( ) ( )|
| ( )
( )
( )
∫ [
( )
( )( ) ] |
| ( )
( )
( )
| | | ∫
| || |
| || | | | | |
∫( )( ) | | | |
∫
( )( ) | | | |
∫
( )( ) | | | |
( )( ) | |
( )
The right hand side is independent of and converges to zero as . This shows that
( ) converges to zero.
Now, let and be an arbitrary but fixed. Choose the real positive number and such that
. Also, let be a complex increment such that | | .
For , we write
( ) ( )
〈 ( ) ( ) 〉
〈 ( ) ( )
[ ( ) ]〉 〈 ( )
( ) 〉
〈 ( ) ( )
[ ( ) ] ( ) 〉
〈 ( ) ( )〉, (1.4)
where,
( )
( )
[ ( ) ] ( )
International Journal in Physical and Applied Sciences (Impact Factor- 4.657)
To proceed, let denote the circle with centre at p and radius , where (
). We may interchange differentiation on with differentiation on and by using the Cauchy’s integral formula,
( ) ( ) ( )
[ ( ) ( ) ( ) ]
( ) ( )
where ( ) is polynomial in and ( ) is polynomial in . Now applying Cauchy’s integral formula, we get
( )
* ∫
( )
∫
( ) +
∫
( ) ( )
( )
( )
∫ [( ) ] ( )
∫
( ) ( )
( )
( )
∫ [
( )( )] ( )
∫
( ) ( )
( )
( )
∫ [( ) ( ) ] ( ) ( )
∫ [
( )( ) ] ( )
( )
( ) ( )
∫ [
( )
( )( ) ]
Now for all and ,
|
( ) ( ) ( ) ( )|
where is a constant independent of and . Moreover, | | and | | ,
*| ( ) | +, consequently
|
( ) ( ) ( )|
| ( ) ( )
( ) ( )
∫ [
( )
( )( ) ] |
| ( ) ( )
( ) ( )
| | | ∫
| ( ) |
| || | | |
| |
∫( )( ) | | | |
∫
( )( ) | | | |
∫
| |
( )( ) | |
( )
The right hand side is independent of and converges to zero as | | . This shows that
( ) converges to zero in as | | , which ends the proof.
Conclusion
In the present paper, we have proved analyticity theorem for the distributional two dimensional Fourier-Mellin transform and generalized two dimensional Fourier-Fourier-Mellin transform in distributional sense.
References
[1] Yunlong Sheng and Henri H. Arsenault: Experiments on pattern recognition using invariant Fourier-Mellin descriptors, J. Opt. Soc. Am. A, Vol. 3, No. 6,June 1986.
[2] V.D. Sharma and A.N. Rangari: Analyticity of Generalized Fourier-Finite Mellin Transform, International Journal of Mathematical Education, Volume 4, pp. 7-12, Number 1 (2014).
[3] Abbas Yaseri, Seyed Mahmoud Anisheh: A Novel Paper Currency Recognition using Fourier Mellin Transform, Hidden Markov Model and Support Vector Machine, International Journal of Computer Applications, Volume 61– No.7, January 2013.
[4] Bertrand, J., Bertrand, P., Ovarlez, J. “The Mellin Transform.”The Transforms and Applications Handbook: Second Edition. Ed. Alexander D. Poularikas Boca Raton: CRC Press LLC, 2000.
[5] Dave Collins: The relationship between Fourier and Mellin transforms, with applications to probability, [email protected].
[6] Anupama Gupta: Fourier Transform and Its Applicationin Cell Phones, International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013.
[7] S. M. Khairnar1, R. M. Pise2 and J. N. Salunkhe3: Study of the mellin integral transform with applications in statistics And probability, Archives of Applied Science Research, 4 (3):1294-1310, 2012. [8] A.H. Zemanian: Distribution theory and Transform Analysis, Mcgraw Hill, New york, 1965
[9] V.D. Sharma and P.D. Dolas: Representation theorem for the distributional two dimensional
Fourier-Mellin Transform, Int. Jr. Matthematical Archieve, Vol. 5, issue 9, Sept. 2014.
[10] A.H. Zemanian: Generalized Integral Transform, Inter Science Publisher, New York, 1968.
