Application of wavelets in numerical evaluation of Hankel Transform

1. START
Application of Wavelets
in solving Hankel
Transformk and future
implications

2. A Systematic Study of a Hankel transform wavelet framework and
future implications
)
Dr.Nagma Irfan
Assistant Professor
(2023)
Date of Presentation:19-06-2023
Department of Mathematics
School of Science & Technology,Himgiri Zee University
Dehradun

3. OUTLINE :
INTRODUCTION
MOTIVATION
OBJECTIVES
FUTURE PROSPECT
REFERENCES

4. INTRODUCTION

5. INTRODUCTION
In this paper, a comprehensive analysis attempted, to
study the application of classical wavelets in Hankel
Transform's numerical computation by systematically
reviewing the previous studies on this subject, which
have been published in recent years, and identifying
areas of research for future development. The research
employed the systematic review approach. The results
for Hankel Transform digital calculation using various
classical wavelets have been explored and several
research gaps have been pointed out and possible
implications for this topic are recommended.

6. Wave:
A wave is a mathematical function
used to divide a given function
or continuous time signal into
different scale components
Wavelet
Wavelets are localised wave that have finite
energy. It is a wave like oscillation with an
Amplitude that starts out at zero, increases &
decreases back to zero.
Mother wavelet: They are defined by

7. OVERVIEW
• It may be remarked that Wavelets are mathematical
functions.
• It Cuts up data into different frequency components , and
then study each component with a resolution matched to its
scale.
• It Analyses discontinuities and sharp spikes of the signal.
• It has wide applications in image compression, human
vision, radar, and earthquake studies.

8. Properties of a function to be a wavelet:
•Function must be oscillatory.
•Must decay to zero quickly.
•Admissiblity condition:









 d
c
0
)
(

9. WAVELET FAMILIES

11. CAS WAVELETS
SINE COSINE WAVELETS
HAAR WAVELETS
DAUBCHIES WAVELETS
LEGENDRE WAVELETS
MORLET WAVELETS

12. Wavelets History:
Fourier analysis (1807): classical, but, limited tool for analyzing signals (problems with
spikes, transients).
Denis Gabor(1946): Windowed Fourier transform.
Jean Morlet(1982): developed idea of wavelet transform.
Morlet & Grossmann(1984): constructed “Mother Wavelet" from translations &
dilations of a single function.
Mallat(1986): image processing.
Mallat & Meyer (1987):mathematical structure Multi-Resolution analysis.
Daubechies(1987): orthogonal transform could be rapidly computed on modern digital
computers
Fourier analysis Wavelet analysis

13. MOTIVATION

14. Application of Wavelets:
Unique properties.
Good approximation properties.
Easy to control wavelet properties eg smoothness.
Smooth functions can be represented extremely
efficiently.
Used in signal & Image compression.
In weather forecasting.
Pattern Recognition.
Promising future in computer animated films.
Multiwavelets used to encode multiple signals
traveling through same line.

15. OBJECTIVES
1 •Hankel Transform
2 •Numerical solution
3 •CAS Wavelets
4 •Sine-cosine Wavelets

16. The Hankel transform is a functional transform that includes a Bessel
kernel in one dimension. It is also the radial solution to an angular
Fourier symmetrical transformation of any dimension, which makes
it extremely useful in several application fields. In the fields of
astronomy, geophysics, fluid mechanics, electrical dynamics,
thermodynamics and acoustics, the NASA Astronomical Data
Service produced over 700 papers, including the word ' Hankel
transform.' In this paper, we explore the basic definitions of a few
terms with their properties. Conceptually, in contrast to the Fourier
transformation, the calculation of such problems with the Hankel
transformation has the advantage of reducing the dimensionality of
the problem in unity regardless of the original dimension. This can
be a useful tool in the study to overcome the transformation
analytically. This simply seeks to improve efficiency numerically.

17. Hankel Transform:
Mathematical Background
Definition
The general Hankel transform pair with the kernel
being is defined as
and Hankel transform is self reciprocal.
Inverse Hankel transform is

)
(p
F 

0
)
(
)
( dr
pr
J
r
rf  ,

)
(r
f 

0
)
(
)
( dp
pr
J
p
pF 
 ,

18. The Bessel function of the first kind are
defined as the solution to the Bessel differential
equation

19.  The Hankel transform arises naturally in the
discussion of problems posed in cylindrical
coordinates (with axial symmetry) and hence, as a
result of separation of variables involving Bessel
functions.
 Analytical evaluations are rare and hence
numerical methods become important. The usual
classical methods like Trapezoidal rule, cotes rule
etc. connected with replacing the integrand by
sequence of polynomials have high accuracy if
integrand is smooth. But r and Jv(pr) are rapidly
oscillating functions for large r and p, respectively.

20. When we are dealing with problems that show circular symmetry, Hankel
transforms may be very useful. Laplace’s partial differential equation in
cylindrical coordinates can be transformed into an ordinary differential equation
by using the Hankel transform. Because the Hankel transform is the two-
dimensional Fourier transform of a circularly symmetric function, it plays an
important role in the optical data processing. Hankel transforms have also proven
to be extremely useful in problems associated with geophysics, electro-scattering,
acoustics, hydrodynamics, image processing, seismology etc Kisselev(2018). The
Hankel transform (HT) becomes very useful in the analysis of wave fields where
it is used in the mathematical handling of radiation, diffraction and field
projection. The Hankel transform has seen applications in many areas of science
and engineering. For example, there are applications in propagation of beams and
waves, the generation of diffusion profiles and diffraction patterns, imaging and
tomographic reconstructions, designs of beams, boundary value problems, etc.
The Hankel transform also has a natural relationship to the Fourier transform
since the Hankel transform of zeroth order is a 2D Fourier transform of a
rotationally symmetric function. Furthermore, the Hankel transform also appears
naturally in defining the 2D Fourier transform in polar coordinates and the
spherical Hankel transform also appears in the definition of the 3D Fourier
transform in spherical polar coordinates Natalie(2019).

21. Problem: & are rapidly oscillating function so
its difficult to get solution with high accuracy as integrand is not smooth.
Solution:
 Fast Hankel Transform:By substitution & Scaling. Problem is transformed in the
space of the logarithmic co-ordinates & Fast Fourier transform in that space. . In this
method smoothing of the function in log space is required
 Filon Quadrature Philosophy: Integrand is separated into product of an
(assumed) slowly varying component and a rapidly oscillating component
.Error is appreciable between 0<p<1.

22. Properties of Sine-Cosine wavelets
Wavelets constitute a family of functions constructed from dilation and translation of a single
function )
(t
 called the mother wavelets. When the dilation parameter is 2 and the translation
parameter is 1 we have the following family of discrete wavelets [30]:
),
2
(
2
)
( 2
n
t
t k
k
kn 
 

where kn
 form a wavelet orthonormal basis for )
(
2
R
L .
Sine-cosine wavelets )
,
,
,
(
)
(
, t
m
k
n
t
m
n 
  have four arguments;
,....
2
,
1
,
0
,
1
2
,....,
2
,
1
,
0 

 k
n k
, the values of m are given in Eq. (4) and t is the normalized
time. They are defined on the interval [0, 1) as
,
0
2
1
2
,
)
2
(
2
)
(
2
1
,




 





otherwise
n
t
n
n
t
f
t k
k
k
m
k
m
n

with












,
2
,
,.........
2
,
1
,
)
)
(
2
sin(
,
.,
,.........
2
,
1
,
)
2
cos(
0
,
)
(
2
1
L
L
L
m
t
L
m
L
m
t
m
m
t
fm


It is clear that the set of Sine-cosine wavelets also forms and orthonormal basis for ])
1
,
0
([
2
L .

23. Orthonormal Basis Functions For Sine-Cosine Wavelets
In this section orthonormal basis functions for sine-cosine wavelets are obtained by fixing
1

k and 2

L :
2
1
0
)
8
sin(
2
)
(
)
4
sin(
2
)
(
)
8
cos(
2
)
(
)
4
cos(
2
)
(
2
)
(
4
,
0
3
,
0
2
,
0
1
,
0
0
,
0
















t
t
t
t
t
t
t
t
t
t









1
2
1
))
1
2
(
4
sin(
2
)
(
))
1
2
(
2
sin(
2
)
(
))
1
2
(
4
cos(
2
)
(
))
1
2
(
2
cos(
2
)
(
2
)
(
4
,
1
3
,
1
2
,
1
1
,
1
0
,
1




















t
t
t
t
t
t
t
t
t
t










24. We have projected a method depending on separating the integrand
  )
(pr
J
r
rf  into two components. First integrand is slowly varying components
 
r
rf and the second one is rapidly oscillating component )
(pr
J . Then function
is extended into Sine-Cosine wavelets orthonormal series and is truncated at
optimal level. The solutions obtained by proposed Sine-Cosine wavelet method
applied on different functions considered here that our algorithm is easy to
implement. We have studied a new efficient algorithm based on compactly
supported orthonormal wavelet bases in this chapter. Numerical result showed
graphically indicates that our method is also computationally attractive.

25. RESULTS AND DISCUSSION
Example 1: Sombrero Function(Zero Order)
A veryimportant, and often used function, is the Circ function that can be defined as [
Circ( a
r/ )=





.
,
0
,
,
1
a
r
a
r
The zeroth-order Hankel transform of Circ( a
r/ ) is the Sombrero function [33], given
by
ap
ap
J
a
p
F
)
(
)
( 1
2
0 

26. CAS WAVELET
0.01 6.675 13.34 20.005 26.67 33.335 40
0.2

0.0857

0.0286
0.1429
0.2571
0.3714
0.4857
0.6
Exact
&
Approx
HT
F0 p
( )
F p
( )
p
Fig.1. The exact transform, )
(
0 p
F (solid line) and the approximate transform, )
( p
F
(dotted- line).
0 6.667 13.333 20 26.667 33.333 40
0.02

0
0.02
0.04
0.06
E0 p
( )
E1 p
( )
E2 p
( )
E3 p
( )
p
Fig.2. Comparison of the errors.
Exact Sol.
CAS Method.
Simpson1/3
Composite Simpson 1/3
Simpson 3/8
Composite Simpson 3/8

27. SINE-COSINE WAVELETS
1 10
2

 6.675 10
0
 1.334 10
1
 2.0005 10
1
 2.667 10
1
 3.3335 10
1
 4 10
1

2
 10
1


8.5714
 10
2


2.8571 10
2


1.4286 10
1


2.5714 10
1


3.7143 10
1


4.8571 10
1


6 10
1


Exact
&
Approx
HT
F0 p
( )
F p
( )
p
Fig.3. The exact transform )
(
0 p
F (solid line) and the approximate transform )
( p
F
(dotted line).
0 10
0
 6.667 10
0
 1.333 10
1
 2 10
1
 2.667 10
1
 3.333 10
1
 4 10
1

4
 10
1


2
 10
1


0 10
0

2 10
1


4 10
1


E0 p
( )
E1 p
( )
E2 p
( )
p
Fig4.Comparison of the Errors
Exact Sol.
SCW Method.
Simpson1/3
Simpson 3/8
Composite Simpson 1/3

28. Example 2: (Zero order) Let  
2
/
1
2
)
1
(
)
arccos(
2
)
( r
r
r
r
f 


 , 1
0 
 r ,
then,



 p
p
p
J
p
F 0
,
)
2
/
(
2
)
( 2
2
1
0
A well known result. The pair  
)
(
),
( 0 p
F
r
f arises in optical diffraction theory.
The function )
(r
f is the optical transfer function of an aberration-free optical
system with a circular aperture, and )
(
0 p
F is the corresponding spread function.
Barakat evaluated )
(
0 p
F numerically using Filon quadrature philosophy but the
associated error is appreciable for 1

p ; whereas our method gives almost zero
error in that range.

29. CAS WAVELETS
0 20 40 60 80
0.05

0
0.05
0.1
0.15
0.2
Exaxt
&
Approx
HT
F1 p
( )
F p
( )
p
Fig.5. The exact transform, )
(
0 p
F (solid line) and the approximate transform, )
( p
F
(dotted- line).
0 6.667 13.333 20 26.667 33.333 40
0.02

0.01

0
0.01
0.02
0.03
E0 p
( )
E1 p
( )
E2 p
( )
E3 p
( )
p
Fig.6. Comparison of the Errors
Exact Sol.
CAS Method.

30. SINE COSINE WAVELETS
1 10
2

 6.675 10
0
 1.334 10
1
 2.0005 10
1
 2.667 10
1
 3.3335 10
1
 4 10
1

2
 10
1


8.5714
 10
2


2.8571 10
2


1.4286 10
1


2.5714 10
1


3.7143 10
1


4.8571 10
1


6 10
1


Exact
&
Approx
HT F0 p
( )
F p
( )
p
Fig7. The exact transform )
(
0 p
F (solid line) and the approximate transform )
( p
F
(dotted line).
0 10
0
 6.667 10
0
 1.333 10
1
 2 10
1
 2.667 10
1
 3.333 10
1
 4 10
1

1
 10
2


0 10
0

1 10
2


2 10
2


3 10
2


E0 p
( )
E1 p
( )
E2 p
( )
p
Fig 8.Comparison of the Errors
Exact Sol.
SCW Method.
Simpson1/3
Simpson 3/8
Composite Simpson 1/3

31. Example 3: (First order) Let 2
/
1
2
)
1
(
)
( r
r
f 
 , 1
0 
r , then,










0
,
0
0
,
2
)
2
/
(
)
(
2
1
1
p
p
p
p
J
p
F

.
Barakat et al., evaluated )
(
1 p
F numerically using Filon quadrature philosophy but
again the associated error is appreciable for 1

p ; whereas our method give almost
zero error in that range.

32. CAS WAVELETS
0 20 40 60 80
0.05

0
0.05
0.1
0.15
0.2
Exact
&
Approx
HT
F1 p
( )
F p
( )
p
Fig.9. The exact transform, )
(
1 p
F (solid line) and the approximate transform, )
( p
F
(dotted- line).
0 6.667 13.333 20 26.667 33.333 40
0.02

0.01

0
0.01
0.02
0.03
E0 p
( )
E1 p
( )
E2 p
( )
E3 p
( )
p
Fig.10. Comparison of the Error
Exact Sol.
CAS Method.
Simpson1/3
Composite Simpson 1/3
Simpson 3/8
Composite Simpson 3/8

33. SINE COSINE WAVELETS
1 10
2

 6.675 10
0
 1.334 10
1
 2.0005 10
1
 2.667 10
1
 3.3335 10
1
 4 10
1

2
 10
1


8.5714
 10
2


2.8571 10
2


1.4286 10
1


2.5714 10
1


3.7143 10
1


4.8571 10
1


6 10
1


Exact
&
Approx
HT
F1 p
( )
F p
( )
p
Fig.11.The exact transform )
(
1 p
F (solid line) and the approximate transform )
( p
F
(dotted line).
0 10
0
 6.667 10
0
 1.333 10
1
 2 10
1
 2.667 10
1
 3.333 10
1
 4 10
1

2
 10
1


1
 10
1


0 10
0

1 10
1


2 10
1


3 10
1


4 10
1


E0 p
( )
E1 p
( )
E2 p
( )
p
Fig 12.Comparision of the Errors
Exact Sol.
SCW Method.
Simpson1/3
Simpson 3/8
Composite Simpson 1/3

34. Example 4: (Higher order) In this example, we choose as a test function the
generalized version of the top-hat function, given as
0
)],
(
)
(
[
)
( 


 a
a
r
H
r
H
r
r
f 
and )
(r
H is the step function given by






0
,
0
0
,
1
)
(
r
r
r
H .
Then,
p
p
J
p
F
)
(
)
( 1

 
 .
Guizar-Sicairos, took 1

a and 4

 for numerical calculations. We take 1

a ,
,
0

 and observe that the error is quite small.

35. CAS WAVELETS
0.01 6.675 13.34 20.005 26.67 33.335 40
0.2

0.0857

0.0286
0.1429
0.2571
0.3714
0.4857
0.6
Exact
&
Approx
HT
F p
( )
F p
( )
p
Fig.13. The exact transform, )
( p
F (solid line) and the approximate transform, )
( p
F
(dotted- line).
0 6.667 13.333 20 26.667 33.333 40
0.02

0
0.02
0.04
0.06
E0 p
( )
E1 p
( )
E2 p
( )
E3 p
( )
p
Fig.14. Comparison of the errors.
Exact Sol.
CAS Method.

36. Conclusion
Since the basis functions used to construct the Sine–Cosine
wavelets are orthogonal and have compact support, it makes them
more useful and simple in actual computations. Also, since the
numbers of mother wavelet’s components are restricted to one, so
they do not lead to the growth of complexity of calculations. The
error associated with Filon quadrature philosophy [9] is
appreciable for small p < 1 compared to other algorithm. Choosing
Sine–Cosine wavelets as basis to expand the input signal rf ( r )
makes our algorithm attractive in the applied phys- ical problems
as they eliminate the problems connected with the Gibbs
phenomenon taking place in [31] . Sine–Cosine wavelet method is
very simple and attractive. The implementation of current approach
in analogy to existed methods is more convenient and the
accuracy is high. The numerical example and the compared
results support our claim. The difference between the exact and
approximate solutions for each example plotted graphically to
determine the accuracy of numerical solutions.

38. SCOPE FOR FUTURE WORK
1. Since computational work is fully supportive of compatibility of
proposed algorithm and hence the same may be extended to
other physical problems also. A very high level of accuracy
explicitly reflects the reliability of this scheme for such
problems. We would like to stress that the approximate solution
includes not only time information but also frequency
information due to the localization property of wavelet basis;
with some change we can apply this method with the help of
other wavelet basis.
2. Projected method can be cast into a general class by expansion of
integral by wavelets or Hybrid function Bernoulli polynomials.

39. We have chosen special kind of wavelets for solving Hankel
transform which were not used in earlier algorithms. So, our choice
of wavelets makes them more attractive and applicable in real
world applications. The performance of our method has been
compared with random noise. It is observed after analyzing that
our method has performed better than any other, since our choice
of wavelet eliminates the difficulties related to Gibbs phenomenon
taking place in [1, 2]. Errors related to Filon quadrature approach are
considerable if in comparison to other methods which are
available in literature.

40. Numerical evaluation of Hankel transform
by using Haar Vilenkin wavelets: it may be a
good idea to develop an algorithm based on Haar
Vilenkin wavelets to numerically evaluate Hankel
transform.
Numerical evaluation of HT with
shearlets: One can develop new numerical
technique to solve Hankel transform using Shearlets.

41.  Other available Wavelets can be applied in solving Hankel transform and they
could be compared to see which is better then other.
 Numerical evaluation of Hankel transform by using Haar Vilenkin
wavelets: it may be a good idea to develop an algorithm based on Haar Vilenkin
wavelets to numerically evaluate Hankel transform.
 Numerical evaluation of HT with shearlets: One can develop new numerical
technique to solve Hankel transform using Shearlets.
 In earlier cases Mathcad and Mathematica software is used to find out
computational graphs and results other mathematical softwares could be used
and compared.
 Error analysis part could be extended to show theoretical framework of results.

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