HUYGENS WAVE EQUATIONS IN THE FIELD OF 2D-CWT
Victor Vermehren Valenzuela1, 2, Hélio Magalhães de Oliveira2
1
Univ. Estadual do Amazonas, Manaus-AM, Brazil, 2Univ.de Federal de Pernambuco, Recife-PE, Brazil
ABSTRACT
hz ( x , y )
In this paper it is shown the performing of an optical
transform to state the scalar diffraction in the
formulation of the wavelet transform and the wave
equations. From there, a bridge is build between
equations of spherical waves presented in 1678 by
Huygens and the continuous wavelet transform. For
such a purpose, wavelets are introduced that meet the
principles of waves and the properties of wavelets. The
following equations are applied in solution to show a
correspondence
between
the
Huygens-Fresnel
diffraction and the wavelet transform.
Index Terms— Diffraction, 2D-CWT, Fresnel,
wavelets, chirp.
e
U
( x, y )
z
j z
U (x , y )
0
0
e
j [( x x0 ) ( y y0 ) ]
z
2
j z ( x y )
2
2
.
(3)
hz ( x , y )
e j(x y
2
)
, yac ,
,
(4)
thus deriving from (5) the following family of wavelets
ha bc ( x , y )
where a =
K
a
e
x b
a
.
z / , b =, c = , and the constant Ka is
Ka
e
j 2 z
j z
a 4
j 22
a
2
(5)
(6)
Note that the propagation distance z can be interpreted
as the wavelet scale factor.
Everything would be solved in the field of transform if
the optical wavelet, as defined in the Equation (4), satisfies
the conditions of wavelets, including the admissibility.
Although the initial field equations E (x0, y0, 0) establishes
recovery via the diffracted field E (x, y, z) by the inverse
optical propagation, integrals do not converge to a solution
even in FFT and IFFT computations, since they produce
distortions in the framework of circular shapes [12].
Another challenge is that these wavelets are not
located, neither in the space-time domain, nor in the
frequency domain. Later, Onural himself called them chirp
scaling functions instead of wavelets in the field of
Fractional Fourier Transform [13].
2
dx dy
0
U ( x 0 , y 0 ) hz x , y ,
0
(1)
The Equation (1) can be rewritten as
U ( x, y )
z
j z e
2
Wavelet and its Continuous and Discrete transformed
(DWT and CWT, respectively) have emerged as a
definitive tool in signal processing [1] and [2]. This is
mainly due to the fact it provides an accurate local and
global information identification of signals [3]. Another
application experiencing great attention is the use of
wavelet analysis techniques in optical processing [4-5],
both treating images and diffraction, which extend wavelet
analysis for two (2D) and three (3D) dimensions. However,
an understanding of the relationship between the very
essence of optics and the concept of wavelet, has been
attempted in recent decades, through the implementation of
optical devices or by representations in own domains [610]. Nevertheless, it had not yet been established with
property the wave propagation carries inherent
characteristics of the two-dimensional wavelet transform.
The scalar theory of diffraction involves the
translation of the wave equation, which is a partial
differential equation, in an integral equation. This can be
used to analyze most imaging systems and phenomena
within its area of validity [11]. The diffraction pattern of a
2-D object, U(x, y), at a distance z according to the theory
of Fresnel diffraction region (approximate) can be written
as
j 2 z
Equation (2) indicates that the Fresnel diffraction,
which is based on the theory of waves of Huygens, exhibits
characteristics of modern mathematical wavelet signals,
namely a set of different diffraction patterns produces a set
of "wavelet transformed images", from the same 2-D
object.
The optical waves defined in Equation (3) are in fact
the monochromatic spherical waves of Huygens. Under the
Fresnel approximation, the Huygens-Fresnel equation can
be interpreted as a wavelet transform. According Onural
[6] it is straightforward to see from Equations (2) and (3)
that the core of the diffraction convolution equation has
indeed the properties shifting and scaling associated with a
family of wavelet itself. This is shown by defining such a
wavelet as follows:
1. INTRODUCTION
e
j 2 z
(2)
where denotes a 2-D convolution, is the wavelength
and the impulse response is expressed by
2. THE PROPOSED WAVELET
It can be observed that the convolution in Equation (2)
must be performed through a family of wavelet function for
a continuous 2-D space that satisfies all conditions of
waves. Classically, the wavelet functions are obtained from
a function, indexed by two coefficients [1]:
a , s ( x)
1
s
,
x a
s
(7)
where a is a shift parameter, s is a scale parameter, and s1/2
is a normalization factor. The spread to higher dimensions
consequently may be obtained and shown as follow,
a b s ( x , y ) 1s
x a
s
, y s b .
(8)
occurs naturally through Fourier transform pair of a
complex exponential signal quadratic, which dual is a
complex Gaussian pulse too [15].
The graphs of chirplet optical wavelet and its
corresponding FT are shown in Figure 1. Note the pulse
characteristic of (w) and the decreasing to zero value
when it tends to the coordinates (0,0) and , as expected.
From the point of view of wave equations and CWT,
there had to be a family of wavelets that perform the same
location or diffraction properties at the CWT. How to make
a chirp function that tends to infinity become "behaved”
(i.e., meet the requirements or properties of wavelets?)
The solution partly comes from a recent family of
wavelets, the chirplets [14]. In part because although they
have become bandwith limited functions due to Gaussian
window applied, these wavelets are not even functions,
since they have a term shift in exponential and increases
with frequency and one end of chirp rate. They wavelets
are building to specific image compression applications.
Then, taking into account the Equations (3-8)
introduced in this paper, a new family of 2D wavelet called
chirplet optics are presented below
( x, y )
1
2
e
( x2 y 2 )
4 2
e
j z ( x 2 y 2 )
,
(9)
where as usual z is the propagation factor and is the
wavelength. Note that the value of 2 should be large
enough, in order to take into account a maximum number
of side lobes involved in the kernel. Furthermore, this
wavelet becomes an even function as its analogous at (3)
all-encompassing with the same representation of families
of wavelets (5) and the continuous phase (6), limited
bandwidth and holding the properties listed below.
The function (x,y) meets the requirements of
wavelets of unity energy,
| ( x, y ) |2 dxdy 1 ,
( x, y )dxdy 0 ,
( x, y ) x n y n dxdy 0 ,
( uw0 cos u ) 2( vw0 sin v ) 2
4( j )
Figure 1. Graphs of the chirplet optical wavelet. a) 3-D wavelet
representation. b) 3-D representation of Fourier transforma of the
chirplet optical wavelet.
3. THE 2D-CWT
(11)
CWT ( a , s )
(12)
with n = 1, 2, 3.... (computational verification up to 35).
The Fourier transform (FT) of the optical chirplet
is expressed as follow:
2D
The continuous wavelet transform of a 1D signal f(x)
is defined as [16],
and the nice property of vanishing moments
b)
(10)
the requirement of zero mean
a)
(u , v ) j e
,
(13)
2
where = 1/4 , = /z and w0 = 2/. The optical
chirplet function is defined as a cosenoidal complex chirp,
whose amplitude is modulated by a Gaussian function and
their frequency has quadratic sweep. Based on this model,
the calculation of expression in the frequency domain
1
s
f ( x ) a , s ( x ) dx .
*
(14)
Note that Equation (14) takes the form of correlation
between the input signal f(x) and dilated and shifted mother
wavelet as(x), which is rewritten in the following
convolution form [16]:
CWT ( a , s )
( f s )( a )
s ( a x ) f ( x ) dx.
(15)
Implementing the 2D-CWT, a pattern g(x,y) is
projected onto the wavelet a,b,s, by the translation of a and
b, in the x and y axes respectively, and scaling by s, the
mother wavelet (x, y), as shown below.
s1 g ( x, y ) abs
( x , y ) dxdy
CWT ( a , b, s )
or
CWT ( a , b, s ) ( g s )( a, b).
(16)
The wavelet transform, as in Equation (15) is therefore
the convolution of the signal with a wavelet function. Thus,
one can employ the convolution theorem to express the
wavelet transform in terms of products of the Fourier
transform of signal, X(), and the wavelet, a,b(), as
shown in [17]
CWT (a, s )
*
X ( ) a ,s ( ) d ,
1
s
( t sa )e
j t
dt .
U ( x,
1
s
(t ') e
j ( st ' a )
y, z )
b.
U ( x,
(19)
U ( x,
y, z )
se
j a
(t ) e
j ( s )t
dt .
(20)
The integral expression in the previous equation is just
the FT of the wavelet at rescaled frequency a.
Consequently one can rewrite Equation (19) as:
a ,s ( )
a ,s ( )
s ( s ) e
j a
.
(21)
z0
s ( s )e
*
j a
.
CWTFT (a, b, s ) s
X (u, v) (au, bv) e
*
j ( au bv )
dudv (23)
4. THE 2D-ICWT
The diffraction involves reverse recovery image of an
object to which the diffraction pattern was measured, for
example, on a plane. In the case of the Fresnel and the
Fraunhofer approximation inverting equations are obtained
directly, only by inverting complexes operators [11]. In the
case of very close fields, i.e., the one that approximates the
illuminated object, it can be used angular spectrum
representation techniques, but with the fulfillment of
certain conditions. In both cases, the mathematical
complexity is rather enormous, since it involves double
integrals, solution typically are not easily convergent, as
can be seen in the sequel.
U (x , y
0
0
j 2 z0
z0
e
, z0 ) e
0
2
2
e
0
(24)
0
0
j z ( x02 y02 )
dx0 dy0
j z ( x 2 y 2 )
j z ( x0 y0 )
j 2z ( x0 x y0 y )
0
U (x , y
0
dx0 dy 0
0
, z0 )
(25)
j 2 ( f x f y )
U ( x0 , y 0 , z 0 ) e
dx0 dy 0
e
x 0
jz0
4 2 4 2 ( fx 2 fy 2 )
2
e
j 2 ( f x x f y y )
y 0
df x df y
(26)
where z is the distance between the observation plain and
the generation plain.
In contrast to the previous equations, there exists now
the 2D-ICWT to reconstruct the input pattern through their
wavelet chirplet optics decompositions:
ICWT ( x, y )
1
sC
2
CWT ( a, b, s )
yb
K s xs a , s da db
(22)
Equation (17) can thus be rewritten in expanded
bidimensional form, yielding:
j 2z ( x0 x y0 y )
The FT of the wavelet function conjugate is merely
*
Angular spectrum
Separating out the constant part of the exponential
function and omitting the prime from the variable t’, one
gets
a ,s ( )
je
e
c.
j 2 z0
Fresnel
y, z )
(18)
s dt ' .
je
e
Making the variable substitution t’= (t – a)/s, and so dt
= sdt’, it is derived
a ,s ( )
Fraunhofer
(17)
where the conjugate of the wavelet function is used, and
=2f, whose the Fourier transform of the dilated and
translated wavelet is
a ,s ( )
a.
(27)
where s= z / denotes the observation scale parameter
and C is constant equal to (some sort of 2D admissibility
condition).
||(uu v,v| )| du dv .
C (u , v )
2
(28)
Whereas C must be a finite number so as to obtain the
inverse wavelet transform, the wavelet chirplet optical
meets this condition, as seen in Figure 1, and
(0,0) = 0.
(29)
The wavelet transform, as in Equation (27), is the
convolution of the CWT with the wavelet function. Thus,
one can also adopt the same procedure of Section 3 to
express the 2D CWT products in terms of the Fourier
transform of the CWT, X(), and wavelet, a,b(), as
shown [12]:
ICWT ( x, y )
CWT
1
sC
FT
(u , v ) ( su , sv ) e
j ( au bv )
dudv (30)
This is a result particularly useful in the same way as
(23), since it assists the calculation of the inverse
continuous wavelet transform into higher complex
mathematical functions.
5. SIMULATION AND EXPERIMENTS
In this section some MathCadTM simulation are
presented to corroborate the validity of the expressions of
chirplet optical wavelet in the 2D-CWT, both under
forward and reverse forms. Figure 2 shows the simulation
the approximated Fresnel diffraction equation and optical
chirplet 2D-CWT applied to a test object with a rectangular
aperture 6 mm 2 with unit amplitude, being illuminated
by a uniform white light source.
a)
b)
6.
ON POTENTIAL APPLICATIONS OF
THE NEW APPROACH
Despite the fact of knowing the foundations of the
diffraction theory, well-established by Huygens-Fresnel for
over 300 years, only in recent decades a renewed impulse
appears with the introduction of its application in the fields
of holography [19], tomography [20], ultrasound [21], and
even in image compression [22].
These applications, especially in the biomedical area,
present serious technological challenges and high-cost
implications in the physical implementation of optical
devices, as an attempt to overcome various barriers that
cause errors and interferences.
Under this new formulation of the wavelet transform,
with their characteristics of multiresolution and location in
time or space-frequency, and while the inverse transform is
a filter (13), now, e.g., can be applied to the 3D image
retrieval in digital holographic microscopy [22]. In this
application while eliminating the effect of the zero-order
images, it is also possible to mitigate the blurry and twin
image without spatial filters. The image of order-zero (DC)
and the twin images are a pair of symmetrical terms in the
zero-order spectrum. After that it is possible to adjust the
filter in order to position the bandpass in range the required
image (+1 order) as shown in Figure 3.
c)
Figura 3. Filtragem com ICWT Chirplet Óptica. a) Representação
das imagens no espectro. b) A transformada de Fourier da Wavelet
Chirplet Óptica ajustada para filtragem da imagem ordem+1.
7.
d)
e)
Figure 2. a) Object of the test at the generator plan. (b) Illustrates
the application of the Fresnel diffraction equation at a distance
z=54000mm and =550nm (c) Illustrates the amplitude pattern
application of the chirplet 2D-CWT direct, at the same conditions
as (b). (d) Shows the result of applying chirplet optical 2D-CWT
reverse in the reconstruction of the object pattern generator and,
(d) the mask that generates the amplitude pattern.
It is noteworthy that the diffraction reverse only the
integrals of 2D-CWT converged to a solution (even the
angular spectrum equations do not achieved any valuable
result).
CONCLUDING REMARKS
It was successfully performed the wavelet optical
chirplet, unprecedented for the 2D-analysis of waves,
which builds a bridge between the equations of spherical
waves introduced in 1678 by Christian Huygens and the
continuous wavelet transform. Wide-ranging wavelets
introduced, which meet the principles of waves and the
properties of wavelets with the notorious fact of its reverse
integral converge to a solution and its filter can be
positioned in required range of application.
This development may product in significant reduction
of costs, because only mathematical operations are
performed in their implementation. An overview of
preliminary experimental results illustrates this fact. At the
same time the effect of zero-order term are eliminated, as
well as the term obfuscation edge image and twin image
without spatial filters holographic microscopy. A major
advantage of the introduced technique is that it can be
applied in the analysis of biological specimen’s live
dynamic images from digital holographic microscopy [23]
more precisely. This analysis also can particularly be
applied to microbial structures such as cancer cell, stem cell
and botanical specimens.
The analysis of holograms to extract information
related to objects can find a useful theoretical framework as
a result of the relationship diffraction-wavelet presented. In
addition, the noise filtering in holographic images is an
extra and vast field of analysis.
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Huygens Wave Equations in the Field of 2D-CWT
arXiv:1503.05820 [math.AP], 2015
In this paper it is shown the performing of an optical transform to state the scalar diffraction in the formulation of the wavelet transform and the 'wave equations'. From there, a bridge is build between equations of spherical waves presented in 1678 by Huygens and the continuous wavelet transform. For such a purpose, wavelets are introduced that meet the principles of waves and the properties of wavelets. The following equations are applied in solution to show a correspondence between the Huygens-Fresnel diffraction and the wavelet transform. ...Read more
