Optics and Photonics Journal, 2013, 3, 250-258
http://dx.doi.org/10.4236/opj.2013.33040 Published Online July 2013 (http://www.scirp.org/journal/opj)
Recognition of Direction of New Apertures from the
Elongated Speckle Images: Simulation
Abdallah Mohamed Hamed
Physics Department, Faculty of Science, Ain Shams University, Cairo, Egypt
Email: Amhamed73@hotmail.com
Received January 31, 2013; revised March 2, 2013; accepted March 9, 2013
Copyright © 2013 Abdallah Mohamed Hamed. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
In this paper, we present an elongated speckle images produced from diffusers using sharp elliptical apertures. The ori-
entation of the elliptic aperture is recognized from the direction of the elongation in the speckle images. The aperture
tilting out of the plane is investigated. Three models of elliptical apertures are considered and the corresponding speckle
images are obtained. The 1st model is composed of two orthogonal ellipses or plus symbol pupil; the 2nd has four sym-
metric ellipses with an angle of 45˚ between each of them or in the form of a snow flake pupil and the 3rd model looks
like an airplane. Also, the autocorrelation profiles of the speckle images corresponding to the diffused airplane are ob-
tained from which the average speckle size is computed. Finally, the reconstructed images of the described elliptical
models and its autocorrelation images, making use of Mat lab code, are obtained.
Keywords: Digital Imaging; Sharp Elliptical Apertures; Plus Symbol and Snow Flake Pupils
1. Introduction
The production of elongated speckle images was ob-
tained using the mechanical scanning of the static spec-
kle pattern [1]. The author presented a technique of spa-
tially oriented speckle-screen encoding to improve the
grating encoding technique for white-light image proc-
essing. Also, an artificial screen composed of small strips
photographed several times on a high resolution film is
designed to obtain elongation of nearly ten times the av-
erage grain size of natural speckle [2]. In a recent publi-
cation by the author [3], numerical elliptical apertures of
small elliptic shapes are analyzed and the Fourier trans-
form is operated giving the speckle images of diffusers
modulated by these elliptic apertures.
An approach for determining the roughness of engi-
neering surfaces is resulted from the speckle elongation
effect. The laser speckle pattern, arising from light scat-
tered from rough surfaces that are illuminated by poly-
chromatic laser light, is detected in the far-field region.
The incoherent superposition of these light intensities
and the angular dispersion cause the effect of speckle
elongation [4,5]. This is characterized by increasing
speckle widths and leads to a radial structure of the spec-
kle patterns. With increasing surface roughness, the elon-
gation is replaced more and more by the de-correlation of
the monochromatic speckle patterns for the different
wavelengths. Such effects were detected with the CCD
technique and analyzed by local autocorrelation func-
tions of intensity fluctuations that were calculated for
different areas of the speckle patterns. Hence, the auto-
correlation method is applied to process laser speckle
patterns. The relation between surface roughness, speckle
elongation, and correlation length of autocorrelation
function can be obtained. Consequently, the measured
surface roughness can be achieved [6]. An oriented pho-
tographic diffuser was used to record an elongated spec-
kle pattern. It is found that the contrast transfer, when
gratings are imaged through the slits in the diffuser, is
considerably higher compared to imaging through a cir-
cular pinhole of comparable dimensions [7]. An autocor-
relation algorithm for speckle size evaluation has been
investigated [8-11]. The authors measured the average
speckle size from the auto-covariance function of the
digitized intensity speckle pattern. The spatial character-
istics such as speckle size can be used to measure the
roughness of surfaces [12,13]. An important remark must
be taken into consideration during the recording of speckle
data which states that the speckle size must be greater
than the pixel size of the CCD camera in order to resolve
variations in speckle intensity [14]. Recently, a Fast cal-
culation method for optical diffraction on tilted plane
was investigated [15].
C
opyright © 2013 SciRes. OPJ
A. M. HAMED 251
In this work, two sharp elliptical apertures with dif-
ferent orientations are proposed and the corresponding
elongated speckle images are obtained. Also, three dif-
ferent new models in the form of the plus symbol, a
snowflake and an airplane, are numerically simulated and
the corresponding speckle images are computed and
plotted. Finally, the reconstructed images of the models
and the autocorrelation of these models are plotted. It is
noted that the sharp ellipses considered in this study have
a semi-major axis ten times larger than the semi-minor
axis as compared with the recent work [3] with the semi-
major axis nearly equal to the semi-minor axis. These
new models allow the recognition of the different elliptic
models from the speckle images as compared with the
circular aperture.
2. Formation of Speckle Images Using
Diffusers Modulated by Different
Sharp Elliptic Apertures
In this work, a simple ellipse of semi-major axis ten
times the semi-minor axis is investigated. We assume
that this sharp ellipse is represented as follows:

22
22
,1; 1with0.1
xy
f
xyb a
ab
  (1)
The numerical ellipse is made from pixels of dimen-
sions 1024 1024 and is represented as follows:

,1; ,1024,1024fmxny mn 
(2)
x and y are unity.
A diffuser of random phase changes, has the same di-
mensions like the above described elliptical aperture, is
written as follows:
 
2
,exp and,;
k
dxyjr xyk
f






(3)
In matrix form, the simulated diffuser becomes:
 
,exp and,
k
d m xn yjrm xn y
f

 




,
(3)
Consider coherent imaging system, with laser uniform
illumination incident upon the object (the diffuser) fol-
lowed by the simulated sharp elliptic aperture. In this
case, we can write the complex amplitude transmittance
as follows:

,,cmxnyf mxnydmxny  (4)
Hence, the Fourier transform of Equation (4), realized
by a converging lens, gives in its front focal plane this
complex amplitude:
,
mn mn
cf d
where

.. ,
f
FTfmx ny


..d ,T mxny
,
dF
, and .

.,cFTcmxny

The symbol
stands for the convolution product.
Equation (5) means that the modulated elongated
speckles are formed from the convolution product of
both of the Fourier spectrum of the diffuser and the sharp
elongated aperture. The direction of speckle is elongated
normal to the semi-major axis of the elliptic aperture.
2.1. Effect of the Aperture Tilting upon the
Elongated Speckles
Assume that the aperture is tilted making an angle
with the aperture plane (x, y) which is not a simple trans-
lation. In this case, the sharp elliptic aperture is decom-
posed into two components one along the optic axis z and
the other lies in the plane (x, y) assumed along y-direc-
tion written as follows:
 
exp cos,
tilted
f
jkyfx y


(6)
The concept of aperture tilting which is represented as
a linear phase shift in the Fourier plane resembles the
inclined plane wave which gives a linear shift in its front
focal plane. Hence, the coherent light emitted from laser
beam is incident upon the tilted aperture described by
Equation (6) and the transmitted light is incident upon
the diffuser described in the precedent section Equation
(3) giving this complex amplitude:
 
,,
tilted tilted
cmxnyfmxny dmxny,
   (7)
Substitute from Equation (6) in Equation (7), we re-
write Equation (7) as follows:

,
exp cos,,
tilted
cmxny
jkmyf mxnydmxny




(8)
Applying the Fourier transform operation and making
use of convolution operation we get:

..
cos
,, ,
ctilt apert
mn mnmnf
fd
xyxyx y




 

(9)
represents the Dirac-Delta distribution.
This Delta function displaces the whole speckle pat-
tern by an amount equal to
cosf
in the direction
conjugate to the y direction since f is the focal length of
the Fourier transform lens. Hence, for the tilted aperture
the complex amplitude of the speckle pattern is written as
follows:

cos
.. ,
mn f
ctilt apertcxy





(10)
,
x
yx
 



 

y
(5)
Copyright © 2013 SciRes. OPJ
A. M. HAMED
252
It is noted that the linear shift of the whole speckle
images is large as compared with the field of view of the
whole image. To detect the shift this condition must be
fulfilled: fcos(
< 3
;
is the average speckle size.
2.2. Autocorrelation Algorithm for Speckle Size
Evaluation
The average speckle size of a speckle image is estimated
by calculating the auto-covariance function of the digi-
tized intensity speckle pattern as follows:
If I(x1, y1) and I(x2, y2) represent the intensities of two
points in the imaging plane (x, y), the intensity autocor-
relation function is defined by equation

112 2
,,,
I
cxy IxyIxy

(11)
where 12
x
xx
, 12
yy y
and corresponds
to a spatial average.
The auto-covariance function corresponds to the nor-
malized autocorrelation function of the intensity which
has zero base and its full width at half maximum (FWHM)
provides a reasonable measurement of the average width
of a speckle [6]. In order to use autocorrelation function
method to calculate the average speckle size, it requires
sufficient sampling speckles in an image to give a rea-
sonable statistical evaluation.
A Matlab program is used [8] to compute the auto-
covariance of the speckle image. The calculated auto-
covariance functions are shown and the FWHM of the
calculated function gives the average speckle size of the
speckle pattern.
2.3. The Reconstruction Process and the
Autocorrelation of Elliptic Apertures
Since the complex amplitude of modulated speckle im-
ages using the above different apertures is given by Equ-
ation (5) as follows:


,, ,
,,
mn mn
cuv fd
x
yx
fuv duv
 




 


y
(5)

,, mn
x
y
uv

are the reduced coordinates in the
Fourier plane of speckles.
The reconstruction of the different apertures obtained
by operating the inverse Fourier transform upon Equation
(5) to get:
 
1
,.., ,Rx yFTfuvduv






(12)
,
x
y

are the Cartesian coordinates in the imaging
reconstruction plane.
Making use of the properties of Fourier transform and
convolution, then we get finally this result:

,, Rx yfx ydx y
,


(13)
Hence, in the imaging plane we localize the aperture
image affected by a noise originated from the diffuser
function.
The autocorrelation function of the different apertures
is obtained by operating the Fourier transform upon the
intensity of the speckle image as follows.
Firstly we get the speckle intensity as the modulus
square of the complex amplitude of the speckle Equation
(5') as follows:
 
2
,,
I
uvc uv (14)
Operating the F.T1 over Equation (14), we get the au-
tocorrelation function of the multiplication product as
follows:

 
1
2
1
,..,
.. ,,
, ,,,
cx yFTIuv
FTfu vdu v
f
xydxyfxydxy










 
 



(15
3. Results and Discussion
nges used in the forma-
diffuser is multiplied by the sharp elliptical aper-
tu
)
A diffuser of randomly phase cha
tion of all modulated speckle images is shown in Figure
1.
The
re of semi-major axis ten times the semi-minor axis as
shown in the left of Figure 2. A matrix of dimensions
1024 × 1024 pixels for this diffused aperture is consid-
ered. The elongated speckle pattern is obtained by oper-
ating the FFT algorithm upon the diffused aperture as
100 200 300400 500600 700 800900 1000
100
200
300
400
500
600
700
800
900
1000
Figure 1. A random diffuser d(x,y) = exp[(j 2/f) rand(x
with dimensions 1024 × 1024 pixels.
,y)]
Copyright © 2013 SciRes. OPJ
A. M. HAMED 253
Figure 2. On the left, an elliptical aperture of a semi-major
axis a = 2 and semi-minor axis b = 0.2 with matrix dimen-
shown that the
longation of speckles is normal to the major axis of the
angle
= 45˚ with the x-axis and the
co
α is the angle between the inci-
de
sions of 1024 × 1024 pixels is considered. On the right is the
corresponding speckle image elongated along the y-direc-
tion of matrix dimensions 256 × 256 pixels. The elliptical
aperture is superimposed over the diffuser and its semi-
major axis is extended along x-direction.
shown on the right of Figure 2. It is
e
ellipse shown along x-direction giving the elongation
directed along the y-axis. In my opinion, this can be in-
terpreted if we consider that the elongated elliptical ap-
erture looks like a slit of finite width along y-direction
and hence its Fraunhoffer diffraction pattern located in
the Fourier plane is represented by approximate sinc(y)
function varies along y. Consequently, the convolution
product of the speckle pattern formed from the ordinary
diffuser and the diffraction pattern of the elongated el-
lipse will give the elongated speckle distribution shown
along y-direction as in Figure 2. This reasoning is estab-
lished from the consideration of sharp elliptic aperture of
minor axis equal to the width of the slit along y-direction
while the major axis equal to the length of the slit along
x-direction. Based, on this analogy all the proposed mo-
dels are justified.
Figure 3 shows the diffused sharp ellipse with its ma-
jor axis making an
rresponding elongated speckle with the elongation
orthogonal to the pupil major axis. It is shown that the
direction of the ellipse is recognized from the direction of
the elongated speckle where both of them must be or-
thogonal to each other.
The effect of aperture misalignment is shown in Fig-
ure 4. The tilting angle
nt ray and the normal to the aperture plane. This angle
is taken to be 3˚ and 4˚ as shown in figures (b), (c) and
compared with the case of perfectly aligned aperture
shown in Figure 4(a). This tilting is completely different
as compared with aperture orientation in its plane (results
Figure 3. On the left, an elliptical aperture making an angle
0f 45˚ with the x-axis and of matrix dimensions of 1024 ×
1024 pixels is shown. On the right is the corresponding
speckle image elongated normal to the semi-major axis of
matrix dimensions 256 × 256 pixels. The elliptical aperture
is superimposed over the diffuser.
50100150200250
50
100
150
200
250
50100 150 200 250
50
100
150
200
250
(a) (b)
50 100 150 200 250
50
100
150
200
250
(c)
Figure 4. Three different elongated speckle imes using the
sharp elliptic aperture show Figure 2. (a) On the left is
e three elon-
ated speckle images taking the aligned aperture as a
ag
n in
the elongated speckle for aligned sharp elliptic aperture.
While (b) and (c) are obtained for misaligned aperture tak-
ing the tilting angle α = 3˚ and 4˚ respectively.
shown in Figures 1 and 2). If we compare th
g
reference hence we can predict any misalignment due to
the aperture tilting by examining the speckle pattern. Any
larger inclinations will give different speckle patterns
since the inclination in the pupil plane is transformed into
Copyright © 2013 SciRes. OPJ
A. M. HAMED
254
linear shift in the Fourier plane. Hence, larger aperture
tilting give a difference as compared with the aligned
aperture but small tilting is interesting to check the
alignment of the optical system.
Also, larger aperture tilting give a difference as com-
pared with the aligned aperture but small tilting is inter-
es
ft of
Fi
dimensions 2048
×
orresponding
to
ting to check the alignment of the optical system.
The plus symbol pupil is composed of two orthogonal
ellipses modulated by the diffuser is shown in the le
gure 5 for a matrix of 2048 × 2048 pixels. On the right
is the corresponding speckle image. The elongation of
the speckle image is shown directed along both of the x-
and y-axes. Also, another arrangement of a snow flake or
four ellipses superimposed over the diffuser and the cor-
responding elongated speckle image is plotted as in Fig-
ure 6. This arrangement shows a complicated elonga-
tion since each ellipse gives speckle elongation orthogo-
nal to its elliptical major axis. Hence, the recognition of
the snow flake pupil is attributed to the four different
directions shown for the elongation of the same speckle.
All the elongated speckle images shown in Figures 2-
6 are of dimensions 256 × 256 pixels.
The 3rd model of a pupil in the form of an airplane su-
perimposed over the diffuser of matrix
2048 pixels is numerically constructed as shown in
Figure 7. The corresponding speckle image is obtained
by operating the FFT showed a specific elongation along
the plane (x, y) represented as in Figure 8.
The profile shape of the autocorrelation intensity is
shown as in Figure 9(a), along the x-axis c
the elongated speckles Figure (8) for the airplane pu-
pilof 2048 2048 pixels. Also, the autocorrelation pro-
file along the y-axisis shown in Figure 9(b). The spec-
Figure 5. On the left, a snow flake pupil of two orthogona
ellipses of matrix dimensions 2048 × 2048 pixels is shown
l
.
On the right is the corresponding speckle image elongated
along the x- and y-directions and the matrix dimensions are
256 × 256 pixels. The elliptical aperture is superimposed
over the diffuser.
Figure 6. On the left, a snow flake pupil superimposed ove
the diffuser of matrix dimension 2048 × 2048 pixels i
r
s
shown. On the right is the corresponding speckle image
elongated along the different directions of matrix dimen-
sions 256 × 256 pixels.
Figure 7. A pupil in the form of an airplane superimpose
over the diffuser of matrix dimensions 2048 × 2048 pixels.
d
Figure 8. Speckle elongation in different directions of di-
mensions 256 × 256 pixels corresponding to the airplane
diffused pupil of dimensions 2048 × 2048 pixels.
Copyright © 2013 SciRes. OPJ
A. M. HAMED 255
(a)
(b)
Figure 9. (a) Autocorrelationtensity along the x – axis of
the elongated speckles shown Figure (8) for the airplane
is computed by taking the full width at half
in
in
pupil of 2048 × 2048 pixels. The average speckle size is x =
(4.5 mm/512 pixels) (5 pixels) = 44 mm; (b) Autocorrelation
intensity along the y-axis of the elongated speckles shown in
Figure (8) for the airplane pupil of 2048 × 2048 pixels. The
average speckle size is y = (3.6 mm/512 pixels) (8 pixels) =
56 mm.
kle size
maximum along x-direction, referring to Figure 9(a) as
follows: FWHM = x = 5 pixels and along the y-direc-
tion, referring to Figure 9(b) as: y = 8 pixels. Hence the
average speckle size along x-direction is calculated, as-
suming field of view in the speckle image equal 4.5 mm
3.6 mm, as follows:

4.5 mm512 pixels5 pixels44mm
x

while the average speckle size along the y-direction is
computed as:

3.6 mm512 pixels8 pixels56 mm
y

For comparison, the average speckle size for circular
apert
Fi
ure of radius 128 pixels is obtained (referring to
gures 10-12 as follows):
Figure 10. uniform circular aperture of pixels dimensions
2048 × 2048 and radius 128 pixels.
Figure 11. The field of view is 4.5 mm × 3.6 mm correspond-
ing to 512 × 512 pixels for the speckle image, and is- ob
tained for the diffuser provided with uniform circular ap-
erture shown in Figure 10.
 
ixels10 pixels44m 4.5 mm2048 p
x
 
3.6 mm2048 pixels10 pixels35m
y
Another set of three different speckle images using a
circular aperture are plotted as in Figure (13). On th
is
e left
the speckle for the aligned circular aperture. While the
other two figures (b), (c) are obtained for misaligned cir-
cular apertures, taking the tilting angles α =3˚ and 4˚ re-
spectively. All apertures have equal radii of 64 pixels.
The autocorrelation intensity of the corresponding spec-
kle patterns are shown as in Figures 14(a)-(c). It is shown
that the aligned aperture is recognized from its speckle
pattern Figure 14(a) since it is different from the tilted
aperture Figures 14(b) and (c). Hence, any misalignment
due to aperture tilting is recognized referring to its dif-
ferent speckle pattern.
Copyright © 2013 SciRes. OPJ
A. M. HAMED
Copyright © 2013 SciRes. OPJ
256
(a) (b)
Figure 12. (a) Autocorrelation intensity of the speckle imageown in Figure 10 along the x-axis. The FWM of the speckle
size corresponding to the 10 pixels sh4.5 mm/1024 pixels)(10 pixels) = 44
shH
own on the autocorrelation peak is equal to: s = (
mm;(b) Autocorrelation intensity of the speckle image shown in Figure 9 along y-axis. The FWHM of the speckle size corre-
sponding to the 10 pixels shown on the autocorrelation peak is equal to: s = (3.6 mm/1024 pixels)(10 pixels) = 35 mm.
20 40 60 80100120
20
40
60
80
100
120
20 406080100120
20
40
60
80
100
120
20 406080100120
20
40
60
80
100
120
(a) (b) (c)
Figure 13. Three different speckle images using a circular aperture. On the left is the specklee aligned circular aperture.
While the other two figures (b), (c) ae tilting angles α = 3˚ and 4˚ re-
for th
re obtained for misaligned circular apertures, taking th
spectively. All apertures have equal radii of 64 pixels.
050 100 150 200250 300
-0.01
0
0. 01
0. 02
0. 03
0. 04
0. 05
0. 06
0. 07
050 100150 200 250300
-0.0 1
0
0. 01
0. 02
0. 03
0. 04
0. 05
0. 06
0. 07
pixel s al ong x-axi s
autocorrel atio n i nt ensit y for al i gned ci rcul ar apert
pix els al ong x-ax i s
aut ocorrel a t i on i nten s i ty for m i sal i g ned aper t ure al ph a= 3
(a) (b)
ure
050100 150 200250 300
-0.01
0
0.01
0.02
0.03
0.04
0.05
0.07
0.06
pix el s al ong x-axis
autocorrelation int ensi ty for m i saligned apert ure
(c)
Figure 14. (a) The autocorrelation intensity of the speckle pattern corresponding to the aligned circular aperture. (b) The
autocorrelation intensity of the speckle pattern correspondin misaligned circular aperture of tilting angle
= 3˚. (c) The
autocorrelation intensity of the speckle pattern corresponding to misaligned circular aperture of tilting angle
= 4˚.
alpha= 4
g to
A. M. HAMED 257
The reconstruction of the two crossed ellipses using
the inverse Fourier transform of matrix dimensions 2048
2048 pixels is shown as in Figure 15. Also, the recon- ×
st
nd the autocorrelation of the four
eq
oposed sharp models is for the
on by comparing the different speckle
inary speckle image for uniform cir-
ructed image of the aperture of four crossed ellipses is
shown as in Figure 16.
The autocorrelation of the two orthogonal ellipses ob-
tained from the reconstruction of the speckle intensity is
shown as in Figure 17 a
ually spaced ellipses where the angle between each
two is
= 45˚ are shown as in Figure 18. The autocorre-
lation images shown in Figures 17 and 18 are obtained
using Equation (15).
4. Conclusions
The motivation of the pr
sake of its recogniti
images with the ord
cular aperture and to check its alignment in its plane.
The snowflake and the airplane models showed a rela-
tively complicate elongation since each part give elonga-
Figure 15. Reconstruction of the plus symbol pupil of ma-
trix 2048 × 2048 pixels.
Figure 17. The autocorrelation of plus symbol pupil or the
two orthogonal ellipses.
Figure 18. The autocorrelation of the snow flake pupil or
the four crossed ellipses.
The alignment of optical microscopic systems can be
verified by testing the aperture inclination using speckle
techniques.
A potential application of this work may be extended
to polychromatic illumination using a mixture of He-Ne
laser and Ar ion laser in order to differentiate the colored
parts of the elliptic apertures.
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