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Improvement of signal-to-noise ratio in digital holography using wavelet transform

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Optics and Lasers in Engineering 46 (2008) 42–47
Improvement of signal-to-noise ratio in digital holographyusing wavelet transform
Akshay Sharma
a,
, Gyanendra Sheoran
b
, Z.A. Jaffery
a
, Moinuddin
a
a
Faculty of Engineering and Technology, Department of Electrical Engineering, Jamia Millia Islamia, New Delhi 110025, India
b
Instrument Design Development Centre, Indian Institute of Technology, New Delhi 110016, India
Received 21 April 2007; received in revised form 2 July 2007; accepted 16 July 2007Available online 29 August 2007
Abstract
The basic problem in optical and digital holography is the presence of speckle noise in the reconstruction process, which reduces thesignal-to-noise ratio (SNR). The presence of speckle noise is serious drawback in optical and digital holography since it substantiallyreduces the SNR in the reconstructed image. In this paper, we present wavelet ﬁltering to improve SNR in the reconstructed images fromdigital holograms. Experimental results are presented.
r
2007 Elsevier Ltd. All rights reserved.
Keywords:
Digital holography; Wavelet transform; Improvement of SNR
1. Introduction
The fast development of digital computers and CCDcameras has made possible to implement the idea of Goodman for recording and reconstructing holograms bydigital process [1]. This development has given birth toimportant ﬁeld called digital optics, and particularly digitalholography as it was envisaged by Goodman. Digitalholography has inherited both desirable and undesirablecharacteristics from its optical counterpart. Among thedesirable inherited characteristics digital holography allowsus to determine the phase of light ﬁeld as well as theintensity; i.e., the whole wave ﬁeld can be measured andmay be stored in computer. This capability is a verypowerful scientiﬁc and technological tool. However, someundesirable characteristics such as zero diffraction orderand speckle noise present in the reconstructed images werealso inherited and have an undesirable effect on the signal-to-noise ratio (SNR) of the reconstructed hologram bydigital process. Zero-order diffraction can be removed bysubtracting the average intensity of all pixels of thehologram matrix from the srcinal hologram matrix. It isalso possible to ﬁlter the hologram matrix by high-passﬁlter with low cut-off frequency [2–4]. The presence of undesired bright and dark speckles in the image leads toinaccuracies in interpretation of image. Different methodsinvestigated to reduce the speckle noise are only partiallysuccessful [5–9]. The conventional as well as Fourierﬁltering method to reduce the speckle noise has severelimitation because in practice objects usually contains holes,cracks or shadows in the image ﬁeld. This is basically due tothe reason that Fourier transforms expands the originalfunction in terms of orthonormal basis function of sine andcosine wave of inﬁnite duration. Thus errors are introducedin measurement at boundaries when ﬁltered image is used toevaluate phase. The Symlet wavelet is near symmetrical/linear phase ﬁlter that make it easier to deal withboundaries of images [10–17]. We have applied waveletﬁltering to smoothen/reduce speckle noise as wavelets haveemerged as a powerful tool for image ﬁltering.In this paper, the wavelet ﬁltering is implemented toreduce the speckle noise and thus improve SNR in thereconstructed images from digital holograms. The papercontains the details of procedure to implement waveletbased ﬁltering technique by using MATLAB and achievedexperimental results.
ARTICLE IN PRESS
www.elsevier.com/locate/optlaseng0143-8166/$-see front matter
r
2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.optlaseng.2007.07.004
Corresponding author. Tel.: +911126862988,mobile: +919810688529.
E-mail address:
sharmaakshay1985@yahoo.co.in (A. Sharma).
2. Basic theory of digital holography
In digital holography holograms are recorded opticallyand stored in a digital image processing system. The realimage can be constructed from the digitally sampledhologram if the intensity pattern of the hologram is carriedout by numerical methods.The diffraction of a plane wave at the hologram isdescribed by the Fresnel–Kirchhoff integral and it is givenby [3]
O
ð
X
I
;
Y
I
Þ ¼
i
l
ZZ
H
ð
X
H
;
Y
H
Þ
R
ð
X
H
;
Y
H
Þ
exp
i
2
pl
r
r
12
þ
12cos
y
d
X
H
d
Y
H
with
r
¼
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ð
X
I
X
H
Þ
2
þ ð
Y
I
Y
H
Þ
2
þ
d
2
q
,
ð
1
Þ
where
O
(
X
I
,
Y
I
) is the complex amplitude of the diffractionpattern of hologram in the image plane,
H
(
X
H
,
Y
H
) isintensity of optically recorded hologram and
r
is thedistance between a point in the hologram plane and a pointin the reconstructed plane,
y
is the angle between theoutward normal
^
n
and the vector
r
pointing from a point
P
in the aperture plane to a point
Q
in the image plane asshown in Fig. 1.This study is based on the assumption that the distance
d
is much greater than the maximum dimension of the CCDchip. With this restriction on the separation, the distance
d
satisﬁes the condition:
d
3
b
p
4
l
ð
X
I
X
H
Þ
2
þ ð
Y
I
Y
H
Þ
2
2max
.Hence, Eq. (1) becomes:
O
ð
X
I
;
Y
I
Þ ¼
i
l
dexp
i2
pl
d
exp
i
pl
d
X
2
I
þ
Y
2
I
h i
ZZ
R
ð
X
H
;
Y
H
Þ
H
ð
X
H
;
Y
H
Þ
exp
i
pl
d
X
2
H
þ
Y
2
H
h i
exp i2
pl
d
X
H
X
I
þ
Y
H
Y
I
ð Þ
d
X
H
d
Y
H
.
ð
2
Þ
This equation is called the Fresnel approximation orFresnel transformation. It enables reconstruction of thewaveﬁeld in a real image plane.The factor exp
ð
i
ð
2
p
=
l
Þ
d
Þ
can be omitted, since it onlyaffects the overall phase. It has no effect on the intensityand interference phase of digital hologram. The function
O
(
X
I
,
Y
I
) can be digitized if the hologram intensity
H
(
X
H
,
Y
H
) is sampled on a rectangular raster of N
Npoints, with steps
D
X
H
,
D
Y
H
along the coordinates.
D
X
H
,
D
Y
H
are the distances between neighboring pixels on theCCD in the horizontal and vertical directions. Samplingintervals in the hologram and image plane are relatedthrough the following relations:
D
X
I
¼
l
d N
D
X
H
;
D
Y
I
¼
l
d N
D
Y
H
, (3)which are in compliance with the sampling theorem, whichrequires that the angle between the object beam and thereference beam at any point of the CCD sensor be limited
ARTICLE IN PRESS
Aperture plane Real Image plane X
H
Y
H
Y
I
Numerical wavepropagation dX
I
Reconstructed wave PQ
n
ˆZ
Fig. 1. Schematic diagram of coordinate system used to describe Fresneldiffraction integral.
BSM1M2Laser MO2ObjectMO1L1M3ORCMOSCOMPUTER
Fig. 2. Schematic of experimental arrangement to record digital hologram.
A. Sharma et al. / Optics and Lasers in Engineering 46 (2008) 42–47
43
in such a way that the micro interference fringe spacing islarger than double the pixel size. By using the aboverelationship Eq. (2) converts to
O
ð
p
D
X
I
;
q
D
Y
I
Þ ¼
i
l
d
exp
i
pl
d
p
2
N
2
D
X
2
H
þ
q
2
N
2
D
Y
2
H
X
N
1
r
¼
0
X
N
1
s
¼
0
R
ð
r
;
s
Þ
H
ð
r
;
s
Þ
exp
i
pl
d r
2
D
X
2
H
þ
s
2
D
Y
2
H
h i
exp i2
p
rpN
þ
sqN
h i
.
ð
4
Þ
This is the discrete Fresnel transform.Where
m
and
n
are the number of pixels in the
X
and
Y
direction, respectively.
p
¼
0
;
1
;
2
;
. . .
;
N
1
;
q
¼
0
;
1
;
2
;
. . .
;
N
1.The matrix
O
(
p
D
X
I
,
q
D
Y
I
) is calculated by multiplying
H
(
r
,
s
),
R
(
r
,
s
), the quadratic phase factorexp
½
i
ð
p
=
l
d
Þð
r
2
D
X
2
H
þ
s
2
D
Y
2
H
Þ
and applying an inversediscrete Fourier transform to the product. The calculationis done most effectively using the fast Fourier transform(FFT) algorithm.
O
ð
p
D
X
I
;
q
D
Y
I
Þ ¼
i
l
d
exp
i
pl
d
p
2
N
2
D
X
2
H
þ
q
2
N
2
D
Y
2
H
FFT
R
ð
r
;
s
Þ
H
ð
r
;
s
Þ
exp
n
i
pl
d r
2
D
X
2
H
þ
s
2
D
Y
2
H
h io
.
ð
5
Þ
Fig. 3 shows planes of (
X
O
,
Y
O
), (
X
H
,
Y
H
) and (
X
I
,
Y
I
) incartesian coordinate system as the object, hologram andimage planes, respectively. In the speciﬁc geometry of lensless Fourier transform holography, the effect of thespherical phase factor associated with the Fresnel diffrac-tion pattern of the object is eliminated by use of a sphericalreference wave
R
(
r
,
s
) with the same average curvature [2]
R
ð
r
;
s
Þ ¼ ð
const
Þ
exp i
pl
d r
2
D
X
2
H
þ
s
2
D
Y
2
H
h i
. (6)Use of Eq. (6) into (5) results in a simpler algorithm forlensless Fourier transforms digital holography:
O
ð
p
D
X
I
;
q
D
Y
I
Þ ¼
i
l
d
exp
i
pl
d
p
2
N
2
D
X
2
H
þ
q
2
N
2
D
Y
2
H
FFT
f
H
ð
r
;
s
Þg
.
ð
7
Þ
The cancellation of quadratic phase factor reduces thecomplexity in the algorithm for numerical reconstructionof Fourier holograms. The above algorithm, involving onlyone simple Fourier transform apart from some multi-plicative constants, is fast enough compared to otherexisting methods like Fresnel method [4], convolutionapproach [18,19], etc., which uses combination of severalFourier transforms and complex multiplications.The reconstructed wave ﬁeld
O
(
p
D
X
I
,
q
D
Y
I
) is an arrayof complex optical ﬁeld; the intensity is calculated bytaking the modulus and squaring [3]:
I X
I
;
Y
I
ð Þ ¼
O X
I
;
Y
I
ð Þ
2
¼
Re
2
O X
I
;
Y
I
ð Þ
þ
Im
2
O X
I
;
Y
I
ð Þ
.
ð
8
Þ
The phase is calculated by
f
ð
X
I
;
Y
I
Þ ¼
arctanIm
½
O
ð
X
I
;
Y
I
Þ
Re
½
O
ð
X
I
;
Y
I
Þ
. (9)In Eq. (9), the operators Re and Im denote real andimaginary part of a complex optical ﬁeld.It is apparent from the Eqs. (8) and (9) that digitalholography allows us to calculate the intensity and phase of the reconstructed wave ﬁeld
O
(
p
D
X
I
,
q
D
Y
I
) for a particulardistance
d
from the hologram plane.According to Eq. (3) the pixel distance in the recon-structed image,
D
X
I
,
D
Y
I
are different from those of thehologram matrix. At ﬁrst sight it seems to lose (or gain)resolution by applying the numerical Fresnel transform.On closer examination one recognizes that Eq. (3)corresponds to the diffraction limited resolution of opticalsystem with side
N
D
X
H
.
D
X
I
¼
(
l
d
)/(
N
D
X
H
) is thereforethe diameter of the Airy disk or speckle diameter in theplane of the reconstructed image, which limits theresolution.
ARTICLE IN PRESS
X
H
Y
H
Y
I
Y
O
X
O
Actual wave propagationX
I
Real Image plane Hologram plane Object & Point Reference sourceplane Numerical wave propagationdd
Fig. 3. Schematic diagram of the geometry used for recording and reconstruction of holograms.
A. Sharma et al. / Optics and Lasers in Engineering 46 (2008) 42–47
44
3. Wavelet ﬁlter to improve signal-to-noise ratio in digitalholography
In Digital holography, speckles are generated in theplane of reconstructed image according to the relationgiven in Eq. (3). To remove the speckle noise, a ﬁlteringscheme is needed that can decompose images at differentscales and then remove the unwanted intensity variations.These speckles limit the resolution in the reconstructedimage. The intensity changes occur at different scales in theimage, so that their optimal detection requires the use of operators of different sizes. A sudden intensity changeproduces a peak and a trough in the ﬁrst derivative of theimage. This requires that the vision ﬁlter should have twocharacteristics: ﬁrst, it should be differential operator, andsecond, it should be capable of being tuned to act at anydesired scale [20,21].Wavelet ﬁlters have these properties. Wavelets are newfamilies of orthonormal basis functions, which do not needto have inﬁnite duration [16,20]. When a wavelet decom-position function is dilated, it accesses lower-frequencyinformation, and when compressed, it accesses higher-frequency information. It is computationally efﬁcient andprovides signiﬁcant speckle reduction while maintainingthe sharp features in the image [22,23].Daubechies analyzed [24] the phase of the Haar wavelet.The phase of the Haar wavelet has a discontinuity at
p
. TheDaubechies wavelet (dbN) is a minimum-phase ﬁlter.Shakher et al. [11] demonstrated that the phase introducedby the Symlet wavelet ﬁlter is closer to linear than that of the dbN. The Symlets are compactly supported waveletswith less asymmetry and with larger number of vanishingmoments for a given support width. The associated scalingﬁlters are near-linear phase ﬁlters having support width2
N
1 and ﬁlter length 2
N
. To make a ﬁlter close tosymmetric, the idea is then to juggle its phase so that it isalmost linear. This is done by repeated use of the function
m
0
ð
x
Þ ¼ ð
1
=
ﬃﬃﬃ
2
p Þ
P
n
h
n
e
i
n
x
(
h
being the kernel ﬁlter) intro-duced in the wavelet dbN and considering |
m
0
(
x
)|
2
as afunction of
W
of
z
¼
e
i
x
. We can factor
W
in severaldifferent ways in the form
W
ð
z
Þ ¼
U
ð
z
Þ
U
ð
1
=
z
Þ
. The rootsof
W
with modulus not equal to one appear in pairs. If
z
isa root, then 1/
z
is also a root. By selecting
U
such that themoduli of all its roots are strictly less than one, dbNs arebuilt.The
U
ﬁlter is a minimum-phase ﬁlter. By makinganother choice, we can obtain more symmetrical ﬁlters[15,25]. These are Symlets.The appearance of digitally recorded and numericallyreconstructed hologram is shown in Fig. 4(a). Fig. 4(a)
show the well deﬁned real image of the object, where thezero-order diffraction has been eliminated. However, aspeckle noise as is clear from Eq. (3), over the recon-structed image is still apparent. This reduces the SNR inthe reconstructed image. From this, it is clear that the causeof reduction of SNR of the digitally reconstructedholographic image is the speckle noise. The presence of speckle noise in digital holography arises from coherentimaging [26] and due to ﬁnite size of the pixels in the CCDcamera [27,28]. Thus the reduction of speckle noise can beattempted by two ways. Firstly by means of reducing itfrom recording process itself; by improving the digitalrecording devices. Secondly by means of ﬁltering of thereconstructed images by digital process. Below we willdiscuss the Symlet wavelet ﬁltering technique to reducespeckle noise from the reconstructed image of the digitalholograms.Filtering scheme is tested for their potential to improveSNR in digital holography. One of the parameter forﬁltering scheme is to calculate the speckle index. Thespeckle index is the ratio of standard deviation to mean of intensity in a homogeneous area [17,29,30]. Speckleindex
¼
(standard deviation of intensity/mean of inten-sity):
C
¼
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
var
x
ð Þ
p
E x
ð Þ ¼
s
m
, (10)where
C
is the speckle index,
s
and
m
are the standarddeviation of intensity and mean of intensity, respectively.The SNR is a reciprocal of the speckle index.SNR
¼
1
C
. (11)
4. Experimental
Fig. 2 shows the schematic diagram for recording digitalhologram. A He–Ne laser having out put
30mW is usedas a light source. Light from a laser is divided into twoparts by using a beam splitter. In the object beam, a spatialﬁlter and collimating lens produces a collimated beam,which illuminates the object. The angle of illumination isabout 44
1
. Another spatial ﬁlter in the reference beamproduces spherical reference wave. The interference patternformed by the superposition of the reference beam and thelight scattered from object is recorded by CMOS sensor,image grabber card and stored in a personal computer(PC). The pixel size on the CMOS sensor is 6.7
6.7
m
mand total numbers of pixels are 1280
1022. The dimen-sions of the sensor chip are 8.6
6.9mm. Distance of thefaceplate of the sensor is 50cm from the plane containingthe object and reference source.The hologram of an object (dice) was recorded. Imagereconstructed and the relevant image processing wascarried out in the MATLAB environment (version 6.5).The normal appearance of the reconstructed image of dicefrom the digital hologram is shown in Fig. 4(a). There iswell-deﬁned image of the object along with speckle noiseover the image. The speckle noise can be reduced by usingmedian ﬁlter and wavelet ﬁlter. Figs. 4(b) and (c) shows theﬁltered images by using 3
3 median ﬁlter and 5
5median ﬁlter respectively. Fig. 4(d) shows the ﬁltered imageby Symlet wavelet ﬁltering (sym 5) in MATLAB environ-ment. The ﬁltered images are tested for improvement by
ARTICLE IN PRESS
A. Sharma et al. / Optics and Lasers in Engineering 46 (2008) 42–47
45
calculating speckle index and SNR by using Eqs. (10) and(11), respectively. Calculated values of speckle index andSNR for different images (Fig. 4) are given in Table 1.
Table 1 show that median ﬁltering with higher ordermatrix also reduces the speckle index (increases the SNR)substantially but it also increases the blurring in the image.Fig. 4(d) shows the wavelet ﬁltering signiﬁcantly reducesthe speckle index (improves the SNR) without blurring theedges and holes.
5. Conclusion
The presence of speckle noise is serious drawback indigital holography since it substantially reduces the SNR inthe reconstructed image. This has limited the use of digitalholography in many applications as for example 3D-imagereconstruction of object, microstructure testing, digitalholographic microscopy etc. as speckle reduces theperformance of these systems. It is apparent that waveletﬁltering over reconstructed images with low SNR results inthe processed image with greater SNR than reconstructedimage. Thus digital holography can be made more practicalfor its use in different applications.
Acknowledgments
Authors are thankful to referee for his comments toimprove the manuscript. Authors gratefully acknowledgevarious fruitful discussions with Prof. Chandra Shakherduring the stage of experimental work and many discus-sions to improve the manuscript.
References
[1] Goodman JW, Lawrence RW. Digital image formation fromelectronically detected holograms. Appl Phys Lett 1967;11:77–9.[2] Wagner C, Seebacher S, Osten W, Juptner W. Digital recording andnumerical reconstruction of lensless Fourier holograms in opticalmetrology. Appl Opt 1999;38(22):4812.
ARTICLE IN PRESS
Table 1The speckle index and SNR of different imagesImage name Fig. 4(a) Fig. 4(b) Fig. 4(c) Fig. 4(d)Speckle index (
C
)
C
¼
s
/
m
0.08105 0.02395 0.01259 0.01165SNR
¼
1/
C
12.33806 41.75365 79.42811 85.83690Fig. 4. (a) Digitally reconstructed hologram of dice after removal of zero diffraction order. Speckle index
¼
0.08105. SNR
¼
12.3434. (b) Improvement inSNR by 3
3 median ﬁltering applied to the srcinal reconstructed image shown in Fig. 4(a). Speckle index
¼
0.02395. SNR
¼
41.7596. (c) Improvementin SNR by 5
5 median ﬁltering applied to the srcinal reconstructed image shown in Fig. 4(a). Speckle index
¼
0.01259. SNR
¼
79.4018.(d) Improvement in SNR by Symlet wavelet ﬁltering applied to the srcinal reconstructed image shown in Fig. 4(a). Speckle index
¼
0.01165.SNR
¼
85.8221.
A. Sharma et al. / Optics and Lasers in Engineering 46 (2008) 42–47
46

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