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International Journal of Research in Advent Technology, Vol.3, No.12, December 2015
E-ISSN: 2321-9637
Available online at www.ijrat.org
Implementation of Proposed Threshold for Despeck-
ling in Stationary Wavelet Domain
A.Stella1
, Dr. Bhushan Trivedi2
, Dr. N.N.Jani3
1
Faculty, Kadi Sarva Vishwavidyalaya, 2
Dean, GLS Institute of Computer Technology (MCA)
3
Ex-Dean, Kadi Sarva Vishwavidyalaya
Email:rosystella@gmail.com,bhtrivedi@gmail.com,drnnjcsd@gmail.com
Abstract— Medical images are prone to different types of noise. Such types of noise corrupted images
leads to incorrect diagnosis. Hence, removal of noise is a prerequisite in medical imaging modality.
Speckle noise is widely found in coherent medical images, like in Ultra Sound images and Optical Cohe-
rence Tomography images.In the preprocessing stage, the noise present in the medical image has to be
removed while preserving the edge information and other structural details of the image. Relevant denois-
ing technique has to be chosen based on the nature of the medical image. This research is focused on de-
sign of algorithms for speckle denoising of Ultra Sound images and Optical Coherence Tomography im-
ages in stationary wavelet domain. Standard speckle filters in wavelet domain were analyzed and compared
with the proposed method. Results obtained proved that the proposed method was able to remove speckle
noise while preserving better edges.
Index Terms—Despeckling, Stationary Wavelet Domain, Shrinkage methods, Edge preservation
1. INTRODUCTION
In image processing and computer vision,
the techniques of image denoising from noise conta-
minated version of image to restore the originality of
the image is a continuous research issue, aiming at
arriving more better performance in the applications
such as visual tracking, image classification, segmen-
tation, registration etc. Usually a captured image gets
contamination embedded into an image due to intrin-
sic and extrinsic causes[1]. The researchers has so far
used a wide variety of methodology for the stated
purpose, but the undertaken research has focused on
spatial and transform domain techniques for image
denoising.
Mostly the images captured through cohe-
rence illumination are formed with higher level of
speckle noise. The success ratio of segmentation af-
ter the preprocessing of the image that involves de-
noising depends on the extent of the removal of noise
from the image. Coherent Medical images and Satel-
lite images are usually degraded with noise during
image acquisition and transmission process[2]. Such
types of images are corrupted by speckle noise. The
researchers are making efforts to reduce speckle
noise with highest possible level with the objective
of retaining important features of the image. Synthet-
ic Aperture Radar (SAR) imagery uses microwave
radiation to illuminate the earth surface. Optical Co-
herence Tomography (OCT) and Ultra Sound (US)
medical images are also affected due to speckle
noise[3].
Image processing techniques have been
widely used in medical imaging research. These
techniques provides support in visualization, en-
hancements, segmentation and many more operations
which are useful for processing medical images[4].
The main reason for utilization of these techniques is
to detect any abnormality in the medical images. Few
abnormalities to be mentioned are detection of tu-
mors, finding blocked vessels and even detecting
broken joints. Medical image analysis is performed
in stages like removal of the noise, segmentation of
the suspected parts of the image, feature extraction
and its measurement.
2. REVIEW OF LITERATURE
Jyoti Sahu et al[5] proposed a multivariate
thresholding technique for image denoising using
multiwavelets. The proposed technique is based on
the idea of restoring the spatial dependence of the
noisy pixels in the subbands of wavelet decomposi-
tion. Coefficients with high correlation are consi-
dered for thresholding operation.
Yong Yue et.al[6] introduced a novel Mul-
tiscale Nonlinear Wavelet Diffusion (MNWD) me-
thod for denoising speckle in ultrasound images.
68
Wavelet diffusion is considered as an approximation
to nonlinear diffusion within the framework of the
dyadic wavelet transform. This idea is used in the
design of a speckle suppression filter with an edge
enhancement feature. MNWD takes advantage of the
sparsity and multiresolution properties of wavelet,
and the iterative edge preservation and enhancement
feature of nonlinear diffusion.
David Donoho[7] proposed visushrink. It is
also called as universal threshold. An estimate of the
noise level σ was defined based on the median abso-
lute deviation. VisuShrink does not deal with the
minimization of mean squared error as a result it
over smoothes the image, because it removes too
many coefficients. VishuShrink performs well for
additive noise but not for multiplicative noise.
Iman Elyasi et al[8] proposed Normal
Shrink, following a generalized gaussian distribution
model of the subband in wavelet domain. It produces
best result of minimum MSE and maximum SNR
only when the noise is low. Its performance is better
than bayes shrink in terms of preserving the edges as
well as in removing the noise.
Donoho et al[9] proposed Stein‟s Unbiased
Risk Estimator (SURE). It is referred as subband
dependent threshold because it determines a thre-
shold value for each resolution level in the wavelet
transform. The main advantage of SureShrink is, it
minimize the mean squared error, unlike VisuShrink,
SureShrink reduces the noise by thresholding the
empirical wavelet coefficients. It follows the soft
thresholding rule and it is adaptive in nature.
Chang et al[10] proposed BayesShrink. The
goal of this method is to minimize the Bayesian risk.
It uses soft thresholding and it is also subband-
dependent, like Sure Shrink, which means that thre-
shold level is selected at each subband of resolution
in the wavelet decomposition. The noise variance is
obtained by median estimator in the HH1 subband.
2.1 Review Findings for Shrinkage Methods
• Linear filters causes blurring of edges whereas
nonlinear filters preserves the edges with the
drawback that these filters are sensitive to the
size and shape of the filter window[11].
• Overall most of these techniques do not enhance
edges, as these filters are not directional, and
may not suppress noise near the edges[12].
• Drawback of discrete wavelet transformation is
that it is not translation invariant. It loses lots of
important pixel coefficients during reconstruc-
tion of the denoised signal to all most the origi-
nal signal[13].
• In wavelet transform methods, the noise variance
for threshold computation is obtained from coeffi-
cients of high frequency subband and the same
threshold is used for all the resolution scales. The
level of noise decreases as the scale of resolution
increases. Therefore, noise variance should be es-
timated separately for each subbands[14].
3. MATHEMATICAL MODEL OF SPECKLE NOISE
Speckle Noise is multiplicative in na-
ture. This type of noise is an inherent property of
coherent imaging. It affects the diagnostic value of
imaging modality, because of reduced image resolu-
tion and image contrast[15]. So, speckle noise reduc-
tion is an essential preprocessing step, in coherent
medical images. Mathematically, the speckle noise is
represented with the help of these equations below:
݃ሺ‫,ݔ‬ ‫ݕ‬ሻ = ݂ሺ‫,ݔ‬ ‫ݕ‬ሻ ∗ ‫ݑ‬ሺ‫,ݔ‬ ‫ݕ‬ሻ + ߦሺ‫,ݔ‬ ‫ݕ‬ሻ (1)
Where, gሺx, yሻ is the observed image, uሺx, yሻ is the
multiplicative component and ξሺn, mሻ is the additive
component of the speckle noise. Here ‘x’ and ‘y’
denotes the radial and angular indices of the image
samples. As in coherent imaging, only multiplicative
component of the noise is to be considered and addi-
tive component of the noise has to be ignored.
Hence, equation (1) can be modified as;
݃ሺ‫,ݔ‬ ‫ݕ‬ሻ = ݂ሺ‫,ݔ‬ ‫ݕ‬ሻ ∗ ‫ݑ‬ሺ‫,ݔ‬ ‫ݕ‬ሻ + ߦሺ‫,ݔ‬ ‫ݕ‬ሻ − ߦሺ‫,ݔ‬ ‫ݕ‬ሻ (2)
Therefore,
݃ሺ‫,ݔ‬ ‫ݕ‬ሻ = ݂ሺ‫,ݔ‬ ‫ݕ‬ሻ ∗ ‫ݑ‬ሺ‫,ݔ‬ ‫ݕ‬ሻ (3)
4. PROBLEM FORMULATION
The proposed work focuses on the wavelet
transform filtering method. This method is chosen
because; most of the signal energy is contained in a
few large wavelet coefficients, whereas a small por-
tion of the energy is spread across a large number of
small wavelet coefficient. These coefficients
represent details as well as high frequency noise in
the image. By appropriately thresholding these wave-
let coefficients, image denoising is achieved while
preserving fine structures in the image[16]. All wave-
let transform denoising algorithms involve the fol-
lowing three steps in general.
1. Forward Wavelet Transform: Wavelet coeffi-
cients are obtained by applying the wavelet
transform.
69
2. Estimation: Clean coefficients are estimated
from the noisy ones.
3. Inverse Wavelet Transform: A clean image is
obtained by applying the inverse wavelet
transform.
Discrete Wavelet Transform (DWT) does not pro-
vide shift invariance. This leads to small shifts in the
input waveform which makes major changes in the
wavelet coefficients[17]. To overcome the problem
of DWT, Stationary Wavelet Transform (SWT) of
two dimensions is used in the proposed work. SWT2
performs a multilevel wavelet decomposition using
orthogonal wavelet filters.
The noisy image is read as input. As discrete
stationary wavelet domain is used the size of the im-
age must be strictly a positive integer. The value 2N
must equally divide the row value and column value
of the input image before performing 2D stationary
wavelet transform. But all the input images will not
be having the size which is strictly a positive integer
value. In such cases the image has to be extended
symmetrically to overcome this problem[18].
After the input image is symmetrically ex-
tended, the next step is to decompose the input image
upto 3 levels using “bior 3.1” wavelet filter. The de-
composition results in subdivision of the input image
into four subbands namely LL, LH, HL and HH. The
size of the input image in all the four subbands will
be the same. The proposed threshold function is ap-
plied separately to all the subbands except for LL
subband. The proposed threshold is as follows.
In the proposed threshold technique, in each
subband median and absolute difference between the
median and the pixel is calculated. This calculation is
used to measure the variability between noisy pixel
and the noiseless pixel. In the next step the threshold
(th2) is calculated using tukey’s biweight
function[19]. This function helps in determining the
outlier. Next the threshold value is compared with
the MD(x,y) to determine whether the pixel lies in-
side the outlier value or not. If the variable measure
of the pixel is above the threshold, then the pixel is
removed using soft thresholding technique else the
pixel is not a noisy pixel and hence it is retained.
4.1 Proposed Algorithm
Step-1: Start
Step-2: Read the noisy image.
Step-3: Extend the noisy image. The noisy image
will be extended using symmetric extension in or-
der to improve the boundary problem.
Step-4: Set the level of wavelet decomposition to 3.
Step-5: Choose bior3.1 wavelet filter.
Step-6: Perform decomposition of the input image
using swt2() upto 3 levels.
Step-7: Perform thresholding in LH, HL, HH
subband
Step-7.1: Calculate the median M value of each sub-
band image.
Step-7.2: Calculate ‫ܦܯ‬ሺ‫,ݔ‬ ‫ݕ‬ሻ = ݉݁݀݅ܽ݊ሺ∑ |‫ݔ‬௜,௝ −௡
௜,௝
‫.|ܯ‬ It is a measure to indicate the variability of the
pixel.
Step-7.3: Formulate the threshold (th1) using tukey’s
biweight function
‫ݐ‬ℎ1 = ‫ܫ‬ሺ‫,ݔ‬ ‫ݕ‬ሻ ∗ ቆ൬1 − ቀ
ூሺ௫,௬ሻ
௙
ቁ
ଶ
൰ ˄2ቇ ∗ 0.5
Step- 7.4: If th1 > MD(x,y)
Perform Soft thresholding of the subband
image.
else
Retain the pixel
Step-8: Perform inverse stationary wavelet transform
using ISWT2().
Step-9: Calculate PSNR, RMSE, IQI, SSIM, MSD,
DR, ENL, FOM, CC.
Step-10: Stop
5. IMAGE METRICS
5.1 Peak Signal to Noise Ratio
Peak Signal to Noise Ratio (PSNR)[20] is one of the
most essential statistical parameter for quality mea-
surement of an image or signal. It is used as an esti-
mate to measure the quality of objective difference
between the noisy and the denoised image. The basic
idea is to compute a single number that reflects the
quality of the reconstructed image. Higher PSNR
value provides higher image quality. It is calculated
as;
ܴܲܵܰ = 10 ∗ ݈‫01݃݋‬ ቀ
ଵ
ெௌா
ቁ (4)
5.2 Root Mean Square Error
Root Mean Square Error (RMSE)[21], is an estima-
tor in to quantify the amount by which a noisy image
differs from noiseless image. RMSE is computed by
averaging the squared intensity of the noisy image
and the denoised image, where error is the difference
between desire quantity and estimated quantity. Hav-
ing a RMSE value of zero is ideal.
ܴ‫ܧܵܯ‬ = ඨ∑ ∑ ቀ୤ሺ୶,୷ሻି ୤ሺ୶,୷ሻቁ
మ
౤
౯సభ
ౣ
౮సభ
୫∗୬
(5)
70
5.3 Image Quality Index
The Image Quality Index (IQI)[20] is a measure of
comparison between original and distorted image. It
is divided into three parts: luminance ݈ሺ‫,ݔ‬ ‫ݕ‬ሻ, contrast
ܿሺ‫,ݔ‬ ‫ݕ‬ሻ, and structural comparisons ‫ݏ‬ሺ‫,ݔ‬ ‫ݕ‬ሻ as men-
tioned in equation (6),(7) and (8). The dynamic range
for IOI(x, y) is [-1, 1].
݈ሺ‫,ݔ‬ ‫ݕ‬ሻ =
ଶఓೣఓ೤
µ೉
మ ା µ೤
మ (6)
ܿሺ‫,ݔ‬ ‫ݕ‬ሻ =
ଶ ఙೣ ఙ೤
ఙೣ
మାఙ೤
మ (7)
‫ݏ‬ሺ‫,ݔ‬ ‫ݕ‬ሻ =
ଶఙೣ೤
ఙೣାఙ೤
(8)
‫ܫܳܫ‬ሺ‫,ݔ‬ ‫ݕ‬ሻ = ݈ሺ‫,ݔ‬ ‫ݕ‬ሻ. ܿሺ‫,ݔ‬ ‫ݕ‬ሻ. ‫ݏ‬ሺ‫,ݔ‬ ‫ݕ‬ሻ =
ସఓೣఓ೤ఓೣ೤
ሺµ೉
మ
ା µ೤
మሻሺఙೣ
మାఙ೤
మሻ
(9)
5.4 Structural Similarity Index
The Structural Similarity Index (SSIM)[20] measures
the similarity between two images which is more
consistent with human perception than conventional
techniques. The range of values for the SSIM lies
between −1, for a bad and 1 for a good similarity
between the original and despeckled images, respec-
tively.
ܵܵ‫ܯܫ‬ሺ‫,ݔ‬ ‫ݕ‬ሻ =
൫ଶఓೣఓ೤ା௖ଵ൯ሺଶఙೣ೤ା௖ଶሻ
ሺµ೉
మ
ା µ೤
మ ା௖ଵሻሺఙೣ
మାఙ೤
మା௖ଶሻ
(10)
5.5 Noise Mean Value (NMV), Noise Standard Dev-
iation (NSD)
Noise Variance determines the contents of the
speckle in an image. A lower variance gives a
“cleaner” image as more speckle is reduced, it is not
necessarily that it should depend on the intensity of
the image. The formulas for the NMV and NSD cal-
culation are as follows[22].
ܰ‫ܸܯ‬ =
∑ ௙೏ሺ௥,௖ሻೝ,೎
௥∗௖
(11)
ܰܵ‫ܦ‬ = ට
∑ ሺ௙೏ሺ௥,௖ሻିேெ௏ሻమ
ೝ,೎
௥∗௖
(12)
5.6 Pratt’s Figure of Merit (FOM)
It measures edge pixel displacement between each
filtered image Ifilt and the original image Iorig. It is
defined as[23]:
FOM =
ଵ
୫ୟ୶ ሺே೑೔೗೟, ಿ೚ೝ೔೒ሻ,
∑
ଵ
ଵାௗ೔
మఈ
௡
௜ୀଵ (13)
where Nfilt and Norig are the number edge pixels in
edge maps of Ifilt and Iorig. Parameter α is set to a con-
stant 1/9, and di is the euclidean distance between the
detected edge pixel and the nearest ideal edge pixel.
The FOM metric measures how well the edges are
preserved throughout the filtering process. This me-
tric has a significant relationship with the overall
quality score at 1% significance level.
5.7 Equivalent Number of Looks
Equivalent Numbers of Looks (ENL)[24] is a measure
to estimate the speckle noise level in the image. The
value of ENL depends on the size of the tested region;
theoretically a larger region will produces a higher
ENL value than a smaller region. The formula for the
ENL is
‫ܮܰܧ‬ =
ேெ௏మ
ேௌ஽మ (14)
5.8 Deflection Ratio (DR)
The formula for the deflection ratio[25] calculation
is;
‫ܴܦ‬ =
ଵ
ோ∗஼
∑
ሺ௙೏ሺ௥,௖ሻିேெ௏ሻ
ேௌ஽௥,௖ (15)
After speckle reduction the deflection ratio should be
higher at pixels with stronger reflector points and
lower elsewhere.
5.9 Correlation Coefficient (CC)
For digital images, correlation[26] is a measure of
the strength and direction of a linear relationship
between two variable. A correlation of 1 indicates a
perfect one-to-one linear relationship and -1 indicates
a negative relationship. The square of the correlation
coefficient describes the variance between two va-
riables in a linear fit. The Pearson’s correlation coef-
ficient is defined as;
‫ݎ‬ =
∑ ሺ௙೔ ି௙೘ሻሺ௙̅೔ି௙̅೘ሻ೔
ට∑ ሺ௙೔ ି௙೘ሻ೔
మ ට∑ ሺ௙̅೔ି௙̅೘ሻ೔
మ
(16)
where, ݂௜ and ݂̅௜ are intensity values of ith pixel in
noisy and denoised image respectively. Also, ݂௠ and
71
݂̅௠ are mean intensity values of noisy and denoised
image respectively.
5.10 Execution Time
Execution Time(ET) [27]of a denoising filter, is de-
fined as the time taken by a processor to execute an
algorithm when no other software, except the operat-
ing system (OS), runs on it. Execution time is re-
ferred with respect to the system’s clock time-period.
The execution time taken by a filtering algorithm
should be low for real-time image processing appli-
cations. Hence, when all metrics give the identical
values then a filter with lower execution time is bet-
ter than a filter having higher execution time.
6. RESULTS AND DISCUSSIONS
An objective evaluation of the existing thre-
sholding techniques and the proposed threshold tech-
niques is listed in Table 1. The PSNR value is very
high for the proposed threshold technique. The next
highest PSNR value is generated by proposed thre-
shold technique. The proposed threshold has a very
low RMSE value compared with other thresholding
techniques. It also indicates that the proposed thre-
shold is capable of removing more speckle noise
equally maintaining low error between the original
and denoised image.
High image quality index is exhibited by
Vishu shrink, indicating that the denoised image has
a better variation between the original and denoised
image. If the structural similarity index is equal to
one, then it is an indication that the structural detail
of the original image is preserved even after denois-
ing. Hence proposed threshold has produced a value
which is very close to one.
The NMV value and NSD values of the pro-
posed threshold has produced the same value, com-
paratively less than other thresholding techniques. It
indicates that the speckle noise content of the de-
noised images is very less.
ENL value and DR values of both the proposed
threshold are same and less. When compared with
other thresholding techniques Bayes Shrink has pro-
duced a higher ENL value indicating that the original
and denoised image has more similar features. But
the proposed threshold as exhibited a higher DR val-
ue indicating that there is more deflection along the
edges in the denoised image.
The FOM value is high in proposed threshold
whereas the existing threshold shrinkages were not
evaluated using this parameter. Similarly, the CC
value is high in proposed threshold. The execution
time for the proposed algorithm is 4.173seconds
Table 1. Comparison Of Existing Denoising Filters With Proposed Threshold
Existing Denoising Filters
Assessment
Parameters
Visu Shrink
Normal
Shrink
Bayes Shrink
Sure Shrink
Proposed
PSNR 31.65 29.28 38.70 29.60 68.2510
RMSE 11.68 10.23 12.67 10.81 0.0098
IQI 0.5902 0.3812 0.3938 0.4645 0.1823
SSIM 0.7882 0.8214 0.8532 0.8953 0.9997
NMV 11.56 9.61 21.72 13.48 0.2205
NSD 3.30 2.01 6.75 4.23 0.2164
ENL 1.2685 3.6853 7.6893 3.5742 1.0378
DR 0.0031 0.0381 0.0610 0.0461 0.8023
FOM NA NA NA NA 0.2498
CC NA NA NA NA 0.5847
72
7. CONCLUSION
As a prerequisite, Ultrasound images and Optical
Coherence Tomography images should undergo
denoising before being interpreted by the medical
expert, as an objective to be achieved. The pro-
posed work was tested with Ultrasound images and
Optical Coherence Tomography images. The im-
ages were obtained from online database and the
database of Optical Coherence Tomography images
were collected from hospital.The proposed algo-
rithms were evaluated with several parameters and
the best proposed algorithm was identified. The
proposed threshold gave good results both objec-
tively and subjectively.The proposed threshold
gave good results for ultra sound image than for
optical coherence tomography images.
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Paper id 312201526

  • 1. International Journal of Research in Advent Technology, Vol.3, No.12, December 2015 E-ISSN: 2321-9637 Available online at www.ijrat.org Implementation of Proposed Threshold for Despeck- ling in Stationary Wavelet Domain A.Stella1 , Dr. Bhushan Trivedi2 , Dr. N.N.Jani3 1 Faculty, Kadi Sarva Vishwavidyalaya, 2 Dean, GLS Institute of Computer Technology (MCA) 3 Ex-Dean, Kadi Sarva Vishwavidyalaya Email:rosystella@gmail.com,bhtrivedi@gmail.com,drnnjcsd@gmail.com Abstract— Medical images are prone to different types of noise. Such types of noise corrupted images leads to incorrect diagnosis. Hence, removal of noise is a prerequisite in medical imaging modality. Speckle noise is widely found in coherent medical images, like in Ultra Sound images and Optical Cohe- rence Tomography images.In the preprocessing stage, the noise present in the medical image has to be removed while preserving the edge information and other structural details of the image. Relevant denois- ing technique has to be chosen based on the nature of the medical image. This research is focused on de- sign of algorithms for speckle denoising of Ultra Sound images and Optical Coherence Tomography im- ages in stationary wavelet domain. Standard speckle filters in wavelet domain were analyzed and compared with the proposed method. Results obtained proved that the proposed method was able to remove speckle noise while preserving better edges. Index Terms—Despeckling, Stationary Wavelet Domain, Shrinkage methods, Edge preservation 1. INTRODUCTION In image processing and computer vision, the techniques of image denoising from noise conta- minated version of image to restore the originality of the image is a continuous research issue, aiming at arriving more better performance in the applications such as visual tracking, image classification, segmen- tation, registration etc. Usually a captured image gets contamination embedded into an image due to intrin- sic and extrinsic causes[1]. The researchers has so far used a wide variety of methodology for the stated purpose, but the undertaken research has focused on spatial and transform domain techniques for image denoising. Mostly the images captured through cohe- rence illumination are formed with higher level of speckle noise. The success ratio of segmentation af- ter the preprocessing of the image that involves de- noising depends on the extent of the removal of noise from the image. Coherent Medical images and Satel- lite images are usually degraded with noise during image acquisition and transmission process[2]. Such types of images are corrupted by speckle noise. The researchers are making efforts to reduce speckle noise with highest possible level with the objective of retaining important features of the image. Synthet- ic Aperture Radar (SAR) imagery uses microwave radiation to illuminate the earth surface. Optical Co- herence Tomography (OCT) and Ultra Sound (US) medical images are also affected due to speckle noise[3]. Image processing techniques have been widely used in medical imaging research. These techniques provides support in visualization, en- hancements, segmentation and many more operations which are useful for processing medical images[4]. The main reason for utilization of these techniques is to detect any abnormality in the medical images. Few abnormalities to be mentioned are detection of tu- mors, finding blocked vessels and even detecting broken joints. Medical image analysis is performed in stages like removal of the noise, segmentation of the suspected parts of the image, feature extraction and its measurement. 2. REVIEW OF LITERATURE Jyoti Sahu et al[5] proposed a multivariate thresholding technique for image denoising using multiwavelets. The proposed technique is based on the idea of restoring the spatial dependence of the noisy pixels in the subbands of wavelet decomposi- tion. Coefficients with high correlation are consi- dered for thresholding operation. Yong Yue et.al[6] introduced a novel Mul- tiscale Nonlinear Wavelet Diffusion (MNWD) me- thod for denoising speckle in ultrasound images. 68
  • 2. Wavelet diffusion is considered as an approximation to nonlinear diffusion within the framework of the dyadic wavelet transform. This idea is used in the design of a speckle suppression filter with an edge enhancement feature. MNWD takes advantage of the sparsity and multiresolution properties of wavelet, and the iterative edge preservation and enhancement feature of nonlinear diffusion. David Donoho[7] proposed visushrink. It is also called as universal threshold. An estimate of the noise level σ was defined based on the median abso- lute deviation. VisuShrink does not deal with the minimization of mean squared error as a result it over smoothes the image, because it removes too many coefficients. VishuShrink performs well for additive noise but not for multiplicative noise. Iman Elyasi et al[8] proposed Normal Shrink, following a generalized gaussian distribution model of the subband in wavelet domain. It produces best result of minimum MSE and maximum SNR only when the noise is low. Its performance is better than bayes shrink in terms of preserving the edges as well as in removing the noise. Donoho et al[9] proposed Stein‟s Unbiased Risk Estimator (SURE). It is referred as subband dependent threshold because it determines a thre- shold value for each resolution level in the wavelet transform. The main advantage of SureShrink is, it minimize the mean squared error, unlike VisuShrink, SureShrink reduces the noise by thresholding the empirical wavelet coefficients. It follows the soft thresholding rule and it is adaptive in nature. Chang et al[10] proposed BayesShrink. The goal of this method is to minimize the Bayesian risk. It uses soft thresholding and it is also subband- dependent, like Sure Shrink, which means that thre- shold level is selected at each subband of resolution in the wavelet decomposition. The noise variance is obtained by median estimator in the HH1 subband. 2.1 Review Findings for Shrinkage Methods • Linear filters causes blurring of edges whereas nonlinear filters preserves the edges with the drawback that these filters are sensitive to the size and shape of the filter window[11]. • Overall most of these techniques do not enhance edges, as these filters are not directional, and may not suppress noise near the edges[12]. • Drawback of discrete wavelet transformation is that it is not translation invariant. It loses lots of important pixel coefficients during reconstruc- tion of the denoised signal to all most the origi- nal signal[13]. • In wavelet transform methods, the noise variance for threshold computation is obtained from coeffi- cients of high frequency subband and the same threshold is used for all the resolution scales. The level of noise decreases as the scale of resolution increases. Therefore, noise variance should be es- timated separately for each subbands[14]. 3. MATHEMATICAL MODEL OF SPECKLE NOISE Speckle Noise is multiplicative in na- ture. This type of noise is an inherent property of coherent imaging. It affects the diagnostic value of imaging modality, because of reduced image resolu- tion and image contrast[15]. So, speckle noise reduc- tion is an essential preprocessing step, in coherent medical images. Mathematically, the speckle noise is represented with the help of these equations below: ݃ሺ‫,ݔ‬ ‫ݕ‬ሻ = ݂ሺ‫,ݔ‬ ‫ݕ‬ሻ ∗ ‫ݑ‬ሺ‫,ݔ‬ ‫ݕ‬ሻ + ߦሺ‫,ݔ‬ ‫ݕ‬ሻ (1) Where, gሺx, yሻ is the observed image, uሺx, yሻ is the multiplicative component and ξሺn, mሻ is the additive component of the speckle noise. Here ‘x’ and ‘y’ denotes the radial and angular indices of the image samples. As in coherent imaging, only multiplicative component of the noise is to be considered and addi- tive component of the noise has to be ignored. Hence, equation (1) can be modified as; ݃ሺ‫,ݔ‬ ‫ݕ‬ሻ = ݂ሺ‫,ݔ‬ ‫ݕ‬ሻ ∗ ‫ݑ‬ሺ‫,ݔ‬ ‫ݕ‬ሻ + ߦሺ‫,ݔ‬ ‫ݕ‬ሻ − ߦሺ‫,ݔ‬ ‫ݕ‬ሻ (2) Therefore, ݃ሺ‫,ݔ‬ ‫ݕ‬ሻ = ݂ሺ‫,ݔ‬ ‫ݕ‬ሻ ∗ ‫ݑ‬ሺ‫,ݔ‬ ‫ݕ‬ሻ (3) 4. PROBLEM FORMULATION The proposed work focuses on the wavelet transform filtering method. This method is chosen because; most of the signal energy is contained in a few large wavelet coefficients, whereas a small por- tion of the energy is spread across a large number of small wavelet coefficient. These coefficients represent details as well as high frequency noise in the image. By appropriately thresholding these wave- let coefficients, image denoising is achieved while preserving fine structures in the image[16]. All wave- let transform denoising algorithms involve the fol- lowing three steps in general. 1. Forward Wavelet Transform: Wavelet coeffi- cients are obtained by applying the wavelet transform. 69
  • 3. 2. Estimation: Clean coefficients are estimated from the noisy ones. 3. Inverse Wavelet Transform: A clean image is obtained by applying the inverse wavelet transform. Discrete Wavelet Transform (DWT) does not pro- vide shift invariance. This leads to small shifts in the input waveform which makes major changes in the wavelet coefficients[17]. To overcome the problem of DWT, Stationary Wavelet Transform (SWT) of two dimensions is used in the proposed work. SWT2 performs a multilevel wavelet decomposition using orthogonal wavelet filters. The noisy image is read as input. As discrete stationary wavelet domain is used the size of the im- age must be strictly a positive integer. The value 2N must equally divide the row value and column value of the input image before performing 2D stationary wavelet transform. But all the input images will not be having the size which is strictly a positive integer value. In such cases the image has to be extended symmetrically to overcome this problem[18]. After the input image is symmetrically ex- tended, the next step is to decompose the input image upto 3 levels using “bior 3.1” wavelet filter. The de- composition results in subdivision of the input image into four subbands namely LL, LH, HL and HH. The size of the input image in all the four subbands will be the same. The proposed threshold function is ap- plied separately to all the subbands except for LL subband. The proposed threshold is as follows. In the proposed threshold technique, in each subband median and absolute difference between the median and the pixel is calculated. This calculation is used to measure the variability between noisy pixel and the noiseless pixel. In the next step the threshold (th2) is calculated using tukey’s biweight function[19]. This function helps in determining the outlier. Next the threshold value is compared with the MD(x,y) to determine whether the pixel lies in- side the outlier value or not. If the variable measure of the pixel is above the threshold, then the pixel is removed using soft thresholding technique else the pixel is not a noisy pixel and hence it is retained. 4.1 Proposed Algorithm Step-1: Start Step-2: Read the noisy image. Step-3: Extend the noisy image. The noisy image will be extended using symmetric extension in or- der to improve the boundary problem. Step-4: Set the level of wavelet decomposition to 3. Step-5: Choose bior3.1 wavelet filter. Step-6: Perform decomposition of the input image using swt2() upto 3 levels. Step-7: Perform thresholding in LH, HL, HH subband Step-7.1: Calculate the median M value of each sub- band image. Step-7.2: Calculate ‫ܦܯ‬ሺ‫,ݔ‬ ‫ݕ‬ሻ = ݉݁݀݅ܽ݊ሺ∑ |‫ݔ‬௜,௝ −௡ ௜,௝ ‫.|ܯ‬ It is a measure to indicate the variability of the pixel. Step-7.3: Formulate the threshold (th1) using tukey’s biweight function ‫ݐ‬ℎ1 = ‫ܫ‬ሺ‫,ݔ‬ ‫ݕ‬ሻ ∗ ቆ൬1 − ቀ ூሺ௫,௬ሻ ௙ ቁ ଶ ൰ ˄2ቇ ∗ 0.5 Step- 7.4: If th1 > MD(x,y) Perform Soft thresholding of the subband image. else Retain the pixel Step-8: Perform inverse stationary wavelet transform using ISWT2(). Step-9: Calculate PSNR, RMSE, IQI, SSIM, MSD, DR, ENL, FOM, CC. Step-10: Stop 5. IMAGE METRICS 5.1 Peak Signal to Noise Ratio Peak Signal to Noise Ratio (PSNR)[20] is one of the most essential statistical parameter for quality mea- surement of an image or signal. It is used as an esti- mate to measure the quality of objective difference between the noisy and the denoised image. The basic idea is to compute a single number that reflects the quality of the reconstructed image. Higher PSNR value provides higher image quality. It is calculated as; ܴܲܵܰ = 10 ∗ ݈‫01݃݋‬ ቀ ଵ ெௌா ቁ (4) 5.2 Root Mean Square Error Root Mean Square Error (RMSE)[21], is an estima- tor in to quantify the amount by which a noisy image differs from noiseless image. RMSE is computed by averaging the squared intensity of the noisy image and the denoised image, where error is the difference between desire quantity and estimated quantity. Hav- ing a RMSE value of zero is ideal. ܴ‫ܧܵܯ‬ = ඨ∑ ∑ ቀ୤ሺ୶,୷ሻି ୤ሺ୶,୷ሻቁ మ ౤ ౯సభ ౣ ౮సభ ୫∗୬ (5) 70
  • 4. 5.3 Image Quality Index The Image Quality Index (IQI)[20] is a measure of comparison between original and distorted image. It is divided into three parts: luminance ݈ሺ‫,ݔ‬ ‫ݕ‬ሻ, contrast ܿሺ‫,ݔ‬ ‫ݕ‬ሻ, and structural comparisons ‫ݏ‬ሺ‫,ݔ‬ ‫ݕ‬ሻ as men- tioned in equation (6),(7) and (8). The dynamic range for IOI(x, y) is [-1, 1]. ݈ሺ‫,ݔ‬ ‫ݕ‬ሻ = ଶఓೣఓ೤ µ೉ మ ା µ೤ మ (6) ܿሺ‫,ݔ‬ ‫ݕ‬ሻ = ଶ ఙೣ ఙ೤ ఙೣ మାఙ೤ మ (7) ‫ݏ‬ሺ‫,ݔ‬ ‫ݕ‬ሻ = ଶఙೣ೤ ఙೣାఙ೤ (8) ‫ܫܳܫ‬ሺ‫,ݔ‬ ‫ݕ‬ሻ = ݈ሺ‫,ݔ‬ ‫ݕ‬ሻ. ܿሺ‫,ݔ‬ ‫ݕ‬ሻ. ‫ݏ‬ሺ‫,ݔ‬ ‫ݕ‬ሻ = ସఓೣఓ೤ఓೣ೤ ሺµ೉ మ ା µ೤ మሻሺఙೣ మାఙ೤ మሻ (9) 5.4 Structural Similarity Index The Structural Similarity Index (SSIM)[20] measures the similarity between two images which is more consistent with human perception than conventional techniques. The range of values for the SSIM lies between −1, for a bad and 1 for a good similarity between the original and despeckled images, respec- tively. ܵܵ‫ܯܫ‬ሺ‫,ݔ‬ ‫ݕ‬ሻ = ൫ଶఓೣఓ೤ା௖ଵ൯ሺଶఙೣ೤ା௖ଶሻ ሺµ೉ మ ା µ೤ మ ା௖ଵሻሺఙೣ మାఙ೤ మା௖ଶሻ (10) 5.5 Noise Mean Value (NMV), Noise Standard Dev- iation (NSD) Noise Variance determines the contents of the speckle in an image. A lower variance gives a “cleaner” image as more speckle is reduced, it is not necessarily that it should depend on the intensity of the image. The formulas for the NMV and NSD cal- culation are as follows[22]. ܰ‫ܸܯ‬ = ∑ ௙೏ሺ௥,௖ሻೝ,೎ ௥∗௖ (11) ܰܵ‫ܦ‬ = ට ∑ ሺ௙೏ሺ௥,௖ሻିேெ௏ሻమ ೝ,೎ ௥∗௖ (12) 5.6 Pratt’s Figure of Merit (FOM) It measures edge pixel displacement between each filtered image Ifilt and the original image Iorig. It is defined as[23]: FOM = ଵ ୫ୟ୶ ሺே೑೔೗೟, ಿ೚ೝ೔೒ሻ, ∑ ଵ ଵାௗ೔ మఈ ௡ ௜ୀଵ (13) where Nfilt and Norig are the number edge pixels in edge maps of Ifilt and Iorig. Parameter α is set to a con- stant 1/9, and di is the euclidean distance between the detected edge pixel and the nearest ideal edge pixel. The FOM metric measures how well the edges are preserved throughout the filtering process. This me- tric has a significant relationship with the overall quality score at 1% significance level. 5.7 Equivalent Number of Looks Equivalent Numbers of Looks (ENL)[24] is a measure to estimate the speckle noise level in the image. The value of ENL depends on the size of the tested region; theoretically a larger region will produces a higher ENL value than a smaller region. The formula for the ENL is ‫ܮܰܧ‬ = ேெ௏మ ேௌ஽మ (14) 5.8 Deflection Ratio (DR) The formula for the deflection ratio[25] calculation is; ‫ܴܦ‬ = ଵ ோ∗஼ ∑ ሺ௙೏ሺ௥,௖ሻିேெ௏ሻ ேௌ஽௥,௖ (15) After speckle reduction the deflection ratio should be higher at pixels with stronger reflector points and lower elsewhere. 5.9 Correlation Coefficient (CC) For digital images, correlation[26] is a measure of the strength and direction of a linear relationship between two variable. A correlation of 1 indicates a perfect one-to-one linear relationship and -1 indicates a negative relationship. The square of the correlation coefficient describes the variance between two va- riables in a linear fit. The Pearson’s correlation coef- ficient is defined as; ‫ݎ‬ = ∑ ሺ௙೔ ି௙೘ሻሺ௙̅೔ି௙̅೘ሻ೔ ට∑ ሺ௙೔ ି௙೘ሻ೔ మ ට∑ ሺ௙̅೔ି௙̅೘ሻ೔ మ (16) where, ݂௜ and ݂̅௜ are intensity values of ith pixel in noisy and denoised image respectively. Also, ݂௠ and 71
  • 5. ݂̅௠ are mean intensity values of noisy and denoised image respectively. 5.10 Execution Time Execution Time(ET) [27]of a denoising filter, is de- fined as the time taken by a processor to execute an algorithm when no other software, except the operat- ing system (OS), runs on it. Execution time is re- ferred with respect to the system’s clock time-period. The execution time taken by a filtering algorithm should be low for real-time image processing appli- cations. Hence, when all metrics give the identical values then a filter with lower execution time is bet- ter than a filter having higher execution time. 6. RESULTS AND DISCUSSIONS An objective evaluation of the existing thre- sholding techniques and the proposed threshold tech- niques is listed in Table 1. The PSNR value is very high for the proposed threshold technique. The next highest PSNR value is generated by proposed thre- shold technique. The proposed threshold has a very low RMSE value compared with other thresholding techniques. It also indicates that the proposed thre- shold is capable of removing more speckle noise equally maintaining low error between the original and denoised image. High image quality index is exhibited by Vishu shrink, indicating that the denoised image has a better variation between the original and denoised image. If the structural similarity index is equal to one, then it is an indication that the structural detail of the original image is preserved even after denois- ing. Hence proposed threshold has produced a value which is very close to one. The NMV value and NSD values of the pro- posed threshold has produced the same value, com- paratively less than other thresholding techniques. It indicates that the speckle noise content of the de- noised images is very less. ENL value and DR values of both the proposed threshold are same and less. When compared with other thresholding techniques Bayes Shrink has pro- duced a higher ENL value indicating that the original and denoised image has more similar features. But the proposed threshold as exhibited a higher DR val- ue indicating that there is more deflection along the edges in the denoised image. The FOM value is high in proposed threshold whereas the existing threshold shrinkages were not evaluated using this parameter. Similarly, the CC value is high in proposed threshold. The execution time for the proposed algorithm is 4.173seconds Table 1. Comparison Of Existing Denoising Filters With Proposed Threshold Existing Denoising Filters Assessment Parameters Visu Shrink Normal Shrink Bayes Shrink Sure Shrink Proposed PSNR 31.65 29.28 38.70 29.60 68.2510 RMSE 11.68 10.23 12.67 10.81 0.0098 IQI 0.5902 0.3812 0.3938 0.4645 0.1823 SSIM 0.7882 0.8214 0.8532 0.8953 0.9997 NMV 11.56 9.61 21.72 13.48 0.2205 NSD 3.30 2.01 6.75 4.23 0.2164 ENL 1.2685 3.6853 7.6893 3.5742 1.0378 DR 0.0031 0.0381 0.0610 0.0461 0.8023 FOM NA NA NA NA 0.2498 CC NA NA NA NA 0.5847 72
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