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PERFORMANCE ANALYSIS OF NEUTROSOPHICSET APPROACH OF MEDIAN FILTERING FOR MRIDENOISING
J. Mohan1, V. Krishnaveni1, Yanhui Guo2
E-mail addresses:
[email protected], [email protected], [email protected]
1Department of Electronics and Communication Engineering, PSG College ofTechnology, Coimbatore, Tamilnadu, India 641 004
2Department of Radiology, University of Michigan, Ann Arbor, MI 48109, USA
* Corresponding Author: J. Mohan, Research Scholar, Department of Electronics andCommunication Engineering, PSG College of Technology, Coimbatore, Tamilnadu,India 641 004
Tel.: (+91) 98 407 13417E-Mail: [email protected] (J. Mohan)
Abstract
In this paper, the performance analysis of the neutrosophic set (NS) approach of
median filtering for removing Rician noise from magnetic resonance image is
presented. A Neutrosophic Set (NS), a part of neutrosophy theory, studies the origin,
nature, and scope of neutralities, as well as their interactions with different
ideational spectra. Now, we apply the neutrosophic set into image domain and
define some concepts and operators for image denoising. The image is transformed
into NS domain, described using three membership sets: True (T), Indeterminacy (I)
and False (F). The entropy of the neutrosophic set is defined and employed to measure
the indeterminacy. The -median filtering operation is used on T and F to decrease the
set indeterminacy and remove noise. The experiments have conducted on simulated
MR images from Brainweb database and clinical MR images corrupted by Rician
noise. The results show that the NS median filter produces better denoising results in
terms of qualitative and quantitative measures compared with other denoising methods,
such as the anisotropic diffusion filter and the total variation minimization.
Keywords:
Denoising, magnetic resonance imaging, median, neutrosophic set, PSNR, Rician
distribution, SSIM.
1. Introduction
Magnetic resonance imaging (MRI) plays an important role in modern medical
diagnosis because of their noninvasive and high resolution techniques [1]. However,
the incorporated noise during image acquisition degrades the human interpretation or
computer-aided analysis of the images. Time averaging of image sequences aimed at
improving the signal-to-noise ratio (SNR) would result in additional acquisition time
and reduce the temporal resolution [2]. Therefore, denoising which is an important
preprocessing step should be performed to improve the image quality for more accurate
diagnosis.
Noise in Magnetic resonance (MR) images obeys a Rician distribution [3]. In
contrast to Gaussian additive noise, Rician noise is signal dependent and is therefore
more difficult to separate from the signal. For low SNR, the Rician distribution tends to
the Rayleigh distribution. For high SNR the Rician distribution tends to the Gaussian
distribution [3].
Many MRI denoising methods have been proposed in prior research. Some popular
approaches include the anisotropic diffusion filter [2, 4-7], wavelet based denoising
techniques [8-13], total variation minimization scheme [14], the non local means filter
[15-23] and hybrid approaches [24, 25]. This paper deals the performance analysis of
the neutrosophic set (NS) approach of median filtering to remove the Rician noise in
MRI. First the noisy MRI is transformed into the neutrosophic set and the - median
filtering is employed to reduce the indetermination degree of the image, which is
measured by the entropy of the indeterminate subset, after filtering, the noise will be
removed.
2. Methodology
Neutrosophy, a branch of philosophy introduced in [26] as a generalization of
dialectics, studies the origin, nature and scope of neutralities, as well as their
interactions with different ideational spectra. In neutrosophy theory, every event has
not only a certain degree of the truth, but also a falsity degree and an indeterminacy
degree that have to be considered independently from each other [26]. Thus, a theory,
event, concept, or entity, A is considered with its opposite AAnti and the
neutrality ANeut . ANeut is neither A nor AAnti . The ANeut and
AAnti are referred to as ANon . According to this theory, every idea A is
neutralized and balanced by AAnti and ANon [24]. NS provides a powerful tool
to deal with indeterminacy.
Neutrosophic set theory has been applied to image thresholding, image
segmentation and image denoising applications [27-32]. Cheng and Guo [27] proposed
a thresholding algorithm based on neutrosophy, which could select the thresholds
automatically and effectively. The NS approach of image segmentation for real images
discussed in [28]. In [29], some concepts and operators were defined based on NS and
applied for image denoising. It can process not only noisy images with different levels
of noise, but also images with different kinds of noise well. The method proposed in
[29] adapted and applied for MRI denoising [30]. In this paper, the NS median filter’s
performance have been analyzed with simulated and clinical MR images and compared
with the anisotropic diffusion filter (ADF) and the total variation (TV) minimization
scheme.
2.1 Neutrophic Set
Let U be a Universe of discourse and a neutrosophic set A is included in U . An
element x in set A is noted as FITx ,, . T, I and F are called the neutrosophic
components. The element FITx ,, belongs to A in the following way. It is %t true in
the set, %i indeterminate in the set, and %f false in the set, where t varies inT , ivaries
in I and f varies in F [29].
2.2 Transform the image into neutrosophic set
Let U be a Universe of discourse and W is a set of U , which is composed by
bright pixels. A neutrosophic image NSP is characterized by three membership sets
FIT ,, . Then the pixel P in the image is described as FITP ,, and belongs to W in
the following way: It is t true in the set, iindeterminate in the set, and f false in the
set, where t varies in T , ivaries in I and f varies in F . Then the pixel ),( jiP in the
image domain is transformed into the neutrosophic set domain
)},(),,(),,({),( jiFjiIjiTjiPNS . ),(),,( jiIjiT and ),( jiF are the probabilities
belong to white pixels set, indeterminate set and non white pixels set respectively [29],
which are defined as:
minmax
min,),(
gg
gjigjiT
(1)
2
2
2
2,
1,
wi
wim
wj
wjnnmg
wwjig (2)
minmax
min,),(
ji
jiI (3)
)),(),((, jigjigabsji (4)
),(1),( jiTjiF (5)
where ),( jig is the local mean value of the pixels of the window. ),( ji is the absolute
value of difference between intensity ),( jig and its local mean value ),( jig .
2.3 Neutrosophic image entropy
For an image, the entropy is utilized to evaluate the distribution of the gray levels.
If the entropy is the maximum, the intensities have equal probability. If the entropy is
small, the intensity distribution is non-uniform [29]. Neutrosophic entropy of an image
is defined as the summation of the entropies of three subsets IT , and F :
FITNS EnEnEnEn (6)
)(ln)(}max{
}min{ipipEn T
T
TiTT
(7)
)(ln)(}max{
}min{ipipEn I
I
IiII
(8)
)(ln)(}max{
}min{ipipEn F
F
FiFF
(9)
where TEn , IEn and FEn are the entropies of sets IT , and F respectively. )(ipT , )(ipI
and )(ipF are the probabilities of elements in IT , and F respectively, whose values
equal to i.
2.4 - median filtering operation
The values of ),( jiI is employed to measure the indeterminate degree of element
),( jiPNS . To make the set I correlated with T and F , the changes in T and F
influence the distribution of element in I and vary the entropy of I [29, 30].
A median filtering operation for NSP , NSP̂ , is defined as:
)ˆ,ˆ),(ˆ(ˆ FITPPNS (10)
IT
ITT ˆˆ (11)
),(,ˆ,),(
nmTmedianjiTjiSnm
(12)
IF
IFF ˆˆ (13)
),(,ˆ,),(
nmFmedianjiFjiSnm
(14)
minˆmaxˆ
minˆˆ ,,ˆ
TT
TTji
jiI
(15)
)),(ˆ),(ˆ(),(ˆ jiTjiTabsjiT
(16)
2
2
2
2,ˆ1
,ˆ wi
wim
wj
wjnnmT
wwjiT (17)
where ),(ˆ jiT is the absolute value of difference between intensity ),(ˆ jiT and its local
mean value ),(ˆ jiT at ),( ji after - median filtering operation.
The summary of neutrosophic set approach of median filtering for MRI denoising
[30] is described as below (see Fig. 1):
1. Transform the image into NS domain;
2. Use - median filtering operation on the true subsetT to obtain T̂ ;
3. Compute the entropy of the indeterminate subset )(,ˆˆ iEnII ;
4. if
)(
)()1(
ˆ
ˆˆ
iEn
iEniEn
I
II, go to 5; Else TT ˆ , go to 2;
5. Transform subset T̂ from the neutrosophic domain into the gray level domain.
3. Materials
The experiments were conducted on two MRI datasets. The first data set consists of
simulated MR images obtained from the Brainweb database [31]. The second data set
consists of clinical MRI collected from PSG Institute of Medical Sciences and
Research (PSG IMS & R), Coimbatore, Tamilnadu, India. Simulated MR images are
used collectively as the reference to evaluate and compare the validity of the proposed
technique. It evicts the data dependency enabling precise comparative studies. The
data set consists of T1 weighted Axial, T2 weighted Axial and T1 weighted Axial with
multiple sclerosis (MS) lesion volumes of 181×217×181 voxels (voxels resolution is
1mm3), which are corrupted with different level of Rician noise (1% to 15 % of
maximum intensity).
In the clinical data sets, the images are acquired using Siemens Magnetom Avanto 1.5T
Scanner.
a. T2 weighted Coronal MR image normal brain with TR = 4000ms, TE = 104
ms, 5mm Thick and 512×512 resolution.
b. T1 weighted Axial MR image of granulomatous lesion pathology with TR =
550ms, TE = 8.7 ms, 5mm Thick and 512×512 resolution.
Rician noise was generated by Gaussian noise to real and imaginary parts and then
computing the magnitude of the image
Fig. 1. Neutrosophic Set approach of MRI denoising
4. Validation Strategies
The performance of the denoising algorithm is measured by using the quality metrics
such as peak-signal-to-noise ratio (PSNR), the Structural Similarity (SSIM) index [32].
The peak signal to noise ratio in decibel (dB) is measured using the following formula:
2
10
10
2
255
,,log10
WH
jiIjiIPSNR
Hii
Wjj d (24)
where jiI , and jiI d , represent the intensities of pixels ji, in the original image
and denoised image respectively. The higher the PSNR, the better the denoising
algorithm is. SSIM gives the measure of the structural similarity between the original
and the denoised images and are in the range of 0 to 1. The SSIM works as follows: Let
xand y be two non negative images, where as one has perfect quality. Then, the SSIM
can serve as a quantitative measure of the similarity of the second image. The system
separates the task of similarity measurement into three comparisons: luminance,
contrast and structure. It can be defined as
2
221
22
21 22,
CC
CCyxSSIM
yxyx
xyyx
(25)
where x and y are the estimated mean intensity and x and y are the standard
deviations respectively. xy can be estimated as
yi
N
ixixy yx
N
11
1(26)
1C and 2C in Eqn. 25 are constants and the values are given as 2
11 LKC and
2
22 LKC where 1, 21 KK is a small constant and L is the dynamic range of the
pixel values (255 for 8 bit gray scale images).
The residual image is also obtained by subtracting the denoised image from the
noisy image [18]. The residual image is required to verify the traces of anatomical
information removed during denoising. Hence, this reveals the excessive smoothing
and the blurring of small structural details contained in the image.
5. Results and Discussions
5.1 Performance Comparison
The performance of the proposed method has been compared with the anisotropic
diffusion filter (ADF) and total variation (TV) minimization scheme.
5.1.1 Anisotropic diffusion filter
Perona and Malik [33] introduced the anisotropic diffusion filtering method. In this
approach the image u is only convolved in the direction orthogonal to the gradient of
the image which ensures the preservation of edges. The iterative denoising process of
initial image 0u can be expressed as
)()0,(
)),(),((),(
0 xuxu
txutxcdivt
txu(27)
where ),( txu is the image gradient at voxel xand iteration t , ttxu ),( is the partial
temporal derivation of ),( txu and
2
),(
),((),( K
txu
etxugtxc
(28)
where K is the diffusivity parameter.
5.1.2. Total Variation minimization scheme
The difficult task to preserve edges while correctly denoising constant areas has been
addressed also by Rudin, Osher and Fatemi. They proposed to minimize the TV norm
subject to noise constraints [34], that is
dxtxuuu
),(argˆ min3
(29)
subject to
22
0033
)()(0))()((
dxxuxuanddxxuxu (30)
where 0u is the original noisy image, u is the restored image and is the standard
deviation of the noise. In this model, the TV minimization tends to smooth inside the
image structures while keeping the integrity of boundaries. The TV minimization
scheme can be expressed as an unconstrained problem
3 33
2
0 )()()(argˆ min dxxuxudxxuuu
(31)
where is a Lagrange multiplier which controls the balance between the TV norm and
the fidelity term. Thus acts as the filtering parameter. Indeed, for high values for
the fidelity term is encouraged. For small values for the regularity term is desired.
All these methods were implemented using MATLAB 2010a (The Math Works, Inc)
in Windows XP 64-bit Edition (Pentium Dual core 2.4 GHz with 4GB of RAM). For
AD filter, the parameter K varies from 0.05 to 1 with a step of 0.05 and the number of
iterations varies from 1 to 15. For TV minimization, the parameter varies from 0.01 to
1 with step of 0.01 and the number of iterations varies from 1 to 10.
5.2 Experiment on Simulated data set
The detailed images and the residual images of the denoising results obtained for
the T1 weighted and T2 weighted axial images corrupted by 9% and 15% Rician noise
respectively shown in Figs. 2 and 3. In Fig. 4, the denoising images and its residual
images of the different denoising methods for the part of the T1 weighted with MS
lesion corrupted by 9% Rician noise is shown. These figures are provided the visual
comparison of the results. The residual images reveal the excessive smoothing and the
blurring of small structural details contained in the image. From these results, the
proposed method surpassed all other methods at high noise levels on the three types of
data in terms of producing more detailed denoised image in which all the distinct
features and small structural details are well preserved.
The PSNR and the SSIM values obtained for the T1 weighted, T2 weighted axial
images with different noise levels (1% to 15%) using the aforementioned denoising
techniques are given in Fig. 5. As the level of noise increases, the performance of the
proposed NS wiener filter shows significant improvement over the other denoising
methods. Table 1 shows a comparison of the experimental results for the denoising
methods based on PSNR and SSIM for the T1 weighted, T2 weighted and T1 weighted
with MS lesion axial images corrupted by 7%, 9% and 15% respectively. Higher the
value of PSNR in dB and higher the value of SSIM shows that the proposed filter
perform superior that the other denoising methods.
5.3 Experiment on Clinical data set
The detailed images and the residual images of the denoising results obtained for the
T2 weighted coronal MR image of normal brain and T1 weighted axial brain with
granulomatous lesion pathology corrupted by 9% and 15%of the Rician noise level
respectively are shown in Figs. 6 and 7. The detailed features and edges in the image
are well preserved by the proposed method compared with other denoising methods.
As the noise level increases, significant change in the performance of the denoising
results can be observed from Fig. 8. The comparison of the denoising techniques based
on the similarity metrics between original and denoised images such as PSNR and
SSIM values for T2 weighted Coronal MRI of normal brain and the T1 weighted axial
brain with granulomatous lesion pathology corrupted by 9% and 15% of the Rician
noise level respectively are tabulated in Table 2. This confirms that NS median is
superior with respect to PSNR and SSIM.
Table 1
Comparison of the denoising techniques based on the performance metrics for
simulated MR images
MR Image DenoisingMethods
Performance MetricsPSNR(dB) SSIM
T1- Weighted brain MRIcorrupted by 7% Rician noise
TV 22.61 0.9538
ADF 23.64 0.9662
NS Median 29.85 0.9839
T2-Weighted brain MRIcorrupted by 9% Rician noise
TV 20.4 0.9455
ADF 19.85 0.9546
NS Median 24.95 0.9767
T1- Weighted brain MRI withMS lesion corrupted by 15%Rician noise
TV 16.15 0.7914
ADF 16.17 0.8425
NS Median 23.09 0.9634
Original Noisy
TV ADF NS Median
TV Residual ADF Residual NS Median Residual
Fig. 2. Denoising results of simulated T1 weighted axial MRI corrupted by 9% of
Rician noise.
Original Noisy
TV ADF NS Median
TV Residual ADF Residual NS Median Residual
Fig. 3. Denoising results of simulated T2 weighted axial MRI corrupted by 15% of
Rician noise
Original Noisy
TV ADF NS Median
TV Residual ADF Residual NS Median
Residual
Fig. 4. Denoising results of small part of the T1 weighted axial MRI with MS lesion
corrupted by 9% of Rician noise.
Fig. 5. Comparison results for Brainweb simulated MR images. Left: PSNR of the
compared methods for different image types and noise levels. Right: SSIM of the
compared methods for different image types and noise levels
0 5 10 1515
20
25
30
35
40
45
% of Noise level
PSNR
in d
B
T1w
TVADFNS Median
0 5 10 150.75
0.8
0.85
0.9
0.95
1
% of Noise level
SSIM
T1w
TVADFNS Median
0 5 10 1515
20
25
30
35
40
45
% of Noise level
PSNR
in d
B
T2w
TVADFNS Median
0 5 10 150.85
0.9
0.95
1
% of Noise level
SSIM
T2w
TVADFNS Median
Original Noisy
TV ADF NS Median
TV Residual ADF Residual NS Median Residual
Fig. 6. Denoising results of clinical T2 weighted Coronal MRI corrupted by 9% of
Rician noise.
Original Noisy
TV ADF NS Median
TV Residual ADF Residual NS Median Residual
Fig. 7. Denoising results of clinical T1 weighted axial MRI with granulomatous
lesion corrupted by 9% of Rician noise.
Clinical T1 weighted axial MRI with granulomatous lesion pathology
Clinical T2 weighted Coronal MRI
Fig. 8. Comparison results for clinical MR images. Left: PSNR of the compared
methods for different noise levels. Right: SSIM of the compared methods for different
noise levels
0 5 10 1520
25
30
35
40
45
50
55
% of Noise level
PSNR
in d
BTVADFNS Median
0 5 10 150.65
0.7
0.75
0.8
0.85
0.9
0.95
1
% of Noise level
SSIM
TVADFNS Median
0 5 10 1520
25
30
35
40
45
50
% of Noise level
PSNR
in d
B
TVADFNS Median
0 5 10 150.7
0.75
0.8
0.85
0.9
0.95
1
% of Noise level
SSIM
TVADFNS Median
Table 2
Comparison of the denoising techniques based on the performance metrics for
clinical MR images
6. Conclusion
The performance analysis of the neutrosophic set approach median filtering for
removing Rician noise from MR images is presented in this paper. The image is
described as a neutrosophic set using three membership sets T, I and F. The entropy in
neutrosophic image domain is defined and employed to measure the indetermination.
The median filter is applied to reduce the set’s indetermination and to remove the noise
in the MR image.
The performance of the proposed denoising filter is compared with ADF and TV and
NLM based on PSNR and SSIM. The experimental results demonstrate that the
proposed approach can remove noise automatically and effectively. This filtering
method tends to produce good denoised image not only in terms of visual perception but
MR Image DenoisingMethods
Performance MetricsPSNR(dB) SSIM
T2- Weighted Coronal brainMRI corrupted by 9% Riciannoise
TV 29.7 0.7357
ADF 30.17 0.8019
NS Median 37.27 0.9674
T1-Weighted axial brainMRI with granulomatouslesion pathology corruptedby 15% Rician noise
TV 24.97 0.6911
ADF 25.02 0.7791
NS Median 34.37 0.9702
also in terms of the quality metrics. In the clinical MRI with granulomatous lesion
pathology, the filter preserves the major visual signature of the given pathology.
Acknowledgements
The authors would like to thank Dr. V. Maheswaran of PSG Institute of Medical
Sciences and Research (PSG IMS&R) Coimbatore, Tamilnadu, India for providing us
the clinical MRI data and for providing his opinion on the diagnostic details of the
denoised images.
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