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PERFORMANCE ANALYSIS OF NEUTROSOPHIC SET APPROACH OF MEDIAN FILTERING FOR MRI DENOISING J. Mohan 1 , V. Krishnaveni 1 , Yanhui Guo 2 E-mail addresses: [email protected], [email protected], [email protected] 1 Department of Electronics and Communication Engineering, PSG College of Technology, Coimbatore, Tamilnadu, India 641 004 2 Department of Radiology, University of Michigan, Ann Arbor, MI 48109, USA * Corresponding Author: J. Mohan, Research Scholar, Department of Electronics and Communication Engineering, PSG College of Technology, Coimbatore, Tamilnadu, India 641 004 Tel.: (+91) 98 407 13417 E-Mail: [email protected] (J. Mohan)

PERFORMANCE ANALYSIS OF NEUTROSOPHIC SET APPROACH … · 1. Introduction Magnetic resonance imaging (MRI) plays an important role in modern medical diagnosis because of their noninvasive

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Page 1: PERFORMANCE ANALYSIS OF NEUTROSOPHIC SET APPROACH … · 1. Introduction Magnetic resonance imaging (MRI) plays an important role in modern medical diagnosis because of their noninvasive

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)

Page 2: PERFORMANCE ANALYSIS OF NEUTROSOPHIC SET APPROACH … · 1. Introduction Magnetic resonance imaging (MRI) plays an important role in modern medical diagnosis because of their noninvasive

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.

Page 3: PERFORMANCE ANALYSIS OF NEUTROSOPHIC SET APPROACH … · 1. Introduction Magnetic resonance imaging (MRI) plays an important role in modern medical diagnosis because of their noninvasive

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

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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].

Page 5: PERFORMANCE ANALYSIS OF NEUTROSOPHIC SET APPROACH … · 1. Introduction Magnetic resonance imaging (MRI) plays an important role in modern medical diagnosis because of their noninvasive

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

Page 6: PERFORMANCE ANALYSIS OF NEUTROSOPHIC SET APPROACH … · 1. Introduction Magnetic resonance imaging (MRI) plays an important role in modern medical diagnosis because of their noninvasive

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)

Page 7: PERFORMANCE ANALYSIS OF NEUTROSOPHIC SET APPROACH … · 1. Introduction Magnetic resonance imaging (MRI) plays an important role in modern medical diagnosis because of their noninvasive

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

Page 8: PERFORMANCE ANALYSIS OF NEUTROSOPHIC SET APPROACH … · 1. Introduction Magnetic resonance imaging (MRI) plays an important role in modern medical diagnosis because of their noninvasive

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

Page 9: PERFORMANCE ANALYSIS OF NEUTROSOPHIC SET APPROACH … · 1. Introduction Magnetic resonance imaging (MRI) plays an important role in modern medical diagnosis because of their noninvasive

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

Page 10: PERFORMANCE ANALYSIS OF NEUTROSOPHIC SET APPROACH … · 1. Introduction Magnetic resonance imaging (MRI) plays an important role in modern medical diagnosis because of their noninvasive

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.

Page 11: PERFORMANCE ANALYSIS OF NEUTROSOPHIC SET APPROACH … · 1. Introduction Magnetic resonance imaging (MRI) plays an important role in modern medical diagnosis because of their noninvasive

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.

Page 12: PERFORMANCE ANALYSIS OF NEUTROSOPHIC SET APPROACH … · 1. Introduction Magnetic resonance imaging (MRI) plays an important role in modern medical diagnosis because of their noninvasive

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.

Page 13: PERFORMANCE ANALYSIS OF NEUTROSOPHIC SET APPROACH … · 1. Introduction Magnetic resonance imaging (MRI) plays an important role in modern medical diagnosis because of their noninvasive

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

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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.

Page 15: PERFORMANCE ANALYSIS OF NEUTROSOPHIC SET APPROACH … · 1. Introduction Magnetic resonance imaging (MRI) plays an important role in modern medical diagnosis because of their noninvasive

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

Page 16: PERFORMANCE ANALYSIS OF NEUTROSOPHIC SET APPROACH … · 1. Introduction Magnetic resonance imaging (MRI) plays an important role in modern medical diagnosis because of their noninvasive

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.

Page 17: PERFORMANCE ANALYSIS OF NEUTROSOPHIC SET APPROACH … · 1. Introduction Magnetic resonance imaging (MRI) plays an important role in modern medical diagnosis because of their noninvasive

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

Page 18: PERFORMANCE ANALYSIS OF NEUTROSOPHIC SET APPROACH … · 1. Introduction Magnetic resonance imaging (MRI) plays an important role in modern medical diagnosis because of their noninvasive

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.

Page 19: PERFORMANCE ANALYSIS OF NEUTROSOPHIC SET APPROACH … · 1. Introduction Magnetic resonance imaging (MRI) plays an important role in modern medical diagnosis because of their noninvasive

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.

Page 20: PERFORMANCE ANALYSIS OF NEUTROSOPHIC SET APPROACH … · 1. Introduction Magnetic resonance imaging (MRI) plays an important role in modern medical diagnosis because of their noninvasive

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

Page 21: PERFORMANCE ANALYSIS OF NEUTROSOPHIC SET APPROACH … · 1. Introduction Magnetic resonance imaging (MRI) plays an important role in modern medical diagnosis because of their noninvasive

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

Page 22: PERFORMANCE ANALYSIS OF NEUTROSOPHIC SET APPROACH … · 1. Introduction Magnetic resonance imaging (MRI) plays an important role in modern medical diagnosis because of their noninvasive

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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