animagedenoisingmethodbasedonbm4dandganin3d...

Research ArticleAn Image Denoising Method Based on BM4D and GAN in 3DShearlet Domain

Shengnan Zhang 1 Lei Wang 1 Chunhong Chang 1 Cong Liu1 Longbo Zhang1

and Huanqing Cui 2

1School of Computer Science and Technology Shandong University of Technology Zibo 255000 China2Shandong Key Laboratory of Wisdom Mine Information Technology Shandong University of Science and TechnologyQingdao 266590 China

Correspondence should be addressed to Lei Wang wanglei0511sduteducn and Huanqing Cui cuihqsdusteducn

Received 14 January 2020 Revised 15 March 2020 Accepted 31 March 2020 Published 28 April 2020

Guest Editor Chunjia Han

Copyright copy 2020 Shengnan Zhang et al-is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

To overcome the disadvantages of the traditional block-matching-based image denoising method an image denoising methodbased on block matching with 4D filtering (BM4D) in the 3D shearlet transform domain and a generative adversarial network isproposed Firstly the contaminated images are decomposed to get the shearlet coefficients then an improved 3D block-matchingalgorithm is proposed in the hard threshold and wiener filtering stage to get the latent clean images the final clean images can beobtained by training the latent clean images via a generative adversarial network (GAN)Taking the peak signal-to-noise ratio(PSNR) structural similarity (SSIM for short) of image and edge-preserving index (EPI for short) as the evaluation criteriaexperimental results demonstrate that the proposed method can not only effectively remove image noise in high noisy envi-ronment but also effectively improve the visual effect of the images

1 Introduction

-e typical image denoising methods can be commonlyclassified into three schemes the filtering-based methodsthe decomposition-based methods and dictionary learning-basedmethods [1]-e classical filtering includes themedianfilter [2] and the wiener filter [3] -e basic principle ofdecomposition-based methods is to decompose the con-taminated images into low-pass and high-pass subbands andthen separate the image noise by manipulating the obtainedcoefficients And the wavelet transform is the typical de-composition tool -e basic principle of the dictionarylearning-based methods is to sparsely represent the noisyimage by overcomplete atoms and only the large repre-sentation coefficients are used to reconstruct the originalimage while the small coefficients are discarded [4ndash6]Another famous strategy for denoising is based on the self-similarity of the image such as the block-matching and 3Dfiltering (BM3D) algorithm [7] -e above methods usuallyperform well but they will lose the edge texture and other

details and result in blur and block effect when the noise isheavy

Nowadays since the adaptive space estimation strategyfor nonlocal mean [8]can effectively alleviate the highcomplexity and low efficiency of nonlocal mean filtering thenonlocal similarity becomes an effective feature for imagedenoising [9] Its main disadvantage however is the loss ofthe directions So the geometric regularity of images cannotbe effectively captured to sparsely represent the features ofthe original images [10] Besides the computation of thenonlocal similarity is implemented only in the spatial do-main As reported the multiscale geometric transforma-tions such as contourlet transform can greatly help tosuppress the heavy noise in the frequency domain [11] Butthis scheme is limited to the mathematical properties of theselected transformation On the other hand the deeplearning technologies such as the convolutional neuralnetwork [12] and the generative adversarial network [13]have achieved great success in the areas of image classifi-cation target recognition [14] and image fusion [15]

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 1730321 11 pageshttpsdoiorg10115520201730321

-erefore a novel image denoising method based onblock matching with 4D filtering in the 3D shearlet trans-form domain and the generative adversarial network isproposed -e 3D shearlet transform provides bettermathematical properties than the commonly used wavelet orcontourlet to capture the anisotropic features of images indifferent scales and directions In addition the traditionalBM3D is extended into four 4D space which can effectivelyimprove the edge and texture details of the images -eoutput of the BM4D is used as the input to the designedgenerative adversarial network to make full use of its goodlearning ability

-e remainder of this paper is organized as follows -edetails of the proposed method are presented in Section 2Experimental results and discussions are shown in Section 3Finally the whole paper is concluded in Section 4

2 Methodology

21 e Architecture of Proposed Method -e overallstructure of the proposed method is shown in Figure 1-rough multiscale decomposition and directional parti-tion the 3D shearlet coefficients are obtained -en thecoefficients are input into the hard threshold and wienerfiltering contained in the BM4D model For the hardthreshold the similar cubes are extracted from volumeobservation and then they are combined together If thedistance between the cubes is smaller than the settingthreshold the collaborative filters are carried out to apply onthe similar cubes In the aggregation process the basicvolume estimation is generated by the adaptive convexcombination For the process of the wiener filtering theaggregation is implemented by the inversion of the shearlettransformation Finally the clean results are obtained by thedesigned generative adversarial network

22e 3D Shearlet Transform In the 3D space the shearletregion Pi(i 1 2 3) is obtained by combining the functionsystem associated with the pyramid region in the 3D Fourierspace R3

For d 1 2 3 l (l1 l2) isin Z2 the 3D shear systemassociated with Pd in the shearlet region is defined as a set

ψdjlk detA(d)

11138681113868111386811138681113868

11138681113868111386811138681113868j2ψ(d)

B(d)l A

(d)1113872 1113873

jx minus k1113874 1113875 j isin Z1113882

l l1 l2( 1113857 isin Λj k isin Z3 d 1 2 31113967

(1)

where ψ(d) isin L2(R3) (l1 l2) minus 2j le l1 l2 le 2j1113864 1113865 andΛj sub Z2Ad is the specific anisotropic expansion matrix and Bd

l is thespecific shearmatrixMore details can be found in the literature[16 17] In Figure 2 a shearlet in 3D space is shown

23 e Improved 3D Block-Matching Algorithm Letz X⟶ R be the noise form

z(x) y(x) + η(x) x isin X (2)

where y is the original unknown volume signal x sub Z3 is the3D signal and η(middot) sim N(0 σ2) is an independent and

uniformly distributed Gaussian noise whose standard de-viation is noted as σ

-e goal of the improved 3D block-matching algorithm[18]was to obtain the estimation 1113954y of y from the noiseobservations -e implementation of the improved algo-rithm was divided into two cascade stages the hardthreshold and wiener filtering [19] each of which includes 3steps grouping collaborative filtering and aggregation

For the hard threshold let CzxR represent a cube whose

volume is L times L times L where L isin N was extracted from z -esimilarity between two cubes is measured by photometricdistance

d Czxi

Czxj

1113874 1113875 Cz

ximinus Cz

xj

2

2L3

(3)

where middot represents the summation of the squareddifferences between the corresponding intensities of thetwo cubes and the denominator L3 is the normalizationfactor No prefiltering is performed before cube matchingso the similarity of noise observations can be directlytested

In the grouping step cubes similar to each other areextracted from z and combined to form a group for eachcube Cz

xR If the distance between two cubes was not larger

than the predefined threshold τhtmatch the two cubes are

considered to be similar Similar to CzxR we here firstly define

a set that contains the indexes for the cubes as follows

SzxR

xi isin X d CzxR

Czxi

1113872 1113873le τhtmatch1113966 1113967 (4)

-en a four-dimensional group is built by the aboveformula

GzSz

xR

∐xiisinSz

xR

Czxi

(5)

where the reference cube (represented by R) matches a set ofsimilar cubes located in the 3D data Particularly the co-ordinates xR and xi correspond to the tail and head of thearrow connecting the cubes in formula (4) respectivelyNote that since the distance from any cube to itself is always0 according to the definition of formula (4) each formula(5) must contain at least its reference cube

In the collaborative filtering step a joint four-dimen-sional transformation Tht

4D was applied to each dimension ofequation (5) respectively -en by a hard threshold op-erator cht with the threshold σλ4D the obtained four-di-mensional group spectrum is

cht

Tht4D G

zSz

xR

1113874 11138751113874 1113875 (6)

Representing the filter group it is transformed into thefollowing form

Thtminus1

4D cht

Tht4D G

zSz

xR

1113874 11138751113874 11138751113874 1113875 1113954Gy

SzxR

∐xiisinSz

xR

1113954Cy

xi (7)

For each unknown volume data y the estimated 1113954Cy

xiof

the original Cyxiwas extracted separately Formula (7) was an

overcompleted representation of the denoising data because

2 Mathematical Problems in Engineering

there may be overlap between the cubes in the same groupand different groups

In the aggregation step the redundancy is used to generatea basic volume estimation by adaptive convex combination

1113954y 1113936xRisinX 1113936xiisinSz

xR

whtxR

1113954Cy

xi1113874 1113875

1113936xRisinX 1113936xiisinSzxR

whtxRχxi

1113874 1113875

(8)

where whtxR

is the group-related weight and χxi X⟶ 0 1

is the feature (indicator) function of the 1113954Cy

xidomain that is

χxi 1 at the coordinates of χxi

0 -e weight is defined as

whtxR

1

σ2NhtxR

(9)

where σ is the standard deviation of noise in z and NhtxR

is thenumber of nonzero coefficients in formula (6) Since thecoefficient always remains the same after doing the thresholdoperation that is Nht

xRge 1 the denominator of equation (9)

is never equal to zero -e numerical NhtxR

has two functionson the one hand it measures the sparsity of the thresholdspectrum in (5) and on the other hand it approximates thetotal residual noise variance of the group estimation in (6)As a result the groups that are in a high degree of correlationwill be given more weight while other groups with largerresidual noise are punished with less weight

For theWiener filtering the cube matching is performedwithin the basic estimation of 1113954yht In fact since the noiselevel in 1113954yhtis much smaller than that in the noise observationz it is expected to obtain the more reliable match to makethe packet data more sparse Formally for each reference

cube C1113954y

ht

xRextracted from the basic estimated 1113954yht its cube-like

coordinate set is constructed as follows

S1113954y

ht

xR xi isin X d C

1113954yht

xR C

1113954yht

xi1113874 1113875lt τwiematch1113882 1113883 (10)

-e collaborative filtering here is implemented as anempirical Wiener filter Similar to formula (6) it firstly uses

the coordinate set (9) to extract a set of G1113954y

ht

S1113954y

ht

xR

from 1113954yht and

then defines the empirical Wiener filter coefficients as

WS1113954y

ht

xR

Twie4D G

1113954yht

S1113954y

ht

xR

⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868

2

Twie4D G

1113954yht

S1113954y

ht

xR

⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868

2

+ σ2 (11)

where σ is the standard deviation of noise and Twie4D is a

transformation operator that is composed of four one-di-mensional linear transformations Such transformations areusually different from those in Tht

4D Subsequently the same

6080

4020

0

6080

4020

0

30

40

20

10

0

(a)

70

60

50

40

30

20

10

070 60 50 40 30 20 10 0

(b)

Figure 2 -e shearlet in 3D space and its projection (a) -e 3D shearlet and (b) the top view of the 3D shearlet

3D shearlettransform

3D shearlet coefficient BM4D

Hard threshold

Wiener filtering

Grouping

Collaborative filtering

Aggregation(adaptive convex

combination)

Grouping


Aggregation(3D shearlet inverse

transform)

GAN

Clean-clean reconstruction

Noise-clean reconstruction

Adversarial training

Multiscale decomposition

Directional subdivision

Figure 1 -e architecture of the proposed image denoising method

Mathematical Problems in Engineering 3

set of formula (10) is used to extract the second noise groupfrom the observed z noted as Gz

S1113954y

ht

xR

An element multipli-

cation is implemented between the spectrum of the noisegroup and the wiener filter coefficient formula (11) as thecoefficient shrinkage rate -e grouprsquos estimations are

1113954Gy

S1113954y

ht

xR

Twieminus1

4D WS1113954y

ht

xR

middot Twie4D G

z

S1113954y

ht

xR

⎛⎝ ⎞⎠⎛⎝ ⎞⎠ (12)

-en the inversion of the 3D shearlet transform [14] isapplied to shrink the spectrum -e final estimation of 1113954ywie

is generated by convex combination which is similar toformula (8) and formula (4) is replaced by formula (10)-e aggregation weight of the specific group estimation(11) is defined by the energy of the wiener filter coefficient(12)

wwiexR

σminus 2W

S1113954y

ht

xR

2

2

(13)

In this way each formula (13) provides an estimation ofthe total residual noise variance of the corresponding for-mula (12)

24 e Generative Adversarial Network for TrainingAfter obtaining the intermediate results we can obtain the finaldenoising image by training the generative adversarial network(GAN) [20] -e training process is shown in Figure 3

241 e Generator Network G -e generated networkgenerates a fake image from the noise image as shown inFigure 4 -e generation network consists of 11 cascadedconvolution layers which are trained to learn the labelimage and the residual image of the input image Internalconnection is introduced into each block to save infor-mation and reduce the training time In order to maintaingood performance and reduce computational complexitythe network adopts a bottleneck structure in which thenumber of the first feature mapping the middle layer andthe last layers are 64 layers According to suggestion fromreference [21 22] for low-level computer vision problemsa 3times3 convolution kernel is used in each convolution layerand the linear unit (ReLU) is used as the activationfunction

242 e Discriminator Network D As shown in Figure 5the discriminator network D is trained to distinguish thefake image and the real image It has four convolution blocksand two fully connected layers Each convolution blockconsists of the convolution layer the batch normalizationlayer and the ReLU activation function -e size of thecore K is 3times3 and the number of filters N increases from 64to 256-e step size S of each convolution layer is 2 to reducethe resolution of the image -e probability that the in-putting image is noiseless is generated by a fully connectedlayer of 1024 neurons

243 Adversarial Training -e aim is to use the adversarialstrategy to train a model to remove the image noisingAdversarial training is a way to train the generator networkGto generate samples from real data x sim pdata -e generatoris input into a noise variable zwith a distribution pZ andthen trained to learn the mapping to the data space -edistribution of the generator model is

pg sim G z θg1113872 1113873 (14)

where θg is the parameter of the generator network Whentraining a generator the essentially exception is to maximizethe probability that the samples match the data which can benoted as pdata(G(z θg))

To guarantee the above probability the discriminatornetwork D whose input is the data sample x and the outputis the D(x θd) should learn to distinguish the generatedsamples from real samples It must maximize the probabilityvalue assigned to the actual data samples and minimize theprobability value assigned to the generated samples that is

maxθd

Exsimpdata[log(D(x))] + EzsimpZ

[log(1 minus D(G(z)))]

(15)

Both the generator and discriminator networks are al-ternately trained and they try to cheat each other Finallywhen the generator has successfully learned how to generatethe samples from pdata the whole process is converged

In Figure 6 the experiments on four groups of colorimages show the necessity and effectiveness of using the GAN

3 Experimental Results and Discussions

In this section two groups of experiments are implementedto show the performance of the proposed method -eplatform is the Dell workstation M4800 with the Intel CPU25GHz and 32G RAM operating underMatlab and PythonPSNR [23] SSIM [24] and the edge-preserving index (EPIfor short) [25] are used as objective evaluation measure-ments PSNR can be computed by the following formula

PSNR(y 1113954y) 10 log10D2| 1113957X|

1113936xisin1113957X(1113954y(x) minus y(x))2

⎛⎝ ⎞⎠ (16)

where D is the peak of y 1113957X x isin X y(x)gt 10 middot D2551113864 1113865and | 1113957X| is the base of 1113957X

-e SSIM is defined as

SSIM(x y) l(x y)α

middot c(x y)β

middot s(x y)c (17)

where l(x y) c(x y) and s(x y) can be computed as

l(x y) 2uxuy + c1

u2x + u2

y + c1

c(x y) 2σxσy + c2

σ2x + σ2y + c2

s(x y) 2σxy + c3

σxy + c3

(18)


in which x and y are the reference image and the image to betested respectively ux and uyare the mean values of the twoimages σx and σy are the standard deviation σxy is thecovariance of x and y and c1 c2 and c3 are the smallconstants whose values are positive It is mainly to avoid the

instability when the denominator is 0 in the above formulaWhen α β c 1 and c3 c22 then we can get

SSIM(x y) 2uxuy + c11113872 1113873 2σxσy + c21113872 1113873

u2x + u2

y + c11113872 1113873 σ2x + σ2y + c21113872 1113873 (19)

Noiseimage

Fakeimage

Realimage

Generator network

Discriminatornetwork

Predictedlables

1 real 0 fake

Figure 3 Image denoising by training the GAN

Conv3 times 3 times 64


Conv3 times 3 times 16 Block2Conv

3 times 3 times 16 +ReLU Block3 Block4 Conv3 times 3 times 64

Block1

Noise image Fake image

Figure 4 -e structure of the generator network

Conv

Block2K = 3 N = 64 S = 2

ReLUBatch

specification layer

Block1Noise image

Real image

Block3K = 3 N = 128S = 2

Block4K = 3 N = 128S = 2

Full(1024) ReLU Full

(1) Probability

Figure 5 -e structure of the discriminator network

Noise image Without GAN (3113db) Proposed (3227db)

(a)


(b)


(c)

Noised image Without GAN (3244db) Proposed (3317db)

(d)

Figure 6 -e compared experiments with and without GAN under the Gaussian noise of σ 25


(a) (b) (c) (d) (e)

(f ) (g) (h) (i) (j)

Figure 7 -e images used in the first experiment (a) Cameraman (b) Barbara (c) man (d) couple (e) hill (f ) Lena (g) house (h) F16 (i)peppers and (j) baboon

Table 1 -e results of the proposed method for Gaussian white noise with different standard deviations

ImagePSNR σ 5 σ 10 σ 15 σ 20 σ 25 σ 30 σ 35 σ 40 σ 50Cameraman 3910 3479 3301 3143 3012 2937 2869 2780 2678Barbara 3847 3508 3415 3259 3127 3011 2920 2805 2798Man 3801 3549 3276 3159 3030 2912 2893 2781 2682Couple 3858 3539 3362 3136 3045 2973 2899 2840 2701Hill 3847 3488 3290 3163 3081 3016 2959 2877 2796Lena 3881 3599 3394 3323 3227 3159 3091 3011 2988House 3902 3647 3517 3397 3320 3268 3209 3112 3079F16 3970 3685 3539 3451 3317 3201 3176 3099 3051Peppers 3739 3428 3301 3191 3167 3073 3088 2970 2915Baboon 3596 3129 2818 2760 2616 2592 2547 2501 2489

Table 2 -e results of the different method for standard Gaussian white noise with σ 15

PSNR SSIM EPI Times BM3D EPLL TNRD WNNM WSNM Proposed

Cameraman

3191 3181 3218 3217 3215 3301072 071 076 080 079 086039 033 039 044 047 059051 2986 088 13675 17214 351

Barbara

3369 3181 3215 4326 3427 3415077 075 077 083 087 086041 040 043 050 059 061305 17825 164 82550 98636 762

Lena

3494 3291 3344 3354 3353 3394070 077 078 082 088 086040 041 043 048 055 059046 3031 090 13441 16501 490

House

3494 3414 3455 3515 3514 3517069 078 081 081 083 087044 043 051 058 058 060078 3232 087 13616 17027 474

Peppers

3270 3258 3303 3297 3297 3301066 081 083 084 085 084032 035 042 055 058 058046 2955 091 13265 16062 389

Baboon

2778 2785 2789 2793 2789 2818064 077 079 080 084 084028 028 044 050 053 056149 11198 122 49038 65431 711



PSNRSSIMEPITimes

BM3D EPLL TNRD WNNM WSNM Proposed

Cameraman

2945 2923 2971 2963 2970 3012070 070 074 071 075 076033 031 035 045 045 050054 3143 092 26172 28996 457

Barbara

3118 2898 2963 3183 3184 3127075 071 075 077 082 085037 033 040 044 056 060314 17930 164 170684 235022 1062

Lena

3060 3030 3077 3088 3089 3227068 072 075 080 080 082032 036 040 046 047 048047 3478 091 25941 28763 622

House

3286 3204 3254 3323 3323 3328066 076 080 083 080 083029 040 050 056 056 057149 3116 092 26304 29806 643

Peppers

3016 3007 3055 3040 3041 3167064 077 081 081 082 080026 031 040 040 044 055044 3225 087 25973 29105 488

Baboon

2523 2534 2539 2542 2544 2616062 071 077 077 078 080026 025 034 046 046 050162 10628 123 98696 129627 960

(a) (b) (c) (d)

(e) (f ) (g)

Figure 8 -e denoising results of different methods with the noise level σ 25 (a) Noise image (2017 db) (b) BM3D (2523 db) (c) EPLL(2534 db) (d) TNRD (2539 db) (e) TNNM (2542 db) (f ) WSNM (2544 db) and (g) proposed (2616 db)


Based on the edge contrast the edge-preserving index(EPI) is defined as

EPI 1113936 ps(i j) minus ps(i + 1 j)

11138681113868111386811138681113868111386811138681113868 + ps(i j) minus ps(i j + 1)

111386811138681113868111386811138681113868111386811138681113960 1113961

1113936 po(i j) minus po(i + 1 j)1113868111386811138681113868

1113868111386811138681113868 + po(i j) minus po(i j + 1)1113868111386811138681113868

11138681113868111386811138681113960 1113961

(20)

where ps(i j) is a pixel from the testing image po(i j) is apixel from the original image ps and po are located in theedge region i is the number of rows and j is the number ofcolumns -e range of EPI is 0 to 1 When EPI equals to 1the edge of the image is completely maintained When theEPI equals to 0 it means the image becomes a plane withoutany change -e larger the EPI value the stronger the edgeholding ability

31 Experiment 1 Adding to the Gaussian white noise withdifferent standard deviations the first experiment isimplemented under some popularly used natural images

shown in Figure 7 Five famous methods are used to be thecomparative methods including BM3D [26] EPLL [12]TNRD [27]WNNM [28] andWSNM [29]and the proposedmethod (proposed for short) All the results are reported inTables 1ndash3 respectively

In addition the visual performance of different methodswhen the noise level σ 25 is shown in Figure 8 to providethe subjective comparison A small region marked by the redrectangle is enlarged to clearly display the visual difference

32 Experiment 2 -e second experiment is implementedunder Gaussian noise and Rician noise -e testing volumedata is a T1 brain network voxel with a size of181times 217times181 -e slice thickness is 1mm Without loss ofgenerality it is assumed that all the images are normalized toa real-valued signal in the intensity range [0 1] -e ex-perimental results are shown in Figures 9 and 10

In Figures 9 and 10 the first row is one slice of theoriginal volume data In each column it shows the lateral

Original

Gauss σ = 005 σ = 011 σ = 015

Lateral

Coronal

Sagittal

Cross

Figure 9 -e results of removing the Gauss noise from the left to right the original image the noisy image and the denoising result

Rician σ = 005 σ = 011 σ = 015

Lateral

Coronal

Sagittal

Cross

Figure 10 -e results of removing the Rician noise from the left to right the original image the noisy image and the denoising result


coronal sagittal and cross results when the standard de-viation is 005 011 and 0015 respectively -e results ofunder OB-NLM3D [30] OB-NLM3D-WM [31] ODCT3D[32] PRI-NLM3D [33] BM4D [34] and the proposedmethod are reported in Tables 4 and 5 All the quantitativeresults are shown in Tables 4 and 5 respectively

By the comparison of all the above experimental resultsit strongly demonstrates the efficiency of the proposed fu-sion methods both visually and quantitatively Speciallyamong the algorithms considered the PSNR and SSIM

performance of the proposed method shows that the betterresults can be obtained when the noise level increases Inaddition better visual effects can be obtained from thesmoothness of flat areas the preservation of details along theedges and the accurate restoration of phantom intensity

-e time complexity of the proposed method isO(N log2 N) -e cost of computing time is indeed anannoying problem at present -e main reason is that theprocess of our method contains more steps than othermethods -e good news is that our method is not the

Table 4 -e results of the different methods for Gaussian noise

PSNR SSIM EPI Filterσ

1 3 5 7 9 11 13 15 17 19

Gauss noise

Noisy data4000 3046 2602 2310 2091 1917 1772 1648 1539 1442097 081 066 053 043 036 030 025 022 019020 019 017 017 014 013 010 009 006 004

OB-NLM3D4247 3757 3473 3282 3142 3032 2940 2861 2791 2728099 097 095 092 090 087 084 082 079 077029 028 026 025 023 020 019 017 015 014

OB-NLM3D-WM4252 3775 3501 3313 3173 3061 2968 2888 2818 2755099 097 095 093 090 088 085 083 080 078032 030 029 028 026 026 024 022 020 018

ODCT3D4378 3753 3489 3318 3191 3090 3007 2935 2873 2818099 097 095 093 091 089 088 086 085 083047 046 045 042 040 037 036 031 028 025

PRI-NLM3D4404 3826 3551 3367 3237 3129 3040 2965 2899 2840099 098 096 094 092 090 089 087 085 084066 064 062 060 059 056 055 055 053 051

BM4D4409 3839 3595 3438 3321 3228 3150 3082 3023 2970099 098 096 095 093 092 091 090 088 087070 069 068 067 060 058 056 056 055 054

Proposed4508 3931 3679 3512 3379 3331 3216 3158 3079 3053099 097 096 095 093 092 091 090 088 088075 072 068 067 063 061 060 059 057 055

Table 5 -e results of the different methods for Rician noise


1 3 5 7 9 11 13 15 17 19

Rician noise

Noisy data4000 3049 2609 2320 2104 1932 1788 1665 1557 1460097 081 066 053 043 036 030 025 021 018020 019 017 017 014 013 010 009 006 004

OB-NLM3D4241 3745 3454 3251 3097 2971 2862 2764 2674 2591099 097 094 091 088 085 081 078 074 070029 028 026 025 023 021 019 017 015 014

OB-NLM3D-WM4244 3754 3466 3261 3101 2969 2853 2750 2657 2571099 097 095 092 088 085 081 077 074 070032 030 029 028 027 027 026 023 023 020

ODCT3D4296 3738 3470 3290 3153 3041 2948 2867 2795 2730099 097 095 093 090 088 086 084 082 080047 047 045 044 042 039 037 035 034 033

PRI-NLM3D4397 3819 3534 3337 3194 3074 2975 2888 2810 2739099 098 096 094 091 089 087 085 082 080066 065 063 060 057 055 054 052 048 046

BM4D4408 3834 3583 3417 3289 3182 3090 3006 2929 2857099 098 096 094 093 091 089 088 086 084071 070 069 067 062 060 059 057 056 055

Proposed4510 3949 3697 3538 3361 3292 3218 3161 3042 2971099 097 096 094 093 091 090 089 087 086076 071 068 067 064 063 062 060 058 057


slowest We think it will be effectively solved in the future byimproving the hardware conditions In addition someparallel computing methods (CUDA will be used in ourplan) can also be used to reduce the time cost-is is also thework we will deal in the future

4 Conclusion

In this paper an image denoising method that is based onthe BM4D in the 3D shearlet transform domain and GAN isproposed -e proposed method can make full use of thesparse representation of the 3D shearlet transform and thegood deeply learning ability of the generative adversarialnetwork All the experimental results prove the effectivenessand accuracy of the proposed method in the form of sub-jective comparison and objective quantification stronglydemonstrating the superiority of the proposed method overthe traditional image denoising methods

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by a Project of Shandong ProvinceHigher Educational Science and Technology Program(J18KA362) the National Natural Science Foundation ofChina (61502282 61902222) the Natural Science Founda-tion of Shandong Province (ZR2015FQ005) the TaishanScholars Program of Shandong Province (tsqn201909109)and the Open Fund Project of Shandong Key Laboratory ofInformation Technology of Intelligent Mine in ShandongUniversity of Science and Technology

References

[1] T Yongpeng J Yu and X Cong ldquoImage denoising algorithmbased on grouped dictionary and variational modelrdquo Com-puter Application vol 39 no 2 pp 551ndash555 2019

[2] S Yongkui T Wen and C Tongwen ldquoA method to removechattering alarms using median filtersrdquo ISA Transactionsvol 73 no 1 pp 201ndash207 2018

[3] L Zhiliang S Jia G Lihui et al ldquoExperimental study onactive noise control based on wiener filteringrdquo TechnologyWind vol 5 no 15 pp 216-217 2018

[4] M Xiaole H Shaohai and Y Dongsheng ldquoSAR image de-noising based on residual image fusion and sparse repre-sentationrdquo KSII Transactions on Internet amp InformationSystems vol 13 no 7 pp 3620ndash3637 2019

[5] H Hao and L Wu ldquoA new complex valued dictionarylearning method for group-sparse representationrdquo Optikvol 196 p 163150 2019

[6] L Shuaiqi L Ming L Peifei et al ldquoSAR image denoising viasparse representation in shearlet domain based on continuouscycle spinningrdquo IEEE Transactions on Geoscience and RemoteSensing vol 55 no 5 pp 2985ndash2992 2017

[7] C Bo D Ning and B Jing ldquoApplication of BM3D denoisingalgorithms in instrument image recognitionrdquo ZhejiangElectric Power vol 38 no 3 pp 54ndash58 2019

[8] Z Wenwen and H Yusheng ldquoLow rank sparse imagedenoising based on non-local self-similarityrdquo ComputerApplication vol 38 no 9 pp 2696ndash2700+2746 2018

[9] X Su and Z Yingyue ldquoNon-local mean denoising based onimage segmentationrdquo Computer Application vol 37 no 7pp 2078ndash2083 2017

[10] L Yu and C Sheng ldquoSummary of medical image segmen-tation methodsrdquo Electronic Technology vol 30 no 8pp 169ndash172 2017

[11] W Rui and Z Youchun ldquoWavelet threshold denoising undernew threshold functionrdquo Computer Engineering and Appli-cation vol 49 no 15 pp 215ndash218 2013

[12] X Wang Q Tao L Wang et al ldquoDeep convolutional ar-chitecture for natural image denoisingrdquo International Con-ferenceWireless Communications and Signal Processing vol 5no 53 pp 1ndash4 2015

[13] Y Min W Xintong L Yao et al ldquoPlausibility-promotinggenerative adversarial network for abstractive text summa-rization with multi-task constraintrdquo Information Sciencesvol 521 pp 46ndash61 2020

[14] P Siddharth R Pranshu and T Jing ldquoAn image augmen-tation approach using two-stage generative adversarial net-work for nuclei image segmentationrdquo Biomedical SignalProcessing and Control vol 57 p 101782 2020

[15] M Jiayi Y Wei L Pengwei et al ldquoFusionGAN a generativeadversarial network for infrared and visible image fusionrdquoInformation Fusion vol 48 pp 11ndash26 2019

[16] B Lei Z Xiongwei Z Yunfei et al ldquoVideo saliency detectionusing 3D shearlet transformrdquo Multimedia Tools amp Applica-tions vol 75 no 13 pp 7761ndash7778 2016

[17] D Labate and P Negi ldquo3D discrete shearlet transform andvideo denoisingrdquo Proceedings of Spie the International Societyfor Optical Engineering vol 8138 no 3 pp 815ndash822 2011

[18] D Lijuan W Chunli E Qing et al ldquoSuper-resolution al-gorithm of depth residual network image based on waveletdomainrdquo Journal of Software vol 30 no 4 pp 941ndash953 2019

[19] L Sun B Jenbo Y Zheng et al ldquoA novel weighted cross totalvariation method for hyperspectral image mixed denoisingrdquoIEEE Access vol 6 no 1 pp 172ndash188 2017

[20] C Zailiang Z Ziyang S Hailan et al ldquoDN-GAN Denoisinggenerative adversarial networks for speckle noise reduction inoptical coherence tomography imagesrdquo Biomedical SignalProcessing and Control vol 55 p 101632 2020

[21] Z Kai Z Wangmeng C Yunjin et al ldquoBeyond a Gaussiandenoiser residual learning of deep CNN for image denoisingrdquoIEEE Transactions on Image Processing vol 26 no 7pp 3142ndash3155 2017

[22] L Zhiyu Z Chengkun and H Min ldquoA nonsubsampledcountourlet transform based CNN for real image denoisingrdquoSignal Processing Image Communication vol 82 p 1157272020

[23] F Fan Y Ma C Li X Mei J Huang and J Ma ldquoHyper-spectral image denoising with superpixel segmentation andlow-rank representationrdquo Information Sciences vol 397-398pp 48ndash68 2017

[24] M Hongqiang M Shiping X Yuelei et al ldquoImage denoisingbased on improved stack sparse denoising self-encoderrdquoComputer Engineering and Application vol 54 no 4pp 199ndash204 2018


[25] H R Shahdoosti and Z Rahemi ldquoEdge-preserving imagedenoising using a deep convolutional neural networkrdquo SignalProcessing vol 159 pp 20ndash32 2019

[26] X Jia Z Junhua and M Lihui ldquoDenoising of salt and peppernoise based on improved BM3D algorithmrdquo Computer En-gineering and Application vol 54 no 21 pp 170ndash175 2018

[27] H Burger C Schuler and S Harmeling Learning How toCombine Internal and External Denoising Methods PatternRecognition Springer Berlin Germany pp 121ndash130 2013

[28] G Shuhang Z Lei Z Wangmeng et al ldquoWeighted nuclearnorm minimization with application to image denoisingrdquoComputer Vision and Pattern Recognition vol 3 pp 2862ndash2869 2014

[29] X Yuan G Shuhang L Yan et al ldquoWeighted schattenp-norm minimization for image denoising and backgroundsubtractionrdquo IEEE Trans Image Processvol 8 no 10pp 4842ndash4857 2016

[30] A Welinton S Tiago B Gabriel et al ldquoImproving non-localvideo denoising with local binary patterns and image quan-tizationrdquo in Proceedings of the 2016 29th SIBGRAPI Confer-ence on Graphics pp 1ndash9 Sao Paulo Brazil October 2016

[31] P Coupe P Hellier S Prima C Kervrann and C Barillotldquo3D wavelet subbands mixing for image denoisingrdquo Inter-national Journal of Biomedical Imaging vol 2008 Article ID590183 11 pages 2008

[32] L Xiangbo and Q Tianshuang ldquoDenoise MRI images usingsparse 3D transformation domain collaborative filteringrdquo inProceedings of the 2011 4th International Conference onBiomedical Engineering and Informatics (BMEI) vol 1pp 233ndash236 Shanghai China October 2011

[33] Y Daitianyi ldquoApplication of PRI_NLM3D denoising algo-rithms in CBCT 3D imagesrdquo New National RadiographicDigital Imaging and CT Technology vol 10 no 8 pp 99ndash1082012

[34] X Ping C Bingqiang L Xue et al ldquoA new MNF-BM4Ddenoising algorithm based on guided filtering for hyper-spectral imagesrdquo ISA Transactions vol 2 no 3 pp 1ndash10 2019


-erefore a novel image denoising method based onblock matching with 4D filtering in the 3D shearlet trans-form domain and the generative adversarial network isproposed -e 3D shearlet transform provides bettermathematical properties than the commonly used wavelet orcontourlet to capture the anisotropic features of images indifferent scales and directions In addition the traditionalBM3D is extended into four 4D space which can effectivelyimprove the edge and texture details of the images -eoutput of the BM4D is used as the input to the designedgenerative adversarial network to make full use of its goodlearning ability

-e remainder of this paper is organized as follows -edetails of the proposed method are presented in Section 2Experimental results and discussions are shown in Section 3Finally the whole paper is concluded in Section 4

2 Methodology

21 e Architecture of Proposed Method -e overallstructure of the proposed method is shown in Figure 1-rough multiscale decomposition and directional parti-tion the 3D shearlet coefficients are obtained -en thecoefficients are input into the hard threshold and wienerfiltering contained in the BM4D model For the hardthreshold the similar cubes are extracted from volumeobservation and then they are combined together If thedistance between the cubes is smaller than the settingthreshold the collaborative filters are carried out to apply onthe similar cubes In the aggregation process the basicvolume estimation is generated by the adaptive convexcombination For the process of the wiener filtering theaggregation is implemented by the inversion of the shearlettransformation Finally the clean results are obtained by thedesigned generative adversarial network

22e 3D Shearlet Transform In the 3D space the shearletregion Pi(i 1 2 3) is obtained by combining the functionsystem associated with the pyramid region in the 3D Fourierspace R3

For d 1 2 3 l (l1 l2) isin Z2 the 3D shear systemassociated with Pd in the shearlet region is defined as a set

ψdjlk detA(d)

11138681113868111386811138681113868

11138681113868111386811138681113868j2ψ(d)

B(d)l A

(d)1113872 1113873

jx minus k1113874 1113875 j isin Z1113882

l l1 l2( 1113857 isin Λj k isin Z3 d 1 2 31113967

(1)

where ψ(d) isin L2(R3) (l1 l2) minus 2j le l1 l2 le 2j1113864 1113865 andΛj sub Z2Ad is the specific anisotropic expansion matrix and Bd

l is thespecific shearmatrixMore details can be found in the literature[16 17] In Figure 2 a shearlet in 3D space is shown

23 e Improved 3D Block-Matching Algorithm Letz X⟶ R be the noise form

z(x) y(x) + η(x) x isin X (2)

where y is the original unknown volume signal x sub Z3 is the3D signal and η(middot) sim N(0 σ2) is an independent and

uniformly distributed Gaussian noise whose standard de-viation is noted as σ

-e goal of the improved 3D block-matching algorithm[18]was to obtain the estimation 1113954y of y from the noiseobservations -e implementation of the improved algo-rithm was divided into two cascade stages the hardthreshold and wiener filtering [19] each of which includes 3steps grouping collaborative filtering and aggregation

For the hard threshold let CzxR represent a cube whose

volume is L times L times L where L isin N was extracted from z -esimilarity between two cubes is measured by photometricdistance

d Czxi

Czxj

1113874 1113875 Cz

ximinus Cz

xj

2

2L3

(3)

where middot represents the summation of the squareddifferences between the corresponding intensities of thetwo cubes and the denominator L3 is the normalizationfactor No prefiltering is performed before cube matchingso the similarity of noise observations can be directlytested

In the grouping step cubes similar to each other areextracted from z and combined to form a group for eachcube Cz

xR If the distance between two cubes was not larger

than the predefined threshold τhtmatch the two cubes are

considered to be similar Similar to CzxR we here firstly define

a set that contains the indexes for the cubes as follows

SzxR

xi isin X d CzxR

Czxi

1113872 1113873le τhtmatch1113966 1113967 (4)

-en a four-dimensional group is built by the aboveformula

GzSz

xR

∐xiisinSz

xR

Czxi

(5)

where the reference cube (represented by R) matches a set ofsimilar cubes located in the 3D data Particularly the co-ordinates xR and xi correspond to the tail and head of thearrow connecting the cubes in formula (4) respectivelyNote that since the distance from any cube to itself is always0 according to the definition of formula (4) each formula(5) must contain at least its reference cube

In the collaborative filtering step a joint four-dimen-sional transformation Tht

4D was applied to each dimension ofequation (5) respectively -en by a hard threshold op-erator cht with the threshold σλ4D the obtained four-di-mensional group spectrum is

cht

Tht4D G

zSz

xR

1113874 11138751113874 1113875 (6)

Representing the filter group it is transformed into thefollowing form

Thtminus1

4D cht

Tht4D G

zSz

xR

1113874 11138751113874 11138751113874 1113875 1113954Gy

SzxR

∐xiisinSz

xR

1113954Cy

xi (7)

For each unknown volume data y the estimated 1113954Cy

xiof

the original Cyxiwas extracted separately Formula (7) was an

overcompleted representation of the denoising data because





xR

whtxR

1113954Cy

xi1113874 1113875


whtxRχxi

1113874 1113875

(8)

where whtxR



xidomain that is



whtxR

1

σ2NhtxR

(9)







cube C1113954y

ht



S1113954y

ht

xR xi isin X d C

1113954yht

xR C

1113954yht

xi1113874 1113875lt τwiematch1113882 1113883 (10)



ht

S1113954y

ht

xR

from 1113954yht and


WS1113954y

ht

xR

Twie4D G

1113954yht

S1113954y

ht

xR

⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868

2

Twie4D G

1113954yht

S1113954y

ht

xR

⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868

2

+ σ2 (11)




6080

4020

0

6080

4020

0

30

40

20

10

0

(a)

70

60

50

40

30

20

10

070 60 50 40 30 20 10 0

(b)




Hard threshold

Wiener filtering

Grouping



combination)

Grouping



transform)

GAN









S1113954y

ht

xR



1113954Gy

S1113954y

ht

xR

Twieminus1

4D WS1113954y

ht

xR

middot Twie4D G

z

S1113954y

ht

xR

⎛⎝ ⎞⎠⎛⎝ ⎞⎠ (12)



wwiexR

σminus 2W

S1113954y

ht

xR

2

2

(13)






pg sim G z θg1113872 1113873 (14)



maxθd



(15)





PSNR(y 1113954y) 10 log10D2| 1113957X|


⎛⎝ ⎞⎠ (16)



SSIM(x y) l(x y)α

middot c(x y)β

middot s(x y)c (17)


l(x y) 2uxuy + c1

u2x + u2

y + c1

c(x y) 2σxσy + c2

σ2x + σ2y + c2

s(x y) 2σxy + c3

σxy + c3

(18)





u2x + u2

y + c11113872 1113873 σ2x + σ2y + c21113872 1113873 (19)

Noiseimage

Fakeimage

Realimage

Generator network


Predictedlables

1 real 0 fake






Block1



Conv

Block2K = 3 N = 64 S = 2

ReLUBatch

specification layer

Block1Noise image

Real image

Block3K = 3 N = 128S = 2

Block4K = 3 N = 128S = 2


(1) Probability



(a)


(b)


(c)


(d)



(a) (b) (c) (d) (e)

(f ) (g) (h) (i) (j)






Cameraman

3191 3181 3218 3217 3215 3301072 071 076 080 079 086039 033 039 044 047 059051 2986 088 13675 17214 351

Barbara

3369 3181 3215 4326 3427 3415077 075 077 083 087 086041 040 043 050 059 061305 17825 164 82550 98636 762

Lena

3494 3291 3344 3354 3353 3394070 077 078 082 088 086040 041 043 048 055 059046 3031 090 13441 16501 490

House

3494 3414 3455 3515 3514 3517069 078 081 081 083 087044 043 051 058 058 060078 3232 087 13616 17027 474

Peppers

3270 3258 3303 3297 3297 3301066 081 083 084 085 084032 035 042 055 058 058046 2955 091 13265 16062 389

Baboon

2778 2785 2789 2793 2789 2818064 077 079 080 084 084028 028 044 050 053 056149 11198 122 49038 65431 711



PSNRSSIMEPITimes


Cameraman

2945 2923 2971 2963 2970 3012070 070 074 071 075 076033 031 035 045 045 050054 3143 092 26172 28996 457

Barbara

3118 2898 2963 3183 3184 3127075 071 075 077 082 085037 033 040 044 056 060314 17930 164 170684 235022 1062

Lena

3060 3030 3077 3088 3089 3227068 072 075 080 080 082032 036 040 046 047 048047 3478 091 25941 28763 622

House

3286 3204 3254 3323 3323 3328066 076 080 083 080 083029 040 050 056 056 057149 3116 092 26304 29806 643

Peppers

3016 3007 3055 3040 3041 3167064 077 081 081 082 080026 031 040 040 044 055044 3225 087 25973 29105 488

Baboon

2523 2534 2539 2542 2544 2616062 071 077 077 078 080026 025 034 046 046 050162 10628 123 98696 129627 960

(a) (b) (c) (d)

(e) (f ) (g)





11138681113868111386811138681113868111386811138681113868 + ps(i j) minus ps(i j + 1)

111386811138681113868111386811138681113868111386811138681113960 1113961

1113936 po(i j) minus po(i + 1 j)1113868111386811138681113868

1113868111386811138681113868 + po(i j) minus po(i j + 1)1113868111386811138681113868

11138681113868111386811138681113960 1113961

(20)







Original

Gauss σ = 005 σ = 011 σ = 015

Lateral

Coronal

Sagittal

Cross


Rician σ = 005 σ = 011 σ = 015

Lateral

Coronal

Sagittal

Cross









1 3 5 7 9 11 13 15 17 19

Gauss noise

Noisy data4000 3046 2602 2310 2091 1917 1772 1648 1539 1442097 081 066 053 043 036 030 025 022 019020 019 017 017 014 013 010 009 006 004

OB-NLM3D4247 3757 3473 3282 3142 3032 2940 2861 2791 2728099 097 095 092 090 087 084 082 079 077029 028 026 025 023 020 019 017 015 014

OB-NLM3D-WM4252 3775 3501 3313 3173 3061 2968 2888 2818 2755099 097 095 093 090 088 085 083 080 078032 030 029 028 026 026 024 022 020 018

ODCT3D4378 3753 3489 3318 3191 3090 3007 2935 2873 2818099 097 095 093 091 089 088 086 085 083047 046 045 042 040 037 036 031 028 025

PRI-NLM3D4404 3826 3551 3367 3237 3129 3040 2965 2899 2840099 098 096 094 092 090 089 087 085 084066 064 062 060 059 056 055 055 053 051

BM4D4409 3839 3595 3438 3321 3228 3150 3082 3023 2970099 098 096 095 093 092 091 090 088 087070 069 068 067 060 058 056 056 055 054

Proposed4508 3931 3679 3512 3379 3331 3216 3158 3079 3053099 097 096 095 093 092 091 090 088 088075 072 068 067 063 061 060 059 057 055



1 3 5 7 9 11 13 15 17 19

Rician noise

Noisy data4000 3049 2609 2320 2104 1932 1788 1665 1557 1460097 081 066 053 043 036 030 025 021 018020 019 017 017 014 013 010 009 006 004

OB-NLM3D4241 3745 3454 3251 3097 2971 2862 2764 2674 2591099 097 094 091 088 085 081 078 074 070029 028 026 025 023 021 019 017 015 014

OB-NLM3D-WM4244 3754 3466 3261 3101 2969 2853 2750 2657 2571099 097 095 092 088 085 081 077 074 070032 030 029 028 027 027 026 023 023 020

ODCT3D4296 3738 3470 3290 3153 3041 2948 2867 2795 2730099 097 095 093 090 088 086 084 082 080047 047 045 044 042 039 037 035 034 033

PRI-NLM3D4397 3819 3534 3337 3194 3074 2975 2888 2810 2739099 098 096 094 091 089 087 085 082 080066 065 063 060 057 055 054 052 048 046

BM4D4408 3834 3583 3417 3289 3182 3090 3006 2929 2857099 098 096 094 093 091 089 088 086 084071 070 069 067 062 060 059 057 056 055

Proposed4510 3949 3697 3538 3361 3292 3218 3161 3042 2971099 097 096 094 093 091 090 089 087 086076 071 068 067 064 063 062 060 058 057



4 Conclusion


Data Availability




Acknowledgments


References








































xR

whtxR

1113954Cy

xi1113874 1113875


whtxRχxi

1113874 1113875

(8)

where whtxR



xidomain that is



whtxR

1

σ2NhtxR

(9)







cube C1113954y

ht



S1113954y

ht

xR xi isin X d C

1113954yht

xR C

1113954yht

xi1113874 1113875lt τwiematch1113882 1113883 (10)



ht

S1113954y

ht

xR

from 1113954yht and


WS1113954y

ht

xR

Twie4D G

1113954yht

S1113954y

ht

xR

⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868

2

Twie4D G

1113954yht

S1113954y

ht

xR

⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868

2

+ σ2 (11)




6080

4020

0

6080

4020

0

30

40

20

10

0

(a)

70

60

50

40

30

20

10

070 60 50 40 30 20 10 0

(b)




Hard threshold

Wiener filtering

Grouping



combination)

Grouping



transform)

GAN









S1113954y

ht

xR



1113954Gy

S1113954y

ht

xR

Twieminus1

4D WS1113954y

ht

xR

middot Twie4D G

z

S1113954y

ht

xR

⎛⎝ ⎞⎠⎛⎝ ⎞⎠ (12)



wwiexR

σminus 2W

S1113954y

ht

xR

2

2

(13)






pg sim G z θg1113872 1113873 (14)



maxθd



(15)





PSNR(y 1113954y) 10 log10D2| 1113957X|


⎛⎝ ⎞⎠ (16)



SSIM(x y) l(x y)α

middot c(x y)β

middot s(x y)c (17)


l(x y) 2uxuy + c1

u2x + u2

y + c1

c(x y) 2σxσy + c2

σ2x + σ2y + c2

s(x y) 2σxy + c3

σxy + c3

(18)





u2x + u2

y + c11113872 1113873 σ2x + σ2y + c21113872 1113873 (19)

Noiseimage

Fakeimage

Realimage

Generator network


Predictedlables

1 real 0 fake






Block1



Conv

Block2K = 3 N = 64 S = 2

ReLUBatch

specification layer

Block1Noise image

Real image

Block3K = 3 N = 128S = 2

Block4K = 3 N = 128S = 2


(1) Probability



(a)


(b)


(c)


(d)



(a) (b) (c) (d) (e)

(f ) (g) (h) (i) (j)






Cameraman

3191 3181 3218 3217 3215 3301072 071 076 080 079 086039 033 039 044 047 059051 2986 088 13675 17214 351

Barbara

3369 3181 3215 4326 3427 3415077 075 077 083 087 086041 040 043 050 059 061305 17825 164 82550 98636 762

Lena

3494 3291 3344 3354 3353 3394070 077 078 082 088 086040 041 043 048 055 059046 3031 090 13441 16501 490

House

3494 3414 3455 3515 3514 3517069 078 081 081 083 087044 043 051 058 058 060078 3232 087 13616 17027 474

Peppers

3270 3258 3303 3297 3297 3301066 081 083 084 085 084032 035 042 055 058 058046 2955 091 13265 16062 389

Baboon

2778 2785 2789 2793 2789 2818064 077 079 080 084 084028 028 044 050 053 056149 11198 122 49038 65431 711



PSNRSSIMEPITimes


Cameraman

2945 2923 2971 2963 2970 3012070 070 074 071 075 076033 031 035 045 045 050054 3143 092 26172 28996 457

Barbara

3118 2898 2963 3183 3184 3127075 071 075 077 082 085037 033 040 044 056 060314 17930 164 170684 235022 1062

Lena

3060 3030 3077 3088 3089 3227068 072 075 080 080 082032 036 040 046 047 048047 3478 091 25941 28763 622

House

3286 3204 3254 3323 3323 3328066 076 080 083 080 083029 040 050 056 056 057149 3116 092 26304 29806 643

Peppers

3016 3007 3055 3040 3041 3167064 077 081 081 082 080026 031 040 040 044 055044 3225 087 25973 29105 488

Baboon

2523 2534 2539 2542 2544 2616062 071 077 077 078 080026 025 034 046 046 050162 10628 123 98696 129627 960

(a) (b) (c) (d)

(e) (f ) (g)





11138681113868111386811138681113868111386811138681113868 + ps(i j) minus ps(i j + 1)

111386811138681113868111386811138681113868111386811138681113960 1113961

1113936 po(i j) minus po(i + 1 j)1113868111386811138681113868

1113868111386811138681113868 + po(i j) minus po(i j + 1)1113868111386811138681113868

11138681113868111386811138681113960 1113961

(20)







Original

Gauss σ = 005 σ = 011 σ = 015

Lateral

Coronal

Sagittal

Cross


Rician σ = 005 σ = 011 σ = 015

Lateral

Coronal

Sagittal

Cross









1 3 5 7 9 11 13 15 17 19

Gauss noise

Noisy data4000 3046 2602 2310 2091 1917 1772 1648 1539 1442097 081 066 053 043 036 030 025 022 019020 019 017 017 014 013 010 009 006 004

OB-NLM3D4247 3757 3473 3282 3142 3032 2940 2861 2791 2728099 097 095 092 090 087 084 082 079 077029 028 026 025 023 020 019 017 015 014

OB-NLM3D-WM4252 3775 3501 3313 3173 3061 2968 2888 2818 2755099 097 095 093 090 088 085 083 080 078032 030 029 028 026 026 024 022 020 018

ODCT3D4378 3753 3489 3318 3191 3090 3007 2935 2873 2818099 097 095 093 091 089 088 086 085 083047 046 045 042 040 037 036 031 028 025

PRI-NLM3D4404 3826 3551 3367 3237 3129 3040 2965 2899 2840099 098 096 094 092 090 089 087 085 084066 064 062 060 059 056 055 055 053 051

BM4D4409 3839 3595 3438 3321 3228 3150 3082 3023 2970099 098 096 095 093 092 091 090 088 087070 069 068 067 060 058 056 056 055 054

Proposed4508 3931 3679 3512 3379 3331 3216 3158 3079 3053099 097 096 095 093 092 091 090 088 088075 072 068 067 063 061 060 059 057 055



1 3 5 7 9 11 13 15 17 19

Rician noise

Noisy data4000 3049 2609 2320 2104 1932 1788 1665 1557 1460097 081 066 053 043 036 030 025 021 018020 019 017 017 014 013 010 009 006 004

OB-NLM3D4241 3745 3454 3251 3097 2971 2862 2764 2674 2591099 097 094 091 088 085 081 078 074 070029 028 026 025 023 021 019 017 015 014

OB-NLM3D-WM4244 3754 3466 3261 3101 2969 2853 2750 2657 2571099 097 095 092 088 085 081 077 074 070032 030 029 028 027 027 026 023 023 020

ODCT3D4296 3738 3470 3290 3153 3041 2948 2867 2795 2730099 097 095 093 090 088 086 084 082 080047 047 045 044 042 039 037 035 034 033

PRI-NLM3D4397 3819 3534 3337 3194 3074 2975 2888 2810 2739099 098 096 094 091 089 087 085 082 080066 065 063 060 057 055 054 052 048 046

BM4D4408 3834 3583 3417 3289 3182 3090 3006 2929 2857099 098 096 094 093 091 089 088 086 084071 070 069 067 062 060 059 057 056 055

Proposed4510 3949 3697 3538 3361 3292 3218 3161 3042 2971099 097 096 094 093 091 090 089 087 086076 071 068 067 064 063 062 060 058 057



4 Conclusion


Data Availability




Acknowledgments


References






































S1113954y

ht

xR



1113954Gy

S1113954y

ht

xR

Twieminus1

4D WS1113954y

ht

xR

middot Twie4D G

z

S1113954y

ht

xR

⎛⎝ ⎞⎠⎛⎝ ⎞⎠ (12)



wwiexR

σminus 2W

S1113954y

ht

xR

2

2

(13)






pg sim G z θg1113872 1113873 (14)



maxθd



(15)





PSNR(y 1113954y) 10 log10D2| 1113957X|


⎛⎝ ⎞⎠ (16)



SSIM(x y) l(x y)α

middot c(x y)β

middot s(x y)c (17)


l(x y) 2uxuy + c1

u2x + u2

y + c1

c(x y) 2σxσy + c2

σ2x + σ2y + c2

s(x y) 2σxy + c3

σxy + c3

(18)





u2x + u2

y + c11113872 1113873 σ2x + σ2y + c21113872 1113873 (19)

Noiseimage

Fakeimage

Realimage

Generator network


Predictedlables

1 real 0 fake






Block1



Conv

Block2K = 3 N = 64 S = 2

ReLUBatch

specification layer

Block1Noise image

Real image

Block3K = 3 N = 128S = 2

Block4K = 3 N = 128S = 2


(1) Probability



(a)


(b)


(c)


(d)



(a) (b) (c) (d) (e)

(f ) (g) (h) (i) (j)






Cameraman

3191 3181 3218 3217 3215 3301072 071 076 080 079 086039 033 039 044 047 059051 2986 088 13675 17214 351

Barbara

3369 3181 3215 4326 3427 3415077 075 077 083 087 086041 040 043 050 059 061305 17825 164 82550 98636 762

Lena

3494 3291 3344 3354 3353 3394070 077 078 082 088 086040 041 043 048 055 059046 3031 090 13441 16501 490

House

3494 3414 3455 3515 3514 3517069 078 081 081 083 087044 043 051 058 058 060078 3232 087 13616 17027 474

Peppers

3270 3258 3303 3297 3297 3301066 081 083 084 085 084032 035 042 055 058 058046 2955 091 13265 16062 389

Baboon

2778 2785 2789 2793 2789 2818064 077 079 080 084 084028 028 044 050 053 056149 11198 122 49038 65431 711



PSNRSSIMEPITimes


Cameraman

2945 2923 2971 2963 2970 3012070 070 074 071 075 076033 031 035 045 045 050054 3143 092 26172 28996 457

Barbara

3118 2898 2963 3183 3184 3127075 071 075 077 082 085037 033 040 044 056 060314 17930 164 170684 235022 1062

Lena

3060 3030 3077 3088 3089 3227068 072 075 080 080 082032 036 040 046 047 048047 3478 091 25941 28763 622

House

3286 3204 3254 3323 3323 3328066 076 080 083 080 083029 040 050 056 056 057149 3116 092 26304 29806 643

Peppers

3016 3007 3055 3040 3041 3167064 077 081 081 082 080026 031 040 040 044 055044 3225 087 25973 29105 488

Baboon

2523 2534 2539 2542 2544 2616062 071 077 077 078 080026 025 034 046 046 050162 10628 123 98696 129627 960

(a) (b) (c) (d)

(e) (f ) (g)





11138681113868111386811138681113868111386811138681113868 + ps(i j) minus ps(i j + 1)

111386811138681113868111386811138681113868111386811138681113960 1113961

1113936 po(i j) minus po(i + 1 j)1113868111386811138681113868

1113868111386811138681113868 + po(i j) minus po(i j + 1)1113868111386811138681113868

11138681113868111386811138681113960 1113961

(20)







Original

Gauss σ = 005 σ = 011 σ = 015

Lateral

Coronal

Sagittal

Cross


Rician σ = 005 σ = 011 σ = 015

Lateral

Coronal

Sagittal

Cross









1 3 5 7 9 11 13 15 17 19

Gauss noise

Noisy data4000 3046 2602 2310 2091 1917 1772 1648 1539 1442097 081 066 053 043 036 030 025 022 019020 019 017 017 014 013 010 009 006 004

OB-NLM3D4247 3757 3473 3282 3142 3032 2940 2861 2791 2728099 097 095 092 090 087 084 082 079 077029 028 026 025 023 020 019 017 015 014

OB-NLM3D-WM4252 3775 3501 3313 3173 3061 2968 2888 2818 2755099 097 095 093 090 088 085 083 080 078032 030 029 028 026 026 024 022 020 018

ODCT3D4378 3753 3489 3318 3191 3090 3007 2935 2873 2818099 097 095 093 091 089 088 086 085 083047 046 045 042 040 037 036 031 028 025

PRI-NLM3D4404 3826 3551 3367 3237 3129 3040 2965 2899 2840099 098 096 094 092 090 089 087 085 084066 064 062 060 059 056 055 055 053 051

BM4D4409 3839 3595 3438 3321 3228 3150 3082 3023 2970099 098 096 095 093 092 091 090 088 087070 069 068 067 060 058 056 056 055 054

Proposed4508 3931 3679 3512 3379 3331 3216 3158 3079 3053099 097 096 095 093 092 091 090 088 088075 072 068 067 063 061 060 059 057 055



1 3 5 7 9 11 13 15 17 19

Rician noise

Noisy data4000 3049 2609 2320 2104 1932 1788 1665 1557 1460097 081 066 053 043 036 030 025 021 018020 019 017 017 014 013 010 009 006 004

OB-NLM3D4241 3745 3454 3251 3097 2971 2862 2764 2674 2591099 097 094 091 088 085 081 078 074 070029 028 026 025 023 021 019 017 015 014

OB-NLM3D-WM4244 3754 3466 3261 3101 2969 2853 2750 2657 2571099 097 095 092 088 085 081 077 074 070032 030 029 028 027 027 026 023 023 020

ODCT3D4296 3738 3470 3290 3153 3041 2948 2867 2795 2730099 097 095 093 090 088 086 084 082 080047 047 045 044 042 039 037 035 034 033

PRI-NLM3D4397 3819 3534 3337 3194 3074 2975 2888 2810 2739099 098 096 094 091 089 087 085 082 080066 065 063 060 057 055 054 052 048 046

BM4D4408 3834 3583 3417 3289 3182 3090 3006 2929 2857099 098 096 094 093 091 089 088 086 084071 070 069 067 062 060 059 057 056 055

Proposed4510 3949 3697 3538 3361 3292 3218 3161 3042 2971099 097 096 094 093 091 090 089 087 086076 071 068 067 064 063 062 060 058 057



4 Conclusion


Data Availability




Acknowledgments


References








































u2x + u2

y + c11113872 1113873 σ2x + σ2y + c21113872 1113873 (19)

Noiseimage

Fakeimage

Realimage

Generator network


Predictedlables

1 real 0 fake






Block1



Conv

Block2K = 3 N = 64 S = 2

ReLUBatch

specification layer

Block1Noise image

Real image

Block3K = 3 N = 128S = 2

Block4K = 3 N = 128S = 2


(1) Probability



(a)


(b)


(c)


(d)



(a) (b) (c) (d) (e)

(f ) (g) (h) (i) (j)






Cameraman

3191 3181 3218 3217 3215 3301072 071 076 080 079 086039 033 039 044 047 059051 2986 088 13675 17214 351

Barbara

3369 3181 3215 4326 3427 3415077 075 077 083 087 086041 040 043 050 059 061305 17825 164 82550 98636 762

Lena

3494 3291 3344 3354 3353 3394070 077 078 082 088 086040 041 043 048 055 059046 3031 090 13441 16501 490

House

3494 3414 3455 3515 3514 3517069 078 081 081 083 087044 043 051 058 058 060078 3232 087 13616 17027 474

Peppers

3270 3258 3303 3297 3297 3301066 081 083 084 085 084032 035 042 055 058 058046 2955 091 13265 16062 389

Baboon

2778 2785 2789 2793 2789 2818064 077 079 080 084 084028 028 044 050 053 056149 11198 122 49038 65431 711



PSNRSSIMEPITimes


Cameraman

2945 2923 2971 2963 2970 3012070 070 074 071 075 076033 031 035 045 045 050054 3143 092 26172 28996 457

Barbara

3118 2898 2963 3183 3184 3127075 071 075 077 082 085037 033 040 044 056 060314 17930 164 170684 235022 1062

Lena

3060 3030 3077 3088 3089 3227068 072 075 080 080 082032 036 040 046 047 048047 3478 091 25941 28763 622

House

3286 3204 3254 3323 3323 3328066 076 080 083 080 083029 040 050 056 056 057149 3116 092 26304 29806 643

Peppers

3016 3007 3055 3040 3041 3167064 077 081 081 082 080026 031 040 040 044 055044 3225 087 25973 29105 488

Baboon

2523 2534 2539 2542 2544 2616062 071 077 077 078 080026 025 034 046 046 050162 10628 123 98696 129627 960

(a) (b) (c) (d)

(e) (f ) (g)





11138681113868111386811138681113868111386811138681113868 + ps(i j) minus ps(i j + 1)

111386811138681113868111386811138681113868111386811138681113960 1113961

1113936 po(i j) minus po(i + 1 j)1113868111386811138681113868

1113868111386811138681113868 + po(i j) minus po(i j + 1)1113868111386811138681113868

11138681113868111386811138681113960 1113961

(20)







Original

Gauss σ = 005 σ = 011 σ = 015

Lateral

Coronal

Sagittal

Cross


Rician σ = 005 σ = 011 σ = 015

Lateral

Coronal

Sagittal

Cross









1 3 5 7 9 11 13 15 17 19

Gauss noise

Noisy data4000 3046 2602 2310 2091 1917 1772 1648 1539 1442097 081 066 053 043 036 030 025 022 019020 019 017 017 014 013 010 009 006 004

OB-NLM3D4247 3757 3473 3282 3142 3032 2940 2861 2791 2728099 097 095 092 090 087 084 082 079 077029 028 026 025 023 020 019 017 015 014

OB-NLM3D-WM4252 3775 3501 3313 3173 3061 2968 2888 2818 2755099 097 095 093 090 088 085 083 080 078032 030 029 028 026 026 024 022 020 018

ODCT3D4378 3753 3489 3318 3191 3090 3007 2935 2873 2818099 097 095 093 091 089 088 086 085 083047 046 045 042 040 037 036 031 028 025

PRI-NLM3D4404 3826 3551 3367 3237 3129 3040 2965 2899 2840099 098 096 094 092 090 089 087 085 084066 064 062 060 059 056 055 055 053 051

BM4D4409 3839 3595 3438 3321 3228 3150 3082 3023 2970099 098 096 095 093 092 091 090 088 087070 069 068 067 060 058 056 056 055 054

Proposed4508 3931 3679 3512 3379 3331 3216 3158 3079 3053099 097 096 095 093 092 091 090 088 088075 072 068 067 063 061 060 059 057 055



1 3 5 7 9 11 13 15 17 19

Rician noise

Noisy data4000 3049 2609 2320 2104 1932 1788 1665 1557 1460097 081 066 053 043 036 030 025 021 018020 019 017 017 014 013 010 009 006 004

OB-NLM3D4241 3745 3454 3251 3097 2971 2862 2764 2674 2591099 097 094 091 088 085 081 078 074 070029 028 026 025 023 021 019 017 015 014

OB-NLM3D-WM4244 3754 3466 3261 3101 2969 2853 2750 2657 2571099 097 095 092 088 085 081 077 074 070032 030 029 028 027 027 026 023 023 020

ODCT3D4296 3738 3470 3290 3153 3041 2948 2867 2795 2730099 097 095 093 090 088 086 084 082 080047 047 045 044 042 039 037 035 034 033

PRI-NLM3D4397 3819 3534 3337 3194 3074 2975 2888 2810 2739099 098 096 094 091 089 087 085 082 080066 065 063 060 057 055 054 052 048 046

BM4D4408 3834 3583 3417 3289 3182 3090 3006 2929 2857099 098 096 094 093 091 089 088 086 084071 070 069 067 062 060 059 057 056 055

Proposed4510 3949 3697 3538 3361 3292 3218 3161 3042 2971099 097 096 094 093 091 090 089 087 086076 071 068 067 064 063 062 060 058 057



4 Conclusion


Data Availability




Acknowledgments


References





































(a) (b) (c) (d) (e)

(f ) (g) (h) (i) (j)






Cameraman

3191 3181 3218 3217 3215 3301072 071 076 080 079 086039 033 039 044 047 059051 2986 088 13675 17214 351

Barbara

3369 3181 3215 4326 3427 3415077 075 077 083 087 086041 040 043 050 059 061305 17825 164 82550 98636 762

Lena

3494 3291 3344 3354 3353 3394070 077 078 082 088 086040 041 043 048 055 059046 3031 090 13441 16501 490

House

3494 3414 3455 3515 3514 3517069 078 081 081 083 087044 043 051 058 058 060078 3232 087 13616 17027 474

Peppers

3270 3258 3303 3297 3297 3301066 081 083 084 085 084032 035 042 055 058 058046 2955 091 13265 16062 389

Baboon

2778 2785 2789 2793 2789 2818064 077 079 080 084 084028 028 044 050 053 056149 11198 122 49038 65431 711



PSNRSSIMEPITimes


Cameraman

2945 2923 2971 2963 2970 3012070 070 074 071 075 076033 031 035 045 045 050054 3143 092 26172 28996 457

Barbara

3118 2898 2963 3183 3184 3127075 071 075 077 082 085037 033 040 044 056 060314 17930 164 170684 235022 1062

Lena

3060 3030 3077 3088 3089 3227068 072 075 080 080 082032 036 040 046 047 048047 3478 091 25941 28763 622

House

3286 3204 3254 3323 3323 3328066 076 080 083 080 083029 040 050 056 056 057149 3116 092 26304 29806 643

Peppers

3016 3007 3055 3040 3041 3167064 077 081 081 082 080026 031 040 040 044 055044 3225 087 25973 29105 488

Baboon

2523 2534 2539 2542 2544 2616062 071 077 077 078 080026 025 034 046 046 050162 10628 123 98696 129627 960

(a) (b) (c) (d)

(e) (f ) (g)





11138681113868111386811138681113868111386811138681113868 + ps(i j) minus ps(i j + 1)

111386811138681113868111386811138681113868111386811138681113960 1113961

1113936 po(i j) minus po(i + 1 j)1113868111386811138681113868

1113868111386811138681113868 + po(i j) minus po(i j + 1)1113868111386811138681113868

11138681113868111386811138681113960 1113961

(20)







Original

Gauss σ = 005 σ = 011 σ = 015

Lateral

Coronal

Sagittal

Cross


Rician σ = 005 σ = 011 σ = 015

Lateral

Coronal

Sagittal

Cross









1 3 5 7 9 11 13 15 17 19

Gauss noise

Noisy data4000 3046 2602 2310 2091 1917 1772 1648 1539 1442097 081 066 053 043 036 030 025 022 019020 019 017 017 014 013 010 009 006 004

OB-NLM3D4247 3757 3473 3282 3142 3032 2940 2861 2791 2728099 097 095 092 090 087 084 082 079 077029 028 026 025 023 020 019 017 015 014

OB-NLM3D-WM4252 3775 3501 3313 3173 3061 2968 2888 2818 2755099 097 095 093 090 088 085 083 080 078032 030 029 028 026 026 024 022 020 018

ODCT3D4378 3753 3489 3318 3191 3090 3007 2935 2873 2818099 097 095 093 091 089 088 086 085 083047 046 045 042 040 037 036 031 028 025

PRI-NLM3D4404 3826 3551 3367 3237 3129 3040 2965 2899 2840099 098 096 094 092 090 089 087 085 084066 064 062 060 059 056 055 055 053 051

BM4D4409 3839 3595 3438 3321 3228 3150 3082 3023 2970099 098 096 095 093 092 091 090 088 087070 069 068 067 060 058 056 056 055 054

Proposed4508 3931 3679 3512 3379 3331 3216 3158 3079 3053099 097 096 095 093 092 091 090 088 088075 072 068 067 063 061 060 059 057 055



1 3 5 7 9 11 13 15 17 19

Rician noise

Noisy data4000 3049 2609 2320 2104 1932 1788 1665 1557 1460097 081 066 053 043 036 030 025 021 018020 019 017 017 014 013 010 009 006 004

OB-NLM3D4241 3745 3454 3251 3097 2971 2862 2764 2674 2591099 097 094 091 088 085 081 078 074 070029 028 026 025 023 021 019 017 015 014

OB-NLM3D-WM4244 3754 3466 3261 3101 2969 2853 2750 2657 2571099 097 095 092 088 085 081 077 074 070032 030 029 028 027 027 026 023 023 020

ODCT3D4296 3738 3470 3290 3153 3041 2948 2867 2795 2730099 097 095 093 090 088 086 084 082 080047 047 045 044 042 039 037 035 034 033

PRI-NLM3D4397 3819 3534 3337 3194 3074 2975 2888 2810 2739099 098 096 094 091 089 087 085 082 080066 065 063 060 057 055 054 052 048 046

BM4D4408 3834 3583 3417 3289 3182 3090 3006 2929 2857099 098 096 094 093 091 089 088 086 084071 070 069 067 062 060 059 057 056 055

Proposed4510 3949 3697 3538 3361 3292 3218 3161 3042 2971099 097 096 094 093 091 090 089 087 086076 071 068 067 064 063 062 060 058 057



4 Conclusion


Data Availability




Acknowledgments


References






































PSNRSSIMEPITimes


Cameraman

2945 2923 2971 2963 2970 3012070 070 074 071 075 076033 031 035 045 045 050054 3143 092 26172 28996 457

Barbara

3118 2898 2963 3183 3184 3127075 071 075 077 082 085037 033 040 044 056 060314 17930 164 170684 235022 1062

Lena

3060 3030 3077 3088 3089 3227068 072 075 080 080 082032 036 040 046 047 048047 3478 091 25941 28763 622

House

3286 3204 3254 3323 3323 3328066 076 080 083 080 083029 040 050 056 056 057149 3116 092 26304 29806 643

Peppers

3016 3007 3055 3040 3041 3167064 077 081 081 082 080026 031 040 040 044 055044 3225 087 25973 29105 488

Baboon

2523 2534 2539 2542 2544 2616062 071 077 077 078 080026 025 034 046 046 050162 10628 123 98696 129627 960

(a) (b) (c) (d)

(e) (f ) (g)





11138681113868111386811138681113868111386811138681113868 + ps(i j) minus ps(i j + 1)

111386811138681113868111386811138681113868111386811138681113960 1113961

1113936 po(i j) minus po(i + 1 j)1113868111386811138681113868

1113868111386811138681113868 + po(i j) minus po(i j + 1)1113868111386811138681113868

11138681113868111386811138681113960 1113961

(20)







Original

Gauss σ = 005 σ = 011 σ = 015

Lateral

Coronal

Sagittal

Cross


Rician σ = 005 σ = 011 σ = 015

Lateral

Coronal

Sagittal

Cross









1 3 5 7 9 11 13 15 17 19

Gauss noise

Noisy data4000 3046 2602 2310 2091 1917 1772 1648 1539 1442097 081 066 053 043 036 030 025 022 019020 019 017 017 014 013 010 009 006 004

OB-NLM3D4247 3757 3473 3282 3142 3032 2940 2861 2791 2728099 097 095 092 090 087 084 082 079 077029 028 026 025 023 020 019 017 015 014

OB-NLM3D-WM4252 3775 3501 3313 3173 3061 2968 2888 2818 2755099 097 095 093 090 088 085 083 080 078032 030 029 028 026 026 024 022 020 018

ODCT3D4378 3753 3489 3318 3191 3090 3007 2935 2873 2818099 097 095 093 091 089 088 086 085 083047 046 045 042 040 037 036 031 028 025

PRI-NLM3D4404 3826 3551 3367 3237 3129 3040 2965 2899 2840099 098 096 094 092 090 089 087 085 084066 064 062 060 059 056 055 055 053 051

BM4D4409 3839 3595 3438 3321 3228 3150 3082 3023 2970099 098 096 095 093 092 091 090 088 087070 069 068 067 060 058 056 056 055 054

Proposed4508 3931 3679 3512 3379 3331 3216 3158 3079 3053099 097 096 095 093 092 091 090 088 088075 072 068 067 063 061 060 059 057 055



1 3 5 7 9 11 13 15 17 19

Rician noise

Noisy data4000 3049 2609 2320 2104 1932 1788 1665 1557 1460097 081 066 053 043 036 030 025 021 018020 019 017 017 014 013 010 009 006 004

OB-NLM3D4241 3745 3454 3251 3097 2971 2862 2764 2674 2591099 097 094 091 088 085 081 078 074 070029 028 026 025 023 021 019 017 015 014

OB-NLM3D-WM4244 3754 3466 3261 3101 2969 2853 2750 2657 2571099 097 095 092 088 085 081 077 074 070032 030 029 028 027 027 026 023 023 020

ODCT3D4296 3738 3470 3290 3153 3041 2948 2867 2795 2730099 097 095 093 090 088 086 084 082 080047 047 045 044 042 039 037 035 034 033

PRI-NLM3D4397 3819 3534 3337 3194 3074 2975 2888 2810 2739099 098 096 094 091 089 087 085 082 080066 065 063 060 057 055 054 052 048 046

BM4D4408 3834 3583 3417 3289 3182 3090 3006 2929 2857099 098 096 094 093 091 089 088 086 084071 070 069 067 062 060 059 057 056 055

Proposed4510 3949 3697 3538 3361 3292 3218 3161 3042 2971099 097 096 094 093 091 090 089 087 086076 071 068 067 064 063 062 060 058 057



4 Conclusion


Data Availability




Acknowledgments


References







































11138681113868111386811138681113868111386811138681113868 + ps(i j) minus ps(i j + 1)

111386811138681113868111386811138681113868111386811138681113960 1113961

1113936 po(i j) minus po(i + 1 j)1113868111386811138681113868

1113868111386811138681113868 + po(i j) minus po(i j + 1)1113868111386811138681113868

11138681113868111386811138681113960 1113961

(20)







Original

Gauss σ = 005 σ = 011 σ = 015

Lateral

Coronal

Sagittal

Cross


Rician σ = 005 σ = 011 σ = 015

Lateral

Coronal

Sagittal

Cross









1 3 5 7 9 11 13 15 17 19

Gauss noise

Noisy data4000 3046 2602 2310 2091 1917 1772 1648 1539 1442097 081 066 053 043 036 030 025 022 019020 019 017 017 014 013 010 009 006 004

OB-NLM3D4247 3757 3473 3282 3142 3032 2940 2861 2791 2728099 097 095 092 090 087 084 082 079 077029 028 026 025 023 020 019 017 015 014

OB-NLM3D-WM4252 3775 3501 3313 3173 3061 2968 2888 2818 2755099 097 095 093 090 088 085 083 080 078032 030 029 028 026 026 024 022 020 018

ODCT3D4378 3753 3489 3318 3191 3090 3007 2935 2873 2818099 097 095 093 091 089 088 086 085 083047 046 045 042 040 037 036 031 028 025

PRI-NLM3D4404 3826 3551 3367 3237 3129 3040 2965 2899 2840099 098 096 094 092 090 089 087 085 084066 064 062 060 059 056 055 055 053 051

BM4D4409 3839 3595 3438 3321 3228 3150 3082 3023 2970099 098 096 095 093 092 091 090 088 087070 069 068 067 060 058 056 056 055 054

Proposed4508 3931 3679 3512 3379 3331 3216 3158 3079 3053099 097 096 095 093 092 091 090 088 088075 072 068 067 063 061 060 059 057 055



1 3 5 7 9 11 13 15 17 19

Rician noise

Noisy data4000 3049 2609 2320 2104 1932 1788 1665 1557 1460097 081 066 053 043 036 030 025 021 018020 019 017 017 014 013 010 009 006 004

OB-NLM3D4241 3745 3454 3251 3097 2971 2862 2764 2674 2591099 097 094 091 088 085 081 078 074 070029 028 026 025 023 021 019 017 015 014

OB-NLM3D-WM4244 3754 3466 3261 3101 2969 2853 2750 2657 2571099 097 095 092 088 085 081 077 074 070032 030 029 028 027 027 026 023 023 020

ODCT3D4296 3738 3470 3290 3153 3041 2948 2867 2795 2730099 097 095 093 090 088 086 084 082 080047 047 045 044 042 039 037 035 034 033

PRI-NLM3D4397 3819 3534 3337 3194 3074 2975 2888 2810 2739099 098 096 094 091 089 087 085 082 080066 065 063 060 057 055 054 052 048 046

BM4D4408 3834 3583 3417 3289 3182 3090 3006 2929 2857099 098 096 094 093 091 089 088 086 084071 070 069 067 062 060 059 057 056 055

Proposed4510 3949 3697 3538 3361 3292 3218 3161 3042 2971099 097 096 094 093 091 090 089 087 086076 071 068 067 064 063 062 060 058 057



4 Conclusion


Data Availability




Acknowledgments


References











































1 3 5 7 9 11 13 15 17 19

Gauss noise

Noisy data4000 3046 2602 2310 2091 1917 1772 1648 1539 1442097 081 066 053 043 036 030 025 022 019020 019 017 017 014 013 010 009 006 004

OB-NLM3D4247 3757 3473 3282 3142 3032 2940 2861 2791 2728099 097 095 092 090 087 084 082 079 077029 028 026 025 023 020 019 017 015 014

OB-NLM3D-WM4252 3775 3501 3313 3173 3061 2968 2888 2818 2755099 097 095 093 090 088 085 083 080 078032 030 029 028 026 026 024 022 020 018

ODCT3D4378 3753 3489 3318 3191 3090 3007 2935 2873 2818099 097 095 093 091 089 088 086 085 083047 046 045 042 040 037 036 031 028 025

PRI-NLM3D4404 3826 3551 3367 3237 3129 3040 2965 2899 2840099 098 096 094 092 090 089 087 085 084066 064 062 060 059 056 055 055 053 051

BM4D4409 3839 3595 3438 3321 3228 3150 3082 3023 2970099 098 096 095 093 092 091 090 088 087070 069 068 067 060 058 056 056 055 054

Proposed4508 3931 3679 3512 3379 3331 3216 3158 3079 3053099 097 096 095 093 092 091 090 088 088075 072 068 067 063 061 060 059 057 055



1 3 5 7 9 11 13 15 17 19

Rician noise

Noisy data4000 3049 2609 2320 2104 1932 1788 1665 1557 1460097 081 066 053 043 036 030 025 021 018020 019 017 017 014 013 010 009 006 004

OB-NLM3D4241 3745 3454 3251 3097 2971 2862 2764 2674 2591099 097 094 091 088 085 081 078 074 070029 028 026 025 023 021 019 017 015 014

OB-NLM3D-WM4244 3754 3466 3261 3101 2969 2853 2750 2657 2571099 097 095 092 088 085 081 077 074 070032 030 029 028 027 027 026 023 023 020

ODCT3D4296 3738 3470 3290 3153 3041 2948 2867 2795 2730099 097 095 093 090 088 086 084 082 080047 047 045 044 042 039 037 035 034 033

PRI-NLM3D4397 3819 3534 3337 3194 3074 2975 2888 2810 2739099 098 096 094 091 089 087 085 082 080066 065 063 060 057 055 054 052 048 046

BM4D4408 3834 3583 3417 3289 3182 3090 3006 2929 2857099 098 096 094 093 091 089 088 086 084071 070 069 067 062 060 059 057 056 055

Proposed4510 3949 3697 3538 3361 3292 3218 3161 3042 2971099 097 096 094 093 091 090 089 087 086076 071 068 067 064 063 062 060 058 057



4 Conclusion


Data Availability




Acknowledgments


References






































4 Conclusion


Data Availability




Acknowledgments


References





































animagedenoisingmethodbasedonbm4dandganin3d...

Documents