High-Resolution SAR Image Despeckling Based on Nonlocal ...

Research ArticleHigh-Resolution SAR Image Despeckling Based on NonlocalMeans Filter and Modified AA Model

QiaoKe1 SunZeng-guo 2YangLiu3WeiWei4MarcinWozniak 5 andRafałScherer 6

1School of Software Northwestern Polytechnical University Xirsquoan 710129 China2School of Computer Science Shaanxi Normal University Xirsquoan 710119 China3College of Computer Science and Technology Huaqiao University Xiamen 361021 China4School of Computer Science and Engineering Xirsquoan University of Technology Xirsquoan 710048 China5Institute of Mathematics Silesian University of Technology Kaszubska 23 Gliwice 44-100 Poland6Czestochowa University of Technology Al Armii Krajowej 36 Czestochowa 42-200 Poland

Correspondence should be addressed to Sun Zeng-guo sunzgsnnueducn

Received 7 September 2020 Revised 8 October 2020 Accepted 7 November 2020 Published 29 November 2020

Academic Editor Manjit Kaur

Copyright copy 2020Qiao Ke et al+is is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A new speckle suppression algorithm is proposed for high-resolution synthetic aperture radar (SAR) images It is based on thenonlocal means (NLM) filter and themodified Aubert and Aujol (AA)model+ismethod takes the nonlocal Dirichlet function asa linear regularization item which constructs the weight by measuring the similarity of images +en a new despeckling model isintroduced by combining the regularization item and the data item of the AA model and an iterative algorithm is proposed tosolve the new model +e experiments show that compared with the AA model the proposed model has more effectiveperformance in suppressing speckle namely ENL and DCV measures are 2175 and 45 higher respectively than for NLMMoreover it also has better performance in keeping the edge information

1 Introduction

Synthetic aperture radar (SAR) is widely used in manyaspects such as ecology hydrology ocean monitoring andtopographic mapping for its advantageous of all-day all-weather multiangle of view and penetration of groundobjects Because of the coherent imaging system of SARimages speckle inevitably appears in the imaging processEspecially imaging of high-resolution SAR images is morecomplex and demanding However the existence of speckleseriously affects the quality of images makes the interpre-tation and subsequent processing of images difficult andcannot correctly reflect the characteristics of the object+erefore the speckle suppression in high-resolution SARimages is of great significance [1ndash3]

+ere are two main aims of speckle suppression in SARimages One is to effectively suppress speckle in homoge-neous regions and the other is to preserve edges and finedetails as much as possible SAR image despeckling

algorithm has been deeply studied Traditional filtering al-gorithms based on local statistics [4] derive local statisticsbased on homogeneous regions so the image structure is notpreserved enough the edge is blurred to some extent andthe point target is filtered out+e despecklingmethod basedon wavelet transform [5ndash7] is the filter based on a fixedwindow and image edge information can produce Gibbsphenomenon [8] In recent years the method based onpartial differential equation (PDE) has become the focus ofspeckle suppression in SAR images due to its good edgepreservation [9 10] Many PDE methods are limited toimages with additive noise Based on this Aubert and Aujolpropose a multiplier speckle suppression model namely AAmodel in which speckle obeys gamma distribution [11] +eAA model can suppress speckle to a certain extent but theeffect is not very ideal in image edge and texturepreservation

Buades et al [12] propose a nonlocal means (NLM)filtering algorithm +e algorithm extends the local feature

HindawiSecurity and Communication NetworksVolume 2020 Article ID 8889317 8 pageshttpsdoiorg10115520208889317

statistics of traditional speckle suppression algorithm tononlocal domain and uses structural similarity to measurethe difference between pixels Compared with traditionalfiltering algorithm the NLM filter can preserve image detailsand texture information well [13ndash15] In addition based onthe PDE method Kindermann et al [16] propose a generalform of energy functional regularization item based onnonlocal means Some authors use clustering [17] or neuralnetworks [18] for processing satellite images

Gilboa and Osher [19] construct weights and regulari-zation items by measuring the similarity of images andpropose a nonlocal Dirichlet function as a regularizationitem In this paper a new despeckling model of high-res-olution SAR images is introduced by combining the regu-larization item and the data item of AA model and aniterative algorithm is proposed to solve the new modelDespeckling experiments on different kinds of SAR imagesdemonstrate that compared with the AA model the pro-posed model has more effective performance in suppressingspeckle and it also has better performance in keeping theedge information

2 AA Model

For SAR images let u be the restored image f be the ob-served image and ] be the speckle +en

f uv (1)

Based on the Bayesian framework assuming that thespeckle ] obeys Gamma distribution the probability densityfunction is written as follows

gv(v) L

L

Γ(L)v

Lminus1e

minusLv (2)

where L denotes the number of looks of images and Γ(middot)

denotes the gamma function By reasoning Aubert andAujol proposed a multiplicative noise denoising modelnamely the AA model as follows

argminu

TV(u) +λ2

1113946Ωlogu(x) +

f(x)

u(x)dx1113888 1113889 (3)

where regularization item TV(u) is the total variation of uwhich guarantees the smoothness of the restored imageData item1113938Ωlogu(x) + (f(x)u(x))dx is used to ensure thatthe restored image u retains the main features of the ob-served image f and λ is a scale parameter used to balance theregularization item and the data item

+e AA model is a problem of minimizing the totalvariation and is solved discretely +e discrete iteration formis obtained as follows

un+1 un + Δtdivnablau

|nablau|1113888 1113889 minus Δtλ

un minus u0

u2n

1113888 1113889 (4)

where Δt is the time step and div(middot) is the divergenceFigure 1 shows the despeckling results of the AAmodel for a

real high-resolution SAR image and the SAR image is ac-quired by the Sandia National Laboratories in the UnitedStates When the maximum peak signal-to-noise ratio(PSNR) of the speckle suppression image is reached theiteration is stopped It can be seen that the AA modelsuppresses speckle well in homogeneous regions but theeffect is not ideal in image edge and texture preservationwhich blurs image edge structure information to a certainextent

3 NLM Filter

Buades et al proposed a NLM filter [12] Its basic idea is toopen a window centered on each pixel i in the image and useevery pixel j in the window to estimate the value of the pixel i+e estimation uses two smaller window similarities asevaluation criteria +e centers of the two windows are i andj respectively Andweights are calculated using the Gaussianweighted Euclidean distance between the two windowsWhen the current pixel is estimated the weight of the pixelwhich is similar to the central pixel structure in the localstructure is larger and noise can be effectively removed bythe weighted mean +e mathematical expression of theNLM filter is written as follows

NLM(u)(i) 1

C(i j)1113944jisinΩ

w(i j)u(j) (5)

where NLM(u)(i) is the weighted filtering result w( i j)

eminus d(ij)h2 is a filtering weight d(i j) is a similarity distancefunction of pixels i and j h is the filtering parameterC(i j) 1113936jisinΩw(i j) is a normalized function and Ω is aneighborhood size of pixel i Parameters h and Ω affect thefinal denoising effect

+e SAR image speckle obeys themultiplicativemodel andthe NLM filtering algorithm is deduced based on the additiveGaussian noise +erefore original SAR images are usuallytransformed logarithmically before processing them Figure 2shows the despeckling results of the NLM filter for a real high-resolution SAR image and SAR images were acquired by theSandia National Laboratories in the United States It can beseen that the NLM filter has a weak ability to suppress specklein homogeneous regions but has stronger ability to preserveedge and point targets than the AA model in texture regions

4 ModifiedModel Based on the NLM Filter andthe AA Model

+e above results show that the AAmodel has more effectiveperformance in suppressing speckle in homogeneous re-gions but it has weak performance in keeping the edgeinformation In contrast the NLM filter has weak perfor-mance in suppressing speckle in homogeneous regions butit has more effective performance in keeping the edge in-formation +us in this paper a modified model based onthe NLM filter and the AA model is proposed

2 Security and Communication Networks

41 Establishment of the Model Gilboa and Osher con-structed weights and regularization terms by measuring thesimilarity of images based on the NLM filter and proposed anonlocal Dirichlet function as a regularization item

D(u) 14

1113946

iisinΩ

nablaNLu1113868111386811138681113868

11138681113868111386811138682(i)di

14BΩtimesΩ

(u(i) minus u(j))2w(i j)didj

(6)

where w(i j) eminusd(f(i)f(j))h2 is a weight function d(f(i)

f(j)) is a similarity distance function of pixel i and pixel jand h is a filtering parameter +e weight function is used tocalculate the similarity between the speckle in the windowand the unsuppressed speckle

In this paper a modified despeckling model of SARimages is proposed by combining the regularization itemand data item of the AA model

argminu

14BΩtimesΩ

(u(i) minus u(j))2w(i j)didj +

λ2

1113946

Ω

logu(i) +f(i)

u(i)1113888 1113889di

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(7)

42 Solution of the Model Using variational method theEulerndashLagrange equation corresponding to model (7) isshown as follows

1113946

Ω

(u(i) minus u(j))(w(i j))dy + λu(i) minus f(j)

u(i)21113888 1113889 0 (8)

Using the steepest descent method the steepest descentflow is shown as follows

(a) (b)

Figure 1 Despeckling results of SAR image (a) SAR image (b) AA model

(a) (b)

Figure 2 Despeckling results of SAR image (a) SAR image (b) NLM filter

Security and Communication Networks 3

ut 1113946

Ω

(u(i) minus u(j))w(i j)dy + λu(i) minus f(j)

u(i)21113888 1113889 (9)

where scale parameter λ can be seen as a Lagrange multiplierand obtained by the following formula

λ 1

|Ω|σ21113946

Ω

(u(i) minus f(i)) 1113946

Ω

(u(j) minus u(i))w(i j)dj⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦di

(10)

+e iteration form of the discretized steepest descendingflow (9) is as follows

un+1i u

ni + Δt 1113944

j

wij uni minus u

nj1113872 1113873 + λΔt

uni minus f

0i1113872 1113873

uni( 1113857

2 (11)

where ui denotes the value of pixel i uj denotes a pixel valuein the neighborhood of pixel i wij is the weight functionw(i j) and Δt is the time step In the actual implementationof the algorithm a small value ε is added to the denominatorto avoid the denominator being zero then the discretizediteration form becomes as follows

un+1i u

ni + Δt 1113944

j

wij uni minus u

nj1113872 1113873 + λΔt

uni minus f

0i1113872 1113873

uni( 1113857

2+ ε

(12)

+e flow chart of the improved filtering algorithm isshown in Figure 3 Firstly in high-resolution SAR imagespixel i is taken and its neighborhood Ω and a block to beestimated N(i) are determined Secondly in neighborhoodΩ of pixel i the filter weights of N(i) and N(i)are calculatedby taking the similar block N(j) of pixel j And then filterweights of all pixels in the neighborhood and pixel i arecalculated by traversing whole neighborhood Ω Finally thedespeckling result is obtained by putting an iterative formulaof the improved model after discretization

5 Despeckling Experiments

Figure 4 shows the despeckling results of a real high-resolution SAR image to illustrate the speckle suppressioneffect of the proposed algorithm SAR images are acquired bythe Sandia National Laboratories in the United States Inorder to verify the filtering performance of the proposedalgorithm more objectively some objective evaluation in-dicators are used to compare the AA model the NLM filterand the proposed algorithm +e experiment results areshown in Table 1 An equivalent number of looks (ENL) anddifference of the coefficient of variation (DCV) are indexesto evaluate the filtering effect [20ndash22] ENL is the mostcommonly used criterion for SAR image speckle suppres-sion and its calculation range is homogeneous regions ENLis defined as the ratio of the square of the mean value of apixel to the square of the standard deviation in a homo-geneous region +e larger the ENL value is the stronger thespeckle suppression ability of the despeckling algorithm is+e calculation range of DCV is edge regions and it isdefined as the difference of the coefficient of variation

between the real image and speckle suppression image in oneedge region +e closer the DCV value is to 0 the strongerthe preservation ability of speckle suppression algorithm foredge information is

As can be seen from Figure 4 the AA model achieves abetter speckle suppression effect in homogeneous regionsbut largely blurs the edge of the image In Table 1 the DCVvalue of the AA model for speckle suppression is the largestwhich also confirms that the edge preservation ability of theAA model is the worst By converting speckle into an ad-ditive noise model the NLM filter is superior to the AAmodel in preserving edge structure information and itscorresponding DCV value is closer to 0 than that of the AAmodel However the NLM filter has a worse ability tosuppress speckle in homogeneous regions than the AAmodel and its corresponding ENL value is also smaller than

Take pixel i and determine its neighborhood Ω and block to be estimated N(i)

Begin

Calculate the filter weights of N(i) and N(j)

Take a similar block N(i) of pixel jin neighborhood Ω of pixel i

Is j the last pixel in neighborhood Ω

Substitute into (12) and calculate filtering results aer iteration

End

Yes

No

Figure 3 Flow chart of the proposed method

Table 1 Quantitative measures evaluating the performance ofvarious methods in Figure 4

ENL DCVAA model 58427 01935NLM filter 35489 00723Proposed algorithm 78692 00756


(a) (b)

(c) (d)

Figure 4 Despeckling results of a SAR image (a) SAR image (b) AA model (c) NLM filter (d) the proposed method

(a) (b)

Figure 5 Continued


that of the AA model In contrast the proposed algorithmcan suppress speckle in homogeneous regions more thor-oughly and preserve edge details better It can also be seen

from Table 1 that the ENL value of the proposed algorithm isthe largest which shows that the proposed algorithm has amore thorough ability of the speckle suppression in

(c) (d)

Figure 5 Despeckling results of SAR image (a) SAR image (b) AA model (c) NLM filter (d) the proposed method

(a) (b)

(c)

Figure 6 Speckle patterns of various methods in Figure 5 (a) AA model (b) NLM filter (c) the proposed method


homogeneous regions than the traditional AAmodel and theNLM filter At the same time the DCV value of the proposedalgorithm is closer to 0 than that of the traditional AAmodel which shows that the edge preservation ability of theproposed algorithm is stronger

In order to further illustrate the speckle suppressionability and edge preservation ability of the proposed algo-rithm for SAR images the results of the speckle suppressionfor a SAR image of farmland and the corresponding speckleimage are given in Figures 5 and 6 respectively [23ndash33] +especkle image is defined by the ratio of the observed image toimage after speckle suppression +e amount of edge in-formation on the speckle image reflects the degree ofpreservation of edge structure information by the specklesuppression method+e less the edge structure informationon speckle image the stronger the edge preservation abilityof the corresponding speckle suppression method +e idealspeckle suppression method can completely suppressspeckle without losing any edge structure information Inaddition the speckle is not only in homogeneous regions butalso in edge regions [33ndash47]

From the results of the speckle suppression in Figure 5 itcan be seen directly that the proposed algorithm is superiorto the AA model and the NLM filter in speckle suppressionand preserves the edge information of the image Further-more the speckle image in Figure 6 shows that the speckleimage corresponding to the proposed algorithm is obviouslycovered by granular speckle which shows that the proposedalgorithm has better speckle suppression ability than the AAmodel and the NLM filter In addition the AA model hasmore edge information on the speckle image while theproposed algorithm has less edge information on the speckleimage +is shows that the AA model has a poor edgepreservation ability and the proposed algorithm has a strongedge preservation ability

6 Conclusions

In this paper a new speckle suppression algorithm is pro-posed for high-resolution SAR images +is method takesthe nonlocal Dirichlet function as a linear regularizationitem which constructs the weight by measuring the simi-larity of images +en an improved despeckling model isintroduced by combining the regularization item and thedata item of the AA model and an iterative algorithm isproposed to solve the new model Moreover we comparedthe proposed method with the AAmodel and the NLM filteralgorithm+e experiments show that the proposed model ismore effective in suppressing speckle in homogeneous re-gions and it also has a better performance in keeping theedge information +us it turned out to be an effectivespeckle suppression method for SAR images

+e limitation of our work is a limited number of imagesIn the future we plan to perform broader experimentsBesides we plan to finetune the method parameters

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+e authors would like to thank the US Sandia NationalLaboratories for providing some SAR images to performdespeckling experiments in this paper +is work wassupported by the China Postdoctoral Science Foundation(Nos 2020TQ0247 and 2020M683567) Development Pro-gram of Shaanxi Province (No 2018ZDXM-GY-036)Shaanxi Key Laboratory of Intelligent Processing for BigEnergy Data (No IPBED7) the Fundamental ResearchFunds for the Central Universities (No GK201903085) andthe Key Laboratory of Land Satellite Remote Sensing Ap-plication Center Ministry of Natural Resources of thePeoples Republic of China (No KLSMNR-202004)

References

[1] Y-N Zhang and L I Yinglte Key Technology of SAR ImageProcessing Publishing House of Electronics Industry BeijingChina 2014

[2] J I A O Li-Cheng and B Hou Intelligent SAR ImageProcessing and Interpretation Science Press Beijing China2008

[3] P I Yi-Ming J-Y Yang F U Yu-Sheng et al SyntheticAperture Radar Imaging Principle University of ElectronicScience and Technology Press Chengdu China 2007

[4] V Bhateja A Tripathi A Gupta et al An Improved LocalStatistics Filter for Denoising of SAR Images Springer Inter-national Publishing Berlin Germany 2014

[5] R Tao H Wan and Y Wang ldquoArtifact-free despeckling ofSAR images using contourletrdquo IEEE Geoscience and RemoteSensing Letters vol 9 no 5 pp 980ndash984 2012

[6] J Ji X Li S-X Xu H Liu and J-J Huang ldquoSAR imagedespeckling by sparse reconstruction based on shearletsrdquoActaAutomatica Sinica vol 41 no 8 pp 1495ndash1501 2015

[7] S Lang X Liu B Zhao X Chen and G Fang ldquoFocusedsynthetic aperture radar processing of ice-sounding datacollected over the east antarctic ice sheet via the modifiedrange migration algorithm using curveletsrdquo IEEE Transac-tions on Geoscience and Remote Sensing vol 53 no 8pp 4496ndash4509 2015

[8] J-J Xu SAR Image Despeckling Based on Nonlocal MeansFiltering Xidian University Xirsquoan China 2010

[9] Y Zhao J G Liu B Zhang W Hong and Y-R WuldquoAdaptive total variation regularization based SAR imagedespeckling and despeckling evaluation indexrdquo IEEE Trans-actions on Geoscience and Remote Sensing vol 53 no 5pp 2765ndash2774 2015

[10] Q Liu Z Yao and Y Ke ldquoSolutions of fourth-order partialdifferential equations in a noise removal modelrdquo ElectronicJournal of Differential Equations vol 22 no 3 pp 249ndash2662007

[11] G Aubert and J-F Aujol ldquoA variational approach to re-moving multiplicative noiserdquo SIAM Journal on AppliedMathematics vol 68 no 4 pp 925ndash946 2008

[12] A Buades B Coll and J M Morel ldquoA review of imagedenoising algorithms with a new onerdquoMultiscale Modeling ampSimulation vol 4 no 2 pp 490ndash530 2005


[13] Y I Zi-Lin D Yin and H U An-Zhou ldquoSAR image des-peckling based on non-local means filterrdquo Journal of Elec-tronics and Information Technology vol 34 no 4pp 950ndash955 2012

[14] Z-M Zhao Y-J Zhao and N I U Chao-Yang ldquoImprovedpolarimetric SAR speckle filter based on non-local meansrdquoJournal of Image and Graphics vol 18 no 8 pp 1038ndash10442013

[15] Y Jin J Jost and G Wang ldquoA new nonlocal H 1 model forimage denoisingrdquo Journal of Mathematical Imaging and Vi-sion vol 48 no 1 pp 93ndash105 2014

[16] S Kindermann S Osher and P W Jones ldquoDeblurring anddenoising of images by nonlocal functionalsrdquo MultiscaleModeling amp Simulation vol 4 no 4 pp 1091ndash1115 2005

[17] G Capizzi G L Sciuto M Wozniak and R DamaseviciusldquoA clustering based system for automated oil spill detection bysatellite remote sensingrdquoArtificial Intelligence and SoftComputing Berlin Germany Springer pp 613ndash623 2016

[18] G Chen C Li W Wei et al ldquoFully convolutional neuralnetwork with augmented atrous spatial pyramid pool and fullyconnected fusion path for high resolution remote sensing imagesegmentationrdquo Applied Sciences vol 9 no 9 p 1816 2019

[19] G Gilboa and S Osher ldquoNonlocal linear image regularizationand supervised segmentationrdquo Multiscale Modeling amp Sim-ulation vol 6 no 2 pp 595ndash630 2007

[20] A Lopes R Touzi and E Nezry ldquoAdaptive speckle filters andscene heterogeneityrdquo IEEE Transactions on Geoscience andRemote Sensing vol 28 no 6 pp 992ndash1000 1990

[21] R Touzi ldquoA review of speckle filtering in the context of es-timation theoryrdquo IEEE Transactions on Geoscience and Re-mote Sensing vol 40 no 11 pp 2392ndash2404 2002

[22] A Lapini t Bianchi F Argenti and L Alparone ldquoBlindspeckle decorrelation for SAR image despecklingrdquo IEEETransactions on Geoscience and Remote Sensing vol 52 no 2pp 1044ndash1058 2014

[23] S U N Zeng-Guo and H A N Chong-Zhao ldquoCombineddespeckling algorithm of synthetic aperture radar imagesbased on region classification adaptive windowing andstructure detectionrdquo Acta Physica Sinica vol 59 no 5pp 3210ndash3220 2010

[24] P Zheng Y Qi Y Zhou P Chen J Zhan and M R-T LyuldquoAn automatic framework for detecting and characterizingthe performance degradation of software systemsrdquo IEEETransactions on Reliability vol 63 no 4 pp 927ndash943 2014

[25] H Dou Y Qi W Wei and H Song ldquoA two-time-scale loadbalancing framework for minimizing electricity bills of In-ternet Data Centersrdquo Personal and Ubiquitous Computingvol 20 no 5 pp 681ndash693 2016

[26] P Wang Y Qi and X Liu ldquoPower-aware optimization forheterogeneous multi-tier clustersrdquo Journal of Parallel andDistributed Computing vol 74 no 1 pp 2005ndash2015 2014

[27] Y-n Qiao Q Yong and H Di ldquoTensor Field Model forhigher-order information retrievalrdquo Journal of Systems andSoftware vol 84 no 12 pp 2303ndash2313 2011

[28] J Yan Y Qi and Q Rao ldquoDetecting malware with an ensemblemethod based on deep neural networkrdquo Security And Commu-nication Networks vol 2018 16 pages Article ID7247095 2018

[29] X Wang Y Qi Z Wang et al ldquoDesign and implementationof SecPod a framework for virtualization-based securitysystemsrdquo IEEE Transactions on Dependable and SecureComputing vol 16 no 1 pp 44ndash57 2019

[30] W Wei and Y Qi ldquoInformation potential fields navigation inwireless Ad-Hoc sensor networksrdquo Sensors vol 11 no 5pp 4794ndash4807 2011

[31] X Fan H Song X Fan and J Yang ldquoImperfect informationdynamic stackelberg game based resource allocation usinghidden markov for cloud computingrdquo IEEE Transactions onServices Computing vol 11 2018

[32] H Song W Li P Shen and A Vasilakos ldquoGradient-drivenparking navigation using a continuous information potentialfield based on wireless sensor networkrdquo Information Sciencesvol 408 pp 100ndash114 2017

[33] Q Xu L Wang X H Hei P Shen W Shi and L ShanldquoGIGeom1 queue based on communication model for meshnetworksrdquo International Journal of Communication Systemsvol 27 no 11 pp 3013ndash3029 2013

[34] Z Sun H Song H Wang and X Fan ldquoEnergy balance-basedsteerable arguments coverage method inWSNsrdquo IEEE Accessvol 99 2017

[35] H Song H Wang and X Fan ldquoResearch and simulation ofqueue management algorithms in ad hoc network underDDoS attackrdquo IEEE Access vol 5 2017

[36] X Fan H Song and HWang ldquoVideo tamper detection basedon multi-scale mutual informationrdquo Multimedia Tools ampApplications vol 78 pp 1ndash18 2019

[37] X L Yang B Zhou J Feng and P Y Shen ldquoCombinedenergy minimization for image reconstruction from fewviewsrdquo Mathematical Problems in Engineering vol 2012Article ID 154630 15 pages 2012

[38] X L Yang P Y Shen and B Zhou ldquoHoles detection in an-isotropic sensornets topological methodsrdquo International Journalof Distributed Sensor Networks vol 8 no 10 p 135054 2012

[39] W Wei Y Qiang and J Zhang ldquoA bijection between lattice-valued filters and lattice-valued congruences in residuatedlatticesrdquo Mathematical Problems in Engineering vol 2013Article ID 908623 6 pages 2013

[40] HM Srivastava Y Zhang LWang P Shen and J Zhang ldquoAlocal fractional integral inequality on fractal space analogousto Andersonrsquos inequalityrdquo Abstract and Applied Analysisvol 2014 Article ID 797561 14 pages 2014

[41] S Liu W Li and D Du ldquoFractal Intelligent Privacy Pro-tection in Online Social Network Using Attribute-BasedEncryption Schemesrdquo IEEE Transactions on ComputationalSocial Systems vol 6 no 3 pp 736ndash747 2019

[42] X Fan M Wozniak H Song W Li Y Li and P Shen ldquoHinfincontrol of network control system for singular plantrdquo In-formation Technology And Control vol 47 no 1 pp 140ndash1502018

[43] Q Ke J Zhang H Song and Y Wan ldquoBig data analyticsenabled by feature extraction based on partial independencerdquoNeurocomputing vol 288 pp 3ndash10 2018

[44] J Zhang D P WeiWei M Wozniak L Kosmider andR Damasevıcius ldquoA neuro-heuristic approach for recognition oflung diseases fromX-ray imagesAuthor links open overlay panelrdquoExpert Systems with Applications vol 126 pp 218ndash232 2019

[45] J zhang W Wei R Damasevicius and M WozniakldquoAdaptive independent subspace analysis (AISA) of brainmagnetic resonance imaging (MRI) datardquo IEEE Access vol 7no 1 pp 12252ndash12261 2019

[46] Q Ke J Zhang M Wozniak and W Wei ldquo+e phase andshift-invariant feature by adaptive independent subspaceanalysis for cortical complex cellsrdquo Information Technologyand Control vol 48 2019

[47] J Su H Song H Wang and X Fan ldquoCDMA-based anti-collision algorithm for EPC global C1 Gen2 systemsrdquo Tele-communication Systems vol 67 no 3 pp 1ndash9 2018


statistics of traditional speckle suppression algorithm tononlocal domain and uses structural similarity to measurethe difference between pixels Compared with traditionalfiltering algorithm the NLM filter can preserve image detailsand texture information well [13ndash15] In addition based onthe PDE method Kindermann et al [16] propose a generalform of energy functional regularization item based onnonlocal means Some authors use clustering [17] or neuralnetworks [18] for processing satellite images

Gilboa and Osher [19] construct weights and regulari-zation items by measuring the similarity of images andpropose a nonlocal Dirichlet function as a regularizationitem In this paper a new despeckling model of high-res-olution SAR images is introduced by combining the regu-larization item and the data item of AA model and aniterative algorithm is proposed to solve the new modelDespeckling experiments on different kinds of SAR imagesdemonstrate that compared with the AA model the pro-posed model has more effective performance in suppressingspeckle and it also has better performance in keeping theedge information

2 AA Model

For SAR images let u be the restored image f be the ob-served image and ] be the speckle +en

f uv (1)

Based on the Bayesian framework assuming that thespeckle ] obeys Gamma distribution the probability densityfunction is written as follows

gv(v) L

L

Γ(L)v

Lminus1e

minusLv (2)

where L denotes the number of looks of images and Γ(middot)

denotes the gamma function By reasoning Aubert andAujol proposed a multiplicative noise denoising modelnamely the AA model as follows

argminu

TV(u) +λ2

1113946Ωlogu(x) +

f(x)

u(x)dx1113888 1113889 (3)

where regularization item TV(u) is the total variation of uwhich guarantees the smoothness of the restored imageData item1113938Ωlogu(x) + (f(x)u(x))dx is used to ensure thatthe restored image u retains the main features of the ob-served image f and λ is a scale parameter used to balance theregularization item and the data item

+e AA model is a problem of minimizing the totalvariation and is solved discretely +e discrete iteration formis obtained as follows

un+1 un + Δtdivnablau

|nablau|1113888 1113889 minus Δtλ

un minus u0

u2n

1113888 1113889 (4)

where Δt is the time step and div(middot) is the divergenceFigure 1 shows the despeckling results of the AAmodel for a

real high-resolution SAR image and the SAR image is ac-quired by the Sandia National Laboratories in the UnitedStates When the maximum peak signal-to-noise ratio(PSNR) of the speckle suppression image is reached theiteration is stopped It can be seen that the AA modelsuppresses speckle well in homogeneous regions but theeffect is not ideal in image edge and texture preservationwhich blurs image edge structure information to a certainextent

3 NLM Filter

Buades et al proposed a NLM filter [12] Its basic idea is toopen a window centered on each pixel i in the image and useevery pixel j in the window to estimate the value of the pixel i+e estimation uses two smaller window similarities asevaluation criteria +e centers of the two windows are i andj respectively Andweights are calculated using the Gaussianweighted Euclidean distance between the two windowsWhen the current pixel is estimated the weight of the pixelwhich is similar to the central pixel structure in the localstructure is larger and noise can be effectively removed bythe weighted mean +e mathematical expression of theNLM filter is written as follows

NLM(u)(i) 1

C(i j)1113944jisinΩ

w(i j)u(j) (5)

where NLM(u)(i) is the weighted filtering result w( i j)

eminus d(ij)h2 is a filtering weight d(i j) is a similarity distancefunction of pixels i and j h is the filtering parameterC(i j) 1113936jisinΩw(i j) is a normalized function and Ω is aneighborhood size of pixel i Parameters h and Ω affect thefinal denoising effect

+e SAR image speckle obeys themultiplicativemodel andthe NLM filtering algorithm is deduced based on the additiveGaussian noise +erefore original SAR images are usuallytransformed logarithmically before processing them Figure 2shows the despeckling results of the NLM filter for a real high-resolution SAR image and SAR images were acquired by theSandia National Laboratories in the United States It can beseen that the NLM filter has a weak ability to suppress specklein homogeneous regions but has stronger ability to preserveedge and point targets than the AA model in texture regions

4 ModifiedModel Based on the NLM Filter andthe AA Model

+e above results show that the AAmodel has more effectiveperformance in suppressing speckle in homogeneous re-gions but it has weak performance in keeping the edgeinformation In contrast the NLM filter has weak perfor-mance in suppressing speckle in homogeneous regions butit has more effective performance in keeping the edge in-formation +us in this paper a modified model based onthe NLM filter and the AA model is proposed



D(u) 14

1113946

iisinΩ

nablaNLu1113868111386811138681113868

11138681113868111386811138682(i)di

14BΩtimesΩ


(6)




argminu

14BΩtimesΩ


λ2

1113946

Ω

logu(i) +f(i)

u(i)1113888 1113889di

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(7)


1113946

Ω


u(i)21113888 1113889 0 (8)


(a) (b)


(a) (b)



ut 1113946

Ω


u(i)21113888 1113889 (9)


λ 1

|Ω|σ21113946

Ω

(u(i) minus f(i)) 1113946

Ω


⎤⎥⎥⎥⎥⎥⎥⎦di

(10)


un+1i u

ni + Δt 1113944

j

wij uni minus u

nj1113872 1113873 + λΔt

uni minus f

0i1113872 1113873

uni( 1113857

2 (11)


un+1i u

ni + Δt 1113944

j

wij uni minus u

nj1113872 1113873 + λΔt

uni minus f

0i1113872 1113873

uni( 1113857

2+ ε

(12)







Begin





End

Yes

No





(a) (b)

(c) (d)


(a) (b)

Figure 5 Continued




(c) (d)


(a) (b)

(c)






6 Conclusions



Data Availability




Acknowledgments


References



















































D(u) 14

1113946

iisinΩ

nablaNLu1113868111386811138681113868

11138681113868111386811138682(i)di

14BΩtimesΩ


(6)




argminu

14BΩtimesΩ


λ2

1113946

Ω

logu(i) +f(i)

u(i)1113888 1113889di

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(7)


1113946

Ω


u(i)21113888 1113889 0 (8)


(a) (b)


(a) (b)



ut 1113946

Ω


u(i)21113888 1113889 (9)


λ 1

|Ω|σ21113946

Ω

(u(i) minus f(i)) 1113946

Ω


⎤⎥⎥⎥⎥⎥⎥⎦di

(10)


un+1i u

ni + Δt 1113944

j

wij uni minus u

nj1113872 1113873 + λΔt

uni minus f

0i1113872 1113873

uni( 1113857

2 (11)


un+1i u

ni + Δt 1113944

j

wij uni minus u

nj1113872 1113873 + λΔt

uni minus f

0i1113872 1113873

uni( 1113857

2+ ε

(12)







Begin





End

Yes

No





(a) (b)

(c) (d)


(a) (b)

Figure 5 Continued




(c) (d)


(a) (b)

(c)






6 Conclusions



Data Availability




Acknowledgments


References


















































ut 1113946

Ω


u(i)21113888 1113889 (9)


λ 1

|Ω|σ21113946

Ω

(u(i) minus f(i)) 1113946

Ω


⎤⎥⎥⎥⎥⎥⎥⎦di

(10)


un+1i u

ni + Δt 1113944

j

wij uni minus u

nj1113872 1113873 + λΔt

uni minus f

0i1113872 1113873

uni( 1113857

2 (11)


un+1i u

ni + Δt 1113944

j

wij uni minus u

nj1113872 1113873 + λΔt

uni minus f

0i1113872 1113873

uni( 1113857

2+ ε

(12)







Begin





End

Yes

No





(a) (b)

(c) (d)


(a) (b)

Figure 5 Continued




(c) (d)


(a) (b)

(c)






6 Conclusions



Data Availability




Acknowledgments


References


















































(a) (b)

(c) (d)


(a) (b)

Figure 5 Continued




(c) (d)


(a) (b)

(c)






6 Conclusions



Data Availability




Acknowledgments


References




















































(c) (d)


(a) (b)

(c)






6 Conclusions



Data Availability




Acknowledgments


References





















































6 Conclusions



Data Availability




Acknowledgments


References


















































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