research article through-wall image enhancement using

Research ArticleThrough-Wall Image Enhancement Using Fuzzy andQR Decomposition

Muhammad Mohsin Riaz1 and Abdul Ghafoor2

1 Centre for Advanced Studies in Telecommunication (CAST) Comsats Islamabad Pakistan2Department of Electrical Engineering College of Signals National University of Sciences andTechnology (NUST) Islamabad Pakistan

Correspondence should be addressed to Abdul Ghafoor abdulghafoor-mcsnustedupk

Received 5 August 2013 Accepted 2 January 2014 Published 23 February 2014

Academic Editors S Bourennane and J Marot

Copyright copy 2014 M M Riaz and A Ghafoor This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

QR decomposition and fuzzy logic based scheme is proposed for through-wall image enhancement QR decomposition is lesscomplex compared to singular value decomposition Fuzzy inference engine assigns weights to different overlapping subspacesQuantitative measures and visual inspection are used to analyze existing and proposed techniques

1 Introduction

Mapping of scenes behind obstacles (including building wallrubbers grass etc) using through-wall imaging (TWI) isan unfolded research domain Different military and com-mercial applications (including antiterrorism hostage rescueand surveillance etc [1]) can benefit from TWI Besideother challenges minimization of unwanted artifacts (clut-tersnoise) has enjoyed special importance over last few years[2ndash13] These unwanted artifacts significantly decrease targetdetection and recognition capabilities

Existing TWI image enhancement (clutter removal) tech-niques include background scene subtraction (only feasibleif with and without target images are available) [2] spatialfiltering (assuming wall homogeneity at low frequencies) [3]wall modeling and subtraction (requiring complex processfor inhomogeneous walls) [4 5] Doppler filtering (applicableformoving targets only) [6] image fusion (requiringmultipledata of the same scene) [7] and statistical techniques [8ndash13]

In this paper a TWI image enhancement (clutter reduc-tion) technique using QR decomposition (QRD) and fuzzylogic is presented (preliminary results presented in [13])Weights are assigned to different QRD subspaces using fuzzyinference engine Simulation results evaluated using meansquare error (MSE) peak signal to noise ratio (PSNR)improvement factor (IF) and visual inspection (based on

miss detection (MD) and false detection (FD)) are used toverify the proposed scheme

2 Proposed Image Enhancement Using QRD

Let the input image119872 (having dimensions119866times119867) be decom-posed into different subspaces (119872cl 119872tar and 119872no) usingsingular value decomposition (SVD) as

119872 =

1198971

sum

119892=1

119904119892119906119892V119879

119892

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872cl

+

1198972

sum

119892=1198971+1

119904119892119906119892V119879

119892

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872tar

+

119866

sum

119892=1198972+1

119904119892119906119892V119879

119892

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872no

(1)

where 119880 and 119881 are singular vector matrices and 119878 containssingular values As discussed in [13] conventional SVD forTWI image enhancement assumes that the target is limitedto the second spectral component only that is

119872SVD = 11990421199062V119879

2 (2)

Besides the high computational complexity of SVD which is41198662119867+8119866119867

2+91198673 [14] the statement of target containment

in the second spectral component is not always true To caterthe above issues QRD and fuzzy logic based scheme isproposed The image119872 can be decomposed into an orthog-onal unitary matrix 119876 (having dimensions 119866 times 119867 and

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 487506 6 pageshttpdxdoiorg1011552014487506

2 The Scientific World Journal

Table 1 Comparison analysis of QRD algorithms for TWI

Algorithms Accuracy Complexity Stability119872minus119876119877 119876

119879119876 minus 119868 119876

119879119872minus 119877 119872119877

minus1minus 119876

CGS QR 1091 times 10minus16

1310 times 10minus10



21198661198672 Unstable

MGS QR 1075 times 10minus16




21198661198672 Stable

HT QR 1291 times 10minus15




41198662119867 + 2119866119867

2+2

31198673 Stable

Givens QR 7532 times 10minus16




81198662119867 + 2119866119867

2+2

31198673 Stable

column vectors 119902119892) and an upper triangular matrix 119877 (havingdimensions 119866 times 119867 and row vectors 119903119892) that is

119872 = 119876119877 (3)

Table 1 shows the accuracy stability and complexity analysisof different QRD algorithms (classical and modified Gram-Schmidt (CGS MGS) Givens decomposition Householdertransformation (HT) etc [14]) for TWI

Identical to SVD the first subspace1198721 = 11990211199031 representswall clutters and rest subspaces contain targets and noiseNote that due to overlapping boundaries of targets and noiseit is difficult to extract target subspaces accurately Foregoingin view a weighting QRD based scheme is proposed toenhance targets The enhanced image119872tar is

119872tar =119866

sum

119892=2

119908119892119902119892119903119892 (4)

where 119908119892 are weights applied to different subspaces Fuzzylogic is used for the automatic weight assignment [15]

21 Input and Output MFs Let 120585119892 = 119903119892 and Δ120585119892 = 119903119892 minus

119903119892+1 be norms and norm differences respectively Note thathigh value of 120585119892 and Δ120585119892 the corresponding subspace 119902119892119903119892

more likely contains target(s) and is therefore enhanced byapplying heavy weights (and vise versa)

Three Gaussian membership functions (MFs) 120577119883119909(1199091) =

exp(minus((1198881 minus 119888(119909)

1)120590(119909)

1)2

) and (119909 isin HighMedium Low)are defined for 120585119892 Similarly 120577119884119910(1199092) = exp(minus((1198882 minus 119888

(119910)

2)

120590(119910)

2)2

) and (119910 isin HighMedium Low) are defined for Δ120585119892where 1198881 1198882 isin [0 1] 119888(119909)

1 119888(119910)2

and 120590(119909)

1 120590(119910)2

are means andvariances of fuzzy sets respectively

119870-means algorithm [16] is used to adjust the fuzzyparameters 120585119896 and Δ120585ℎ are first clustered into three groupsbased on respective histograms The means and variancesrespectively of each group are used as centers 119888(119909)

1 119888(119910)2

andspreads 120590

(119909)

1 120590(119910)

2of MFs Five equally spaced output

MFs 120577119885119911(119889) = minus((119889 minus 119889(119911)

)984858(119911))

2

(119911 isin Very HighHighMedium LowVery Low) wheremean119889

(119911)

and variance 984858(119911)are used

22 Product Inference Engine (PIE) Gaussian fuzzifier mapsthe input 120585119892 and Δ120585119892 as

120577119883119884 (1198881 1198882) = expminus(1198881 minus 120585119892

V1)

2

expminus(1198882 minus Δ120585ℎ

V2)

2

(5)

where V1 and V2 are parameters used for input noise suppres-sion and are chosen as V1 = 2max119909120590

(119909)

1and V2 = 2max3

119910120590(119910)

2

[15]Fuzzy IF-THEN rules for image enhancement are the

following

Rule 1 IF 120585ℎ is 119883High and Δ120585ℎ is 119884

High THEN 119908PIEℎ

is119885Very High

Rule 2 IF 120585ℎ is 119883Med and Δ120585ℎ is 119884


is119885High


Med THEN 119908PIEℎ

is119885High



is119885Med


Low THEN 119908PIEℎ

is119885Med

Rule 6 IF 120585ℎ is 119883Low and Δ120585ℎ is 119884


is119885Med



is119885Low



is119885Low



is119885Very Low

The output of PIE using individual rule based inferenceMamdani implication algebraic product for 119905-norm andmaxoperator for 119904-norm [15] is

1205771198851015840 (119889119892)= max119909119910119911

[

[

sup1198881 1198882

120577119883119884 (1198881 1198882) 120577119883119909 (1198881) 120577119884119910 (1198882) 120577119885119911 (119889119892)]

]

(6)

The Scientific World Journal 3

The weights 119908PIE119892

are then computed as

119908PIE119892

=

sum119911119889(119911)

120603(119911)

119892

sum119911= 1120603(119911)

ℎ

(7)

where 120603(119911)119892

is the height of 1205771198851015840(119889119892) in output MFs [15]

23 Takagi-Sugeno (TS) Inference In contrast to PIE TSinference engine adjusts the output MFs using adaptive andor optimization techniques [17]TheTS rule-base (IF-THEN)for computing weights 119908TS

119892is

IF 120585119892 is 1198831198951 AND Δ120585119892 is 119884

1198952

THEN119901(1198951+1198952minus1)

= (1

1 + exp minus120585119892 + exp minusΔ120585119892)

1198951+1198952minus1

(8)

Note that the output reduces for large 1198951 + 1198952 (which is desir-able) The aggregated weights 119908TS

119892are

119908TS119892

=

sum3

1198951=1

sum3

1198952=1

119901(1198951+1198952minus1)

119905 1205771198831198951 (120585119896) 1205771198841198952 (Δ120585119892)

sum3

1198951=1

sum3

1198952=1

119905 1205771198831198951 (120585119892) 1205771198841198952 (Δ120585119892)

(9)

where 119905 represents algebraic product (intersection operator)

3 Simulation and Results

Experimental setup for TWI is constructed using Agilentrsquosvector network analyzer (VNA) which generates steppedfrequency waveforms between 2GHz and 3GHz (1GHzband width (BW)) having step size of Δ119891 = 5MHz and stepsize 119873119891 = 201 Maximum range is 119877max = 30m and rangeresolution is Δ119877 = 015m

Broadband horn antenna which is mounted on two-dimensional scanning frame (having dimensions 24m times

3m (width times height) and can slide along cross rangeand height) operates in monostatic mode with 12 dB gainThickness of the wall is 5 cm and relative permittivity andpermeability are 23 and 1 respectively The frame is placed003m away from wall and scanning is controlled by amicrocontroller basedmechanismThe scattering parametersare recorded at each step and transferred to a local computerfor image reconstruction and processing Received data isconverted into time domain and beamforming algorithm isused for image reconstruction Existing and proposed image

Table 2 MSE PSNR IF MD and FD comparison

Scenario Scheme MSE PSNR IF MD FD

Two targetsSVD [11] 02726 56442 81258 1 0

Fuzzy QRD (PIE) 01970 70553 112587 0 0Fuzzy QRD (TS) 01726 76296 115870 0 0

Three targetsSVD [11] 02814 55068 71265 1 1


enhancement algorithms are simulated in MATLAB andquantitative analysis is performed using MSE PSNR IF FDMD and visual inspection

MSE =1

119866 times 119867

119866

sum

119892=1

119867

sum

ℎ=1

(119872119887119904(119892 ℎ) minus 119872tar(119892 ℎ))2

PSNR (dB) = 10 log10

1

MSE

IF (dB) = 10 log10[

119875119872tar 119905times 119875119872119888

119875119872119905 times 119875119872tar119888

]

(10)

where 119872119887119904 is a reference image obtained by the differenceof image with and without target 119875119872tar 119905

and 119875119872tar119888are

average pixel values of target and clutter in enhanced imagerespectively 119875119872119905 and 119875119872119888 are average pixel values of targetand clutter in the original image respectively

MD is defined as ldquotarget was present in the original imagebut was not detected in enhanced imagerdquo FD is defined asldquotarget was not present in the original image but was detectedin enhanced imagerdquo For calculating FD and MD a thresholdis calculated using global thresholding algorithm [18]

Figure 1 shows the original B-scan containing two targetsthe background subtracted reference and enhanced imagesusing existing SVD and proposed QRD based schemes Itcan be observed that proposed schemes detect both targetswhereas SVD based scheme is unable to locate both targetsaccurately

Figure 2 shows another example containing three targetsThe proposed scheme detects all targets and provides a bettertarget to background ratio compared to SVDbased scheme Itis further noted that the proposed TS inference based schemeprovides better results compared to PIE

Table 2 shows that proposed fuzzy QRD schemes arebetter (as compared to the SVD image enhancement scheme)in terms of MSE PSNR IF MD and FD

4 Conclusion

QRD and fuzzy logic based image enhancement scheme isproposed for TWI Compared with SVD QRD provides lesscomputational complexity PIE and TS inference engines areused to assign weights to different QRD subspaces Simu-lation results compared on visual and quantitative analysisshow the significance of the proposed scheme


0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(a)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(b)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(c)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(d)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(e)

Figure 1 Image with two targets (a) Original image (b) SVD [11] (c) Proposed fuzzy QRD (PIE) (d) Proposed fuzzy QRD (TS)(e) Background subtracted reference image


0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(a)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(b)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(c)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(d)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(e)

Figure 2 Image with three targets (a) Original image (b) SVD [11] (c) Proposed fuzzy QRD (PIE) (d) Proposed fuzzy QRD (TS)(e) Background subtracted reference image


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M G AminThrough theWall Radar Imaging CRC Press BocaRaton Fla USA 2011

[2] J Moulton S Kassam F AhmadM Amin and K YemelyanovldquoTarget and change detection in synthetic aperture radar sens-ing of urban structuresrdquo in Proceedings of the IEEE Radar Con-ference (RADAR rsquo08) pp 1ndash6 May 2008

[3] Y S Yoon andMGAmin ldquoSpatial filtering forwall-cluttermit-igation in through-the-wall radar imagingrdquo IEEE Transactionson Geoscience and Remote Sensing vol 47 no 9 pp 3192ndash32082009

[4] M Dehmollaian and K Sarabandi ldquoRefocusing through build-ing walls using synthetic aperture radarrdquo IEEE Transactions onGeoscience and Remote Sensing vol 46 no 6 pp 1589ndash15992008

[5] G E Smith and B G Mobasseri ldquoRobust through-the-wallradar image classification using a target-model alignment pro-cedurerdquo IEEE Transactions on Image Processing vol 21 no 2pp 754ndash767 2012

[6] S S Ram C Christianson Y Kim and H Ling ldquoSimula-tion and analysis of human micro-dopplers in through-wallenvironmentsrdquo IEEE Transactions on Geoscience and RemoteSensing vol 48 no 4 pp 2015ndash2023 2010

[7] C DebesAdvances in detection and classification for through thewall radar imaging [PhD Dissertation] Technische UniversityDarmstadt 2010

[8] F H C Tivive M G Amin and A Bouzerdoum ldquoWall cluttermitigation based on eigen-analysis in through-the-wall radarimagingrdquo in Proceedings of the 17th International Conference onDigital Signal Processing (DSP rsquo11) pp 1ndash8 July 2011

[9] M M Riaz and A Ghafoor ldquoPrinciple component analysis andfuzzy logic based through wall image enhancementrdquo Progress inElectromagnetic Research vol 127 pp 461ndash478 2012

[10] MM Riaz and A Ghafoor ldquoThrough wall image enhancementbased on singular value decompositionrdquo International Journal ofAntenna and Propagation vol 2012 Article ID 961829 20 pages2012

[11] P K Verma A N Gaikwad D Singh and M J Nigam ldquoAnal-ysis of clutter reduction techniques for through wall imaging inUWB rangerdquo Progress in Electromagnetics Research B vol 17 pp29ndash48 2009

[12] A N Gaikwad D Singh andM J Nigam ldquoApplication of clut-ter reduction techniques for detection of metallic and lowdielectric target behind the brick wall by stepped frequencycontinuous wave radar in ultra-wideband rangerdquo IET RadarSonar and Navigation vol 5 no 4 pp 416ndash425 2011

[13] M M Riaz and A Ghafoor ldquoQR decomposition based imageenhancement for through wall imagingrdquo in Proceedings of theIEEE Radar Conference pp 978ndash983 2012

[14] GHGolub andC F LoanMatrixComputation JohnHopkinsBaltimore Md USA 1998

[15] L X Wang A Course in Fuzzy Systems and Control PrenticeHall Englewood Cliffs NJ USA 1997

[16] T Kanungo D M Mount N S Netanyahu C D Piatko RSilverman and A Y Wu ldquoAn efficient k-means clustering

algorithms analysis and implementationrdquo IEEETransactions onPattern Analysis andMachine Intelligence vol 24 no 7 pp 881ndash892 2002

[17] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[18] R C Gonzalez and R EWoodsDigital Image Processing Pren-tice Hall Englewood Cliffs NJ USA 3rd edition 2007

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RoboticsJournal of

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Active and Passive Electronic Components

Control Scienceand Engineering

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Propagation




Navigation and Observation



DistributedSensor Networks



Table 1 Comparison analysis of QRD algorithms for TWI

Algorithms Accuracy Complexity Stability119872minus119876119877 119876

119879119876 minus 119868 119876

119879119872minus 119877 119872119877

minus1minus 119876

CGS QR 1091 times 10minus16




21198661198672 Unstable

MGS QR 1075 times 10minus16




21198661198672 Stable

HT QR 1291 times 10minus15




41198662119867 + 2119866119867

2+2

31198673 Stable

Givens QR 7532 times 10minus16




81198662119867 + 2119866119867

2+2

31198673 Stable

column vectors 119902119892) and an upper triangular matrix 119877 (havingdimensions 119866 times 119867 and row vectors 119903119892) that is

119872 = 119876119877 (3)

Table 1 shows the accuracy stability and complexity analysisof different QRD algorithms (classical and modified Gram-Schmidt (CGS MGS) Givens decomposition Householdertransformation (HT) etc [14]) for TWI

Identical to SVD the first subspace1198721 = 11990211199031 representswall clutters and rest subspaces contain targets and noiseNote that due to overlapping boundaries of targets and noiseit is difficult to extract target subspaces accurately Foregoingin view a weighting QRD based scheme is proposed toenhance targets The enhanced image119872tar is

119872tar =119866

sum

119892=2

119908119892119902119892119903119892 (4)

where 119908119892 are weights applied to different subspaces Fuzzylogic is used for the automatic weight assignment [15]

21 Input and Output MFs Let 120585119892 = 119903119892 and Δ120585119892 = 119903119892 minus

119903119892+1 be norms and norm differences respectively Note thathigh value of 120585119892 and Δ120585119892 the corresponding subspace 119902119892119903119892

more likely contains target(s) and is therefore enhanced byapplying heavy weights (and vise versa)

Three Gaussian membership functions (MFs) 120577119883119909(1199091) =

exp(minus((1198881 minus 119888(119909)

1)120590(119909)

1)2

) and (119909 isin HighMedium Low)are defined for 120585119892 Similarly 120577119884119910(1199092) = exp(minus((1198882 minus 119888

(119910)

2)

120590(119910)

2)2

) and (119910 isin HighMedium Low) are defined for Δ120585119892where 1198881 1198882 isin [0 1] 119888(119909)

1 119888(119910)2

and 120590(119909)

1 120590(119910)2

are means andvariances of fuzzy sets respectively

119870-means algorithm [16] is used to adjust the fuzzyparameters 120585119896 and Δ120585ℎ are first clustered into three groupsbased on respective histograms The means and variancesrespectively of each group are used as centers 119888(119909)

1 119888(119910)2

andspreads 120590

(119909)

1 120590(119910)

2of MFs Five equally spaced output

MFs 120577119885119911(119889) = minus((119889 minus 119889(119911)

)984858(119911))

2

(119911 isin Very HighHighMedium LowVery Low) wheremean119889

(119911)

and variance 984858(119911)are used

22 Product Inference Engine (PIE) Gaussian fuzzifier mapsthe input 120585119892 and Δ120585119892 as

120577119883119884 (1198881 1198882) = expminus(1198881 minus 120585119892

V1)

2

expminus(1198882 minus Δ120585ℎ

V2)

2

(5)

where V1 and V2 are parameters used for input noise suppres-sion and are chosen as V1 = 2max119909120590

(119909)

1and V2 = 2max3

119910120590(119910)

2

[15]Fuzzy IF-THEN rules for image enhancement are the

following



is119885Very High



is119885High



is119885High



is119885Med



is119885Med



is119885Med



is119885Low



is119885Low



is119885Very Low

The output of PIE using individual rule based inferenceMamdani implication algebraic product for 119905-norm andmaxoperator for 119904-norm [15] is

1205771198851015840 (119889119892)= max119909119910119911

[

[

sup1198881 1198882

120577119883119884 (1198881 1198882) 120577119883119909 (1198881) 120577119884119910 (1198882) 120577119885119911 (119889119892)]

]

(6)




119908PIE119892

=

sum119911119889(119911)

120603(119911)

119892

sum119911= 1120603(119911)

ℎ

(7)

where 120603(119911)119892



119892is

IF 120585119892 is 1198831198951 AND Δ120585119892 is 119884

1198952

THEN119901(1198951+1198952minus1)

= (1


1198951+1198952minus1

(8)


119892are

119908TS119892

=

sum3

1198951=1

sum3

1198952=1

119901(1198951+1198952minus1)

119905 1205771198831198951 (120585119896) 1205771198841198952 (Δ120585119892)

sum3

1198951=1

sum3

1198952=1

119905 1205771198831198951 (120585119892) 1205771198841198952 (Δ120585119892)

(9)








Two targetsSVD [11] 02726 56442 81258 1 0





MSE =1

119866 times 119867

119866

sum

119892=1

119867

sum

ℎ=1

(119872119887119904(119892 ℎ) minus 119872tar(119892 ℎ))2


1

MSE

IF (dB) = 10 log10[

119875119872tar 119905times 119875119872119888

119875119872119905 times 119875119872tar119888

]

(10)


and 119875119872tar119888are






4 Conclusion



0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(a)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

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Rang

e (m

)

(b)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(c)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

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Rang

e (m

)

(d)

0

0

1

1

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08

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25

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Cross range (m)

Rang

e (m

)

(e)



0

0

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1

08

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02

05

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25

15

15

Cross range (m)

Rang

e (m

)

(a)

0

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1

08

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05

2

25

15

15

Cross range (m)

Rang

e (m

)

(b)

0

0

1

1

1

08

06

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02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(c)

0

0

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08

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25

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15

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Rang

e (m

)

(d)

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25

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Cross range (m)

Rang

e (m

)

(e)





References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119908PIE119892

=

sum119911119889(119911)

120603(119911)

119892

sum119911= 1120603(119911)

ℎ

(7)

where 120603(119911)119892



119892is

IF 120585119892 is 1198831198951 AND Δ120585119892 is 119884

1198952

THEN119901(1198951+1198952minus1)

= (1


1198951+1198952minus1

(8)


119892are

119908TS119892

=

sum3

1198951=1

sum3

1198952=1

119901(1198951+1198952minus1)

119905 1205771198831198951 (120585119896) 1205771198841198952 (Δ120585119892)

sum3

1198951=1

sum3

1198952=1

119905 1205771198831198951 (120585119892) 1205771198841198952 (Δ120585119892)

(9)








Two targetsSVD [11] 02726 56442 81258 1 0





MSE =1

119866 times 119867

119866

sum

119892=1

119867

sum

ℎ=1

(119872119887119904(119892 ℎ) minus 119872tar(119892 ℎ))2


1

MSE

IF (dB) = 10 log10[

119875119872tar 119905times 119875119872119888

119875119872119905 times 119875119872tar119888

]

(10)


and 119875119872tar119888are






4 Conclusion



0

0

1

1

1

08

06

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05

2

25

15

15

Cross range (m)

Rang

e (m

)

(a)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(b)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(c)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(d)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(e)



0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(a)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(b)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(c)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(d)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(e)





References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(a)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(b)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(c)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(d)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(e)



0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(a)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(b)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(c)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(d)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(e)





References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(a)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(b)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(c)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(d)

0

0

1

1

1

08

06

04

02

05

2

25

15

15

Cross range (m)

Rang

e (m

)

(e)





References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article through-wall image enhancement using

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