Iterative Block Tensor Singular Value Thresholding for Extraction of Low Rank Component of Image Data

0.98 MB·

by  Longxi Chen

ITERATIVEBLOCKTENSORSINGULARVALUETHRESHOLDINGFOREXTRACTION
OFLOWRANKCOMPONENTOFIMAGEDATA
LongxiChen YipengLiu CeZhu
SchoolofElectronicEngineering/CenterforRobotics/CenterforInformationinMedicine
UniversityofElectronicScienceandTechnologyofChina(UESTC),Chengdu,611731,China
emails: [email protected],{yipengliu,eczhu}@uestc.edu.cn
7
1
ABSTRACT
0
2 Tensorprincipalcomponentanalysis(TPCA)isamulti-linearextensionofprincipalcomponentanalysiswhichconvertsasetofcor-
related measurements into several principal components. In this paper, we propose a new robust TPCA method to extract the principal
n
componentsofthemulti-waydatabasedontensorsingularvaluedecomposition. Thetensorissplitintoanumberofblocksofthesame
a
size. The low rank component of each block tensor is extracted using iterative tensor singular value thresholding method. The principal
J
componentsofthemulti-waydataaretheconcatenationofallthelowrankcomponentsofalltheblocktensors. Wegivetheblocktensor
5 incoherenceconditionstoguaranteethesuccessfuldecomposition. Thisfactorizationhassimilaroptimalitypropertiestothatoflowrank
1
matrixderivedfromsingularvaluedecomposition. Experimentally,wedemonstrateitseffectivenessintwoapplications,includingmotion

separationforsurveillancevideosandilluminationnormalizationforfaceimages.
]
V
C IndexTerms— tensorprincipalcomponentanalysis,tensorsingularvaluedecomposition,lowranktensorapproximation,blocktensor
.
s
1. INTRODUCTION
c
[
Thehigh-dimensionaldata,alsoreferredtoastensors,arisenaturallyinanumberofscenarios,includingimageandvideoprocessing,and

1 datamining[1]. However,mostofthecurrentprocessingtechniquesaredevelopedfortwo-dimensionaldata[2]. Theprincipalcomponent
v analysis(PCA)isoneofthemostwidelyusedoneintwo-dimensionaldataanalysis[3].
3 TherobustPCA(RPCA),asanextensionofPCA,isaneffectivemethodinmatrixdecompositionproblems[4]. Supposewehavea
4 matrixX ∈Rn1×n2 ,whichcanbedecomposedasX =L0+S0,whereL0isthelowrankcomponentofthematrixandS0isthesparse
0 component. TheRPCAmethodhasbeenappliedtoimagealignment[5],surveillancevideoprocessing[6],illuminationnormalizationfor
4 faceimages[7]. Inmostapplications,theRPCAmethodshouldflattenorvectorizethetensordatasoastosolvetheprobleminthematrix.
0 Itdoesn’tusethestructuralfeatureofthedataeffectivelysincetheinformationlossinvolvesintheoperationofmatricization.
.
1 Tensor robust principal component analysis (TRPCA) has been studied in [8, 9] based on the tensor singular value decomposition (t-
0 SVD).Theadvantageoft-SVDovertheexistingmethodssuchascanonicalpolyadicdecomposition(CPD)[10]andTuckerdecomposition
7 [11]isthattheresultinganalysisisvernyclosetothatofmatrixanalysis[12]. Similarly,supposewearegivenatensorX ∈Rn1×n2×n3 and
3
1 itcanbedecomposedintolowrankcomponentandsparsecomnponent.Wecnanwriteitas n
n 3 n 3 n 3
v: n = n X =+Ln0+S0, + +n (1)
i 1 c c c
X whereL denotesthelowrankcomponent,andS isthespa1rsecomponen2tofthetensor. Fnig. 1istheillustrationforTRPCA.In[8]the
0 0
r problem(1)istransformedtotheconvexonptimizationmodel:
2
a
min(cid:107)L (cid:107) +λ(cid:107)S (cid:107) , s.t.X =L +S , (2)
0 ∗ 0 1 0 0
L0,S0
where(cid:107)L (cid:107) isthetensornuclearnorm(seesection2forthedefinition),(cid:107)S (cid:107) denotesthe(cid:96) -norm. Inthepaper[9]theproblem(1)is
0 ∗ 0 1 1
transformedtoanotherconvexoptimizationmodelas:
min(cid:107)L (cid:107) +λ(cid:107)S (cid:107) , s.t.X =L +S , (3)
0 ∗ 0 1,1,2 0 0
L0,S0
= +
Fig.1:IllustrationofTRPCA.
I I  J ,I  J
I  J 1 3 3 1 1
3 3
I  J
2 2 J I
1 1
I I J
2 2 2
Fig.2:Illustrationoftheconcatenationofblocktensors.
1,1 1,2 1,q
n
3

n 2,1 2,2 2,q
1
n
2
z,1 z,2 z,q
Fig.3:Illustrationoftheblocktensordecompositionmodel.
where(cid:107)S (cid:107) isdefinedasΣ (cid:107)S (i,j,:)(cid:107) .Thetwomethodssolvethetensordecompositionproblembasedonthet-SVD.
0 1,1,2 i,j 0 F
The lowrank andsparse matrix decompositionhas beenimproved by the[13]. The mainidea is incorporating multi-scale structures
withlowrankmethods.Theadditionalmulti-scalestructurescanobtainamoreaccuraterepresentationthanconventionallowrankmethods.
Inspiredbythiswork,wenoticethatthesparsecomponentinmatrixisblock-distributedinsomeapplications,e.g. shadowandmotionin
videos.Fortheseimageswefinditismoreeffectivetoextractthelowrankcomponentsinaanothersmallerscaleofimagedata.Herewetry
toextractlowrankcomponentsinblocktensordatathatisstackedbysmallscaleofimagedata.Andwhenwedecomposethetensordatainto
manysmallblocks,itiseasytoextracttheprincipalcomponentinsomeblocksthathavefewsparsecomponents. Wemodelourtensordata
astheconcatenationofblocktensorsinsteadofsolvingtheRPCAproblemasawholebigtensor.Fig.2istheillustrationofconcatenationof
blocktensors.
Basedontheabovemotivation,wedecomposethewholetensorintoconcatenationofblocksofthesamesize,thenweextractlowrank
componentofeachblockbyminimizingthetubalrankofeachblocktensor. Fig. 3istheillustrationofourmethod. Andwegetlowrank
componentofthewholetensorbyconcatenatingallthelowrankcomponentsoftheblocktensors.Theproposedmethodcanbeusedtosome
conventionalimageprocessingproblems,includingmotionseparationforsurveillancevideos(Section4.1)andilluminationnormalization
forfaceimages(Section4.2).Theresultsofnumericalexperimentsdemonstratethatourmethodoutperformstheexistingmethodsintermof
accuracy.
2. NOTATIONSANDPRELIMINARIES
Inthissection,wedescribethenotationsanddefinitionsusedthroughoutthepaperbriefly[14,15,16,17].
Athird-ordertensorisrepresentedasA,andits(i,j,k)-thentryisrepresentedasA .A(i,j,:)denotesthe(i,j)-thtubalscalar.A(i,:
i,j,k
,:),A(:,j,:)andA(:,:,k)arethei-thhorizontal,j-thlateralandk-thfrontalslices,respectively.(cid:107)A(cid:107) = (cid:113)(cid:80) |a |2 and(cid:107)A(cid:107) =
F i,j,k ijk ∞
max |a |tensorkindsoftensornorms.
i,j,k ijk
Wecanviewathree-dimensionaltensorofsizen ×n ×n asann ×n matrixoftubes. Aˆisatensorwhichisobtainedbytaking
1 2 3 1 2
thefastFouriertransform(FFT)alongthethirdmodeofA.ForacompactnotationwewilluseAˆ=fft(A,[],3)todenotetheFFTalongthe
thirddimension.Inthesameway,wecanalsocomputeAfromAˆusingtheinverseFFT(IFFT).
Definition2.1(t-product)[12]Thet-productE =A∗Bof A∈Rn1×n2×n3 andB∈Rn2×n4×n3 isan n1×n4×n3tensor.The(i,j)-th
tubeof E isgivenby
(cid:88)n2
E(i,j,:)= A(i,k,:)•B(k,j,:), (4)
k=1
where•denotesthecircularconvolutionbetweentwotubesofsamesize.
Definition2.2(conjugatetranspose)[14]TheconjugatetransposeofatensorAofsizen ×n ×n isthen ×n ×n tensorAT
1 2 3 2 1 3
obtainedbyconjugatetransposingeachofthefrontalsliceandthenreversingtheorderoftransposedfrontalslicesfrom2ton .
3
Definition2.3(identitytensor)[14]TheidentitytensorI ∈Rn×n×n3 isatensorwhosefirstfrontalsliceisthen×nidentitymatrixand
allotherfrontalslicesarezero.
Definition2.4(orthogonaltensor)[14]AtensorQisorthogonalifitsatisfies
QT∗Q=Q∗QT =I. (5)
Definition2.5(f-diagonaltensor)[14]Atensoriscalledf-diagonalifeachofitsfrontalslicesisadiagonalmatrix.
Definition2.6(t-SVD)[14]ForA∈Rn1×n2×n3 ,thet-SVDof Aisgivenby
A=U∗S∗VT (6)
whereU andVareorthogonaltensorsofsizen ×n ×n andn ×n ×n respectively,andSisaf-diagonaltensorofsizen ×n ×n .
1 1 3 2 2 3 1 2 3
We can obtain this decomposition by computing matrix singular value decomposition (SVD) in the Fourier domain, as it shows in
Algorithm1.Fig.4illustratesthedecompositionforthethree-dimensionalcase.
Algorithm1:t-SVDfor3-waydata
Input:A∈Rn1×n2×n3
D←fft(A,[],3),
fori=1ton ,do
3
[U,S,V]=svd(D(:,:,i)),
Uˆ(:,:,i)=U, Sˆ(:,:,i))=S, Vˆ(:,:,i)=V,
endfor
U ←ifft(Uˆ,[],3), S ←ifft(Sˆ,[],3), V ←ifft(Vˆ,[],3),
Output:U,S,V
n n n
n
3 3 3
3
= * * n
n n n 2
1 1 1 n
n n n 2
2 1 2
Fig.4:Illustrationofthet-SVDofann ×n ×n tensor.
1 2 3
Definition2.7(tensormulti-rankandtubalrank)[9]Thetensormulti-rankof A ∈ Rn1×n2×n3 isavectorr ∈ Rn3 withitsi-thentry
astherankofthei-thfrontalsliceof Aˆ,i.e.r =rank(Aˆ(:,:,i)).Thetensortubalrank,denotedbyrank (A),isdefinedasthenumberof
i t
nonzerosingulartubesof S,whereSisfrom A=U∗S∗VT,i.e.
rank (A)=#{i:S(i,i,:)(cid:54)=0}=maxr (7)
t i
i
Definition2.8(tensornuclearnorm: TNN)[9]ThetensornuclearnormofA∈Rn1×n2×n3 denotedby(cid:107)A(cid:107)∗ isdefinedasthesumof
thesingularvaluesofallthefrontalslicesof Aˆ. TheTNNof A isequaltotheTNNof blkdiag(Aˆ). Hereblkdiag(Aˆ)isablockdiagonal
matrixdefinedasfollows:
 Aˆ(1) 
 Aˆ(2) 
blkdiag(Aˆ)= ... , (8)
Aˆ(n3)
whereAˆ(i)isthei-thfrontalsliceof Aˆ,i=1,2,...,n .
3
Definition2.9(standardtensorbasis)[12]Thecolumnbasis,denotedas˚e ,isatensorofsizen×1×n withits(i,1,1)-thentryequaling
i 3
to1andtherestequalingto0.Naturallyitstranspose˚eTiscalledrowbasis.
i
3. ITERATIVEBLOCKTENSORSINGULARVALUETHRESHOLDING
Wedecomposethewholetensorwhichsatisfiestheincoherenceconditionsintomanysmallblocksofthesamesize.Andthethirddimension
of the block size should be the same as the third dimension of the tensor. That is to say, given an input tensor X ∈ Rn1×n2×n3 and its
correspondingblocksize, weproposeamulti-blocktensormodelingthatmodelsthetensordataX astheconcatenationofblocktensors.
Andeachblocktensorcanbedecomposedintotwocomponents,i.e. X =L +S , p=1,··· ,P,whereL andS denotethelowrank
p p p p p
componentandsparsecomponentofblocktensorX respectively.
p
AsobservedinRPCA,thelowrankandsparsedecompositionisimpossibleinsomecases[4]. Similarly,wearenotabletoidentifythe
lowrankcomponentandsparsecomponentifthetensorisofbothlowrankandsparsity.Similartothetensorincoherenceconditions[8],we
assumetheblocktensordataL ineachblocksatisfiessomeblocktensorincoherenceconditionstoguaranteesuccessfullowrankcomponent
p
extraction.
Definition 3.1 (block tensor incoherence conditions) For Lp ∈ Rn×n×n3, assume that rankt(Lp) = r and it has the t-SVD Lp =
Up∗Sp∗VpT,whereUp ∈Rn×r×n3 andVp ∈Rn×r×n3 satisfyUpT∗Up =IandVpT∗Vp =I,andSp ∈Rr×r×n3 isanf-diagonaltensor.
ThenL satisfiesthetensorincoherenceconditionswithparameterµif
p
(cid:114)
µr
max (cid:107)UT∗˚e (cid:107) (cid:54) (9)
i=1,...,n p i F nn3
(cid:114)
µr
max (cid:107)VT∗˚e (cid:107) (cid:54) (10)
j=1,...,n p j F nn3
and
(cid:114) µr
(cid:107)U ∗VT(cid:107) (cid:54) (11)
p p ∞ n2n2
3
Theincoherenceconditionguaranteesthatforsmallvaluesofµ,thesingularvectorsarenotsparse. ThenthetensorLp ∈ Rn×n×n3 canbe
decomposedintolowrankcomponentandsparsecomponent.
Forextractingthelowrankcomponentfromeveryblock,weprocessthetensornuclearnormofL ,i. e. (cid:107)L (cid:107) =(cid:107)blkdiag(Lˆ )(cid:107) .
p p TNN p ∗
HerewecanusesingularvaluethresholdingoperatorintheFourierdomaintoextractthelowrankcomponentoftheblocktensor[18,19].
Theproposedmethodiscallediterativeblocktensorsingularvaluethresholding(IBTSVT).Thethresholdingoperatorusedhereisthesoft
oneD asfollows:
τ
D (L )=sign(blkdiag(Lˆ ))(|blkdiag(Lˆ )|−τ) , (12)
τ p p p +
where“() ”keepsthepositivepart.
+
AfterweextractthelowrankcomponentL = L (cid:1)L (cid:1)···(cid:1)L ,where(cid:1)denotesconcatenationoperation,wecangetthesparse
1 2 P
componentofthetensorbycomputingtheS =X −L.SeeAlgorithm2indetails.
Algorithm2:IBTSVT
Input:tensordataX ∈Rn1×n2×n3
Initialize:givenµ,η,(cid:15),τ,and
blocktensorsX ofsizen×n×n ,p=1,··· ,P
p 3
whilenotconvergeddo
1.Updateη:=η×µ,
2.Updateτ :=τ/η,
3.ComputeX :=D (X ),p=1,··· ,P.
p τ p
endwhile:
(cid:107)X −X (cid:107) /(cid:107)X (cid:107) ≤(cid:15)at(k+1)-thstep.
k+1 k F k F
Output:L=X (cid:1)X (cid:1)···(cid:1)X
1 2 P
Inourmethod,theblocksizecan’tbetoolarge.Thelargesizeoftheblockwillmakethesparsepartcontainsomelowrankcomponent.
Andifthesizeoftheblockistoosmall,thecomputationaltimewillbelong. Becausethenumberoft-SVDsislarge. Generally,wecan
chooseourblocksize2×2×n .
3
Inouralgorithm,wechooseµ = 1.8,η = 1,(cid:15) = 10−2. Butthethresholdingparameterτ isdifficulttodetermine. Herewecanget
√
avaluebyexperience. Asdiscussedin[8],thethresholdingparametercouldbeτ = 1/ nn foreveryblock. Thisvalueisfordenoising
3 √
probleminimagesorvideos,wherethenoiseisuniformlydistributed. Butfordifferentapplications,itshouldbedifferentfrom1/ nn .
3
Because in these applications, the sparse component in data is not uniformly distributed, such as shadow in face images and motion in
surveillancevideos.
4. EXPERIMENTALRESULTS
Inthissection,weconductnumericalresultstoshowtheperformanceofthemethod.WeapplyIBTSVTmethodontwodifferentrealdatasets
thatareconventionallyusedinlowrankmodel: motionseparationforsurveillancevideos(Section4.1)andilluminationnormalizationfor
faceimages(Section4.2).
4.1. MotionSeparationforSurveillanceVideos
Insurveillancevideo,thebackgroundonlychangesitsbrightnessoverthetime,anditcanberepresentedasthelowrankcomponent. And
theforegroundobjectsarethesparsecomponentinvideos. Itisoftendesiredtoextracttheforegroundobjectsfromthevideo. Weusethe
proposedIBTSVTmethodtoseparatetheforegroundcomponentfromthebackgroundone.
We use the surveillance video data used in [6]. Each frame is of size 144×176 and we use 20 frames. The constructed tensor is
√
X ∈R144×176×20andtheselectedblocksizeis2×2×20.Thethresholdingparameterisτ =20/ nn .
3
Fig. 5 shows one of the results. We can find that IBTSVT method correctly recovers the background, while the sparse component
correctlyidentifiesthemovingpedestrians.Itshowstheproposedmethodcanrealizemotionseparationforsurveillancevideos.
Fig.5: IBTSVTonasurveillancevideo. (a)originalvideo; (b)lowrankcomponentthatisvideobackground; (c)sparsecomponentthat
representstheforegroundobjectsofvideo.
4.2. Illuminationnormalizationforfaceimages
Thefacerecognitionalgorithmsaresensitivetoshadoworocclusionsonfaces[7],andit’simportanttoremoveilluminationvariationsand
shadowonthefaceimages.Thelowrankmodelisoftenusedforfaceimages[20].
Inourexperiments,weusetheYaleBfacedatabase[7].Eachfaceimageisofsize192×168with64differentlightingconditions.We
√
constructthetensordataX ∈R192×168×64andchoosetheblocksize2×2×64.Wesetthethresholdingparameterτ =20/ nn .
3
Wecomparetheproposedmethodwithmulti-scalelowrankmatrixdecompositionmethod[13]andlowrank+sparsemethod[4]. Fig.
6showsoneofthecomparisonresults. TheIBTSVTmethodcanresultinalmostshadow-freefaces. Incontrast,theothertwomethodscan
onlyrecoverthefaceswithsomeshadow.
Inordertofurtherillustratetheeffectofshadoweliminationintherecoveredfaceimages,wecarryonfacedetectionwiththerecovered
datafromdifferentmethods. Inourexperiments,weemploythefacedetectionalgorithmViola-Jonesalgorithm[21]todetectthefacesand
theeyes.TheViola-Jonesalgorithmisaclassicalalgorithmwhichcanbeusedtodetectpeople’sfaces,noses,eyes,mouths,andupperbodies.
InthefirstexperimentweputallfaceimagesintooneimageofJPGformat. Thenweusethealgorithmtodetectfacesinthenewlyformed
image.Inthesecondexperiment,weusethealgorithmtodetecttheeyesofeveryfaceimage.ThesecondandthirdcolumnsofTable1show
thedetectionaccuracyratiosofViola-Jonesalgorithmwithdifferentrecoveredfaceimages. Wetesthowlongthethreemethodsprocessthe
64faceimagesascanbeseeninthefourthcolumn.TheIBTSVTcanimprovetheefficiencybyparallelprocessingoftheblocktensors.From
theresultofTable1,wecanfindourmethodgivesthebestdetectionperformance,becauseremovingshadowoffaceimagesishelpfulfor
facedetection.
facedetection eyedetection Time(s)
Originalimage 0.297 0.58 NULL
Lowrank+sparsity 0.375 0.70 10
Multiscalelowrank 0.359 1.00 4472
IBTSVT 0.844 1.00 715
Table1:TheaccuracyratiosoffacesandeyesdetectionbyViola-Jonesalgorithmandthecomputationaltimetoprocessfaceimages.
Fig.6: Threemethodsforfacewithunevenillumination: (a)originalfaceswithshadows;(b)lowrank+sparsemethod;(c)multi-scalelow
rankdecomposition;(d)IBTSVT.
5. CONCLUSIONS
Inthispaper,weproposedanovelIBTSVTmethodtoextractthelowrankcomponentofthetensorusingt-SVD.TheIBTSVTisagoodway
toutilizethestructuralfeatureoftensorbysolvingTPCAprobleminblocktensorform.Wehavegiventhetensorincoherenceconditionsfor
blocktensordata.Forapplications,weconsideredmotionseparationforsurveillancevideosandilluminationnormalizationforfaceimages,
andnumericalexperimentsshoweditsperformancegainscomparedwiththeexistingmethods.
6. ACKNOWLEDGMENT
ThisworkissupportedbyNationalHighTechnologyResearchandDevelopmentProgramofChina(863,No. 2015AA015903),National
NaturalScienceFoundationofChina(NSFC,No. 61602091,No. 61602091),theFundamentalResearchFundsfortheCentralUniversities
(No. ZYGX2015KYQD004, No. ZYGX2015KYQD004), and a grant from the Ph.D. Programs Foundation of Ministry of Education of
China(No.20130185110010).
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