Effect of speckle on APSCI method and Mueller Imaging

0.68 MB·

by  Debajyoti Upadhyay

Effect of Speckle on APSCI method and
Mueller Imaging
DebajyotiUpadhyay1,MichaelRichert2,EricLacot3,AntonelloDe
Martino2andXavierOrlik1*
1ONERA,TheoreticalandAppliedOpticsDepartment,31055Toulouse,France
4 2LPICM,EcolePolytechnique,CNRS,91128Palaiseau,France
1 3LaboratoiredeSpectrome´triePhysique(UMR5588),Universite´JosephFourier,CNRS,
0 38402StMartind’He`res,France
2 *[email protected]

n
a
J Abstract: The principle of the polarimetric imaging method called

9 APSCI (Adapted Polarization State Contrast Imaging) is to maximize
the polarimetric contrast between an object and its background using
]
s specific polarizationstates of illumination and detection.We performhere
c a comparativestudy of the APSCI methodwith existing Classical Mueller
i
t Imaging(CMI)associatedwithpolardecompositioninthepresenceoffully
p
andpartiallypolarizedcircularGaussianspeckle.Theresultsshowanotice-
o
. able increaseofthe Bhattacharyyadistanceused asourcontrastparameter
s
c fortheAPSCI method,especiallywhenthe objectandbackgroundexhibit
i severalpolarimetricpropertiessimultaneously.
s
y
© 2014 OpticalSocietyofAmerica
h
p OCIScodes:(120.5410)Polarimetry;(260.5430)Polarization;(110.5405)Polarimetricimag-
[ ing;(110.2970)Imagedetectionsystems

1
v
3 Referencesandlinks
3 1. M.Richert,X.Orlik,A.DeMartino,“AdaptedPolarizationstatecontrastimaging,” Opt.Express17,14199-
9 14210(2009).
1 2. S.Y.LuandR.A.Chipman,“InterpretationofMuellermatricesbasedonpolardecomposition,” J.Opt.Soc.Am.
. A13,1106–1113(1996).
1
3. H.Poincare´,“The´oriemathe´matiquedelalumie`re”(GABAY,1892).
0 4. A.Bhattacharyya,“Onameasureofdivergencebetweentwostatisticalpopulationsdefinedbyprobabilitydis-
4 tributions,”Bull.CalcuttaMath.Soc.35,99–109(1943).
1 5. J.W.Goodman,“SpecklePhenomenainoptics,”(Roberts&CompanyPub.,Colorado,2007)pp.48-50.
: 6. P.Drude,U¨berOberfla¨chenschichten.Ann.derPhysik,36,86597(1889)
v
7. I.Jung&al.“CharacterizationofThermallyReducedGrapheneOxidebyImagingEllipsometry,”J.Phys.Chem.
i
X C,112,23,8499-8506(2008)
8. M.P.Rowe,E.N.Jr.Pugh,,J.S.TyoandN.Engheta“Polarization-difference imaging:abiologically inspired
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techniqueforobservationthroughscatteringmedia,” Opt.Lett.20,608–610(1995)
a
9. J. S. Tyo, M. P. Rowe, E. N. Pugh, Jr., and N. Engheta, “Target detection in optically scattering media by
polarization-differenceimaging,” Appl.Opt.35,1855–1870(1996)
1. Introduction
ThepolarimetricimagingmethodAPSCI[1]hasbeenshowntoreachbeyondthelimitofcon-
trastachievablefromtheClassicalMuellerImaging(CMI)withpolardecomposition[2].The
process utilises a selective polarimetric excitation of the scene in order to provoke a scatte-
ringfromthe objectandbackgroundcharacterizedbyStokesvectorsasfaraspossible inthe
Poincare´ Sphere [3]. Then along with an optimal polarimetric detection method specifically
adaptedtoeachsituation,ithasbeendemonstratedthatthecontrastbetweenanobjectandits
backgroundcouldbeincreasedtoahigherorderofmagnitudewithrespecttothecontrastfrom
CMIwithpolardecomposition[1].
We propose here to study the performance of the APSCI method taking into account the
shot noise of the detector and the speckle noise in the case of a monochromaticillumination
givingrise toanadditionalcircularGaussianspecklenoise,wherepartialdepolarizationmay
occur. Moreover, we consider the numerical propagation of errors in the calculation of the
polarimetricdatafromtheacquiredrawdata.Forvarioussituations,wherethesceneexhibits
differentpolarimetricpropertiessuchasdichroism,birefringenceordepolarization,weperform
a comparative study of contrast level, which is quantified by the Bhattacharyya distance [4]
calculatedfromthesignificantparametersoftheCMI,thepolardecompositionandtheAPSCI
method.
2. BriefreviewoftheAPSCImethod
Letusassumeascenewithahomogeneouscircularobjectsurroundedbyahomogeneousback-
ground.Then,thescenecanbemodeledintotwomutuallyexclusiveregionsO andB having
polarimetric propertiescharacterized by their Mueller matrices M and M , respectively for
O B
the object and the background. As the scene is considered to be a priori unknown, we need
aninitialestimationofMuellermatricesoftheobjectM andofthebackgroundM byCMI
O B
before implementing APSCI method. During the Mueller imaging process, we consider that
e e
eachpixelofthedetectorindexedby(u,v)receivesanintensityI(u,v)perturbedbyaPoisson
distribution in order to take into account the shot noise. The Mueller matrix M(u,v) at each
pixelisthencalculatedfromthenoisydetectedintensitiesI(u,v).
e
LetusassumeatotallypolarizedStokesvector~SisusedtoilluminatethesceneafterCMI.
e
TheestimationsoftheStokesvectorsofthefieldscatteredbytheobject~S andbackground~S
O B
canbeexpressedas:
e e
~S = S , S , S , S T =M ~S, ~S = S , S , S , S T =M ~S (1)
O O0 O1 O2 O3 O B B0 B1 B2 B3 B
e (cid:2) (cid:3) e e (cid:2) (cid:3) e
We definethe measureof separationof~S and~S in thePoincare´ spherebythe Euclidean
O B
distanceDbetweentheirlastthreeparameters.Then,wedeterminenumericallyusingasimplex
e e
searchalgorithmthespecificincidentStokesvector~SinthatmaximizesthisEuclideandistance.
ItisworthytoemphasizethattheStokesvectors~S and~S arenotnormalizedandcanexhibit
O B
differentrateofdepolarization.Asaconsequence,themaximizationoftheEuclideandistance
e e
mentionedabovetakesintoaccounttwophysicalentities:thepolarizationstateandtheinten-
sityofthepolarizedpartofthescatteredfield.
Finally, we utilize a Two Channel Imaging (TCI) system that projects the scattered field
resultingfromtheselectiveexcitation~Sin,into2statesofpolarization~Sout1 and~Sout2,thatare
defined to maximize respectively (I −I ) and (I −I ), where I and I are the evaluations
O B B O O B
of the mean intensity detected respectively from the object and backgroundscattering. From
e e e e e e
simplecalculationitcanbeshownthat:
T T
~Sout1= 1, D ~ST / D ~S , ~Sout2= 1, −D ~ST / D ~S (2)
h (cid:13) (cid:13)i h (cid:13) (cid:13)i
(cid:13) (cid:13) (cid:13) (cid:13)
where (cid:13) (cid:13) (cid:13) (cid:13)
T
D ~S= Sout−Sout, Sout−Sout, Sout−Sout , (3)
O1 B1 O2 B2 O3 B3
h i
e e e e e e
with
T out T out
Sout,Sout,Sout,Sout =~S =M ~Sin, Sout,Sout,Sout,Sout =~S =M ~Sin. (4)
O0 O1 O2 O3 O O B0 B1 B2 B3 B B
h i h i
e e e e e e e e e e e e
TheAPSCIparameteristhendefinedforeachpixelofthedetectorindexedbythecoordinates
(u,v)as:
I (u,v)−I (u,v)
1 2
APSCI(u,v)= , (5)
I (u,v)+I (u,v)
1 2
whereI (u,v)andI (u,v)arethedetectedintensityafterprojectionrespectivelyonthe2states
1 2
ofpolarization~Sout1and~Sout2.
In this study, we use the Bhattacharyya distance as a contrast parameter for each physical
quantityunderinvestigationthatcanbe,forcomparisonpurposes,eithertheAPSCIparameter
asdefinedabove,eitherthemorepertinentparametersextractedfromthepolardecomposition
of the Mueller matrices of the object and background. A more detailed discussion of the
APSCImethodisproposedin[1].
3. CharacteristicsoftheSpecklenoise
Wehavechosentostudyanunfavourablesituationofimagingregardingboththespecklegrain
sizeanditscontrast.Thus,weassumeaspecklegrainwithasizesimilartothatofthepixelof
thedetector.Ontheexperimentalpointofview,thissituationcorrespondstoacontrastthatis
notdecreasedbytheintegrationofseveralgrainsintoasinglepixel.
Moreover,wechoosetostudytheeffectofacompletelypolarizedanddevelopedcircularGaus-
sian speckle becauseitexhibitsa strongcontrastandso is susceptibleto decreasethe perfor-
manceoftheAPSCImethod.Aswillbepointedoutlaterinthisarticle,biologicalapplications
of the APSCI method seem very promising. So, in order to take into account some possible
movementof the object, we consider a dynamicalspeckle: each intensity acquisition is then
submittedtoadifferentspecklepattern.
TheeffectofapartiallypolarizedspeckleisalsoofinterestregardingtheAPSCI methodbe-
cause it combines 2 antagonist effects : a decrease of the speckle contrast that increases the
BhattacharyyadistanceoftheAPSCIparameterandaloweramountofpolarizedlightusable
bytheAPSCImethodfortheoptimizationthat,onthecontrary,isexpectedtodecreasesignal
tonoiseratioandhencethisdistance.Inoursimulations,thespeckleistakenintoaccountby
amodulationofintensityattheimageplanethatisconsideredindependentofthestateofpo-
larization scattered by the object and background.This modulationof intensity is performed
accordingtotheprobabilitydensityfunctionofintensity pI(I)ofacompletelydevelopedcir-
cularGaussianspecklethatdependsonthedegreeofpolarizationP[5]:
1 2 I 2 I
pI(I)= PI(cid:20)exp(cid:18)−1+PI(cid:19)−exp(cid:18)−1−PI(cid:19)(cid:21) (6)
whereI istheaverageintensity.
4. Resultsandanalysis
WehavechosentostudyinFig.1theeffectofacompletelydevelopedcircularGaussianspeckle
onthreedifferentsituationswheretheobjectandbackgroundaredefinedtohaveadifference
of10%inonepolarimetricproperty:thecases(a)and(b)exhibitthisdifferenceinthescalar
birefringence,thecases(c)and(d)inthescalardichroismandthecases(e)and(f)inthede-
greeoflinearpolarization.Thesituations(a),(c)and(e)consideronlytheshotnoisewhereas
(b),(d)and(f)take intoaccountanadditionalspecklenoise.Foreachofthese situations,we
calculate between the object and background region, the Bhattacharyya distance of the AP-
SCI parameterand of the other pertinentparametersextractedfromthe polar decomposition.
For comparison purposes, the Bhattacharyya distances are plotted versus the signal to noise
ratio(SNR)forasamenumberofintensityacquisition.Wewouldliketopointoutthat,dueto
theirdifferentdichroism,the energyscatteredbythe objectandbackgroundandfocalizedby
imagingelementstowardsthedetectorcanbedifferent,andhencetheircorrespondingclassical
SNR’s. Thus,wechoosetodefinehereaglobalSNRbyconsideringtheshotnoisegenerated
by the amount of energy received by the detector without the use of any polarizer and after
theback-scatteringonavirtualperfectlylambertianandnonabsorbingobject,whosesizeand
positionaresimilartothatofthesceneunderinvestigation.
In Fig. 1 B(M) is defined to be the selected element between M and M that providesthe
O B
best Bhattacharyya distance over the 16 possible elements. In a similar way, B(M ), B(M )
R D
e e
and B(MD ) representrespectivelythe best Bhattacharyyadistance obtained fromthe selected
elementofthebirefringence,thedichroismandthedepolarizationmatricesextractedfromM
O
andM usingtheforwardpolardecomposition.
B
e
The Bhattacharyya distances corresponding to scenes exhibiting a difference of scalar bire-
e
fringence,scalar dichroismand in their ability to depolarizelinear polarizedlight are plotted
respectivelyasB(R),B(D)andB(DOP ).Finally,B representstheBhattacharyyadistance
L APSCI
oftheAPSCIparameterasdefinedinsection2.
Considering the situations (a), (c) and (e) that take into account only the shot noise, we ob-
servethatfromaSNRthreshold,thepolardecompositionthatisolatesthepropertyofinterest
(redandblackcurves)bringsalwaysbetterBhattacharyyadistancesthanB(M) (greencurve)
selectedfromtherawdataofM andM .Belowthisthreshold,thenoiseintroducedbythis
O B
decompositionworsen the situation (as can be seen in case (a)) because M and M are in-
O B
e e
sufficientlydetermined.Secondly,weobservethattheparameterB (bluecurve)exhibits
APSCI
e e
thehighestBhattacharyyadistancesforalltheSNRstudiedin(a)(c)and(e).However,forthe
case(c),wenoticethatitexhibitsalsohigheruncertaintybarsassociatedtolowermeanvalues
ofBhattacharyyadistancescomparedtocases(a)and(e).ThislowerperformanceoftheAP-
SCI methodinthecase ofdichroismiscomingfrom2phenomena:theabsorptionofenergy
due to the dichroismeffectand the cartesian distance between the matrices of the objectand
backgrounddefinedhereas the squarerootof the sum of the squareofthe element-wise dif-
ferences.Indeed,aspreviouslydiscussedin[1],a10%differenceinonepolarimetricproperty
betweentheobjectandbackgroundgivesrisetovariouscartesiandistancesinfunctionofthe
scenestudied.Forcases(a),(c)and(e),thecartesiandistancesarerespectively:0.44,0.09and
0.14.Thelowestvaluecorrespondstothedichroismcaseandexplainsthelowerperformance
oftheAPSCImethodinthatcase.
Whenaddingthespecklenoise,weobservein(b)comparedto(a),astrongdegradationofall
theBhattacharyyadistancesunderstudy.APSCI stillremainsthemorepertinentparameterto
distinguishtheobjectfromthebackgroundevenifitsstandarddeviationnoticeablyincreases
duetothepresenceofspeckle.Theeffectofthesamespecklenoiseonsituation(c)isplottedon
(d).WeobservethatB(M )andB(M)falltoverylowvaluesevenforhighSNRandasacon-
D
sequencebecomeunusableforimaging.Asaresult,fromtherawdataoftheMuellermatrices
M andM associatedtothepolardecomposition,onlyB(D)reachesanorderofmagnitude
O B
similartoB .Moreover,wenoticethatthestandarddeviationofB hasconsiderably
APSCI APSCI
e e
increasedduetothespecklenoise.Afteradeeperanalysis,wehaveobservedthatM andM
O B
areparticularlypoorlyestimatedforthecaseofdichroism(forthe2reasonsmentionedabove)
e e
andthattheadditionofspecklenoiseworsennoticeablythissituation.Asaconsequence,se-
lective states of excitationSin spread near all overthe Poincare´ sphere, showingonlya weak
increaseofdensityofprobabilityinthetheoreticaloptimumregion.
We consider now on Fig.1 (e) and (f) the effect of a partially polarized speckle noise with
degreesofpolarizationbeingrespectivelyP =0.78andP =0.71fortheobjectandback-
obj back
groundthatexhibitadifferenceof10%intheirabilitytodepolarizelinearpolarizedlight.We
observeonlyaweakdecreaseofalltheBhattacharyyadistancesduetothefactthatthespeckle,
onlypartiallypolarized,exhibitsa lowercontrast(C =0.90andC =0.87)thanin pre-
obj back
vioussituations. Moreover,the APSCI parametergivesrise to Bhattacharyyadistancesmuch
higherthanusingtheCMIaloneorassociatedwiththepolardecomposition.
Inalltheprevioussituations,wehavestudiedscenesthatexhibitadifferencebetweentheob-
ject and background in only one polarimetric property. However, in such pure cases, due to
thenumericalpropagationoferrors,theinterestofusingMuellerImagingcanbeinappropriate
compared to simpler methods such as ellipsometry [6][7] or polarization difference imaging
methods [8][9]. However, Mueller Imaging can be of great interest in the case of scenes ex-
hibitingseveralpolarimetricpropertiesatthesametime.
Thus,inordertoexaminetheperformanceofAPSCImethodinsuchcase,weconsideramore
complexscenewheretheobjectandbackgroundhave10%differenceinscalarbirefringence,
scalardichroismandinthelineardegreeofpolarizationsimultaneously.InFig.2,weshowthe
BhattacharyyadistancesoftheAPSCIparametercomparedtothebestBhattacharyyadistances
obtainedfromtheCMIassociatedtothepolardecompositionthatis,inthisnewsituation,the
Bhattacharyya distance corresponding to scalar dichroism. We observe that both parameters
show only a weak decrease of performance(around 5%) due to the speckle noise because it
is onlypartiallypolarizedand exhibitsa contrastsignificantlyinferiorto 1. Secondly,we see
clearlythattheBhattacharyyadistancesofthe APSCI parameterexhibitsmuchhighervalues
thantheonesofthescalardichroism.AvisualcomparisonforaSNRof3.2isproposedinFig.
2wheretheobjectcanclearlybeseenonlyusingtheAPSCIparameterbecauseofhaving3.8
timeshigherBhattacharyyadistancescomparedtotheoneofthescalardichroism.
We would like to point out that the APSCI parameter of this complex scene exhibits better
Bhattacharyyadistancesthanthepurecasesofdichroismanddepolarizationstudiedseparately
inFig.1.
However,inspiteofthesameamountofbirefringenceinthesituationsofFig.2andFig.1(a)
andthatthereareadditionalproperties(dichroismanddepolarization)inthemixedcasewhich
couldhelpustodifferentiatetheobjectfromthebackground,theBhattacharyyadistancesob-
tainedinthepurecaseofbirefringencearehigherthaninthemixedcaseforagivenSNR,even
ifwecorrecttheSNRvaluebytakingintoaccountabsorptionthatoccursinthelattercase.This
canbeexplainedbythecartesiandistancesbetweentheobjectandbackgroundmatriceswhich
arerespectively0.44and0.13forthepurebirefringentandmixedcase.Itcanappearsurprising
that addingsome polarimetricdifferencesbetween the objectandbackgroundcan reducethe
cartesiandistancebetweentheirMuellermatricesandsoourabilitytodistinguishthem.
Fig.1.BhattacharyyadistancesobtainedfromdifferentSNRlevelswithout(casea,c,e)
andwith(caseb,d,f)specklenoise.Theobjectandbackgroundhaveadifferenceof10%
onlyinscalarbirefringence,scalardichroismandlineardegreeofpolarizationrespectively
for the cases a & b, c & d and e & f. In case a & b, (R, l , e ) represent respectively
R R
thescalarbirefringence,azimuthandellipticityofthebirefringencevector~R.Similarlyfor
casec&d,(D,l D,eD)representrespectivelythescalardichroism,azimuthandellipticity
ofthedichroismvector~D.Forthecasee&f,eigenaxesofdepolarizationoftheobjectand
backgroundareassumedalignedandD DOPLandD DOPCrepresentrespectivelythediffer-
encebetweentheobjectandbackground, intheirabilitytodepolarizelinearandcircular
polarizedlight.
However,we have to keep in mind that mostof the elements of a Mueller matrix describe
severalpropertiessimultaneouslyandthatfromaqualitativepointofview,theeffectsofbire-
fringenceanddichroismcanproducecounter-effectsthatwilldecreasethemaximumdistance
achievable between the Stokes vectorscattered by the object and background.Such observa-
tionistrueforallkindofpolarimetricmeasurements,however,astheAPSCImethoddoesn’t
considereachpolarimetricpropertyseparately,andrathertakesintoaccountthewholeMueller
matrices,itisexpectedtogivealwaysthebestpolarimetriccontrastachievablefora2channel
imagingsystem.
Fig. 2. Comparison of Bhattacharyya distances vs. SNR curves for the best performing
parameter of CMI(inthiscasethescalardichroism) vsAPSCIparameter with(ws)and
withoutspecklenoise(wos).Thesceneiscomposedofanobjectandabackgroundexhibit-
ing10%differenceinscalarbirefringenceR,scalardichroismDandinthelineardegree
ofpolarisationDOP .AtthesameSNRlevelof3.2,theembeddedimages(i)and(ii)are
L
obtainedrespectivelyusingtheAPSCIparameterandthebestparameteroftheCMI.
5. Conclusion
Wehavestudiedinvarioussituationstheeffectofacompletelyorpartiallypolarizedandfully
developedcircularGaussianspeckleinpresenceofshotnoise,ontheAPSCImethodcompared
to the Classical Mueller Imaging associated to the polar decomposition.In spite of the addi-
tional high contrastspeckle noise, the use of selective polarizationstates of illumination and
detectioninthe APSCImethodimprovesnoticeablythepolarimetriccontrastbetweenanob-
jectanditsbackgroundwithrespecttotheoneachievablefromtheClassicalMuellerimaging,
evenwhenpertinentpolarimetricdataareextractedbythepolardecomposition.Moreover,as
APSCIoptimizesthepolarimetriccontrastcombiningdichroism,birefringenceanddepolariz-
ationpropertiessimultaneously,itexhibitsremarkablyhighperformancecomparedtoClassical
MuellerImagingwhentheobjectandbackgroundexhibitmultiplepolarimetricbehaviour.This
previous remark makes this technique very promising for medical applications as biological
tissues often exhibit several polarimetric propertiessimultaneously in presence of dynamical
specklenoise.Especially,itcanrepresenta powerfulandnoninvasivetechniqueforaccurate
detectionofdisplasicareasincaseoftumourablation.
6. Acknowledgement
We would like to thank the region Midi-Pyre´ne´e for providing us financial support for this
work.