Discriminant analysis of solar bright points and faculae I. Classification method and center-to-limb distribution

1.4 MB·English

by  P. Kobel

Astronomy&Astrophysicsmanuscriptno.1117ms (cid:13)c ESO2010
January28,2010
⋆
Discriminant analysis of solar bright points and faculae
I. Classification method and center-to-limb distribution
P.Kobel1,J.Hirzberger1,S.K.Solanki1,2,A.Gandorfer1,andV.Zakharov1
1 Max-PlanckInstitutfu¨rSonnensystemforschung,Max-Planck-Straße2,37191Katlenburg-Lindau,Germany
e-mail:[email protected]
2 SchoolofSpaceResearch,KyungHeeUniversity,Yongin,Gyeonggi,446-701,Korea
0
1 ABSTRACT
0
2 Context.WhilephotosphericmagneticelementsappearmainlyasBrightPoints(BPs)atthediskcenterandasfaculaenearthelimb,
high-resolutionimagesrevealthecoexistenceofBPsandfaculaeoverarangeofheliocentricangles.Thisisnotexplainedbya“hot
n wall”effectthroughverticalfluxtubes,andsuggeststhatthetransitionfromBPstofaculaeneedstobequantitativelyinvestigated.
a Aims.Toachievethis,wemadethefirstrecordedattempttodiscriminateBPsandfaculae,usingastatisticalclassificationapproach
J
basedonLinearDiscriminantAnalysis(LDA).Thispapergivesadetaileddescriptionofourmethod,andshowsitsapplicationon

8 high-resolutionimagesofactiveregionstoretrieveacenter-to-limbdistributionofBPsandfaculae.
2 Methods.Bright “magnetic” features were detected at various disk positions by a segmentation algorithm using simultaneous G-
band and continuum information. Byusing aselectedsample of thosefeaturestorepresent BPsand faculae, suitablephotometric

] parameterswereidentifiedfortheirdiscrimination.WethencarriedoutLDAtofindauniquediscriminantvariable,definedasthe
R
linear combination of the parameters that best separates theBPsand faculae samples. By choosing an adequate threshold on that
S variable,thesegmentedfeatureswerefinallyclassifiedasBPsandfaculaeatallthediskpositions.
. Results.WethusobtainedaCenter-to-LimbVariation(CLV)oftherelativenumberofBPsandfaculae,revealingthepredominance
h
offaculaeatalldiskpositionsexceptclosetodiskcenter(µ≥0.9).
p
Conclusions.Althoughthepresent datasetsuffersfromlimitedstatistics,ourresultsareconsistent withotherobservations ofBPs
-
o andfaculaeatvariousdiskpositions.TheretrievedCLVindicatesthatathighresolution,faculaeareanessentialconstituentofactive
r regionsallacrossthesolardisk.WespeculatethatthefaculaeneardiskcenteraswellastheBPsawayfromdiskcenterareassociated
t withinclinedfields.
s
a
Keywords.Sun:photosphere-Sun:faculae,plages-Sun:magneticfields-Methods:statistical-Techniques:highangularresolution-
[
Techniques:photometric

1
v
3 1. Introduction Also, much of the interest in faculae has been justified by
4
theirmajorroleinproducingthetotalsolarirradiancevariation
1 When imaged at high spatial resolution, the solar photosphere
(Lean&Foukal 1988; Fliggeetal. 1998; Waltonetal. 2003;
5 revealsa myriad of tiny brightfeatures,primarilyconcentrated
Krivovaetal.2003).
.
1 in active regions and outlining the borders of supergranulesin
0 the quiet Sun. Near disk center, they appear mainly as “Bright ThepeculiarappearanceofBPsandfaculaeaswellastheir
0 Points” (BPs) or “filigree” (Dunn&Zirker 1973; Mehltretter different distribution on the disk raises questions about their
1 1974),i.e.roundishorelongatedbrightfeatureslocatedinthein- physicaloriginandmutualrelationship.Thestandardmodelac-
: tergranulardownflowlanes(Titleetal.1987),particularlybright counting for both these phenomena describes BPs and faculae
v
i when observed in Fraunhofer’s G-band (Muller&Roudier as distinct radiative signatures of strongly evacuated thin flux
X 1984; Bergeretal. 1995; Langhansetal. 2002). Near the limb, tubes,arisingfromdifferentviewingangles(“hot-wall”model,
r they resemble more side-illuminated granules called “faculae” Spruit1976;Kno¨lkeretal.1988,1991;Steiner2005).Thissim-
a
or “faculargrains” (e.g.Muller1975), hereinconsideredas in- plified picture has been verified in its salient points by recent
dividual small-scale elements (Hirzberger&Wiehr 2005). The comprehensive 3D MHD simulations (Vo¨gleretal. 2005). A
close association of BPs and faculae with magnetic field in- major success has been the ability to qualitatively reproduce
dicators such as chromospheric CaII emission suggests that BPsneardiskcenter(Schu¨ssleretal.2003;Shelyagetal.2004)
they are related phenomena (Mehltretter 1974; Wilson 1981), andfaculaeclosertothelimb(Kelleretal.2004;Carlssonetal.
bothassociatedwithsmall-scalekGfluxconcentrations(Stenflo 2004), thereby confirming the basic hot-wall model to first or-
1973). These so-called “magnetic elements” are considered as der. However,imageswith the highestspatial resolution reveal
the basic building blocks of the photospheric magnetic activ- thepresenceofBPsawayfromthediskcenter,andoffacularel-
ity (see Schu¨ssler 1992; Solanki 1993, for reviews), whence ementsevenclosetothediskcenter(Hirzberger&Wiehr2005;
the importance of understanding their fundamental physics. Bergeretal. 2007). Such mixtures of BPs and faculae at sev-
eralheliocentric positionsis notexplainedby the hot-wallpic-
Sendoffprintrequeststo:P.Kobel ture considering vertical flux tubes, and seems not apparent in
⋆ Figures 11-14 are only available in electronic form via thesimulatedsyntheticimages(Kelleretal.2004).Further,itis
http://www.edpsciences.org not clear either whether the BPs and faculae seen at different
2 P.Kobeletal.:DiscriminantanalysisofsolarbrightpointsandfaculaeI.
heliocentric angles are associated with similar magnetic struc- tialresolutioninordertoresolveindividualBPsandfaculae,the
tures, or rather with different structures prone to selection ef- datasetwasrestrictedtotheonetothreebestimagepairsateach
fects(Litesetal.2004;Solankietal.2006).Thisshowsthatthe diskposition(obtainedatpeaksofseeing),whichwerekeptfor
transitionfromBPstofaculaeisnotclearlyunderstood,andcur- furtherprocessingandanalysis(seeTable1).
rentmodelsaimingatreproducingBPsandfaculaewouldbene- Forthe selectedimage pairs,phase-diversityreconstruction
fitfromaquantitativestudyofthedistributionofthesefeatures allowed a roughly constant angular resolution to be achieved,
onthedisk. close to the diffraction limit (∼ 0.′′1 at 430 nm). The recon-
Totackletheseissues,anecessarystepistosorttheBPsand structedsimultaneousimagepairs(G-bandandcontinuum)were
faculaeobservedatvariousdiskpositions,inordertotreatthem aligned and destretched using cross-correlation and grid warp-
separately. The approach proposed here is the first attempt in ingtechniques(courtesyP.Su¨tterlin).Thedirectionoftheclos-
thisdirection,andreliesonLinearDiscriminantAnalysis(LDA) est limb was found by comparison with roughly co-temporal
(Fischer 1936) as a basis to “classify” featuresas BPs or facu- SOHO/MDI full disk continuum images, and the images were
lae. Our method makesuse of purelyphotometricinformation, divided by the limb darkeningµ-polynomialof Neckel&Labs
so that it only distinguishes the features appearing as BPs or (1994) at the nearest tabulated wavelength (427.9 nm). For
as faculae. We applied this method to high-resolution images each image pair, the contrast C was then defined relative to
ofactiveregions,coveringarangeofheliocentricangleswhere the mean intensity hIi of a quasi-quiet Sun subfield (of area
QS
the transition fromBPs to faculaeis expected.Thisallowed us ranging from 44 to 114 arcsec2, depending on the image) as
to retrieve, for the first time, an estimate of the center-to-limb C = (I−hIi )/hIi .TheG-bandandcontinuumcontrastare
QS QS
variationoftherelativeamountofbothfeatures,andtherebyto hereafterdenotedC andC ,respectively.Toenhancetheseg-
G C
quantitatively grasp how the appearance of magnetic elements mentationprocess(seeSect.2.2),weappliedahigh-passspatial
variesfromcentertolimb. frequencyfilter to remove mediumand large-scale fluctuations
AlthoughDiscriminantAnalysishasbeenfruitfullyused in oftheintensity(withobservedspatialscalesbetween5and30′′),
Astronomy(see the generalreview by Heck&Murtagh1989), presumablyattributabletop-modes,supergranularcellcontrasts,
its application in the framework of solar physics thus far has straylightandresidualflat-fieldeffects.TheFourierfilterwasof
been restricted to the study of the conditions triggering solar the form f(k) = 1−e−a2k2, where k is the modulusof the spa-
flares (the aim of “probabilistic flare forecasting” Smithetal. tialfrequency,andtheparameterawassettohaveacut-offfre-
1996; Leka&Barnes 2003; Barnesetal. 2007), and to the re- quency(F(k) = 0.5)of0.2arcsec−1 and fullpower(F(k) = 1)
sponse at the Earth’s surface to the solar cycle (Tung&Camp at0.65arcsec−1(inaccordancewithHirzberger&Wiehr2005).
2008). Therefore, this paper is intended to give a detailed de- Finally, sunspots and large pores featuring umbral dots were
scription of our classification method, and by the same token maskedout,togetherwiththeirimmediatesurroundinggranules.
providesaconcreteexampleoflineardiscriminantanalysisap- ThispreventsthecontaminationofBPs/faculaestatisticsbyfea-
plied to solar data. Among otherpotentialapplicationsin solar turesofadifferentphysicalnature.Figures1and2showexam-
physics,wementionthetaxonomyofflaresandtheseparationof plesofG-bandimagesathµi=0.97andhµi=0.6,respectively,
chromosphericBPsandcosmicrayspikesonwavelet-analyzed in which the quiet Sun contrast reference and the masked out
images(Antoineetal.2002). sunspotandporeareasareoutlined.
The structure of this paper reflects the path taken to re-
solve the classification problem. Section 2 describes the origi-
naldatasetprocessing,andtheautomatedsegmentationmethod 2.2.Magneticbrighteningsegmentation
by which bright features were detected at each disk position.
Prior to their classification as BPs or faculae, bright magnetic
Section 3 outlines the classification scheme while briefly pre-
features at the different disk positions of our dataset were de-
senting the principles of LDA. It also gives a detailed report
tected by a segmentationalgorithm.The aimsof our algorithm
of how this technique can be applied to a selected sample of
weretwofold:
BPs and faculae in order to derive a single discriminant vari-
able, based on which a simple classification rule can be built. 1. Detect magnetic brightenings photometrically by compari-
Section4thendealswiththeactualclassificationofalltheseg- sonoftheircontrastinG-bandandcontinuum.
mentedfeatures,aswellasthediscussionoftheseresultsfroma 2. Decompose groups of BPs and striated faculae into in-
methodologicalandphysicalpointofview.Finally,Sect.5sum- dividual elements by using Multi-Level-Tracking (MLT,
marizestheobtainedresultsandgivesfuturedirectionsforsuch Bovelet&Wiehr2001,2003,2007).
work. The second point significantly increases the statistics of the
study, and relies on the assumption that these elements corre-
spond to distinct magnetic features. This has been justified for
2. Imageprocessingandsegmentation intergranular BPs at disk center (Berger&Title 2001), while
observations of the dynamic behavior of striated faculae sug-
2.1.Datasetprocessing
gesta correspondencewith those BPs (DePontieuetal. 2006),
The original dataset consists of simultaneous G-band (430.5 thedarkstriationsbeingassociatedwithsitesoflowermagnetic
± 0.5 nm) and nearby continuum (436.3 ± 0.5 nm) images fieldstrength(Bergeretal.2007;Carlssonetal.2004).
recorded at the 1m Swedish Solar Telescope (SST, La Palma), Theprinciplebehindpoint1isbestillustratedbyC vs.C
G C
on 7th and 8th September 2004. They cover active regions at scatterplotsofaplagearea,asshowninFig.3(seeFigs.1and2
sevendiskpositionsintherange0.56 ≤ hµi ≤ 0.97,whereµ ≡ forthelocationofthechosenplagesubfields).Ascanbeseenin
cosθ,θistheheliocentricangleandhµicorrespondstothecenter Fig.3,thescatterplotsplitsintotwoclearlydistinctpixeldistri-
oftherespectivefield ofview(FOV),equivalentto themeanµ butions.Asimilarpatternappearsinthediagnosticsofradiative
overthe whole FOV (cf. Table 1). This range of disk positions MHDsimulationsofShelyagetal.(2004),wheretheupperdis-
containsbothBPsandfaculae,andisthuswell-suitedtoinvesti- tribution is shown to be associated with strong flux concentra-
gatetheirtransition.Becauseourstudyrequiresthehighestspa- tions,whereastheloweronecorrespondstoweaklymagnetized
P.Kobeletal.:DiscriminantanalysisofsolarbrightpointsandfaculaeI. 3
Fig.1.G-bandimage ofNOAA 0669at hµi= 0.97recordedon7thSeptember,2004.The dashedlinesoutlinethe spotandpore
areas masked out for the segmentation. The “QS Reference” indicates the quasi-quiet Sun subfield chosen as reference for the
contrast.The“Plage”subfieldistheoneselectedfortheC vs.C diagramshowninFig.3.Thearrowindicatesthedirectionofthe
G C
closestlimb.
4 P.Kobeletal.:DiscriminantanalysisofsolarbrightpointsandfaculaeI.
Fig.2. G-band image of NOAA 0671 hµi = 0.6 recordedon 8th September,2004. The “Plage” area was used for theC vs. C
G C
diagramplottedinFig.3.
P.Kobeletal.:DiscriminantanalysisofsolarbrightpointsandfaculaeI. 5
Table1.Datasetspecifications:(µ ,µ )indicatestheµcoverageoftheimages,aftertheirrotationalongthediskradiusvector
min max
pointingtowardstheclosestlimb.FOV istheeffectivefieldofviewoncethespotsandporeshavebeenmaskedout.Thenumber
eff
ofimagepairsselectedforprocessingandanalysisaregiveninthelastcolumn.
Date NOAA hµi (µ ,µ ) FOV [arcsec2] NumberofPairs
min max eff
07-Sept-2004 0669 0.97±0.003 (0.963,0.977) 2034 1
07-Sept-2004 0671 0.78±0.008 (0.75,0.8) 2135 1
08-Sept-2004 0670 0.97±0.003 (0.963,0.976) 1906 1
08-Sept-2004 0667 0.937±0.003 (0.928,0.945) 1027 2
08-Sept-2004 – 0.9±0.005 (0.882,0.916) 2400 1
08-Sept-2004 0671 0.63±0.01 (0.58,0.67) 1923 3
08-Sept-2004 0671 0.6±0.01 (0.55,0.64) 1763 2
08-Sept-2004 0671 0.56±0.01 (0.51,0.6) 1904 1
13-Aug-2006 0671 0.77±0.008 (0.763,0.776) 566 1
granules(seealsoSa´nchezAlmeidaetal.2001,forthecompar- sual count over a portion of the images yielded an estimate of
isonofdifferent1DLTEatmospheres).We canthusselectpix- theremainingfractionoffalsedetections,approximatively4%.
elswhichareG-bandbrightandlikelytobeofmagneticorigin Underthereasonableassumptionthatthesefalsedetectionswere
by imposing two thresholds: a G-band thresholdC selecting alsothefaintest,weadjustedtheG-bandthresholdconsistently
G,t
thebrightportionofthediagram(dashedlinesinFig.3),anda at all hµi to eliminate the ∼ 4 % faintest features (the chosen
thresholdC onthecontrastdifferenceC ≡(C −C )(see value of the threshold was actually rounded up, and was con-
diff,t diff G C
Bergeretal.1998,formoredetails). stantthroughoutthefieldofviewateachhµi).Thevaluesofthe
features maximum G-band contrast C as well as the hµi-
To achieve point 2, we chose a set of closely-spaced MLT G,max
levels between C and C = 0.7. The interlevel spacing was dependentG-bandthresholdsareplottedvs.hµiinFig.4.
G,t G
tunedto0.02(similartoBovelet&Wiehr2007,forBPsatdisk Unlikethe G-bandthreshold,the differencethresholdCdiff,t
center)byvisualcomparisonofthesegmentationmapswiththe can be set to a unique value for all disk positions, inasmuch
originalimages.ThisspacingallowedchainsofBPsandfaculae asthe“non-magnetic”distributionhasasloperoughlyequalto
striationstoberesolved,whileavoidingover-segmentation.The unity at all hµi. To set Cdiff,t properly, we made use of “test”
structureswerethenextendeddowntoC =0withtwointerme- data consisting of a single G-band/continuum image pair ob-
G
diatelevelsatC =0.1andC =0.05.Thisextensionincreases tained with the same setup as our original dataset (and pro-
G G
thesegmentedareaoffaculaecomparedtoBPs,allowingitsfur- cessed as in Sect. 2.1, except for speckle reconstruction), but
theruseasdiscriminantparameter(seeSect.3.3).Theinterme- supplemented with SOUP (Lockheed Solar Optical Universal
diate levels preventthe mergingof BPs with adjacentgranules Polarimeter)Stokes V and I maps, recordedin the wing of the
andtheclumpingofgranularfragmentswhenthecontrastofin- FeI6302.5Ålinewithadetuningof75mÅ.Giventhevalueof
tergranularlanesdoesnotdropbelowC =0.Sinceanecessary theG-bandthresholdforthatdiskposition,thedifferencethresh-
G
conditionforafeaturetobeselectedistohaveitscontrastmax- old was tuned such as to minimize the fraction of “false” de-
imumaboveC ,nootherlevelswereincludedbetweenC =0 tections. By considering false detections as having less than 5
G,t G
andC toavoidoversegmentation.Likewise,structuresofless pixelswith|V/I|≥0.075(wellabovethenoiselevel∼10−2),the
G,t
than5pixelsinarea(correspondingtotheareaofaroundishfea- optimaldifferencethresholdwasfoundasC =0.08.Thecor-
diff,t
turewithadiameterof0.′′1,i.e.roughlyequaltothediffraction respondingfractionoffalsedetectionsamountstoroughly2%.
limit)wereremovedateachMLTlevel. Thesetestimageswere,however,notusedfurtherbecausethey
werefocusedonalargesunspotandhencecontainaverysmall
The segmentation algorithm then proceeded in two steps:
effectivefieldofview(seeTable1,13-Aug-2006).
First,MLTwasappliedtothespatially-filteredG-bandimages.
In a second step, structures corresponding to “magnetic” fea- Withoutinformationaboutthemagneticfielditself,ourseg-
tures were selected by requiring them to contain a minimum mentation has to rely on purely photometric thresholds, and
of 5 pixels satisfying C > C and C > C . A binary hence cannot detect all the magnetic features. The combined
G G,t diff diff,t
mapofsegmentedfeatureswasultimatelyobtainedforeachG- thresholds only aim at detecting a sample of bright features
band/continuumimagepair. thatis least biased by non-magneticones. However,the use of
thresholdsalways implies the drawbackof selection effects. In
From Fig. 3, one notices that the “magnetic” and “non-
particular,theG-bandthresholdwillneglectfainterfeatures,es-
magnetic” pixel distributions overlap more at hµi = 0.6 than
pecially low-contrast BPs near disk center (see Title&Berger
at hµi = 0.97, a tendency that was generally observed for de-
1996;Bovelet&Wiehr2007;Shelyagetal.2004).
creasinghµi.Toavoidthefalsedetectionofgranules,theG-band
thresholdmustthenberaisedashµidecreases,inasmuchasthe
“magnetic” pixel distribution extends towards larger values of
CG while the “non-magnetic”one reaches lower values (as the 3. Discriminantanalysisofbrightpointsand
rmscontrastofgranulesdecreasestowardsthelimb).Todothis
faculae
consistently, we determined a CLV of maximum G-band con-
trastsCG,maxoffeaturessegmentedindependentlyoftheG-band 3.1.Generalschemeandtrainingset
threshold. Specifically, taking one image pair at each disk po-
sition, the features were segmented solely by a safe difference To develop an algorithmic classification method for BPs and
thresholdC = 0.1, and the ones having fewer than 20 pix- faculae, we adopted the following scheme, that uses Linear
diff,t
els above this threshold were removed (as most non-magnetic Discriminant Analysis (LDA, a statistical technique first intro-
detectionscontainonlyafewpixels,Berger&Title2001).Avi- ducedbyFischer1936)onareferencesampleoffeatures:
6 P.Kobeletal.:DiscriminantanalysisofsolarbrightpointsandfaculaeI.
Fig.3.C vs.C scatterplotsforselectedplagesubfieldsofareaapproximately10×12arcsecs2 athµi = 0.97(left)andhµi = 0.6
G C
(right). The exact locations of these subfields within their respective images are outlined in Figs. 1 and 2 (rectangles denoted
“Plage”).ThesolidlinecorrespondstothedifferencethresholdC andthedashedlinetotheG-bandthresholdC .
diff,t G,t
classificationcanbefoundinMurtagh&Heck(1987)andHand
(1981).
Ourtrainingsetwaschosenasasampleof200BPsand200
faculae, obtainedby manualselection of 40 featuresof each at
eachoffivediskpositions:{hµi = 0.56,0.63,0.78,0.9,0.97}for
faculaeand{hµi=0.63,0.78,0.9,0.94,0.97}forBPs.Becauseit
isusedasareferencefortheclasses,theselectedsampleshould
be statistically representative of the actual populations of BPs
and faculae (such as would be identified by eye). At each disk
position, care was thus taken to select the most homogeneous
mixtureoffeatureswithvariouscontrastsandsizes,distributed
overthewholefieldofview.
Itshouldbekeptin mindthatBPs andfaculaeare possibly
nottwo distincttypesof objects,buttheradiativesignaturesof
moreorless similarphysicalentities(magneticfluxconcentra-
tions) viewed under different angles. Consequently, there may
wellbenosharpboundarybetweenthetwoclasses,butrathera
continuoustransitionwithaspectrumof“intermediatefeatures”,
Fig.4. Maximum G-band contrast values (small crosses) of all having variousdegrees of “projection” onto the adjacent limb-
features segmented only with a difference threshold (C = ward granules (see Hirzberger&Wiehr 2005, and Sect. 3.2).
diff,t
0.1) at each disk position. The dashed line corresponds to The conceptof classes can nonethelessbe introducedto repre-
the chosen G-band threshold removing approximately the 4% sentthepopulationsoffeaturesthatwouldbereasonablyidenti-
faintest features, which are delimited exactly by the large fiedasBPsandfaculaeuponvisualinspection,buttheapproach
crosses. proposedherecannotclaimtoclassifytheintermediatefeatures
mentionedabove.
1. Training set selection: Extraction of a reference sample of
3.2.Characteristicprofiles
features,visuallyidentifiedasBPsandfaculae.
2. Discriminant parameter definition: Choice of observables Asabasistodefinediscriminantparameters,weconsideredthe
taking sufficiently different values for the BPs and faculae spatialvariationofG-bandcontrastalongacutmadethrougha
of the training set, in order to be of use for the further dis- BPorafacula.Smallmagneticfeaturesareindeedknowntoex-
criminationoftherestoffeatures. hibitmorepronouncedsignaturesintheG-bandthanincontin-
3. LDA:Determinationofauniquevariablebylinearcombina- uum,andsuchcontrastprofileshavecharacteristicshapeswhen
tionofthechosenparameters,suchthatitbestdiscriminates BPs and faculae are cut along specific directions: radially for
betweenthetwoclassesofthetrainingset. limb faculae, and across the intergranular lane for disk center
4. Assignmentrule:Impositionofanadequatethresholdonthe BPs(Bergeretal.1995;Hirzberger&Wiehr2005).
discriminant variable defined by LDA, separating the BPs Thefollowingprocedurewasdevelopedtoretrieveonechar-
andthefaculaeofthetrainingset. acteristic profile per feature, independently of the feature type
anddiskposition:First,eachfeaturewasorientedinalocalco-
Ultimately, all the magnetic brightenings detected by the seg- ordinate frame x/y as illustrated in Fig. 5. The x/y axes were
mentation algorithm can be classified according to the assign- definedsuchastominimizethey-componentofthefeature’sG-
ment rule, by measuring their value of the variable defined by band“contrastmomentofinertia”M ≡PC (x,y)(x−x )2,
G,y G max
LDA. Comprehensive manuscripts about the general topic of where x is the x-location of the contrast maximum C .
max G,max
P.Kobeletal.:DiscriminantanalysisofsolarbrightpointsandfaculaeI. 7
To give optimalresults on the orientation, the summation only theobservationsofBergeretal.(1995)andHirzberger&Wiehr
ranoverpixelshavingC ≥ 0.5C ,thusinvolvingonlythe (2005), respectively, as well as with the synthetic profiles of
G G,max
“corepixels”ofthefeatures1.Inpractice,theminimumofM Kno¨lkeretal.(1988)andSteiner(2005).
G,y
was found by iteratively rotating a small window surrounding Duetofiniteresolution,straylight,andthepartialcompensa-
the feature with 5◦ steps (smaller steps did not yield better re- tionofspatialintensityfluctuationsbythefilter(seesection2.1),
sults,duetothefinitenumberofpixelsconsidered).Next,con- itiscommontofindBPsembeddedin“grey”laneswithpositive
trastprofileswereobtainedalongxandybyaveragingtherows contrast (Bovelet&Wiehr 2007). Upon careful visual analysis
andcolumnsofthatwindowhavingpixelswithC ≥0.9C ofgreylane-BPsprofiles,weidentifiedthesegreylanesascon-
G G,max
(delimitedby blacklines in Fig. 5c). Such x/y profilesare dis- trastdepressionswithalowminimum(C ≤0.1),separating
G,min
playedinFig.6inthecaseofatypicalBP(right)andatypical theBPprofilefromtheadjacentgranuleprofile.Asmostnormal
facula(left).Theseprofileswerefurtherrestrictedtothecontrast BPsprofileshavequasi-linearslopesattheiredges,thesidesof
rangeC > 0aboutC (delimitedbythelower“+”marks), profilesfeaturinggreylaneswerelinearlyextrapolateddownto
G G,max
suchthatallprofilesshareaconsistently-definedreferencelevel C =0.Onlyafterthiscouldthexandyaverageprofilebeprop-
G
C =0.Finally,thesinglecharacteristicprofileforeachfeature erly restricted to positivecontrastvalues, and their smoothness
G
wasfoundtobethesmoothestofthepositivecontrast-restricted comparedfortheadequateretrievalofthecharacteristicprofile.
x/yprofiles(overplottedinthick).Toquantifythesmoothnessof This linear extrapolationis illustrated in Fig. 7 for characteris-
theprofiles,wecountedthenumberoftheirlocalextrema,even- ticprofilesofbothBPsand“intermediatefeatures”,indicatingat
tuallyaddingthenumberofinflexionsifthenumberofextrema thesametimethevarietyoffeatureprofilesthatcanbeobtained.
wasequalinxandy.TheuseofMLTsegmentation(asopposed
to a single-clip) is an essential prerequesite for obtainingthese
characteristicprofiles,byavoidingthatpixelsfromadjacentfea-
turescontaminatethecontrastmomentofinertiaandthusspoil
thefeature’sorientationprocess.
Fig.5.Orientationofanindividualfeatureinitslocalx/ycoordi-
nateframe.a)Zoomwindowsurroundingthefeatureintheorig-
inalG-bandimage.Thecrossindicatesthelocationofthecon-
trastmaximum.b)Isolatedfeatureasdelimitedbythesegmen-
tationmap. ThepixelshavingC ≥ 0.5C are highlighted
G G,max
in grey, and are used to compute the G-band contrast moment
of inertia. c) Windowrotated such that the y-componentof the
G-band contrast moment of inertia M is minimum, thereby
G,y
definingthelocal x/ycoordinateframe.Therowsandcolumns
usedtoretrievetheaverageprofilesalongxandyarecontained
betweenthestraightblacklines.
Owingtothepreviousorientationofthefeatures,thecharac-
teristicprofilesexhibitdifferentshapesforBPsandfaculae,and
consequently proved very useful for the extraction of valuable
discriminantparameters(see Sect. 3.3).Incontrast,profilesre-
trievedalongthediskradiusvector(asperformedinearlystages
Fig.7.Characteristicprofiles(thicklines)ofBPs(a,b) andin-
ofthiswork)havelesscharacteristicshapesandthuslesspower
termediatefeatures(c,d)surroundedbyoneortwo“grey”lanes.
todistinguishBPsfromfaculae,dueto thescatterin theorien-
Onthegreylanesides,thecharacteristicprofileshavebeenlin-
tation of these features with respect to the radial direction. As
early extrapolated from the half-max level (“+”) to C = 0.
can beseen in theexamplesof Fig. 6, the characteristicprofile G
Froma)to d),these profilesillustrate thecontinuoustransition
ofthetypicalBPisnarrowerandsteeperthantheprofileofthe
betweenthetypicalBPsandtypicalfaculae,forwhichexamples
typicalfacula.Inparticular,thecharacteristicprofileofthefac-
areshowninFig.6.
ulaisindistinguishablefromtheadjacentgranule,asthecontrast
variesmonotonouslyfromonetotheother.We mentionthere-
semblanceofthe characteristicprofilesofthe BPand faculato
1 Involvingpixelswithlowercontrastyieldspoorerresults,asthese 3.3.Discriminantparameters
areoftenassociatedwithgranulationinthecaseoffaculae,andthusdo
not carry information about the orientation of the facular brightening In search of adequate discriminant parameters, we carried out
itself. a pilot study by defining a set of parameters. These included
8 P.Kobeletal.:DiscriminantanalysisofsolarbrightpointsandfaculaeI.
Fig.6. Local frame orientation and G-band contrast profiles and of a typical facula (left) and a typical BP (right). Top windows:
Orientationofthefeaturesintheirlocalx/ycoordinateframes,wheretheblacklinesdelimitthepixelshavingC ≥0.9C used
G G,max
for profile averaging.The white contoursobtained from the segmentation map enclose the area A of the features. Lower panels:
AverageG-bandcontrastprofilesalong x and y. The retrievedcharacteristic profile for the BP and the facula is indicated by the
thicklines.The“+”marksintersectingthereferencelevelC =0(dashedline)delimitthepositivecontrastportionoftheprofiles,
G
and the upper “+” marks indicate the half-max levelC = 0.5C on the characteristic profiles. The parameters ∆ and ∇ are
G G,max
illustratedonthecharacteristicprofilesofthefaculaandtheBP,respectively.Allthe x/yprofileswerecubicspline-interpolatedby
afactorof10,inordertoavoidartificialroughnessduetosamplingwhenchoosingthecharacteristicprofile(asthesmoothestof
thex/yprofiles,seeSect.3.2).
thepeak-to-widthratio,areaasymmetryandsecondmomentof ofBPsandfaculaeforthefollowingreasons.Thebestdiscrim-
the characteristic profiles, the local contrast relative to the im- inantparameter,∆,takesgreatervaluesforfaculaeasitencom-
mediate surroundings (similar to Bovelet&Wiehr 2003), and passesthe widthof theadjacentgranularprofile(asthe facular
the contrastof adjacentlanes. By lookingat the distributionof andgranularprofilearemergedtogether,cf.Fig.6),whereasBPs
theparametervaluesfortheBPs andfaculaeofthetrainingset arelimitedtothewidthofintergranularlanes.Theparameter∇
(meanvaluesandstandarddeviationateach hµi, see below)as describeshowsteeplythecontrastdropstowardstheedgesofthe
wellastheircorrelation,threeroughlymutuallyindependentpa- profileandtypicallyhaslargervaluesforBPs,whichshowsteep
rameterswereeventuallyfoundtobegooddiscriminantsforthe andsymmetriccontrastenhancementssqueezedbetweenthead-
trainingsetclasses.TheirdefinitionsareillustratedinFig.6: jacentgranules.Tosupplementthesetwoprofileparameters,the
segmented feature area A has been added to avoid that faculae
– ∆:=widthofthecharacteristicprofileatthereferencelevel with small widths (typically lying on small abnormal granules
C =0[arcsecs]. frequentlyfoundinactiveregions)areclassifiedasBPs.Inarea
G
– ∇ := average slope (from both sides) of the characteris- thesefaculaeappearsignificantlylarger.
tic profile below the half-max level CG,HM = 0.5CG,max As can be seen from Fig. 8, the parameter values do not
[arcsecs−1]. varysignificantlywithhµi,asthedifferencebetweenthelargest
– A:=apparentarea(projectedontotheplaneofthesky)ofthe andsmallestmeanvaluesoverthewholeµrangebarelyexceed
featuredefinedbythesegmentationbinarymap[arcsecs2]. their standard deviations. The relative constancy of width and
area is particularly surprising for faculae, and could be due to
Weemphasizethatthethreechosenparametersaredefinedusing a compensation of granular foreshortening by enhanced radia-
relativecontrastlevels(CG = 0andCG = 0.5CG,max),allowing tiveescapeinthedirectiontowardthefluxconcentration(Steiner
thecomparisonandclassificationoffeatureshavingdifferentab- 2005),aswellastothedistributionintheorientationsoffaculae
solutecontrastvalues(notablyduetotheCLVofcontrast). (tobediscussedinaforthcomingpaper).Therelativeinvariance
Fig.8(leftcolumn)showsthemeanvaluesandstandardde- oftheparametersisneverthelessadvantageous,asitjustifiesper-
viationsoftheparametersA,∆and∇attheµ-valuesofthetrain- formingLDAonthewholetrainingsetatonce(allhµitogether),
ingset.Theseparametersdescribewellthedifferentappearances thusallowingusto findasingle linearcombinationofparame-
P.Kobeletal.:DiscriminantanalysisofsolarbrightpointsandfaculaeI. 9
Fig.8.Leftcolumn:MeanvaluesandstandarddeviationsofthethreeparametersfortheBPs(“⋄”andthinbars)andfaculae(“△”
andthickbars)ofthetrainingsetvs.diskpositionhµi.ThetrainingsetdoesnotcontainBPsathµi=0.56orfaculaeathµi=0.94.
Togiveanideaoftheoutliersandtheµdistributionofthetrainingsetfeaturesateachdiskposition,individualfeaturesvaluesare
overdrawn(small“⋄”forBPsandsmall“△”forfaculae).Theµvaluesoftheindividualfeatureswerecomputedbyusingthehµi
(center of FOV) of the correspondingimages as reference. Right column:Normalized density functions(DFs) histograms of the
log-transformedparametersforBPs(“⋄”)andfaculae(“△”),obtainedbycombiningalldiskpositionsofthetrainingsettogether.
Cubicsplinesareoverplottedforclarityandtheirmaximawereusedforthenormalizationofthehistograms.
tersandasingleBPs/faculaethresholdvalidforallthediskpo- necessity of having a dataset of roughly constant resolution, a
sitions of our dataset. Moreover,combiningall the training set conditionmetbyourselectedimagepairs(seeSect.2.1).
featuresenhancesthesamplingoftheclassesandyieldsamore
accuratethreshold.
3.4.LinearDiscriminantAnalysis
ItshouldbenotedthatthevaluesoftheparametersA,∆and Because LDA distinguishes classes based solely on means
∇dependtosomeextentonspatialresolution.Thispointstothe and covariances (see Equation 1), it works best for param-
10 P.Kobeletal.:DiscriminantanalysisofsolarbrightpointsandfaculaeI.
eters that are normally or at least symmetrically distributed Table 2. J valuesassociated with each discriminantparameter,
(Murtagh&Heck 1987). To verify this condition, we studied and for the variable F obtained by linear combination of the
the densityfunctions(DFs) of ourthreeparametersbyproduc- threeparameters.Thea’sarethecoefficientsofthelinearcom-
inghistogramsofthetrainingset,andestimatedtheskewnesses bination(absolutevalues).
via the third standardized moment (Kenney&Keeping 1962).
Takingthenaturallogarithmwasfoundtoreducetheskewness log(A) log(∆) log(∇) F(3D)
ofallparameters(Limpertetal.2001),andthereforetheywere J 3.17 4.71 1.8 6.27
replacedbytheirlog-transforms2.TheDFsoflog(A),log(∆)and a 0.27 0.59 0.14 ...
log(∇)aredisplayedinFig.8(rightcolumn).
Having the correctparametersin hand, LDA could then be
Due to the merging of BPs into chains and ribbons, their DFs
carried out to find their linear combination that best discrimi-
areparticularlyaffectedandbecomeskewedtowardslargerA,∆
nates the training set classes. Explicitely, we searched for the
andsmaller∇.For∆and∇,thisisprobablyaconsequenceofthe
axisvector athatmaximizesFischer’sseparabilitycriterion(as
b misorientationofmergedfeatureswhenretrievingthecharacter-
introducedintheoriginalworkofFischer1936)intheparameter
space{x=(log(A),log(∆),log(∇))}: isticprofiles.Webelievethatthosetestsconfirmourappropriate
choice of MLT levels for the purposeof further discriminating
[aT(m −m )]2 betweenindividualBPsandfaculae.
bp fac
J(a)= , (1) Westressthattheprocedureoforientingthefeaturespriorto
aT(S +S )a
bp fac theretrievaloftheircontrastprofiles,asdescribedinSect.3.2,is
an essential ingredientto obtain discriminantparametersbased
wherethesuperscriptTdenotestranspose, m and m arethe
bp fac onthoseprofiles.Inearlystagesofthiswork,profileswereonly
class mean vectors and S ,S the covariance matrices. The
bp fac retrieved along the direction perpendicular to the closest limb,
originalparameterscouldthenbeprojectedonto a,therebyob-
b andtheensuingoverlapoftheDFswassignificantlylarger.
taining the desired linear combination defining the single vari-
Finally,itshouldbenotedthatthevaluesofthesephotomet-
ableF ≡ aTx(for“Fischer”variable).The2Dprojectionsofthe
b ricdiscriminantparametersalldependtosomeextentonthespa-
3DtrainingsetvectorsarerepresentedinFig.9a-c,withanover-
tial resolution. ∇ is probablythe most sensitive in that respect,
laidaxiscorrespondingtothedirectionofmaximumseparability
butithastheleastweightinthevariableFduetoitslowervalue
a.Fig.9ddisplaystheDFoftheobtainedvariableF.Themax-
b of J (see Table 2). This points to the requirement of having a
imalvalue of J associated with the variable F (projectiononto
dataset of roughly constant resolution, a condition met by our
a)andthe valuesof J associatedto eachparameter(projection
b selectionofimages(Sect.2.1).
onto the parameters’ axes) are listed in Table 2. These values
giveanideaoftherelative“discriminantpower”ofthethreepa-
rameters,andthe larger J valueofthevariable F demonstrates 4. Classificationresultsanddiscussion
the advantage of their optimal linear combination provided by
LDA. 4.1.Hardthresholdvs.rejectoption
TheDFsofthediscriminantparameters(Fig.8rightcolumn)
To build an assignment rule, we made the usual choice of
andofthevariableF (Fig.9d)alsogiveagoodvisualestimate
a threshold value F at equal “standardized” distance from
t
oftheamountofoverlapbetweenthetrainingsetclasses.Anin-
the class means (Mahalanobis 1936; Murtagh&Heck 1987),
tuitivemeasureofthe discriminantpowerofa parametercould
namely:
thenbegivenbytheratiobetweenthenumberoffeaturescon-
ttraainineidnginstehtefoeavteurrlaeps.pTinhgispararttioofathlreeaDdFystaankdesthaeftaoirtalylnlouwmbvearluoef (baTmbp−Ft)2 = (baTmfac−Ft)2. (2)
aTS a aTS a
of0.07forlog(∆),andgoesdownto0.042forF.However,such b bpb b facb
a measure is statistically poor compared to J, since it mostly
Thisthresholdisdrawnonthedensityfunctionhistogramof F
relies on the outlierscontained in the tails of the DFs (only28
inFig.9d.
and17featuresforlog(∆)andF,respectively),whereasJ takes
Inafirststep,asinglehardthresholdequaltoF wasusedto
t
advantageofthefullparameterdistributions.
classifyallthesegmentedfeaturesasBPsorfaculaebymeasur-
The overlap of the DF of log(A) and the particular skew-
ingtheirvaluesof F.Becausewearenotinterestedinabsolute
ness of the DF of faculae towards small areas arises from our
numbers,butratherintherelativedistributionofBPsandfacu-
MLTsegmentation.ToinvestigatetheinfluenceoftheMLTlev-
lae,wedefinetheclassifiedfractionsofBPsandfaculaeas:
els on the DFs of the dicriminantparameters and on LDA, we
carried out tests with fewer MLT levels over the same training X ≡ Nbp , X ≡ Nfac , (3)
set.WefoundthattheskewnessoftheDFoflog(A)forfaculae bp N +N fac N +N
bp fac bp fac
isin majorpartduetotheirsegmentationintofine striations.It
shouldbenoticedthattheDFoflog(∆)islessskewed,because whereNbpandNfacarethenumberofclassifiedBPsandfaculae,
thecharacteristicprofilesofthesestriatedfaculaearemostlyre- respectively. The CLV of the fractions Xbp and Xfac, classified
trievedalongthelongdimensionofthestriations(owingtotheir usingthethresholdFt,isdepictedinFig.10a.
individual orientation, see Sect. 3.2), which makes ∆ a robust Asalreadystated attheendofSect. 3.1, thereisa continu-
parameter to distinguish them from BPs. However, the coarser ousspectrumofintermediatefeaturesbetweenBPsandfaculae
segmentationof the tests hasthe undesiredeffect thata partof that cannot be reasonably identified as belonging to one class
thefeaturesareundersegmented,whichleadstolowervaluesof ortheother.Theonlywaytoavoidtheerroneousclassification
J for all parameters as well as for the discriminantvariable F. ofthesefeaturesistoexcludethemfromthestatisticsbyintro-
ducing a so-called “reject option” (Hand 1981), in the form of
2 Thiswaspartlyexpected,asthewidthofmagneticbrightpointshas a rejection range in F centered about Ft. Assuming that all in-
beenobservedtobelog-normallydistributed(Bergeretal.1995). termediatefeaturesfallwithin the rejectionrange,therelations