ADAM: a general method for using various data types in asteroid reconstruction

2.3 MB·English

by  Matti Viikinkoski

Astronomy&Astrophysicsmanuscriptno.ADAM (cid:13)cESO2015
January27,2015
ADAM: a general method for using various data types in asteroid
reconstruction
MattiViikinkoski1,MikkoKaasalainen1,andJosefDˇurech2
1 DepartmentofMathematics,TampereUniversityofTechnology,POBox553,33101Tampere,Finland
2 Astronomical Institute, Faculty of Mathematics and Physics, Charles University in Prague, V Holešovicˇkách 2, 18000 Prague,
CzechRepublic
Received;accepted
5 ABSTRACT
1
0 WeintroduceADAM,theAll-DataAsteroidModellingalgorithm.ADAMissimpleanduniversalsinceithandlesalldisk-resolved
2 datatypes(adaptiveopticsorotherimages,interferometry,andrange-Dopplerradardata)inauniformmannerviathe2DFourier
transform,enablingfastconvergenceinmodeloptimization.Theresolveddatacanbecombinedwithdisk-integrateddata(photome-
n try).Inthereconstructionprocess,thedifferencebetweeneachdatatypeisonlyafewcodelinesdefiningtheparticulargeneralized
a
projectionfrom3Dontoa2Dimageplane.Occultationtimingscanbeincludedassparsesilhouettes,andthermalinfrareddataare
J
efficientlyhandledwithanapproximatealgorithmthatissufficientinpracticeduetothedominanceofthehigh-contrast(boundary)
3
pixelsoverthelow-contrast(interior)ones.ThisisofparticularimportancetotherawALMAdatathatcanbedirectlyhandledby
2
ADAM without having to construct the standard image. We study the reliability of the inversion by using the independent shape

supportsoffunctionseriesandcontrol-pointsurfaces.Whenotherdataarelacking,onecancarryoutfastnonconvexlightcurve-only
]
inversion,butanyshapemodelsresultingfromitshouldonlybetakenasillustrativeglobal-scaleones.
P
E Keywords. Methods:analytical,numerical–Minorplanets,asteroids:general,individual:(41)Daphne,2000ET
70
.
h
p
-1. Introduction andoccultationtimings(“sparsesilhouettes”),weneedageneral
o
procedureforusinganydatasourcesinasteroidmodelling.We
rGroundbasedandotherremote-sensingdataonasteroidsareob-
t callthisADAM:All-DataAsteroidModelling.Conciseaccounts
stained with a variety of instruments that essentially sample re-
ofthevariousdatatypesandtheirmodellingaspectsaregivenin
[agions on the surface of the target in various ways. These share Kaasalainen&Lamberg(2006),Kaasalainen&Dˇurech(2013),
somecommonmathematicalcharacteristicsofgeneralizedpro- and Dˇurech et al. (201). This paper is intended as a technical
1jections (Kaasalainen & Lamberg 2006; Kaasalainen 2011; Vi-
companiontothosereviews.
vikinkoski & Kaasalainen 2014). The most abundant source of
8dataforasteroidshapeandspinreconstructionisdisk-integrated
WepresentheretheADAMalgorithminahigh-levelformat
5photometry, because even datasets sparse in time are often suf-
thatincludesallthenecessarymethodsandformulae,eitherwrit-
9ficient for modelling (Kaasalainen 2004; Dˇurech et al. 2006).
5 tenhereorgivenbyreferencestotheliterature.Wediscussand
Lightcurve-inversion procedures (Kaasalainen et al. 2001) are
0 collect here the essential techniques and aspects of a complete
available at, e.g., the Database of Asteroid Models from Inver-
. inversion procedure capable of handling all the major asteroid
1sion Techniques (DAMIT) site1. Due to the inherently limited
data sources and formats. The key point is that complementary
0informationcontentofthedisk-integrateddata,thecorrespond-
datasourcescanfacilitateagoodreconstructionevenwhennone
5
1ingmodelsareusuallymostreliablydescribedinconvexspace ofthemissufficientalone.
(Dˇurech & Kaasalainen (2003), and further discussed below).
:
vHowever,evenpartiallydisk-resolveddataofferarealisticpos- The paper is organized as follows. In Sect. 2 we describe
Xisibility of more detailed modelling. Previously described ap- the various shape supports we use in the reconstruction; some
proachesforsuchreconstructionaretheSHAPEsoftware(Ostro withtheemphasisonglobalfeatures,someconcentratingonlo-
r
aetal.2002)forradarandlightcurvedata,andtheKOALApro- caldetails.Thisisintimatelyconnectedwiththereliabilityesti-
cedure(Kaasalainen&Viikinkoski2012;Carryetal.2012)for mateoftheresult,sinceindependentshaperepresentationshelp
opticalimages,occultationtimings,andlightcurves. torevealwhichfeaturesarethemostprobableones.Sect.3in-
troduces the Fourier transform method necessary for a simple
anduniversalhandlingofdatasourcesofdisk-resolvedtype.In
1.1. Thewholeismorethanthesumofitsparts
Sect.4,wepresentexamplesofsuchtypes(interferometry,radar,
Thebestwaytoreconstructamodelofanasteroidistouseall and optical images). The interferometric data from ALMA are
available data. To combine disk-resolved data (adaptive optics of particular interest here. We also discuss the special case of
or other images, interferometry, and range-Doppler radar data) one-dimensionalprojections(continuous-waveradarandcertain
with disk-integrated data (photometric or infrared lightcurves) typesofinterferometry).InSect.5wesumupeverythinginthe
form of the ADAM algorithm, and conclude in Sect. 6. Some
1 http://astro.troja.mff.cuni.cz/projects/asteroids3D basicADAMfunctionsarelistedinanAppendix.
Articlenumber,page1of11
A&Aproofs:manuscriptno.ADAM
1.2. ADAMsoftwarepackage ing surfaces represented by non-tangled meshes. Thus we have
found it useful to consider two well-regulated but conceptually
Using the methods and algorithm described here and in differentshapesupportsinpractice:octantoidsbasedonspheri-
Kaasalainenetal.(2001),Kaasalainen(2011),andKaasalainen
calharmonics,andsubdivisionsurfaces.
&Viikinkoski(2012),writinganADAMprogramfromscratch
is quite straightforward (for example, convex lightcurve inver-
sion is inherently more complex). We have uploaded free-to- 2.1.1. Functionseries
use ADAM code files and functions written in Matlab and C
An octantoid is a surface given by p ∈ R3 that can be
to a toolbox at the DAMIT site. These can be used for writ-
parametrizedintheform
ingcustomizedinversionsoftware,andforbrowsingandunder-
standing the computational methods. The latter, too numerous
tpoarbtiealddisecruivsasetidvehecrheaiinnsdfeotraigl,raindcielundt-ebatseecdhnoiqputiemsizsautcihona,srathye- p(θ,ϕ)= yx((θθ,,ϕϕ))== ea(θ,ϕ)e+ab((θθ,,ϕϕ))ssiinnθθcsoinsϕϕ,, (1)
tracing procedures, projections and transforms, scattering and  z(θ,ϕ)= ea(θ,ϕ)+c(θ,ϕ)cosθ,
luminositymodels,GPUacceleration,etc.(Notethatwedonot
offeranyusersupport:thefilesarepresentedasis.) where a, b and c are conveniently expressed as linear com-
binations of the (real) spherical harmonic functions Ym(θ,ϕ),
ADAM is a considerably more general package than the l
with coefficients a , b and c , respectively. Note that (θ,ϕ),
KOALA procedure (Kaasalainen & Viikinkoski 2012; Carry lm lm lm
0 ≤ θ ≤ π, 0 ≤ ϕ < 2π, are coordinates on the unit sphere
et al. 2012) that is based on extractable image contours. The
S2 parametrizing the surface but not describing any physical
KOALAcontour-fittingprincipleisnecessaryforincludingoc-
directions such as polar coordinates. As usual, the Laplace se-
cultationdata,soafullADAMprocedureinheritsthisfunction
ries for a,b,c are useful for keeping the number of unknowns;
fromKOALA.Forfittinganypixelimages,werecommendthe
i.e., the coefficients of Ym, small and the surface smooth. If
ADAMFourier-transformfunctionsratherthanKOALA. l
b = c = 0, this representation is the usual starlike one with
We take asteroid reconstruction to mean here that the fol-
the radius exp(a), but we have found that even if the target is
lowingoutputparametersarederivedforminputdata:1)shape
starlike, the octantoid form allows the capture of detail better,
(surface) definition, 2) rotational state (period and spin axis di-
and b and c can be represented with considerably fewer terms
rection;possiblyalsotermsforYORPacceleration,precession,
than the main function a. The number of shape parameters is
or a binary orbit), 3) scattering or other luminosity parameters
(oftenfixedapriori),and4)imageoffset(alignment)andpossi- thusbetweenthe(lmax+1)2ofthestarlikecaseand3(lmax+1)2,
whenl isthelargestdegreeofthefunctionseries.Thedraw-
bleotherauxiliaryornormalizationterms.Withoutlossofgen- max
back of this representation is its globality: one might want less
erality, we do not discuss each item separately, but mostly take
smoothnessregularizationinsomeregionsthaninothers.When
the shape parameters to represent all the free ones since the
morelocalcontrolisdesired(e.g.,afeatureclearlyvisibleinfly-
optimization principle is technically the same for all parame-
byimagesorinradar),therepresentation(1)maybeexpanded
ter types. The speed, convergence, and reliability of gradient-
withsphericalsplinesorsphericalwaveletstoprovidelocalde-
basedoptimizationmethodsareheresuperiortoglobalmethods
tailwithoutaffectingtheglobalshape.Dependingonthedesired
(suchasgeneticalgorithmsorMonteCarlo;seethediscussionin
level of resolution and the non-starlike irregularity of the sur-
Kaasalainenetal.2001).Weemphasizethatthespinparameters,
face, thenumber offree functionseries coefficientsis typically
especiallytheperiod,usuallyhavenumerouslocalminima,soa
between50and300fromlow-tomid-resolution.Functionseries
denseenoughcombofinitialvaluesoftheseisaprerequisitefor
areseldomusefulforhighresolution,whereonemayultimately
agoodfinalreconstruction.
want to adjust each vertex separately by defining individual a,
i
b,andc.
i i
2. Shape
Giventhediverseshapesofasteroidsandthecontinuingprogress 2.1.2. Subdivisioncontrolpoints
ininstrumenttechnology,effectivemethodsforshaperepresen- Subdivisionsurfacesofferlocalcontrolmorethanglobalrepre-
tationarerequiredforageneralreconstructionschemefromob-
sentations like function series. Beginning with an initial set of
servations. In inverse problems it is typically not clear a priori
verticesandcorrespondingtriangles,calledacontrolmesh,the
howwellagivenshapesupportwillperform.Inthissectionwe
surface is iteratively refined by adding new vertices and com-
present shape supports and corresponding regularization func-
putingnewpositionsforoldvertices.Thevertexcoordinatesof
tionswellsuitedforasteroid-likeshapes.
the control mesh form the parameter set defining the surface.
Eachsubdivisionstepsmoothesoutthesurfaceinahigherlevel
2.1. Shapesupports ofresolution.Well-behavingsubdivisionschemesconvergetoa
smoothlimitsurface.
An important part of shape modelling is the choice of shape In this paper we use the Loop subdivision scheme (Loop
representation. Assuming a typical asteroid surface is homeo- 1987). Considering a vertex p with immediate neighbours
morphic to the unit sphere, we can consider each coordinate p ,...,p , the subdivision method first creates new vertices
0 n−1
as a function on the sphere, and choosing a suitable basis for bysplittingeachedge:
functions, expand coordinate functions using this basis. This is
straightforwardtogeneralizetomultiplebodiessuchasbinaries. q = 3p+3pi+3pi−1+pi+1, i=0,...,n−1, (2)
i
Typicalsuchbasesaresphericalharmonics,sphericalwavelets, 8
andsphericalsplines.Ourexperimentssuggestthatparametriza- where the indices should be interpreted as modulo n. After the
tionswhichexpandeachcoordinatefunctionseparatelytendto vertexcreationstep,thepositionofthevertex pisrefined:
produce suboptimal results since they ignore the geometric de- (cid:88)
pendenciesandconstraintsbetweencoordinateswhenconsider- p(cid:48) =(1−nβ)p+β p. (3)
i
Articlenumber,page2of11
M.Viikinkoskietal.:Disk-resolveddatainasteroidmodelling
Themultiplierβisusuallychosentobe
1(cid:34)5 (3+2cos(2π/n))2(cid:35)
β= − , (4)
n 8 64
butotherchoicesarealsopossible.Thelimitsurfaceiscontinu-
ous;C2 at the ordinary vertices (i.e. vertices that have 6 neigh-
bours)andC1atextraordinaryvertices.Thenumberoffreecon-
Fig. 1: Original control mesh (left) with 18 vertices (54 co-
trolpointsformodelrenderingissimilartoorsomewhatlower
ordinates) as the shape parameter set and 32 facets, after two
thanthenumberoffunctionseriescoefficients(foracomparable √
3-subdivision steps (middle), and after four subdivision steps
levelofresolution).
(right).
Themaincomputationalaspectwithsubdivisionmethodsis
thatthenumberoffacetsincreasesexponentiallywiththenum-
ber of divisions. After n subdivision steps, each facet that has
While not strictly necessary, it is convenient to assume that
been divided has produced 4n subfacets. An alternative scheme
√ the triangular mesh representing the shape forms a manifold.
to Loop subdivision is the 3-subdivision (Kobbelt 2000). In-
√ This assumption makes the checking of shadowing and illumi-
steadofsplittingtheedges, 3-schemesubdividesfacetsbyin- nationbothconceptuallyandcomputationallysimpler.Thusitis
serting a new vertex to the facet centroid and connecting i√t to imperativetoavoidself-intersections,astheyintroduceerrorsto
theverticesofthefacet(Fig.1).Themainattractionofthe 3- the fitting process. One approach is to explicitly check for in-
schemecomparedtotheLoopsubdivisionistheslowerincrease tersectingfacetsandretriangulateifrequired.However,triangu-
(3n)offacets,whileperformingsimilarlyinthelimit. lationandintersectiontestsarecostly,andusuallyoptimization
Inpractice,itisusuallyagoodideatochoosetheinitialcon- steps leading to self-intersections are suboptimal. A better ap-
trolmeshtobeanellipsoidorascaledconvexsurfaceobtained proachistopreventself-intersectionsinthefirstplace.
fromlightcurveinversion,withasuitablenumberofverticesfor Regularization based on dihedral angles penalizes large an-
the mesh. The number of subdivision steps should be chosen gles between adjacent facet normals; i.e., the regularization
carefully: while each subdivision increases resolution and sta- prefersplanarregions.Wethuswanttominimize
bility by spreading the influence of each parameter to a larger (cid:88)
number facets, the computational burden grows exponentially. γ = w (1−ν ·ν ), (6)
1 ij i j
Insteadofsubdividingallthefacets,betterperformancemaybe i,j∈T
obtainedwithadaptivesubdivision,whereonlyfacetsbenefiting
from increased resolution are subdivided. However, how to do whereT arethefacetsofthemesh,andνkistheunitnormalvec-
thisautomaticallyduringoptimizationisnotobvious.Aheuristic torcorrespondingtothefacetk.Thesumisoverallthosefacets
inclusionofsurfaceregionstoberefinedbasedonarankingof j that are adjacent to the facet i, and the weights wij are usu-
theimprovementofthefitisonepossibility(cf.theχ2-sensitivity allychosentobeunity.Asaspecialcase,wemaysuppressonly
map of Kaasalainen & Viikinkoski 2012); visual inspection of concave features, obtaining convex regularization (Kaasalainen
themodelfitandagraphicaluserinterfacecanguideinthis. &Viikinkoski2012):
1 (cid:88)
2.2. Regularizationfunctions γ2 = (cid:80) A Aj(1−νi·νj), (7)
j j i,j
Ininverseproblems,findingafeasibleregularizationmethodis
whereA istheareaofthefacetiandthesumisoverthosefacets
typicallythemostdelicatepartofproblemsolving.Ideally,both i
jthatareadjacenttothefacetiandtiltedaboveitsplane.
the shape representation and regularization method should be
To prevent degenerate facets and maintain a homogeneous
chosentocomplementeachother.Theshapesupportshouldbe
mesh,itisadvantageoustoinhibitlargevariationsinfacetareas:
general enough to represent probable shapes, and the regular-
izationshouldpreventunrealisticordegenerateshapeswhile,at
(cid:88)
thesametime,revealthefeaturespresentinthedata.Foroctan- γ = (A −A¯)2, (8)
3 i
toids, the choice is remarkably easy. Assuming the basic shape
i
isgeometricallystarlike,itisintuitivelyobvioustopenalizethe
deviationfromstarlikeness.Tothiseffect,wedefine whereA¯isthemeanfacetareaofthepolyhedron.
(cid:88) In practice, the regularization functions η and γ are suffi-
η= l(b2 +c2 ). (5) 2
lm lm cient for octantoid surfaces, while γ2 and γ3 are useful for the
l,m subdivision surfaces. Unrealistically sharp angles can be pre-
Every starlike surface has a representation for which η = 0, so ventedwithγ1,buttoolargeaweightwillinhibitconvergence.
ηisanaturalquantitytobeincludedinthefinalχ2-functionto In addition to geometric considerations, one can use regu-
beminimized(Sect.5).Theχ2-sumcontainsboththegoodness- larizationbasedonphysicalconstraints,suchastherequirement
of-fitmeasureandtheregularizingfunctionsthatrepresentprior for the rotation axis to be close to the largest principal axis of
assumptionsandexpectationsofthesolution. theinertiatensor(Kaasalainen2011;Kaasalainen&Viikinkoski
Subdivision surfaces have somewhat different smoothness 2012).
properties in this regard. It is well known that the Loop sub-
divisionconvergestoasmoothsurface,soeachsubdivisionstep
2.3. Reliabilityestimates
will produce a smoother result. However, it is computationally
expensivetotakealargenumberofsubdivisionsteps.Therefore Theoctantoidrepresentationorthesubdivisionmeshtendtopro-
itisadvantageoustocombineafew,usuallytwoorthree,subdi- duceaestheticallypleasing,“asteroid-like”surfaces,butitisnot
visionstepswithmesh-basedregularizationmethods. initiallyobviouswhichsurfacefeaturesofthemodelareactually
Articlenumber,page3of11
A&Aproofs:manuscriptno.ADAM
Fig.2:Modelofasteroid(41)Daphnefromadaptiveopticsim- Fig. 3: Asteroid (6489) Golevka reconstructed from disk-
ages,reconstructedasasubdivisionsurface(left)andanoctan- integratedphotometry.Fromlefttoright:Convex,octantoidand
toid(right). subdivisionsurface.
2.4. Inversionwithphotometryonly
present in the data, and which are the side effects of the shape
support and the regularization used. Conventionally, Markov Since ADAM utilizes photometric data in addition to disk-
chainMonteCarlo(MCMC)methodsareusedtoobtainareli- resolved data, we note that ADAM can be used to reconstruct
abilityestimateforthemodelparameters.However,inourcase, amodelusingphotometricdataonly(simplybyusingonlythe
modellingandsystematicerrorsusuallydominate(Kaasalainen photometricfitfunctionfromthetoolbox).Thisiseasyandfast
&Dˇurech2006),renderingtheMCMCapproachinefficientand to do (and the shape rendering is even faster than the standard
inaccuratebecausetheerrordistributionisnotknown(itiscer- convexinversionoflightcurves),buttheresultisinevitablyun-
tainlynotrandomGaussianasinstandardMCMC). reliable:itiswellknownthatevensizablenonconvexshapefea-
turesrequirehighsolarphaseanglestoshowindisk-integrated
Moreover, the posterior distribution of shape parameters data (Kaasalainen et al. 2001; Dˇurech & Kaasalainen 2003;
from MCMC will not really tell anything about the reliability Kaasalainen&Dˇurech2006).Thiscanbeprobedwiththeshape
ofthemodelwithrespecttodata,butonlyaboutthedistribution
reliabilityapproachabove.
oftheestimatewithintheadaptedshapesupport.Wehavefound
Asanexample,weshowreconstructedshapesoftheasteroid
thatthisresultsinanoveroptimisticconceptionofthereliability
GolevkainFig.3,basedonthedatainKaasalainenetal.(2001).
of the result, simply because the acceptable shape results can-
Boththesubdivisionmethodandtheoctantoid-basedmodeldis-
notbeprobedwidelyenoughusingoneshapesupportonly–the
playadditionaldetailnotseenintheconvexmodel.However,the
Monte Carlo procedure focuses on too small regions of shape
detailsarenotsupportedbythedata:theconvexmodelgivesat
variationforbothcomputationalandgeometricreasons.Inaddi-
leastasgoodafitasthenonconvexones,asisalmostalwaysthe
tion, the computation of the model fit is time consuming if the
casewithlightcurves(sofartheonlycaseofabetternonconvex
datasetandparameterspacearelarge,makingMCMCestima-
model fit to photometry is that of the asteroid Eger in Dˇurech
tioncomputationallyexpensive.
etal.2012).Indeed,withGolevkaandothergroundtruthcases
To circumvent these obstacles, we have found the follow- (maps from space probe missions), even the lightcurve fit with
ingapproachfastandrobustinpractice.Anyrealfeatureofthe thecorrectshapeandthescatteringmodelassumedininversion
modelbasedonthedatashouldalsobepresentifanother,inde- isnotbetterthanthatwiththeconvexmodel(Kaasalainenetal.
pendentmodeltypesuchasshapesupportisused.Whenmodel 2001;Kaasalainen&Dˇurech2006).Thisunderlinesthefactthat,
errors dominate, it is thus better to sample the “model space” due to systematic errors, any best-χ2 solution with photometry
within some χ2 than the χ2-space with some fixed model. As onlyislikelytomissthecorrectdetails.
anexample,shapemodelsoftheasteroidDaphnefromadaptive While the convex model yields the best global agreement
opticsimagesandphotometry(Viikinkoski&Kaasalainen2014) with the radar-based Golevka model (see the comparison in
usingboththeoctantoidrepresentationandsubdivisionsurfaces Figs. 3 and 4 in Kaasalainen et al. 2002), the nonconvex ones
areshowninFig.2.Themodelsarequitesimilarandfitthedata portray much of the general sharpness and ruggedness of the
equallywell,andtheirdifferencegivesanideaofthereallevel bodyeventhoughtheirdetailsarenotcorrect.Theconvexshape
ofresolution.MCMCprobingwitheithershapesupportleadsto presents something of a softened error envelope within which
unrealisticallysmalldifferences(insignificantcomparedtothose numerous local shape variations are possible (as if the target
in Fig. 2). Even the shape-support test is likely to produce too were seen unfocused), while the nonconvex representations are
optimistic reliability limits; the model error can be further en- samplingsofthosevariations.Theirdetailscoincideneitherwith
largedby,e.g.,introducingrandomfluctuationsinthescattering eachothernorwiththoseintheradar-basedmodel,buttheyare
properties over the surface. This principle could be developed usefulasillustrationsandforprobingthepotentialshapeoptions
into a meta-level Monte Carlo procedure that probes the space (cf.thenonconvexexamplesinKaasalainenetal.2004).
ofpossiblemodeltypesusinglatentparameters.
We conclude that shape sampling based on a fixed model
3. Fouriertransformandinformationcontent
type, no matter how diligently done with Monte Carlo or other
methods, leads to overoptimistic resolution with artificial de- As discussed in Viikinkoski & Kaasalainen (2014), the Fourier
tails.Atypicalexampleofthisistheradarmodeloftheasteroid transform (FT) facilitates a natural interpretation for the pixel
Itokawa that portrayed imaginary detail at the resolution level size as the maximum frequency present in the data, and makes
expectedfromthedatawhilenotcapturingeventhelarge-scale it easy to incorporate the impulse response function of the
features.Therewasnothingwrongwiththemodelfittothedata imaging system. It also makes the optimization procedure fast
assuch:theinverseproblemwassimplynonunique(orveryun- andstraightforward,withoutthecumbersomeaspectsrelatedto
stable)duetotherestrictedobservinggeometriesandinstrumen- pixellated image fields and binned model image distributions.
talprojection(Sect.4.2),buttheconstrainedshapesupportofthe TheprincipleoftheADAMapproachistocompare,insteadof
programdidnotrevealthis(Ostroetal.2005;Nolanetal.2014). theimagesthemselves,asetofFTsamples(typicallysomethou-
Articlenumber,page4of11
M.Viikinkoskietal.:Disk-resolveddatainasteroidmodelling
sandsdependingonthelevelofresolution)fromthemodelim-
agewiththoseofthedataimage,andtoiterateuntilthebestfit
isfound.ThisisdescribedinSect.5.
Letting T be the set of facets forming a model polyhedron
andPaprojectionoperator,thetwo-dimensionalFouriertrans-
formofaprojectedpolyhedroninthe(ξ,η)-planeis
(cid:34)
(cid:88)
F(u,v)= B I(ξ,η)e−2πı(uξ+vη)dξdη, (9)
i i
Ti∈T PTi
where B istheluminosityvalueofthefaceti,andthefunction
i
I(ξ,η)isunityifthepointprojectedon(ξ,η)isvisibleandzero
otherwise. As shown in Viikinkoski & Kaasalainen (2014), we
Fig.4:TheantennalocationsofALMA(left)andcorresponding
obtainbyGreen’stheorem(dividingafacetintosubfacetsifnec-
uv-plane visibilities (right). Images generated with the CASA
essarysothatwemayassumeIisconstantwithineachsubfacet)
softwarepackage.
(cid:88) (cid:88)
F(u,v)= B I (u,v), (10)
i ij
the instrumental projection plane and the adopted point-spread
i j
function).Inthissection,wepresentdiverseexamplesofshape
where reconstruction with ADAM using both simulated and observed
1 (b−d)u−(a−c)v data.
I (u,v)=
ij 4π2(u2+v2)(a−c)u+(b−d)v
(cid:104) (cid:105)
× e−2πı(au+bv)−e−2πı(cu+dv) (11) 4.1. InterferometryandALMA
for the j-th boundary line segment (oriented counterclockwise) Theinterferometricimagingmethoddiffersradicallyfromatyp-
ofthefaceti,withtheendpoints(a,b)and(c,d). ical telescope; instead of observing the sky brightness directly,
The summation over the interior edges of a projected poly- the interferometer samples the Fourier transform of sky bright-
hedron can be reordered by noting that each polygon edge in ness. Each antenna pair of the interferometric array determines
the interior is shared by two polygons, so a new factor B˜ can one sample on the Fourier plane. The maximum separation be-
be taken to be the difference between the two B, and the edge tween antennas determines the maximum attainable resolution.
i
term is computed only once. This explicitly shows why most Theinterferometermostrelevanttoasteroidshapestudiesisthe
of the information in the image is indeed from the limb and AtacamaLargeMillimeterArray(ALMA)intheChileandesert.
shadow boundary curves discussed in Kaasalainen (2011) and In its full configuration, the interferometer will be capable of
Kaasalainen&Viikinkoski(2012).Thevaluesof B˜ forinterior observingattheresolutionofafewmilliarcsecondsatthewave-
triangleedgesareusuallyclosetozero(indeed,theyvanishfor lengthof0.3mm,correspondingtotheseparationof16kmbe-
thegeometricscatteringB =const.),somostoftheweightison tweenantennas.
i
theboundaryedges.Inpractice,thisisconfirmedbythesimilar GiventhebrightnessdistributionI(ξ,η)ontheplane-of-sky,
results for, e.g., the asteroid Daphne obtained by KOALA and thevisibilityfunctionisdefinedastheintegral
(cid:34)
ADAM. There is little real information in the interior pixels of
adaptiveopticsimages,butontheotherhandtheirerrorsdonot V(u,v)= I(ξ,η)e−2πı(uξ+vη)dξdη, (12)
distorttheresulteither:thedifferencebetweentheKOALAand
ADAMmodels(forthesameinitialvaluesandshapesupport)is whichisatwo-dimensionalFouriertransformofthebrightness
negligible. distribution. Each antenna pair, corresponding to the projected
The role of boundary information can be understood when baseline on the plane-of-sky, samples the visibility function.
comparingtotheextremecaseoflightcurvedata:ifwesumthe When the visibility function is sampled on a sufficiently dense
pixelbrightnessesovertheimageasinphotometry,allthelocal set, the Fourier transform can be inverted to obtain the bright-
shape information in the image is lost, so the remaining infor- ness distribution I(ξ,η). But since the function V(u,v) is mea-
mation is considerably more dependent on the light-scattering sured only at a finite number of points, the observed visibility
propertiesthatareneververywellknown.Butwithimagesthe functionis
boundary contrast is always the largest, so it is sufficient to
have some kind of reasonable scattering (or thermal distribu- V˜(u,v)= F(u,v)V(u,v), (13)
tion)modeltoaccountfortheinteriorpixelcontrasts.Indeed,the
where F(u,v)isasamplingfunctioncorrespondingtothesam-
uniqueness theorems on the image, interferometry, occultation,
pledpointsonthe(u,v)-plane.Thustheobtainedbrightnessdis-
orradardataarebasedontherobustboundarycontourinforma-
tributionisactually
tion (Kaasalainen 2011; Kaasalainen & Viikinkoski 2012; Vi-
ikinkoski&Kaasalainen2014).Withdisk-integrateddataonly, I˜(ξ,η)= f(ξ,η)(cid:63)I(ξ,η), (14)
Minkowski stability is luckily on our side when using convex
models(Kaasalainenetal.2001;Kaasalainenetal.2002). i.e.,aconvolutionofthetruebrightnessdistributionwiththein-
verseFouriertransform f(ξ,η)ofthesamplingfunction.Deduc-
ingthetruebrightnessdistributionIfromthepartiallymeasured
4. Datasources
brightnessI˜isaninverseproblemandthereareseveraliterative
The versatility of the ADAM algorithm allows the handling of algorithmstoinferIfromI˜,see,e.g.,Labeyrieetal.(2006).
differentdatasourceswithonlyminorchangestotheinstrument- While the images obtained from the interferometer are in-
dependentpartoftheprocedure(essentiallyjustthedefinitionof formative,thegreatadvantagewithADAMisthatthealgorithm
Articlenumber,page5of11
A&Aproofs:manuscriptno.ADAM
&Kaasalainen(2014).Thefastanalyticalcomputationsarethen
efficientintheoptimization.Asimplethermophysicalmodelis
sufficientforshapereconstruction,asthemostrelevantinforma-
tion is contained in the boundary data, which are quite robust
withrespecttothethermalmodelused.Thisisincontrasttothe
disk-integrated thermal data that are more sensitive to both the
surfacepropertiesandthethermalmodel.
Forthermalinfraredimaging,ALMAfacilitatesasteroidob-
servationsatresolutionlevelspreviouslyattainedonlybyrange-
Doppler radar. To explore the possibilities of ALMA for shape
modelling, we use the Common Astronomy Software Applica-
tions(CASA)packagedevelopedbyNationalRadioAstronom-
icalObservatory(NRAO)tosimulateobservations.
Consider a hypothetical asteroid with geocentric and helio-
centricdistancesof1.5and1AU,respectively.Thethermalflux
is observed at the 350 GHz band, a frequency located in an at-
mospheric window. There are 11 observation runs, each obser-
vationlasting50swithten-secondintegrationtime.Wechoose
anantennaconfigurationprovidingapproximateresolutionof10
mas,aresolutionwhichiswellwithinthecapabilitiesofALMA.
Theantennaconfigurationandthecorrespondinguv-planesam-
plingpatternareshowninFig.4.Uncorruptedplane-of-skyim-
ages,witharesolutionoffivemilliarcseconds,aredisplayedin
the column on the left in Fig. 5. We use the CASA software to
add realistic atmospheric noise to the observations. The result-
ing dirty images, which are obtained by assuming that the un-
sampledfrequenciesarezero,areshowninthemiddlecolumn.
These images are provided for illustration purposes only, since
ADAMusestheuv-planesamplesdirectly.
To test the ADAM reconstruction method, we use a low-
resolution octantoid representation with 75 shape parameters.
We also fit a scaling term, common to all observations. Usu-
allyitisagoodideatousescalingspecifictoeachobservation,
butinthiscaseweknowthatallthesimulatedobservationsare
done in similar conditions, so the common scaling term is jus-
tified.Thereconstructedshapeisdisplayedintherightcolumn
inFig.5,withthesameobservationgeometriesasforthemodel
images. The small-scale detail is lost, which is to be expected
Fig.5: Simulated,uncorruptedimageswith5maspixelsize(left
duetotheaddedatmosphericnoiseandcoarseinstrumentreso-
column),observeddirtyimagesgeneratedwithCASA(middle)
lution. However, the bifurcated shape is well recovered despite
and the reconstructed low-resolution shape model (right). Note
the noisy data (note that we used ALMA data only). The com-
that the middle-column images are not needed in inversion; we
putationtimeforthisreconstructionwasafewminutes.Forreal
use the direct FT data instead. The images are what would be
observations, complementary data are often provided by other
seen if the raw data were deconvolved for viewing purposes as
observation methods, e.g., disk-integrated photometric data are
isusuallydoneforALMAtargets.Thetestshapemodelisfrom
almostalwaysavailable.
Ostroetal.(2000).
4.2. Radardata
worksdirectlywiththevaluesofthevisibilityfunctionobtained
from the instrument. This approach has several distinct advan- The mathematical principles of the feasibility and uniqueness
tages: of the inversion of range-Doppler images are discussed in Vi-
ikinkoski & Kaasalainen (2014). Here we consider some prac-
– Sparsedatamaybeused(e.g.interferometrywithafewbase- tical issues related to shape reconstruction. While other imag-
lines) ing methods rely on detecting the radiation of the sun that is
– The distribution of antennas does not cause bias, since the reflectedorre-radiatedfromtheasteroid,radarprovidesitsown
Fouriertransformisnotinverted illumination,makingitpossibletoobserveanasteroidregardless
– Possible artefacts caused by the inversion process are of the position of the sun. Moreover, in contrast to the visible
avoided or infrared wavelengths, the frequencies used by the radar are
– Thedependencebetweendifferentobservationsistakenau- not significantly distorted by the atmosphere. Additionally, the
tomaticallyintoaccount propertiesofthewaveformmaybecarefullycontrolledtoreveal
structuraldetailsonthesurfaceoftheasteroid.Theseproperties
To obtain the luminosity values for the model surface (i.e., makeitpossibletoobtaindataresolutiondowntotenmetersor
the brightness factor B for each facet) in the infrared regime lessfornear-Earthasteroids,butthisdoesnotimmediatelytrans-
i
of ALMA, we can use the Fourier-series approximation of latetothesamemodelresolutionbecauseoftheinverseproblem
Nesvorný & Vokrouhlický (2008) as discussed in Viikinkoski (cf.theItokawaexampleinSect.2.3).
Articlenumber,page6of11
M.Viikinkoskietal.:Disk-resolveddatainasteroidmodelling
Range-Dopplerradarresolvesanobjectbothintherangeand
intheline-of-sightvelocitythattranslatestotheDopplershiftof
the reflected pulse. The frequency spectrum may be extracted
bytakingthefastFouriertransformofthepulsescorresponding
to a particular range gate. The actual hardware implementation
andthesignalprocessingarecomplicatedasthedetectedsignals
are below the noise level of the instrument (Ostro et al. 2002). Fig. 6: Mid-resolution shape model of the asteroid 2000 ET
70
Fortunately, the technical specifics are not required for the ac- reconstructed from radar images. Viewing directions are from
tual shape reconstruction, since the radar performance may be thepositivex,y,andzaxes,respectively.
modelledbythepoint-spreadfunctionofthesystem.
Thepoint p=(x,y,z)ontheasteroid’ssurfacecanbetrans-
ferredtotherange-Dopplerframe(r,D)bythelinearmapping TakingtheFouriertransformonbothsides,applyingthecon-
volutiontheorem,andwritingT(u,v)forthesumovertheedges
r = (xcosφ+ysinφ)sinθ+zcosφ (15) ofaFouriertransformedtrianglei asinSect.3,weobtain
D = ωsinθ(xsinφ−ycosφ), (16)
(cid:88)
L(u,v)= B I H (u)H (v)T(u,v), (20)
i i r D i
where ω is the rotation rate of the asteroid around the z-axis
Ti∈T
and (θ,φ) are the spherical radar direction coordinates as seen
fromtheasteroid.Inthismapping,therange-Dopplerradarim- whereH (u)andH (v)are,respectively,theFouriertransforms
r D
agebrightness Lmaybewrittenasanintegralovertheasteroid ofh andh .
r D
surfaceS: Like any images, radar plots are seldom correctly aligned
(cid:34) insome referenceframedueto theerrorsin thecenterof mass
L(r,D)= h (cid:2)r−r(p)(cid:3) h (cid:2)D−D(p)(cid:3) B(p)I(p)dS, (17) prediction,sotheactualpositionofaradarimagewithrespectto
r D
S thetwo-dimensionalprojectionofthemodelmustbedetermined
duringtheoptimization.Thetaskofimagealignmentisfurther
whereh andh arethepoint-spreadfunctionsoftheradarsys-
D r complicated by the peculiar asymmetric structure of radar im-
tem, corresponding to the Doppler-shifted frequency D and the
ages, especially the bright leading edge, other possible ridges
range r, respectively. Here I is the visibility function, which
ofstrongreflectivity,andthefadingfarthest-rangepixels.Ifthe
is unity if the point is visible to the radar and zero otherwise.
alignment information is unknown, it is usually a good idea to
This form is similarly defined for all generalized projections fittheimageoffsetsfirsttoafixedshape,obtainingbetterinitial
(Kaasalainen & Lamberg 2006). The mapping p → (r,D) is
positionsthatcanbeusedintheshapeoptimization.
unique,butitsinverseismany-to-one,sotheinherentinforma-
To demonstrate the reconstruction method, we make a fast
tion content of a range-Doppler image is considerably smaller
ADAMmodelofthenear-Earthasteroid2000ET .Ourgoalis
70
than that of an optical image of similar resolution. Thus, while
togetaquickfirstlookataninitialmodel(toberefinedatwill).
thenominalresolutionprovidedbyradarmaybeunmatchedby
The asteroid was observed during February 2012 at Arecibo
anyotherinstrument,thedrawbackofradarimagingisthediffi-
and Goldstone observatories using 2380 and 8560MHz range-
cultyoftheinterpretationoftheimages.
Doppler radars (Naidu et al. 2013). The images obtained from
TheradarscatteringfunctionisgivenbyB,whichisusually
Arecibohavearesolutionof15minrangeand0.075Hzinfre-
asimplecosinelaw
quency.Goldstoneimageshaveasomewhatlowerresolution,15
B(p)=C(cid:2)µ(p)(cid:3)n, (18) to75mand1Hz,respectively.Ourgoalistoproducemedium-
scaledetailinthereconstructedshape,soatypicalmodelchoice
isanoctantoidwithl ∼10andaround1500facets.Ourexam-
max
where µ is the cosine of the angle between the surface normal
pleis“first-resultoriented”onpurpose,soweassumenoinfor-
andtheradardirection.TheconstantsC andnmeasurethesur-
mationabouttheinstrument-specificdistortions,ormoreimpor-
face reflectivity and the specularity of scattering, respectively.
tantly, knowledge about the point-spread functions determined
ThevalidityofEq.(18)formodellingthemicrowavescattering
bytheinstrumentandtheprocessingroutinesoftheradarsignal.
fromtheasteroid’ssurfaceisaratherconvolutedquestion.While
Thusthepoint-spreadfunctionusedintheshapereconstruction
thecosinelawisquitesimplified,itshouldbenotedthatasthe
issimplythetwo-dimensionaldeltafunction.
reflectedwaveisformedinacomplicatedmannerbythesurface
Foreachdataimage,wefit,inadditiontotheshapeparame-
material whose properties and roughness are usually unknown, ters,theoffsetwithrespecttothemodelcenterofmassandthe
fully realistic modelling of the reflected wave is not computa-
reflectivityterminEq.(18).Thereconstructedmiddle-resolution
tionally feasible. However, as in the other disk-resolved cases,
shapeisshowninFig.6andthemodelfittothedatainFig.7.
most of the information is contained in the boundary contours
Theshapemodelfitstheboundarycontoursoftheradarimages
andisthusindependentofthescatteringmodelused. satisfactorily, but there are some differences in the interior de-
Assumingtheasteroidismodelledasapolyhedronwithtri-
tails.Thisisaconsequenceoftheparametrizationandfacetsize
angularfacetsT,theintegral(17)maycalculatedseparatelyfor
chosen for reconstruction. The interior could be reproduced in
eachfacet,afterprojectingeachtriangleTi asatrianglePTi on greaterdetailbychoosingadifferentparametrization,forexam-
therange-Dopplerplane:
plelocallyadaptivesubdivisionsurfaces, orby refiningthe po-
(cid:90) sitionsofindividualvertices.Themodeldimensions,shapefea-
(cid:88)
L(r,D)= BI h (r−r(cid:48))h (D−D(cid:48))dr(cid:48)dD(cid:48), (19) tures,andspinparametersagreewiththosepublishedbyNaidu
i i r D
Ti∈T PTi edtiffael.re(n2c0e13in)t(htheepospleinlaptiaturadme,ewteersllawreithidinenetrircoarlleimxcitesp)t. for a 2◦
where we have assumed that the visibility I and the scattering Themainpointoftheinitiallowtomiddleresolutionisthat
lawBareconstantwithinatriangle. the speed of ADAM is considerable, and a detailed knowledge
Articlenumber,page7of11
A&Aproofs:manuscriptno.ADAM
oftheinstrumentorthesurfacescatteringphysicsisnotneeded, theoriginequalsthe1-DFToftheprojectionoftheoriginal2-
sooneobtainsafirstmodelveryfastbyjustfeedingintheim- Dfunctionontoalineinthesamedirection;seee.g.Bracewell
ages. The middle-resolution radar-based reconstruction (using 2003),weget
82 radar images) was computed in less than an hour on a stan-
dardlaptop,andGPUprogrammingcanreducethecomputation S(f)=I(f cosγ,−f sinγ)P(f), (22)
timesignificantly.Thismakespossibleabroadsamplingofthe
wherethecalligraphiccharactersdenotetheFourier-transformed
parameter space or real-time experimenting with various mod-
functions. Now it is obvious that S(f) is a slice of a Fourier-
els.Oncealower-resolutionmodelhasbeenadoptedasthefinal
transformedbrightnessdistributionalongalinethroughtheori-
frame,itisstraightforwardtorefineitfurther.However,thisre-
gin, multiplied with the Fourier transform of the point-spread
quires accurate information about the point-spread and scatter
function.Moreover,thismeansthatthesamealgorithmmaybe
functions.
usedtofitbothFGSandadaptiveopticsdata,andsimilarlyboth
CWDopplerdataandtherange-Dopplerimages.Inotherwords,
4.3. Adaptiveopticsandotherimages we extract a one-dimensional Fourier transform from the 2-D
model FT, and compare this with the 1-D FT formed from the
ModelreconstructionfromadaptiveopticsimagesintheFourier
datainthesamewayasinthefull2-Dcase.
approach is extensively covered in Viikinkoski & Kaasalainen
(2014),alongwithanexampleofthereconstructionofthemain
belt asteroid Daphne from adaptive optics images and photom- 5. ADAMalgorithm
etry (Fig. 2). Other imaging data may be incorporated into the
frameworkusingasimilarapproach.Forinstance,flybyimages TheflowchartinFig.8describestheworkingsofADAM.More
are,fromtheviewpointofthereconstructionalgorithm,concep- specifically,thealgorithmmaybedividedinfivedistinctsteps:
tuallyidenticaltotheAOimages.Thisisoneoftheattractions
1. For each data image D and observation geometry E, the
ofADAM:atthebareminimum,theuserdoesnotneedtoknow i i
two-dimensional Fourier transform FD(u,v) of D is sam-
anything about the images except their projection matrix and i i
pled at a set of points {(u ,v )}, j = 1...N, on the spatial
epochs. ij ij i
frequencyplane.Thesizeofthesetischosentocorrespond
We note that the photometric data were actually not even
to the level of resolution. For pixel images, the transform
needed in reconstructing Daphne (except for a better period
canbecomputedbyEq.(10)whenconsideringeachpixelas
valuethanwithAOimagesonly).Theshaperesultswithorwith-
a polygon, or by using fast Fourier transform functions for
outphotometryaresimilar.ThisshowsthatevensparseAOdata
chosengridpoints(butthetimespentforFD(u,v)isirrele-
arewellsufficientformodellingasteroidspinstatesandshapes i
vantasmostofthecomputationsareforthetrialmodels).
indetail.
2. The shape support and resolution level (number of parame-
ters)arechosen.Theparametersareinitializedsuchthatthe
4.4. One-dimensionalprojectionoperators initial shape is a sphere approximately equal in size to the
target.
In the regime between disk-integrated and disk-resolved ob- 3. For each observation geometry E, the Fourier transform
i
servations there are one-dimensional operators that project the FM(u,v) of the corresponding projection image M of the
i i
plane-of-sky onto a line. Typical examples are the continuous- model is calculated as described in the previous sections,
wave (CW) Doppler spectra that measure the distribution of together with the partial derivatives of FM(u,v) with re-
i
the reflected power in frequency only, and the fine guidance specttoalloptimizedparameters.Ray-tracing,scatteringor
sensors (FGS) onboard the Hubble Space Telescope, measur- luminositymodels,andcoordinatetransformsfortheimage
ing the brightness distribution along an instrument axis. One- planearediscussedinKaasalainenetal.(2001),Kaasalainen
dimensional projections are seldom sufficient for actual shape (2011),andViikinkoski&Kaasalainen(2014).
reconstruction, but they may contain useful information about 4. Anobjectivefunctionχ2 isformed,withthesquarenormof
the object’s size or indications about the bifurcated structure thecomplex-valuedFTfiterror:
(Kaasalainen & Viikinkoski 2012), and combined with other
sourcestheyfacilitateshapeinversion.
In both examples, the measurement can be written in the (cid:88)(cid:88)Ni (cid:13)(cid:13)(cid:13)(cid:13)FDi(uij,vij)−e2πı(oixuij+oyivij)Si(uij,vij)FMi(uij,vij)(cid:13)(cid:13)(cid:13)(cid:13)2
form
i j=1
(cid:90) (cid:88)
+ λγ2 =:χ2,
S(x)= I(ξ,η)P(x−ξcosγ−ηsinγ)dξdη, (21) i i
i
(23)
where I(ξ,η) is the plane-of-sky brightness (optical or radar)
distribution of an object, P is the point-spread function of the where(ox,oy)istheoffsetbetweenthedataimageD andthe
i i i
instrument, and the angle γ corresponds to the rotation of the modelimage M,and,bytheconvolutiontheorem,S isthe
i i
sensorintheimageplane.InKaasalainen&Viikinkoski(2012), Fouriertransformofthepoint-spreadfunctionoftheimaging
theintegralwasevaluatedusingaMonteCarlomethod:thepro- system.Theγ representvariousregularizationtermsdefined
i
jectedmodelwassprinkledwithuniformlydistributedsampling above.
points,andtheintegralwasapproximatedasasumoverthevis- Forbrevity,wehavewrittenonlyonedatamodeinEq.(23);
ibleandilluminatedsamplingpoints.Herewedemonstratehow any number of modes with their goodness-of-fit functions
theFouriertransformmethodcanbeusedtointerprettheintegral can be added to the sum. These functions for photometry
asatomographicoperatorontheFourierplane. andsilhouettes(occultations)aregiveninKaasalainenetal.
Taking the Fourier transform on both sides and using the (2001),Kaasalainen(2011),andViikinkoski&Kaasalainen
projection-slicetheorem(asliceofa2-DFTalongalinethrough (2014).Thedeterminationoftheweightsofthedatamodes
Articlenumber,page8of11
M.Viikinkoskietal.:Disk-resolveddatainasteroidmodelling
Fig.7:Examplesofrange-Dopplerimagesoftheasteroid2000ET fromAreciboobservatory(rows1and3)andcorresponding
70
simulated images from the mid-resolution model (rows 2 and 4). The contrast scale of the model image is somewhat modified to
revealinnerimagefeatures.
(as λ for the regularization functions) is discussed in of each M to be a free parameter and use χ2; this is useful
i i
Kaasalainen (2011) and Kaasalainen & Viikinkoski (2012). in the case where the mean intensity of D is corrupted by
i
Weightscanbedeterminedforanysubsetsofdata(e.g.,less excessive noise in the image background (this is typical for
reliableimages)ifnecessary. range-Doppler images). This causes the χ2 -based solution
rel
Inaddition,theintensitylevelofeachdataandmodelimage tohaveaslightlywrongsizetocompensateforthe“diluted”
mustbenormalized.Oftenitisenoughtodividebothmodel normalized intensity level inside the actual object region of
M and data image D by their respective mean intensities. D.
i i i
Equivalently,writing
(cid:88)(cid:13) (cid:13) 5. Theshapeandspinparametersandtheoffsets(ox,oy)aswell
χ2 := (cid:13)(cid:13)Di(uij,vij)−M˜i(uij,vij)(cid:13)(cid:13)2+λγ2, (24) as the possible intensity level factors C minimiizinig χ2 are
i
ij determined with a suitable method such as the Levenberg-
wehave Marquardt algorithm. If there are several hundreds of pa-
rameters, as in the case of fitting all shape vertices directly
χ2 =(cid:88)(cid:13)(cid:13)(cid:13)(cid:13)Di(uij,vij) − M˜i(uij,vij)(cid:13)(cid:13)(cid:13)(cid:13)2+λγ2, (25) (instead of using function series or control points) to pro-
rel (cid:13)(cid:13) (cid:104)(cid:107)D(cid:107)(cid:105) (cid:104)(cid:107)M˜ (cid:107)(cid:105) (cid:13)(cid:13) duce maximal resolution, the conjugate gradient method is
ij i i efficient(Kaasalainenetal.2001).
where the mean (cid:104)·(cid:105) is taken over {(u ,v )}, j = 1...N.
ij ij i
However, sometimes it is better to allow the intensity level
Articlenumber,page9of11
A&Aproofs:manuscriptno.ADAM
Shape ciples to understand the limitations and information content of
parameters thedatasources.
Acknowledgements. This work was supported by the Academy of Finland
project“Modellingandapplicationsofstochasticandregularsurfacesininverse
Surface problems”andtheCoEininverseproblemsresearch.TheworkofJDˇ wassup-
mesh portedbythegrantGACRP209/10/0537oftheCzechScienceFoundation.We
thankJean-LucMargotforprovidingtheradarimagesofasteroid2000ET70.
Transform
Observation tocamera
frame
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Articlenumber,page10of11