Table Of Content**VolumeTitle**
ASPConferenceSeries,Vol.**VolumeNumber**
**Author**
c**CopyrightYear**AstronomicalSocietyofthePacific
(cid:13)
Improving resolutionand depth ofastronomical observations via
modernmathematical methods forimageanalysis
5
1
0
MarcoCastellano1,DanieleOttaviani2,AdrianoFontana1,EmilianoMerlin1,
2
StefanoPilo1,MaurizioFalcone2
n
a 1INAF-Osservatorio AstronomicodiRoma,ViaFrascati33,I–00040,
J
Monteporzio, Italy.
6
1 2Dipartimento diMatematica, Sapienza Universita` diRoma,P.leAldoMoro,2
-00185Roma,Italy
]
M
Abstract. Inthepastyearsmodernmathematicalmethodsforimageanalysishave
I
. led to a revolutionin many fields, from computer vision to scientific imaging. How-
h
ever, some recentlydevelopedimageprocessingtechniquessuccessfullyexploitedby
p
other sectors have been rarely, if ever, experimented on astronomical observations.
-
o We present here tests of two classes of variational image enhancement techniques:
r ”structure-texturedecomposition”and”super-resolution”showingthattheyare effec-
t
s tiveinimprovingthequalityofobservations.Structure-texturedecompositionallowsto
a recoverfaintsourcespreviouslyhiddenbythebackgroundnoise,effectivelyincreasing
[ thedepthofavailableobservations. Super-resolutionyieldsanhigher-resolutionanda
bettersampledimageoutofasetoflowresolutionframes,thusmitigatingproblemat-
1
v icsindataanalysisarisingfromthedifferenceinresolution/samplingbetweendifferent
9 instruments,asinthecaseofEUCLIDVISandNIRimagers.
9
9
3
0
. 1. Structure-texture imagedecomposition
1
0
Ageneralapproach tothedenoising problemisbasedontheassumptionthatanimage
5
1 f canberegardedascomposedofastructuralpartu,(i.e. theobjectsintheimage),and
: atexturalpartvwhichcorrespondstofinestdetailsplusthenoise. Inthispaperwehave
v
i considered image decomposition models based on total variation regularization meth-
X
odsfollowingtheapproachdescribedinAujoletal.(2006). Suchimagedecomposition
r techniqueisbasedontheminimizationofafunctionalwithtwoterms,onebasedonthe
a
total variation and asecond oneon adifferent norm adapted tothetexture component.
Givenanimage f definedinasetΩ,andbeing BV(Ω)thespaceoffunctions withlim-
ited total variation in Ω (i.e. V(f,Ω) = f(x) dx < + ) we can decompose f into
|∇ | ∞
ΩR
itstwocomponentes byminimizing:
inf Du +λ v p , (1)
Z | | k kX!
where . p indicatesthenormofagivenspaceX(e.g. L2(Ω))andtheminimumisfound
kkX
among all functions (u,v) (BV(Ω) X) such that u +v = f. The parameter p is a
∈ ×
naturalexponent, andλistheso-called splitting parameter. Thebestdecomposition is
1
2 M.Castellanoetal.
Figure 1. From left to right: simulated noise-free mosaic, simulated observed
mosaic(CANDELS-GOODSdepth),”Structure”imageobtainedthorughTVL2de-
composition.
found attheλforwhichthecorrelation between uandvreaches aminimum. Wehave
designedaC++code,namedATVD(Astro-TotalVariationDenoiser)whichimplements
three versions of the technique, based respectively on the TV-L2 (X = L2(Ω)), TV-L1
(X = L1(Ω))andTVG(XbeingaBanachspaceasdefinedinAujoletal.(2006))norms.
Testsonsimulated HSTimages. Wehavetested the performance ofthedecom-
positiontechniquemethodona15 15arcminsimulatedimagereproducingtheaverage
×
depthoftheF160W(H160hereafter)CANDELSGOODS-Southmosaic(Groginetal.
2011). The analysis of a simulated image in which the position of sources below and
above then nominal detection limit are known, allows to test the effectiveness of de-
composition techniques at increasing the depth of astronomical observations. We de-
composed the simulated image through our ATVDcode using different norms and we
foundthattheTV-L2oneisthemosteffective. ThebestStructureimage(H160 here-
St
after) obtained is shown in Fig.1 together with the input noisy one (H160 ) and, as
inp
a reference, the simulated image without noise. At a visual inspection the resulting
Structureimageappearsdeeperandricherindetailsthantheinputobservedmosaic. To
quantitatively assesstheadvantages oftheTV-L2 decomposition wehaverunSExtrac-
tor(Bertin&Arnouts1996)onbothH160 andH160 . ByvaryingSExtractorinput
St inp
parametersaffectingsourcedetectionwehaveestimatedcompletenessandpurityofthe
extracted catalogues. Acomparison ofthebestdetection performances onH160 and
St
H160 isshowninFig.2. SourceextractionontheStructurecomponentH160 yields
inp St
toan higher purity ofthe catalogue ( 0.5magsdeeper confidence threshold) at asim-
∼
ilarcompleteness reached ontheoriginal H160 . Alternatively, bypushing detection
inp
to lower S/N, we can obtain a much higher completeness with acceptable contamina-
tionlevels(rightpanelinFig.2)withrespecttoH160 . AsshowninFig.2magnitudes
inp
estimated onH160 arereliable alsoforthosesources uptomag 30whichareunde-
St
∼
tectableintheoriginalmosaic.
2. Super-resolution
A low-resolution, blurred frame can be considered as the result of the application of a
number of operators (downsampling D, warping W, convolution H), to an ideal high
resolution image,plustheaddition ofanoisecomponent (ǫ)(e.g.Irani&Peleg1991):
Improvingresolutionanddepthofastronomicalobservations 3
Figure 2. Top panels: magnitude counts on the original and on the denoised
(Structure)imagemaximisingpurity(left)orcompleteness(right)onthelatter. Bot-
tompanels: ontheleft,comparisonbetweenmagnitudesmeasuredonthedenoised
imageandinputmagnitudes(additionalfaintsourcesnotfoundonthenoisymosaic
areshowningreen). Right: sourcesdetectedonthe Structureimagebutnotin the
inputnoisyoneastheyappearinthenoise-freesimulatedmosaic,inthenoisyimage,
andintheStructureimage.
x = DHWx + ǫ = Kx + ǫ. Following this image formation model, it can be
L H H
shown that given N low resolution frames with pixel sampling z, we can recover an
higher resolution well-sampled image with pixel-sampling z/√(N) by minimizing the
followingenergyfunction:
1
E(x ) = ( x Kx 2+λf(x )) (2)
H 2 k L− Hk2 H
where the last regularization term, which is needed for the problem to be well-posed,
is a given norm of the image gradient. The minimization of Eq. 2 is found iteratively
starting from a first guess on x , e.g. a bilinear or polinomial interpolation of x . We
H L
have designed a FORTRAN90code (SuperResolve)to implement variational super-
resolution reconstruction of astronomical images based on L1 and L2 regularization
followingUngeretal.(2010)andZomet&Peleg(2002)respectively.
Tests on simulated EUCLID images. The Euclid satellite will observe >15000
sq. deg. of the sky with a visible (VIS) and near-infrared imager (NIR) to constrain
4 M.Castellanoetal.
Figure 3. Top: simulation of a VIS image, one NIR frame, super-resolvedNIR
TV-L1 and TV-L2 imagesobtained from 9 NIR frames. Bottom left: PSF-FWHM
measured on: the VIS image, the SR-NIR at the VIS pixel-scale, and on a NIR
coaddedmosaic resampledto the VIS pixel-scale. Right: comparisonbetween the
S/NofsourcesintheSR-TVL2mosaicandinastandardcoaddedNIRmosaic.
cosmological parameters to an unprecedented accuracy (Laureijs&etal. 2011). SR
reconstruction can be of particular interest to match the resolution (sampling) of VIS
(pix-scale=0.1”, PSF-FWHM=0.2”)andNIR(pix-scale=0.3”, PSF-FWHM=0.3”)im-
ages thus improving reliability of photometric analysis. Through a dedicated simu-
lation code we have produced simulated VIS and NIR frames by degrading existing
HST-CANDELS images. We applied SR-L2 and SR-L1 reconstruction to a set of 9
NIR single epoch frames to produce a super-resolved mosaic at the same pixel sam-
pling (0.1”) of the VIS mosaic. As shown in Fig. 3 SR-mosaic show an higher level
of details, and the PSF-FWHM measured on them is not as worsened with respect to
therealvalueasinthecaseofpixelupsamling through interpolation. TheSR-L2tech-
nique is particularly promising since fluxes of the sources are preserved and the final
S/Nreachedisthesameasexpected inastandard coadded mosaic.
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