A direct variational approach to a problem arising in image reconstruction Luigi Ambrosio1 Simon Masnou2

Simon Masnou2

Description

A direct variational approach to a problem arising in image reconstruction Luigi Ambrosio1 Simon Masnou2 Abstract We consider a variational approach to the problem of recovering missing parts in a panchromatic digital image. Representing the image by a scalar function u, we propose a model based on the relaxation of the energy ∫ |?u|(? + ? | div ?u|?u| | p), ?, ? > 0, p ≥ 1 which takes into account the perimeter of the level sets of u as well as the Lp norm of the mean curvature along their boundaries. We investigate the properties of this variational model and the existence of minimizing functions in BV. We also address related issues for integral varifolds with generalized mean curvature in Lp. Keywords: Image processing; image reconstruction; BV; mean curvature; varifolds; relaxation. 1 Introduction Many problems in digital image processing require the ability to recover missing parts of an image or to remove spurious or undesired objects. One can mention for instance the removal of scratches in old photographs and films, the recovery of pixels blocks corrupted during a binary transmission (or analogously the removal of impulse noise) or the removal of undesired publicity, text or subtitles from a photograph. One can also think to special effects for movie postproduction, e.g. the removal of a microphone appearing in a scene.

• image objects


• direct variational approach


• continuation process


• well suited


• between them


• digital image


• objects boundaries between


• panchromatic digital

Sujets

• Digital image

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 10
• Langue: English

Extrait

Adirectvariationalapproachtoaproblemarisinginimage
reconstruction
LuigiAmbrosio
1
SimonMasnou
2

Abstract
Weconsideravariationalapproachtotheproblemofrecoveringmissingpartsina
panchromaticdigitalimage.Representingtheimagebyascalarfunction
u
,wepropose
amodelbasedontherelaxationoftheenergy
ur|r
u
|
(

+

|
div
|
p
)
,,>
0
,p

1
Z|ur|whichtakesintoaccounttheperimeterofthelevelsetsof
u
aswellastheL
p
norm
ofthemeancurvaturealongtheirboundaries.Weinvestigatethepropertiesofthis
variationalmodelandtheexistenceofminimizingfunctionsinBV.Wealsoaddress
relatedissuesforintegralvarifoldswithgeneralizedmeancurvatureinL
p
.
Keywords:
Imageprocessing;imagereconstruction;BV;meancurvature;varifolds;
relaxation.

1Introduction
Manyproblemsindigitalimageprocessingrequiretheabilitytorecovermissingpartsof
animageortoremovespuriousorundesiredobjects.Onecanmentionforinstancethe
removalofscratchesinoldphotographsandlms,therecoveryofpixelsblockscorrupted
duringabinarytransmission(oranalogouslytheremovalofimpulsenoise)ortheremoval
ofundesiredpublicity,textorsubtitlesfromaphotograph.Onecanalsothinktospecial
eectsformoviepostproduction,e.g.theremovalofamicrophoneappearinginascene.
Adigitalimageisusuallymodeledasafunction
u
fromaboundeddomainofIR
N
(
N
=2
forusualsnapshots,
N
=3formedicalimagesormovies,
N
=4formovingmedicalimages)
ontoIR
M
(
M
=1foragrey-levelimage,
M
=3forcolourimages).Sinceitisnowwell
1
ScuolaNormaleSuperiore,PiazzadeiCavalieri7,56126Pisa,Italy,[e-mail]
2
Lab.J.-L.Lions,B.C.187,Univ.PierreetMarieCurie,75252ParisCedex05,France,mas-
[e-mail]

1

admittedthattheessentialfeaturesofanynaturalimagearecontainedinitsgreylevel
representation,weshallconcentrateonthepanchromaticcase
M
=1.Toextendtothe
colourcaseanoperatordesignedforgreylevelimages,itisgenerallyenoughtoprocess
separatelyeachchannelinthecolourrepresentation,e.g.thered-green-bluerepresentation
or,moreappropriately,anyrepresentationwithtwochannelsforthechromaticityandone
channelfortheluminosity(see[9]andthereferencesherein).
AftertheworkofL.RudinandS.Osher[34],theusualrepresentationofapanchromatic
imageisasumoftwocomponents
u
1
∈
BV(IR
N
)and
u
2
∈
L
2
(IR
N
).Thecomponent
u
1
issupposedtodescribethe
geometry
oftheimage,i.e.itsobjectsandtheirboundaries,
while
u
2
containsallinformationabout
texture
and
additivenoise
.Theassumptionthat
thegeometryoftheimagecanbedescribedbyafunctionofboundedvariationsounds
quitenatural,foritmeansthattherecanbediscontinuitiesintheimagebutsupportedon
rectiablecurves.Thenecessityofanothercomponentthatdoesnotnecessarilybelongs
toBVcanbecorroboratedbyanexperimentalprocedurethatseemstoindicatethat,
givenadigitalimage,thesubjacent“real”imagemaybeoftentoooscillatingtobelongto
BV(see[2]forthedetailsand[11]forconnectedtheoreticissues).Thereadermayrefer
to[4,20]foradetailedsurveyofthespaceBV.
Amongthelargeliteraturethathasbeenpublishedinrecentyearsontherecovery
ofmissingpartsinadigitalimage,onecanbasicallydistinguishbetweentwoapproaches
andeachofthemcorrespondsinsomewaytotheprocessingofonecomponentinthe
decompositionabove:
thestochasticapproach,whichisbasedonthemodelingofanimageasarealization
ofarandomprocess.Usually,itisassumedthattheimageintensityderivesfroma
MarkovRandomFieldand,therefore,satisespropertiesoflocalityandstationarity,
i.e.eachpixelisonlyrelatedtoasmallsetofneighboringpixelsanddierentregions
oftheimageareperceivedsimilar.Thismodelingisparticularlyadaptedfortexture
images(thustotheprocessingorthecomponent
u
2
inthepreviousdecomposition)
andhasmotivatednumerousworksontextureanalysisandsynthesis[5,14,15,25,
32,33,42,44],
thedeterministicapproach,whosemainpurposeistorecoverthegeometryofthe
image.Themodelweshalldiscussinthispaperbelongstothiscategory.
ApioneeringworkontherecoveryofplaneimagegeometryisduetoD.Mumford,
M.NitzbergandT.Shiota[31].Theydidnotdirectlyaddresstheproblemofrecovering
missingpartsinanimagebutrathertriedtoidentifyoccludingandoccludedobjectsin
ordertocomputetheimagedepthmap.Theiralgorithmstartswiththedetectionofthe
boundariesofimageobjects.Thenextstepistheidenticationofoccludedandoccluding
objects.Tothisaim,Nitzberg,MumfordandShiotahadtheluminousideatomimic
anaturalabilityofhumanvisiontocompletepartiallyoccludedobjects,theso-called

2

amodalcompletion
processdescribedandstudiedbytheGestaltschoolofpsychologyand
particularlyG.Kanizsa[23].Fromaseriesofperceptualexperiments,Kanizsafoundout
thatourvisionsystemdetectsocclusionataverylowlevel,actuallyassoonasitdetects
T-junctions
,whicharepointswhereanobjectoutlineabruptlyabutsagainsttheoutline
ofanotherobjectandformsajunctionintheshapeoftheletter“T”.Inparticular,our
perceptionofocclusionhasnothingtodowithapriorrecognitionoftheobjects.Being
theT-junctiondetected,ourbrainperformsacontinuationofobjectsboundariesbetween
T-junctions(seegure1).

T−junctions

Figure1:Thisexample,duetoG.Kanizsa[23],illustratestheamodalcompletionprocess.
Startingfromthefourobjectsontheleftcolumn,theadditionofeitherfourwhiterectangles
orawhitecrossproducesT-junctions(middlecolumn),thatconduceourbraintoperceive
occlusionsthat,inreality,donotexist.Thisillustratesperfectlythelinkbetweenthe
presenceofT-junctionsandtheperceptionofocclusions.Then,ourvisualsystemrecovers
thevirtuallyoccludedobjects(fourblackdisksinonecaseandablacksquareintheother)
byconnectingT-junctionswithcompletioncurves,followinga
goodcontinuation
principle.
Wehaverepresentedthosecurveswithdashlinesontherightcolumn.

AspointedoutbyKanizsa,thiscontinuationprocessreliesonmanydierentlaws[23]
andthereisactuallynoobviousmaytomodelit,eveninrelativelysimplesituations[18].
Again,itseemsthatnoprocessofrecognitionbeinvolved(see