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TIME-DELAY REGULARIZATION OF ANISOTROPIC DIFFUSION AND IMAGE
PROCESSING
Abdelmounim Belahmidi, Antonin Chambolle
To cite this version:
Abdelmounim Belahmidi, Antonin Chambolle. TIME-DELAY REGULARIZATION OF ANISOTROPIC DIFFUSION AND IMAGE PROCESSING. ESAIM: Mathematical Modelling and Numerical Analysis, EDP Sciences, 2005, Vol.39, No.2., pp. 231-251. �10.1051/m2an:2005010�. �hal- 00001401�
image proessing
A. Belahmidi, A. Chambolle
January 2004
Abstrat
Westudyatime-delayregularizationoftheanisotropidiusionmodelforimagedenoisingofMalik
andPerona,whihhasbeenproposedbyNitzbergandShiota. Inthetwo-dimensionalase,weshowthe
onvergeneofanumerial approximationandtheexisteneofaweaksolution. Finally,weshowsome
experimentsonimages.
Keywords:Imagerestoration,Variationalmethods,Numerialapproximation,Time-delayregularization,Malikand
Peronaequation.
1 Introdution
In awell-known paper, Malik and Perona [15℄ have proposed a model for image restoration basedon the
followingpartial dierentialequation:
u
t
=div g(jDuj 2
)Du
u(;0)=u
0
: (1)
Here u
0
isthegreylevelintensityofthe originalimage, u(;t)is therestoredversion,that dependsonthe
sale parameter t, and g is a smooth non-inreasing positive funtion with g(0) = 1 and sg(s 2
) ! 0 at
innity. Themainideaisthattherestorationproessobtainedbytheequationisonditional: ifxisanedge
point, where the gradientis large, then the diusion will bestopped and thereforethe edge will bekept.
If x isin homogeneousarea,thegradienthastobesmall, andthediusion will tendto smootharoundx.
Byintroduinganedgestoppingfuntion g(jDuj 2
)inthe diusionproess, themodelhasbeenonsidered
asan importantimprovementof thetheory of edge detetion[17℄. Theexperiments ofMalik andPerona
wereveryimpressive, edgesremained stable overaverylong time. It wasdemonstrated in [16℄ that edge
detetionbasedonthis proesslearlyoutperformstheCannyedgedetetor [3℄.
Unfortunately, theMalik andPeronamodelisill-posed. Indeed,among thefuntions whih Malik and
Peronaadvoatein theirpapers,wendg(s 2
)=1=(1+s 2
) org(s 2
)=e s
2
forwhihnoorret theoryof
equation(1)isavailable. Bywritingtheequationin dimensiontwo:
u
t
=g(jDuj 2
)jDujdiv
Du
jDuj
+ g(jDuj 2
)+2jDuj 2
g 0
(jDuj 2
)
D 2
u
Du
jDuj
; Du
jDuj
; (2)
whereD 2
u(
Du
jDuj
; Du
jDuj
)istheseondderivativeofuinthegradientdiretionandjDujdiv( Du
jDuj
)istheseond
derivativeintheorthogonaldiretion,weobservethatthediusionrunsbakwardsifsg(s 2
)isnon-inreasing.
Then,in theregionswhere thegradientofasolutionislarge,theproessanbeinterpretedasabakward
heatequationwhihisatuallyillposed. Intheontinuoussetting,itmeansthat(1)mayhavenosolutionat
all. Oneouldalsoimagineverylosepituresproduingdivergentsolutions[11℄. Inpratie,theequation
isdisretizedintoa(obviouslywell-posed)nite-dimensionalversionof(1),however,itdoesnotseemorret
to interpretsuhadisretizationasanapproximationoftheill-posedproblem(1).
nd outwhether(1) anbegivenasound interpretation. There areessentiallytwoapproahes: Therst,
motivated by favorable numerial results, onsists in studying the original equation and in establishing
theoretial results that explain the observed behaviour. The seond approah onsists in modifying the
equationbyregularizingthetermg(jDuj 2
)inordertogetawell-posedequation.
2 The Malik and Perona equation and the regularized versions
First,weexpose themainmathematialresultsestablishedontheMalik andPeronamodel. Mostof these
results are restrited to the dimension one; the unique result in dimension two, given by You et al. [21℄
onrms theill{posedness oftheequation. Kawohland Kutev [13℄establish,in 1D,nonexisteneof global
weak solution, and prove the existene and uniqueness of a lassial solution only if the initial data has
everywhereasmallslope. Inthis asethe equation remainsparaboliforall timeand there is noedge to
preserve: the diusionsmoothesthedata, liketheheat equation woulddo. Theyalsoproveaomparison
prinipleunderspeialassumptionsontheinitialdata.
Kihenassamy[14℄showsthatingeneraltheMalikandPeronaequationdoesnothaveaweaksolutionif
theinitialdataisnotanalytiinaneighborhoodofhighgradientregions. Hisargumentisbasedoninterior
regularity properties of paraboliequations. Only in dimension one, he proposes anotion of generalized
solutions,whih arepieewiselinearwithjumps,andshowsexistene.
Adoptinganumerialviewpoint,Esedoglu[7℄studiestheone-dimensionalMalikandPeronasheme. He
establishes byasalingargumenttheonvergeneto anevolutionin theontinuoussetting. Theresulting
evolutionsolvesasystemofheatequationsoupledto eahother throughnonlinearboundaryonditions.
Working in dimension one learly redues the diÆulty by eliminating the rst term of (2) whih is
nothing but the mean urvature motion operator with theoeÆientg(jDuj 2
). As it is known, the mean
urvature motion evolveseah level line fu = Cg with a normal speed proportional to its urvature (see
[8,1℄ formoredetails).
Indimensiontwo,You et al.[21℄ express theanisotropidiusion of Malik andPeronaas thesteepest
desentof anenergysurfaeandanalyzethebehaviourofthemodel. Theyprovethat theill{posedness is
ausedbythefatthattheenergyfuntionalhasaninnitenumberofglobalminimathataredenseinthe
image spae. Eah of these minima orresponds to a pieewise onstant image. This means that slightly
dierentinitialimages mayendupindierentminimaforlarget.
Asmentionned,anotherapproahreliesontheideathattheill{posednessmaybealleviatedthroughthe
introdutionofasmoothversionofg(jDuj 2
). Thereareessentiallytwopropositionswhihweonsiderasa
diret derivationfrom theMalikand PeronaModel. Therstonsistsin aspatial regularization,asin the
followingmodel:
u
t
=div g(jDG
uj
2
)Du
; (3)
wherebyg(jDuj 2
) isreplaed by g(jDG
uj
2
), where G
is aGaussian with variane . In[4℄, Catteet
al.proveexistene,uniquenessandregularityofasolution. It isknownthatG
u(x;t)isnothingbutthe
solutionatsale oftheheatequationwithu(x;t) asinitialdata.
Arstobservationis thatnear asharporner,thediusion oeÆientg(jDG
uj
2
)mayremainvery
large,henethis modelwillbeunabletopreserveorners.
Anotherproblemisthehoieoftheregularizationparameter. Infat,thishoieisritialinthesens
thatthediusionproesswouldbeill{posedif=0,whileimagefeatureswouldbeblurredfortoolargean
. Asproposed byWhitakerandPizer[20℄,theregularizationparametershould beadereasingfuntion
int,byusinglargeinitiallytosuppressnoiseandreduingsothatimagefeaturesarenotfurtherblurred.
Inspiteofthis,thehoieoftheinitialandnalvaluesof remainsanopenquestion.
Theseond proposition is atime-delay regularization, where onereplaesjDuj 2
with an averageof its
valuesfrom0tot. Theng(jDuj )isreplaedwithg(v)with:
v(x;t)= e t
v
0 (x)+
Z
t
0 e
s t
jDu(x;s)j 2
ds; (4)
where v
0
is aninitialdata, for examplev
0
=0orjDu
0 j
2
. Therefore thenewdiusion proessis desribed
bythefollowingsystem:
u
t
= div g(v)Du
u(;0)=u
0
; (5)
v
t
= jDuj 2
v v(;0)=v
0
: (6)
ProposedbyNitzbergandShiota[19℄,thismodelisverylosetotheMalikandPeronaequationsinethere
is nospatial smoothing. Inpartiular, itshould mean that there is nopreviousmovementof the features
in the diusion proess. In [2℄ the authors of the present paper have shown that in any dimension, the
system(5)-(6)admitsauniquelassialsolution(u;v)whihan blowupinnitetime,andthataslongas
thesolution exists,the equationsatises themaximumprinipleanddoesnotreatespurious information
(that is, stritloalextrema). These properties ofthe system(5)-(6)haveenouraged us tostudy it from
a numerial viewpoint. Let us mentione that time-delay regularization has been already used in image
proessingbyCottetandEl Ayyadi[6℄asanisotropidiusiontensors.
This paper is organized asfollows: In setion 3 we propose a naturaldisretization in time of (5)-(6)
withjDuj 2
replaedbyF(jDuj 2
),F beingasortoftrunation. Numeriallythismodiationdoesnothave
any impat on the output images sine the threshold impliitly exists in the numerial sheme. Indeed,
if the disrete sheme satises the maximum priniple, then the disrete gradientis always bounded (for
exampleby(maxu
0
minu
0
)=x, xbeingthegridsize). Theoretially, theintrodutionofF is ahuge
regularization of the system (we will see that it yields existeneof a weak solution for all time). Setion
4proposesanumerialsheme forsolvingthesystem,and setion5showssomeexperiments onsyntheti
and natural images. In setion 6 we establish a priori estimates and regularity results on the proposed
approximationandprovethemain resultofthispaper. Inthesetion7wegivetheproofsoftwotehnial
resultsonelliptiequationsthatareneededinsetion6.
3 Numerial approximation
Thegoalof thispaperistostudyandapproximate numeriallythesystem:
u
t
= div(g(v)Du) u(;0)=u
0
; (7)
v
t
= F(jDuj 2
) v v(;0)=v
0
; (8)
in (0;T)where=(0;1) 2
, 0<T <1. Wewill showthat thesystemadmits aweaksolution,under
thefollowingtehnialassumptions:
- g2C 1
([0;+1))isapositivenon-inreasingfuntion withg(0)=1andg(+1)=0.
- F 2 C 1
([0;+1)) is a smooth version of s ! min(s;M), where M >0 is a (large) real number (in
partiular,weassume0F 0
1).
FixedÆt>0,wedenethesequene(u n
Æt
;v n
Æt )
n
bythesemi-impliitsheme:
(u 0
Æt
;v 0
Æt )=(u
0
;v
0 )2 H
1
()\L 1
()
H
1
()\L 1
()
; v
0
0 and
u n+1
Æt u
n
Æt
Æt
= div(g(v n
Æt )Du
n+1
Æt )
u n+1
Æt
n
=0 (9)
v n+1
Æt v
n
Æt
Æt
= F(jDu n+1
Æt j
2
) v n+1
Æt
: (10)
Wedenethepieewiseonstant(int>0),funtions
u
Æt
(x;t)=u [t=Æt℄+1
Æt
(x);
where [℄ denotes the integerpart. Wealso dene (v
Æt
) in thesame way. Then wean write the disrete
system(9)-(10)intheform( Æt
isdened by Æt
f(;t)=f(;t Æt)):
u
Æt
Æt
u
Æt
Æt
= div(g(
Æt
v
Æt )Du
Æt );
u
Æt
n
=0; (11)
v
Æt
Æt
v
Æt
Æt
= F(jDu
Æt j
2
) v
Æt
: (12)
Themain resultofthis paperisthefollowingtheorem:
Theorem1. LetT >0. Thereexistsasubsequene(u
Æt
j
;v
Æt
j )of(u
Æt
;v
Æt
)and(u;v)aweak solutionofthe
system(7)-(8)in H 1
((0;T))\L 1
((0;T))
H
1
((0;T))\L 1
((0;T))
suhthat,wehave
the onvergenes, asj!+1:
u
Ætj
! u stronglyin L 2
(0;T;H 1
()); (13)
v
Ætj
* v weakly in L 2
(0;T;H 1
()): (14)
Theproofofthistheoremwill begiveninsetion6.
4 Disretization
To disretize (9)-(10) we denote by u n
i;j
(resp. v n
i;j
) the approximation of u (resp. v) at point (ih;jh)
(0 i;j N)and time t =nÆt, wherethe size of theinitial image u
0
is given by NN and h=1=N.
Usingthefollowingnite-dierenesformulas:
x
+ w=w
i+1;j w
i;j
;
x
w=w
i;j w
i 1;j
;
y
+ w=w
i;j+1 w
i;j
et
y
w=w
i;j w
i;j 1
;
theapproximationofdiv g(v)Du
atpoint(ih;jh) andatsalet=(n+1)Ætisgivenby:
1
h 2
x
g(v n
i;j )
x
+ u
n+1
i;j
+ y
g(v n
i;j )
y
+ u
n+1
i;j
:
Thentheequation(9)beomes:
u n+1
i;j u
n
i;j
Æt
= 1
h 2
n
g(v n
i;j ) u
n+1
i+1;j u
n+1
i;j
g(v n
i 1;j ) u
n+1
i;j u
n+1
i 1;j
+g(v n
i;j ) u
n+1
i;j+1 u
n+1
i;j
+g(v n
i;j 1 ) u
n+1
i;j u
n+1
i;j 1
o
(15)
withtheNeumannboundaryondition:
u n+1
i;0 u
n+1
i;1
=0; u n+1
i;N 1 u
n+1
i;N
=0; for 0iN;
u n+1
0;j u
n+1
1;j
=0; u n+1
N 1;j u
n+1
N;j
=0; for 0jN:
u n+1
u n
Æt
+h 2
A(v n
)u n+1
=0;
where thematrixA(v n
)is tridiagonalby bloks, andpositivedened. Bylassialarguments[5℄weknow
that [I+Æth 2
A(v n
)℄isinvertible.
To avoid any additional anisotropy in the sheme, we try to build adisrete gradient of u in (10) as
rotationallyinvariantaspossible. Weusethedisretizationproposed in[4℄and[19℄whih writes:
x
w = (1+2 1
2
) 1
n
w
i+1;j w
i 1;j
+2 1
2
w
i+1;j 1 w
i 1;j 1
+2 1
2
w
i+1;j+1 w
i 1;j+1
o
;
y
w = (1+2 1
2
) 1
n
w
i;j+1 w
i;j 1
+2 1
2
w
i+1;j+1 w
i+1;j 1
+2 1
2
w
i 1;j+1 w
i+1;j 1
o
:
Thedisretizationof(10)isthenwritten(assumingthatinthewholerange
0;max ( x
u) 2
+( y
u) 2
,we
haveF(s)=s)
v n+1
i;j
= 1
1+Æt Æth
2
((
x
u) 2
+( y
u) 2
)+v n
i;j
: (16)
Weannowgiveadisreteversionofthemaximumprinipleandshowthattheproposedalgorithmwill
notreatenewinformation(loal extrema).
Lemma1. Foralln>0and(k;l),0k;lN,wehave:
min
i;j u
0
i;j
:::min
i;j u
n
i;j u
n+1
k ;l
max
i;j u
n
i;j
max
i;j u
0
i;j
; (17)
In partiular, ifu n+1
k ;l
isa stritloal maximum(resp. stritloal minimum) of u n+1
i;j
then
u n+1
k ;l
<u n
k ;l
resp. u n+1
k ;l
>u n
k ;l
: (18)
Proof : Letu n+1
k ;l
aglobalmaximumof u n+1
i;j
, theninpartiular:
u n+1
k ;l u
n+1
k +1;l
0; u
n+1
k ;l u
n+1
k 1;l 0;
u n+1
k ;l u
n+1
k ;l+1
0; u
n+1
k ;l u
n+1
k ;l 1 0:
(19)
Using(15),andthefatthat g>0,weobtain:
u n+1
k ;l u
n
k ;l
;
andwededue:
max
i;j u
n+1
i;j
max
i;j u
n
i;j
max
i;j u
0
i;j :
In thesame way we provethe\min" partof (17), by onsidering u n+1
k ;l
a global minimumof u n+1
i;j
. We
prove(18)byusingthesameargumentandthefatthat wenowhavestritinequalitiesin(19).
5 Experiments
In gure1weompare theperformanes of oursheme to theCatteand al.model[4℄ and in gure 2we
present an example of restoration on a natural image. The experiments have been done with the edge
stoppingfuntion
g(s 2
)= 1
1+(s 2
= 2
) :
to theonventionintheprevioussetion). ThetemporalinrementwehaveusedisÆt=0:1.
Figure1-(a)isasynthetiimage(128128)representingsuperimposedshapeshavingeahoneaonstant
greylevel. Figure1-bshowsimage1-awhere 20%ofgaussiannoiseis added. Werepresentbygures1-()
therestorationofthenoisyimagewiththeCatteetal.model(3) atsales4and8(fromleftto right)and
by gures1-(d) the restorationwith ourshemeat thesamesales. Notie that respetively, the sales4
and 8orrespond tothe stoppingtime t=8and32. As explainedin [4℄by theauthors ofthe model (3),
thesale used intheonvolutiontermG
umustbein takeninrelationto thestoppingtime. Thusin
gures1-()wehaveused=4and8.
As mentionned in setion 2 the threshold introdued by F impliitly exists in the numerial sheme.
Indeed, sine the disrete sheme satises the maximum priniple and the fat that spatial inrement is
assumedtobe1,thenthedisretegradientisalwaysboundedby p
2(maxu
0
minu
0
)andManbehosen
to be2(maxu
0
minu
0 )
2
.
Intheleftimageof1-(b),thenoiseissmoothedinthehomogeneousareasbutiskeptneartheedges. This
drawbakis ausedby thefat that thediusion is inhibitedalso in the neighborhood of edges. Whereas
in theleft imageof 1-(),thenoiseisonly partiallysmoothedbut in auniformway. In therightimage of
1-(b) theedgesandornersareblurred. Indeed,weknowthat kDG
uk
L 1
dereasesforlargevaluesof
onsequentlyforlargevaluesofwediusemorenearedges: inpartiular,ifkDG
uk
L 1
<,thediusion
is never inhibited. Whereas in the rightimage of 1-(), the noise has disappeared and the reonstruted
imageis verylosetotheoriginal.
Figure2-(Right)representsanaturalimage(256256)withoutadditivenoiseandgure2-(Left)repre-
sentsitsrestorationwithourshemeat sale5thatorrespondstothestoppingtime t=12:5. Weremark
that salient edges and textures are preserved (see for example the top of the hat) whereas the noise in
homogeneousareasissmoothed.
6 Numerial analysis
First we hek that our shemes makessense. Indeed, for all Æt > 0, the sequene (u n
Æt
;v n
Æt
) exists and is
unique. Equation(10)allowstowrite v n+1
Æt
expliitly:
v n+1
Æt
= 1
1+Æt
ÆtF(jDu n+1
Æt j
2
)+v n
Æt
: (20)
andbyindutionwend
0v n+1
Æt
1 (1+Æt) (n+1)
M+ 1+Æt
(n+1)
jjv
0 jj
L 1
() :
Wededuethat(v n
Æt
)isuniformlyboundedinL 1
() andsatises:
0v n
Æt
max(M ;jjv
0 jj
L 1
() ):=M
0
; forallnand Æt: (21)
Using the fat that g is a positive non-inreasing funtion, we have 0 < g(M 0
) g(v n
Æt
) 1. Therefore
equation(9)isstritlyelliptiandweknowthatthere existsauniquesolutionu n+1
Æt in H
1
(). Inaddition,
u n+1
Æt
isgivenbytheproblem
min
E
(Æt ;n) (w)=
Z
g(v
n
Æt )jDwj
2
dx+ 1
2Æt Z
jw u
n
Æt j
2
dx:w2H 1
()
: (22)
Bythemaximumpriniple,itislearthatforalmostallx2wehave
infu
0
infu n
Æt u
n+1
Æt
(x)supu n
Æt
supu
0
: (23)
Multiplying byu n+1
Æt
theequation(9)andintegratingonweget
0Æt Z
g(v
n
Æt )jDu
n+1
Æt j
2
dx Z
u
n
Æt u
n+1
Æt dx
Z
ju
n+1
Æt j
2
; (24)
Line: ()Image(b)restoredbytheCatteetal.model[4℄withsales4,and8. Bottom Line: (d)Image
(b)restoredbyourshemewithsales4,and 8.
