A degenerate parabolic equation
arising in image processing
G.Citti M.Manfredini
1 Introduction
We prove here an existence result for solutions for a parabolic equation,with non
local coecients arising in image processing.An image is a bounded function
u:D!R dened on a rectangular region D.If the function u is not regular,
the image is noisy and it is not possible to use it directly in applications,but
is necessary to smooth it by means of a nonlinear evolution problem,with
Neumann boundary data.To this end dierent model have been proposed.
Perona and Malik proposed in [PM] the following anisotropic diusion model:
@
t
u = div(f(jDuj)Du) in D [0;T];
with a suitable decreasing function f.Even though numerical experiments
provide the desired regularization eect,the problem can be ill posed from
an analytic point of view for particular choice of the function f,and really few
is known about its solutions (see [KK],[K]).Then the model was modied in
dierent ways:in [CLMC] the following equation was proposed,
@
t
u = div(f(jDG
uj)Du) in D [0;T];
where G
is a Gaussian kernel depending on a parameter .For the associated
problem with L
2
initial datum,also existence and uniqueness was proved in
[CLMC].In [ALM] and [AE] non divergence versions of the same operator was
proposed,whose simplest form is
@
t
u = f(jDG
uj)jDujdiv
Du
jDuj
+g(u) in D [0;T]:
The existence of solutions was proved with viscosity solutions methods.Equa
tions of this type has received a lot of attention because of its geometrical
interpretation:models dened in terms of motion by mean curvature have been
proposed by [OS],[S],and model related to properties of the principal curvatures
Dipartimento di Matematica,Univ.di Bologna,P.zza di Porta S.Donato 5,40127,
Bologna,ITALY,email citti@dm.unibo.it,manfredi@dm.unibo.it.Investigation supported
by University of Bologna.Founds for selected research topics.
1
are due to [CS],[ST],[SOL] We also refer to [ES],[CGG],[GGIS],[IS],[S],[GG],
for the application of viscosity methods to mean curvature equations.Similar
techniques can be applied to the study of movies,which can be considered as
family (u
)
2[0;1]
of images.
In [AGLM] the authors introduced a new model in an axiomatic way,requir
ing that the solutions satisfy maximum and comparison primciple,and are in
variant with respect to suitable groups of transformations.The resulting model
 which has a viscosity solution by construction  and it is the following one
@
t
u = jDuj
sign(curv(u))acc(u)
+
sign(curv(u)) u(;0) = u
0
;
where curv(u) is the mean curvature of the graph of u,and the acc(u) repre
sents the acceleration of the movie in the direction of the spatial gradient.By
simplicity of notations we will denote
clt(u) = jDuj
sign(curv(u))acc(u)
+
:
In [G] it is proved that clt(u) has the following discretisation,which we will use
here as a denition of it.
clt(u)(x;;t) = min
1
2A
+
x
;
2
2A
x
n
ju(x +
1
; +;t) u(x;;t)j+ (1)
ju(x
2
; ;t) u(x;;t)j +j < DG
u;
1
2
> j
o
where (x;) 2 R
n
R.
A
+
x
= f 2 R
n
:x + 2
;jj 2rg
A
x
= f 2 R
n
:x 2
;jj 2rg:
A new model was introduced in [SMS]:
@
t
u = h(clt(u))div
x
(f(jD
x
Guj)D
x
u) in D[0;R][0;T] u(;;0) = u
0
;(2)
where h is of class C
1
([0;1[;R),f is of class C
2
([0;1[;R) and nonnegative.
Besides
h is nondecreasing and satisfies h(0) = 0;
f is decreasing and satisfies f(0) = 1:
In their paper the authors provide a numerical discretisation of the equation,
and some numerical experiments  see also [LS],[ZSL],[MSL].Here we provide
a rst existence result under the simplied assumptions that
infclt(u
0
) m> 0;and sup
[m;1]
h
0
(s) inf
[m;1]
h(s);(3)
where is small.A possible choice of h is the following:
h(s) =
s
2
+s
2
;
2
where is suciently small.Note that,also in this simplied assumptions,the
equation is degenerate,since its second order termonly depends on the variables
x.Besides the coecients of the equation are non local.We refer the author to
[ALM],[BN],[C],for other results concerning parabolic dierential equations
with nonlocal coecients.
A standard procedure for nding solutions of the Cauchy problem for a
parabolic equation on a square with Neumann boundary conditions is to extend
the initial datum u
0
on all the space by re ections and periodicity and prove
that the resulting Cauchy problem on all the space has periodic solutions.We
then prove the following:
Theorem 1.1
Let u
0
be a periodic,Lipschitz continuous function dened in
R
n
R,satisfying condition (3) and 0 u
0
(x;) 1:Then there exists a
constant T > 0,and a periodic viscosity solution u of problem (2),dened on
R
n
R[0;T],Lipschitz continuous in (x;) and Holder continuous in t.For
any xed ,and any xed 2 [0;1] the function
(x;t)!u
(x;t) = u(x;;t) (4)
is of class C
2+;1+=2
(R
n
]0;T[) in the variables x and t.Besides,there exist
constants K
1
and K
2
such that if u
0
and v
0
are bounded by 1,periodic and
Lipschitz continuous functions on R
n
R,the corresponding solutions u and v
satisfy
jju vjj
L
1
(R
n+1
[0;T])
K
1
e
K
2
T
jju
0
v
0
jj
L
1
(R
n+1
)
:(5)
The structure of the second order termof the operator in (2) can be described
as follows.We call Lip(D) the set of Lipschitz continuous functions on a set D,
Bd(D) theset of bounded functions,and
Lu =
n
X
i=1
a
i
(u)(x;;t)@
i;i
u +
n
X
i=1
b
i
(u)(x;;t)D
i
u;(6)
where
a
i
;b
i
:Lip(D)\Bd(D)!Lip(D)\Bd(D);
for every compact D in R
n
R.There exist constants C
0
;C
1
such that for every
D,for every u 2 Lip(D)\Bd(D),for every i = 1; n,for every (x;;t) 2 D
C
0
a
i
(u)(x;;t) C
1
(jjujj
1
+1);b
i
(u)(x;;t) C
1
(jjujj
1
+1):(7)
The following condition is satised:
j@
h
(a
i
(u))j +j@
h
(b
i
(u))j C
1
supj@
h
uj +C
1
;(8)
where is a suitably small constant,satisfying
2
=
2
exp(6
2
)
C
2
0
128C
2
1
and
2
=
C
0
64
1 +
4C
2
1
C
0
1
:
3
For every Lipschitz continuous functions u and v
ja
i
(u)(x;;t) a
i
(v)(x;;t)j +jb
i
(u)(x;;t) b
i
(v)(x;;t)j C
1
jjuvjj
1
:(9)
Finally,if D = R
n
R,a
i
is invariant with respect to translations:for every
xed we call
a
i
(u(; +
0
;))(x;;t) = a
i
(u)(x; +
0
;t):(10)
Theorem 1.1 is then a consequence of the following more general result:
Theorem 1.2
Let u
0
be a Lipschitz continuous function dened in R
n
R,
satisfying condition 0 u
0
(x;) 1:Then there exists a bounded,Lipschitz
continuous viscosity solution u of problem
u
t
= Lu in R
n+1
[0;T[
u(x;;0) = u
0
(x;) in R
n+1
:
(11)
For any xed ,and any xed 2 [0;1] the function u
dened in (4) is of
class C
2+;1+=2
in the variables x and t.Besides,the stability condition (5) is
veried.
The proof of this result follows essentially the main ideas of the classical exis
tence results,as nd in the [LUS],or the user guide,but,due to the degeneracy
of the operator,we have to organize in an new way the estimate of the gradient
(D
x
u;@
u) of the solution.Indeed,using the Bernstein method,we rst prove
an a priori bound only for the spatial gradient D
x
u.Using this estimate,we
obtain a stability inequality for the solutions,from which we deduce the esti
mate of @
u.The fact that the coecients depend globally on the unknown u
introduce some additional technical diculties.Indeed,even though we use an
elliptic regularisation,and we always work with regular functions,we are forced
to use an approach proposed by [CLM] for studying the viscosity solutions.
Let us give a more detailed sketch of the proof.In Section 2 we consider the
elliptic regularisation of the operator L:
L
"
u =
n
X
i=1
a
i
@
i;i
u +"
2
@
;
u +
n
X
i=1
b
i
(u)@
i
u;(12)
and we rst prove the existence of solutions of the Cauchy problem
@
t
u = L
"
u in Q
u(x;;t) = u
0
(x;) in @
Q
(13)
on the bounded cylinder Q = B
R
[0;T[,with parabolic boundary @
Q.In
particular,with a suitable modication of the Bernstein method we prove that
the gradient of the solution satises the following estimate
jjD
x
ujj
L
1
(Q
R
)
+"jj@
ujj
L
1
(Q
R
)
C
for a constant C independent of R and".Letting R!1we nd a solution on
all Q = R
n+1
[0;T].
4
In Section 3,using the fact that the estimate of the x gradient is independent
of",we prove an uniqueness and stability result for solutions of (13) on R
n+1
[0;T]:
Theorem There exist constants K
1
and K
2
such that if u and v are two
solutions Lipschitz continuous and bouded of (13) with initial data u
0
and v
0
respectively,then
jju vjj
L
1
(R
n+1
[0;T])
K
1
e
K
2
T
jju
0
v
0
jj
L
1
(R
n+1
)
:
Let us note explicitly that this results holds even if the comparison principle
is not satised.Finally we deduce the boundeness of @
u from this estimate.
In Section 4 we see that equation (2) satises these assumptions,and we
conclude the proof of Theorem 1.1,with an other regularisation procedure.
Acknoledgment
We are deeply indebted with F.Sgallari for bringing the problem to our
attention,and with A.Sarti for many useful conversations on the subject of
their model.
2 A priori bound of the spatial gradient
In this section we prove the existence of a solution of the initial value problem
(13).The proof is based on an a priori estimate of the spatial gradient.For
simplicity we will introduce the following notation:
e
@
h
= @
h
;h = 1 ;n;
e
@
n+1
="@
:(14)
Then this operator in (12) will be written as
L
"
=
n+1
X
i=1
a
i
e
@
i;i
u +
n
X
i=1
b
i
(u)
e
@
i
u;(15)
where a
n+1
= 1 and also this last coecients satises assumptions (7),(8),(9)
with the same constants C
0
and C
1
independent of".We consider the Cauchy
problem (13) on the bounded cylinder Q
R
= B
R
[0;T],with initial datum
0 u
0
(x;) 1 in B
R
:(16)
We rst note that the solutions of (13) satisfy the maximum principle,so
that (16) implies
0 u(x;;t) 1 8(x;;t) 2 B
R
[0;T]:
As we noted in the introduction the classical gradient estimate can not be
applied directly,since the coecients depend globally on u.Hence we suitably
modify the Bernstein method in order to apply it to our situation.
5
Theorem 2.1
If u 2 C
2
(Q)\Lip(
Q) is a solution of problem (13) on the
bounded cylinder Q = B
R
[0;T],then there exists a constant
e
C
1
independent
of R and"such that
jjD
x
ujj
L
1
(Q
R
)
+"jj@
ujj
L
1
(Q
R
)
e
e
C
1
T
jjD
x
u
0
jj
L
1
(B
R
)
+"jj@
ujj
L
1
(B
R
)
;
where D
x
is the gradient with respect to the xvariable.A possible choice of
e
C
1
is
e
C
1
= 4
2C
2
1
+1
C
0
;
where C
0
and C
1
are dened in (7) and (8).
Proof If is an increasing function to be chosen later,we can always represent
u in the form:u = (u).Then the function u is a solution of
@
t
u =
n+1
X
i=1
a
i
(u)
e
@
i;i
u +
n+1
X
i=1
a
i
(u)
00
(u)
0
(u)
(
e
@
i
u)
2
+
n
X
i=1
b
i
(u)
e
@
i
u:
Let be a nonnegative function in C
1
0
(Q
R
).Multiplying by
e
@
h
(
e
@
h
u) we get
Z
@
t
u
e
@
h
(
e
@
h
u)dxddt =
Z
n+1
X
i=1
a
i
(u)
e
@
i;i
u
e
@
h
(
e
@
h
u)dxddt+ (17)
+
Z
n+1
X
i=1
a
i
(u)
00
(u)
0
(u)
(
e
@
i
u)
2
e
@
h
(
e
@
h
u)dxddt +
Z
n
X
i=1
b
i
(u)
e
@
i
u
e
@
h
(
e
@
h
u)dxddt:
Let us consider one term at a time.Integrating by parts the rst one we get
Z
@
t
u
e
@
h
(
e
@
h
u)dxddt =
1
2
Z
(
e
@
h
u)
2
@
t
dxddt:(18)
The second becomes
Z
n+1
X
i=1
a
i
(u)
e
@
i;i
u
e
@
h
(
e
@
h
u)dxddt = (19)
Z
n+1
X
i=1
e
@
h
a
i
e
@
i;i
u
e
@
h
u dxddt +
Z
n+1
X
i=1
e
@
i
a
i
e
@
i;h
u
e
@
h
u dxddt+
+
Z
n+1
X
i=1
a
i
e
@
i;h
u
2
dxddt +
1
2
Z
n+1
X
i=1
a
i
e
@
i
(
e
@
h
u)
2
e
@
i
dxddt:
Hence inserting (18) and (19) in (17),summing over h,and denoting
v =
n+1
X
i=1
(
e
@
h
u)
2
;
6
where
e
@
h
is dened in (14),we obtain
1
2
Z
v@
t
+
1
2
Z
n+1
X
i=1
a
i
e
@
i
v
e
@
i
=
Z
F ;
where
F =
n+1
X
i=1
n+1
X
h=1
a
i
e
@
i;h
u
2
+
e
@
h
a
i
e
@
i;i
u
e
@
h
u
e
@
i
a
i
e
@
i;h
u
e
@
h
u+
+
e
@
h
a
i
(u)
00
(u)
0
(u)
(
e
@
i
u)
2
e
@
h
u
!
n
X
i=1
n+1
X
h=1
e
@
h
b
i
(u)
e
@
i
u
e
@
h
u:
Let us estimate F
F
n+1
X
i=1
n+1
X
h=1
a
i
e
@
i;h
u
2
+j
e
@
i;i
uj
2
+
1
j
e
@
h
a
i
j
2
j
e
@
h
uj
2
+j
e
@
i;h
uj
2
+
1
j
e
@
i
a
i
j
2
j
e
@
h
uj
2
+
+j
e
@
h
a
i
j
2
j
e
@
i
uj
2
+
00
(u)
0
(u)
2
(
e
@
i
u)
2
(
e
@
h
u)
2
+
+a
i
(u)
00
(u)
0
(u)
0
(
e
@
i
u)
2
(
e
@
h
u)
2
+(
e
@
i;h
u)
2
+
a
2
i
(u)
00
(u)
0
(u)
2
(
e
@
i
u)
2
(
e
@
h
u)
2
+
+
n
X
i=1
n+1
X
h=1
j
e
@
h
(b
i
(u))
e
@
i
u
e
@
h
uj +j
e
@
i;h
uj
2
+
1
jb
i
(u)j
2
j
e
@
h
uj
2
(if = C
0
=4,where C
0
is dened in (7) and we set b
n+1
= 0 for simplicity of
notations)
n+1
X
i=1
n+1
X
h=1
1
(
e
@
h
a
i
)
2
+(
e
@
i
a
i
)
2
+(
e
@
i
a
i
)
2
+j
e
@
h
b
i
(u)j
2
j@
h
uj
2
+
+
n+1
X
i=1
1
a
2
i
+jb
i
(u)j
2
+1
v +
n+1
X
i=1
00
(u)
0
(u)
2
+a
i
00
(u)
0
(u)
0
+
a
2
i
00
(u)
0
(u)
2
!
v
2
:
If we choose
00
(u)
0
(u)
0
0;
and use the assumptions (7),(8),(9),then the estimate for F becomes:
F
8(n +1)
C
2
1
2
supv(
0
)
2
+C
2
1
v +
(2C
2
1
+1)v
+
+
n+1
X
i=1
00
(u)
0
(u)
2
+C
0
00
(u)
0
(u)
0
+
C
2
1
00
(u)
0
(u)
2
!
v
2
7
e
C
1
v+(n+1)
32
C
0
C
2
1
2
(
0
)
2
+
1+
4C
2
1
C
0
00
(u)
0
(u)
2
v sup v+C
0
(n+1)
00
(u)
0
(u)
0
v
2
;
for a suitable constant
e
C
1
= 4
2C
2
1
+1
C
0
only dependent on the assumptions.We can now make the same choice of as
in [LUS].We set
:[;2]!R (x) =
Z
2
exp(
q
)d
1
Z
2
exp(
q
)d;(20)
where is dened in (8).The assumption made on assure the existence of
constants
e
C
2
and
e
C
3
such that
(n +1)
32
C
0
C
2
1
2
(
0
)
2
+
1 +
4C
2
1
C
0
00
(u)
0
(u)
2
e
C
2
;(21)
2
e
C
2
e
C
3
and
e
C
3
C
0
(n +1)
00
(u)
0
(u)
0
(for reader convenience the computations are collected in Remark 2.1 below).
Then
F
e
C
1
v +
e
C
2
v supv
e
C
3
v
2
:
The estimate for the gradient is a consequence of the following lemma.
Lemma 2.1
Let v be a nonnegative solution of class C
0
(
Q
R
)\C
1
(Q
R
) of the
following nonlinear equation:
1
2
Z
v@
t
dxddt +
1
2
Z
n+1
X
i=1
a
i
@
i
v@
i
dxddt =
Z
F(v) dxddt;
with
F
e
C
1
v +
e
C
2
v supv
e
C
3
v
2
;
and
2
e
C
1
<
e
C
2
.Then
sup
Q
R
v e
e
C
1
T
sup
@
(Q
R
)
v:
Proof If we set!(x;;t) = v(x;;t)e
e
C
1
t
;the function!is a solution of
1
2
Z
!@
t
+
1
2
Z
n+1
X
i=1
a
i
@
i
!@
i
=
Z
e
F ;
with
e
F 2 L
1
and
e
F
e
C
2
!sup!
e
C
3
!
2
exp(
e
C
1
t):
8
Let (x
0
;
0
;t
0
) a maximum point for!in B
R
[0;T],and assume by contradic
tion that
M
0
=!(x
0
;
0
;t
0
) > max
@
Q
R
!= M:
Then we can choose such that
!(x
0
;
0
;t
0
) > M;2 M
0
:
Let us denote (! M
0
+ )
+
its positive part.Let F
j
be a sequence in C
1
converging to
e
F as j!+1,and let!
j
the corresponding solution.Then!
j
uniformly converges to!,and a simple integration by parts ensures that for
every j
Z
!
j
M
0
n+1
X
i=1
a
i
(u)(@
i
!
j
)
2
dxddt =
Z
F
j
(!
j
M
0
+)
+
dxddt:
Letting j go to 1 we obtain
Z
!M
0
n+1
X
i=1
a
i
(u)(@
i
!)
2
dxddt
e
C
2
Z
M
0
!(!M
0
+)
+
e
e
C
1
t
dxddt
Z
e
C
3
!
2
(!M
0
)
+
e
e
C
1
t
dxddt
(since!(x;;t) > M
0
> M
0
=2)
(2
e
C
2
e
C
3
)
Z
!
2
(!M
0
)
+
e
e
C
1
t
dxddt < 0
This contradiction proves the assertion.
For reader convenience we compute explicitly the derivative of the function
introduced in (20),showing that the relation (21) is satised:
Remark 2.1
Let
:[;2]!R
be the function dened in (20).Then
0
=
Z
2
exp(s
2
)ds
1
exp(x
2
);
00
(u)
0
(u)
= 2x:
We can choice
e
C
3
= (n +1)C
0
00
(u)
0
(u)
0
= 2C
0
(n +1):
Since is dened in (8) as
2
=
C
0
64
1 +
4C
2
1
C
0
1
;
9
then
(n +1)
1 +
4C
2
1
C
0
00
(u)
0
(u)
2
(n +1)
1 +
4C
2
1
C
0
16
2
(n +1)C
0
4
=
e
C
2
2
By assumption (8)
2
=
2
exp(6
2
)
C
2
0
128C
2
1
and by a direct computation
(
0
)
2
exp(6
2
)
2
then
(n +1)
32
C
0
C
2
1
2
(
0
)
2
(n +1)
32
C
0
C
2
1
2
exp(6
2
)
2
(n +1)C
0
4
=
e
C
2
2
:
Relation (21) is proved.
It is standard to prove the existence of a solution of problem (13) on the
cylinder Q = B
R
[0;T],using the estimate of the gradient just established.
We refer for example to [LSU] Theorem 1.1 cap VI x1 and cap V x6.Letting
R!1,we immediately deduce
Theorem 2.2
Let u
0
be a Lipschitz continuous function dened in R
n
R,
satisfying condition 0 u
0
(x;) 1:Then for every T > 0 there exists a
solution u of class C
2+;1+=2
in the variables (x;) and t of the problem
u
t
= L
"
u in R
n+1
[0;T[
u(x;;0) = u
0
(x;) in R
n+1
;
(22)
which satises
jjD
x
ujj
1
+"jj@
ujj
1
e
e
C
1
T
jjD
x
u
0
jj
1
+"jj@
ujj
1
;
and
ju(x;;t) u(x;;t
0
)j
e
C
1
jt t
0
j
1=2
;
for every (x;;t),(x;;t
0
),with a constant
e
C
1
independent of".
3 Stability inequality
In this section we prove that the solution found in Theorem 2.2 is unique,and
conclude the proof of Theorem 1.2.Even though the solutions are regular,we
are forced to use a technique introduced for studying the viscosity solutions in
[ALM].However the choice of the main parameters is dierent here,because we
do not have and estimate of the complete gradient,and we do not yet assume
that the solution is periodic.
10
Theorem 3.1
Let u
0
and v
0
be bounded and Lipschitz continuous on R
n
R.
Let u and v be the correspondent viscosity solutions of problem (22).There
exists a constant K such that
jju vjj
L
1
(R
n+1
[0;T])
Kjju
0
v
0
jj
L
1
(R
n+1
)
:
Proof Let and constants to be xed later,and dependent only on jjujj
1
,
jjvjj
1
,jjD
x
ujj
1
and let
(x;y;;t) = u(x;;t) v(y;;t)
jx yj
4
4
t
jxj
2
+jyj
2
+jj
2
R
;(23)
with R > 0.Since u and v are bounded,then has a maximum,at a point
say (x
0
;y
0
;
0
;t
0
).We can always assume that u(0;0;0) v(0;0;0),so that the
maximum of is nonnegative:
(x
0
;y
0
;
0
;t
0
) (0;0;0;0) = u(0;0;0) v(0;0;0) 0:
Let us rst assume that t
0
> 0.Since all the considered functions are of class
C
2
,at the point (x
0
;y
0
;
0
;t
0
) we have
@
t
u(x
0
;
0
;t
0
) @
t
v(y
0
;
0
;t
0
);(24)
@
i
u(x
0
;
0
;t
0
) =
jx
0
y
0
j
2
(x
0
y
0
)
i
+
2(x
0
)
i
R
;
@
i
v(y
0
;
0
;t
0
) =
jx
0
y
0
j
2
(x
0
y
0
)
i
2(y
0
)
i
R
;
and
0
@
D
2
x
u 0 0
0 D
2
y
v 0
0 0 @
;
(u v)
1
A
D
2
x;y;
jx
0
y
0
j
4
4
+t
0
+
jx
0
j
2
+jy
0
j
2
+j
0
j
2
R
:
(25)
If we denote A the right hand side of (25) we have
A =
jx
0
y
0
j
2
4
0
@
I I 0
I I 0
0 0 0
1
A
+
2
R
0
@
I 0 0
0 I 0
0 0 1
1
A
+
+
2
0
@
(x
0
y
0
)
N
(x
0
y
0
) (x
0
y
0
)
N
(x
0
y
0
) 0
(x
0
y
0
)
N
(x
0
y
0
) (x
0
y
0
)
N
(x
0
y
0
) 0
0 0 0
1
A
and
tr(A) C
jx
0
y
0
j
2
+
1
R
:(26)
Multiplying (25) on the right by the matrix
(u) =
0
@
diag
a
1
(u);:::;a
n
(u)
diag
p
a
1
(u)a
1
(v);:::;
p
a
n
(u)a
n
(v)
0
diag
p
a
1
(u)a
1
(v);:::;
p
a
n
(u)a
n
(v)
diag
a
1
(v);:::;a
n
(v)
0
0 0"
2
1
A
11
and considering the trace we get
n+1
X
i=1
a
i
(u)
e
@
i;i
u
n+1
X
i=1
a
i
(v)
e
@
i;i
v
n
X
i=1
a
i
(u)
1=2
a
i
(v)
1=2
2
tr(A) +
"
2
R
(27)
C
jju vjj
1
+jx
0
y
0
j
2
jx
0
y
0
j
2
+
1
R
+
"
2
R
;
where we have used (26) to estimate the the trace of A and the fact that
ja
1=2
i
(u)(x
0
;
0
;t
0
) a
1=2
i
(v)(y
0
;
0
;t
0
)j
(by (7)
C
1
0
ja
i
(u)(x
0
;
0
;t
0
) a
i
(v)(y
0
;
0
;t
0
)j
jju vjj
1
+jjD
x
ujjjx
0
y
0
j
2
;
where jjD
x
ujj
1
is uniformly bounded.Analogously
b
i
(u)@
i
u(x
0
;
0
;t
0
) b
i
(v)@
i
v(y
0
;
0
;t
0
) = (28)
=
b
i
(u)(x
0
;
0
;t
0
)b
i
(v)(y
0
;
0
;t
0
)
4jx
0
y
0
j
2
(x
0
y
0
)
i
+b
i
(u)
2(x
0
)
i
R
+b
i
(v)
2(y
0
)
i
R
jju vjj
1
+jx
0
y
0
j
jx
0
y
0
j
3
+
jx
0
j +jy
0
j
R
:
By (24) we have:
@
t
u(x
0
;
0
;t
0
) @
t
v(y
0
;
0
;t
0
) =
n+1
X
i=1
a
i
(u)
e
@
i;i
u +b
i
(u)
e
@
i
u
n+1
X
i=1
a
i
(v)
e
@
i;i
v b
i
(v)@
i
v
by (27) and (28)
C
jju vjj
2
1
+jx
0
y
0
j
2
jx
0
y
0
j
2
+
1
R
+
"
2
R
+
jju vjj
1
+jx
0
y
0
j
jx
0
y
0
j
3
+
jx
0
j +jy
0
j
R
:
Since (x
0
;
0
;t
0
) 0;for every R > 0 we have
jx
0
y
0
j
4
4
+
jx
0
j
2
+jy
0
j
2
+j
0
j
2
R
u(x
0
;
0
;t
0
) v(y
0
;
0
;t
0
)
jjujj
L
1
(R
n+1
[0;T[)
+jjvjj
L
1
(R
n+1
[0;T[)
e
C:
Hence
jx
0
j
2
+jy
0
j
2
+j
0
j
2
CR;
jx
0
y
0
j
4
4
e
C:(29)
12
Since (x
0
;y
0
;
0
;t
0
) is a maximum point for ,
u(x
0
;
0
;t
0
) v(y
0
;
0
;t
0
)
jx
0
y
0
j
4
4
t
0
jx
0
j
2
+jy
0
j
2
+j
0
j
2
R
=
(x
0
;y
0
;
0
;t
0
) (y
0
;y
0
;
0
;t
0
) u(y
0
;
0
;t
0
)v(y
0
;
0
;t
0
)t
0
2jy
0
j
2
+j
0
j
2
R
:
Thus
jx
0
y
0
j
4
4
u(x
0
;
0
;t
0
) v(y
0
;
0
;t
0
) +
jy
0
j
2
jx
0
j
2
R
e
Ljx
0
y
0
j +
jx
0
y
0
j(jx
0
j +jy
0
j)
R
;
where
e
L is the Lipschitz constant in x for u.In particular we deduce
jx
0
y
0
j
3
4
e
L+
jx
0
j +jy
0
j
R
(by (29))
e
L+
C
p
R
e
L+1:
If we choose
=
3
jju vjj
3
;
inserting in the estimate of we deduce
Cjju vjj
1
+ +1
+C
jju vjj
2
R
1 +
2
+
C
R
1=2
;
and this is a contradiction,if
= 2Cjju vjj
1
+ +1
+2C
jju vjj
2
R
1 +
2
+
2C
R
1=2
:(30)
Hence t
0
= 0,and for every t,for every x,y
u(x;;t) v(y;;t)
jx yj
4
t
(jxj
2
+jyj
2
+jj
2
)
R
sup
n
u
0
(x;) v
0
(y;)
jx yj
4
(jxj
2
+jyj
2
+jj
2
)
R
o
:
If x = y we get
u
(
x;;t
)
v
(
x;;t
)
T
+
2jxj
2
+jj
2
R
+
jj
u
0
v
0
jj
+
sup
r>0
n
L
0
r
r
4
4
o
;
where L
0
is the Lipschitz norm of v
0
= T +
2jxj
2
+jj
2
R
+jju
0
v
0
jj +
3
4
L
4=3
0
1=3
=
for the choice of and ,
= 2CTjju vjj
1
+ +1
+2CT
jju vjj
2
R
1 +
2
+
2CT
R
1=2
+
13
+
2jxj
2
+jj
2
R
+jju
0
v
0
jj +
3
4
L
4=3
0
jju vjj:
Since x and are xed and the constants C;T;R;L
0
; do not depend on R,
letting R go to +1 we get:
u(x;;t) v(x;;t)
2CTjju vjj
1
+ +1
+jju
0
v
0
jj +
3
4
L
4=3
0
jju vjj:
We now conclude,choosing = L
4=3
0
,and T suciently small.
Therefore,if T
1
is an arbitrary interval of time in [0;+1[,and NT T
1
we
deduce,iterating this argument that
jju vjj
1
C
T
1
jju
0
v
0
jj
1
:
for a constant C depending on jjujj
1
,jjvjj
1
,jjD
x
ujj
1
.
Proof of Theorem 1.2 By assumption u
0
is a bounded and Lipschitz
continuous function on R
n
R.For every"> 0 Theorem2.2 provides a solution
(u
"
) of the regularized problem (22),with initial condition u
0
,satisfying
jjD
x
u
"
jj
1
C;
for a constant C only dependent on u
0
and independent of".On the other side,
by (10),if we x
0
2 R,the function
v
"
(x;;t) = u
"
(x; +
0
;t)
is a solution of the same problem,with initial datum
v
0
(x;) = u
0
(x; +
0
):
Then
ju
"
(x;;t) u
"
(x; +
0
;t)j = ju
"
(x;;t) v
"
(x;;t)j
(by Theorem 3.1)
ju
0
(x;) v
0
(x;)j = ju
0
(x;) u
0
(x; +
0
)j
0
:
The Lipschitz continuity is then proved.Letting"!0 we found a viscosity
lipschitz continuous solution of (11).Keeping xed,the function u
can be
considered a solution of an uniformly parabolic equation,with Lipschitz contin
uous coecients.Hence it belongs to C
2+;1+=2
;for every 2]0;1[,uniformly
with respect to .
Remark 3.1
If the initial datum is periodic,the solution of (12) is periodic.
Indeed if u is a solution,also u
h
= u( + h) is a solution of the same Cauchy
problem,so that it coincides with u,by the asserted uniqueness.
14
4 Application to the model
In this section we show how to apply Theorem 1.2 to equation (2) and we
conclude the proof of Theorem 1.1.
In order to write equation (2) in the nondivergence form (6) we set
a
"
i
(u) = (h(clt(u)) +")f(jDG uj) (31)
b
i
(u) = clt
2
(u)f
0
(jDG uj)
n
X
i;j=1
D
2
i;j
G u
D
j
G u
jDG uj
:(32)
Clearly (2) is obtained for (6) for"= 0.
Let us prove that these function satises the assumptions (7),(8),(9).
Lemma 4.1
Let Q be compact in R
n+1
[0;T],and let u be a bounded and
Lipschitz continuous function on Q.Then the function clt(u) dened in (1) is
bounded and Lipschitz continuous in
Q.Precisely
jjclt(u)jj
1
4jjujj
1
:(33)
For every (x;;t) there exists
1
,
2
2 R
n
such that j
1
j;j
2
j 1 and
j@
h
clt(u)(x;;t)j j@
h
u(x +
1
; +%;t)j +j@
h
u(x
2
; %;t)j+ (34)
+2j@
h
u(x;;t)j +jDG @
h
u(x;;t)j
for every t and a.e.(x;) 2 B
R
.Finally,if u and v are bounded and Lipschiz,
jclt(u)(x
0
;
0
;t
0
) clt(v)(y
0
;
0
;t
0
)j Cjju vjj:(35)
Proof
The estimate (33) follows directly by the denition,simply choosing
1
=
2
.
Let now v;u 2 Bd(Q)\Lip(Q),and assume that clt(u) clt(v) 0.Let
1
,
2
be such that
clt(v)(x;;t) = jv(x +
1
; +;t) v(x;;t)j+
+jv(x
2
; +;t) v(x;;t)j +j < DG v;
1
2
> j:
Then by denition of clt(u),
clt(u) clt(v) ju(x +
1
; +;t) u(x;;t)j+
+ju(x
2
; +;t) u(x;;t)j +j < DG u;
1
2
> j
jv(x +
1
; +;t) v(x;;t)j
jv(x
2
; +;t) v(x;;t)j j < DG v;
1
2
> j
ju(x +
1
; +;t) v(x +
1
; +;t)j+
+ju(x
2
; %;t) v(x
2
; %;t)j+
15
+2ju(x;;t) v(x;;t)j +jDG u DG vj:
And this implies (34).
Now we call e
h
a vector of the canonical basis,
u
;h
= u(x +e
h
;;t);for h = 1; ;n
and
u
;n+1
= u(x; +;t):
It then follows that for every 2 C
1
, 0
Z
@
h
clt(u) dxd = lim
!0
Z
clt(u) clt(u
;h
)
dxd
lim
!0
Z
ju(x +
1
; +;t) u
;h
(x +
1
; +;t)j
dxd+
+
Z
ju(x
2
; %;t) u
;h
(x
2
; %;t)j
dxd+
+2
Z
ju(x;;t) u
;h
(x;;t)j
dxd +
Z
jDG u DG u
;h
j
dxd =
=
Z
j@
h
u(x +
1
; +%;t)j +j@
h
u(x
2
; %;t)j+
+2j@
h
u(x;;t)j +jDG @
h
uj
dxd:
An analogous relation,holds for @
h
clt(u) and the thesis is proved.
Fromthis lemma,and the properties of the convolution,it is easy to recognize
that a
"
i
and b
i
satisfy assumptions (7),(8),(9).Let us now conclude the
Proof of Theorem1.1 By Theorem1.2 for every"there exists (u
"
) solution
of
u
"
t
=
n
X
i=1
a
i
(u
"
)(x;;t)@
i;i
u
"
+
u
"
X
i=1
b
i
(u
"
)(x;;t)D
i
u
"
;
satisfying condition (5),and
ju(x;;t) u(x;;0)j Ct
1=2
for a constant C independent of".If infclt(u
0
) > m> 0,
clt(u)(x;;t) clt(u)(x;;0) Ct
1=2
m
2
;
if ct
1=2
m=2.Then condition (7) is satised on [0;
m
2
4C
2
],with a constant
C
0
independent of".Letting"go to 0 we nd a solution u satisfying all the
conditions listed in the thesis.
16
5 References
[AGLM] L.Alvarez,F.Guichard,P.L.Lions,J.M.Morel,Axioms and
Fundamental Equations of Image Processing,Arch.Rat.Mech.Anal,
123,(1993),200257.
[ALM] L.Alvarez,P.L.Lions,J.M.Morel,Image selective smoothing and edge
detection by nonlinear diusion in R
n
.II,SIAM J.of nonlinear analisys
29,3,(1992),845866.
[AM] L.Alvarez,J.M.Morel,Formalization and computational aspect of
image analysis,Acta Num.,??,(1994),159.
[AE] L.Alvarez,J.Escalatin,Image equalization using reaction diusion
equations,SIAM J.of Appl.Math.,57,1,(1997),15375.
[AMa] L.Alvarez,L.Mazorra,Signal and image restoration by using shock
lters and anisotropic diusion,SIAM J.of Appl.Math.,31,2,(1994),
590605.
[BN] P.Biler,T.Nadzieja,A class of nonlocal parabolic problems occurring in
statistical mechanics,Colloq.Math.66,1,(1993),131145.
[C] J.Chabrowski,On the nonlocal problem with a functional for parabolic
equations,Funkc.Ekv.,24,(1984),101123.
[CGG] Y.G.Chen,Y.Giga,S.Goto,Uniqueness and existence of viscosity so
lutions of generalized mean curvature ow equations,SIAMJ.of nonlinear
analisys 29,1,(1992),182193.
[CLMC] F.Catte,P.L.Lions,J.M.Morel,T.Colle,Image selective smoothing
and edge detection by nonlinear diusion,SIAM J.of nonlinear analisys
29,1,(1992),182193.
[CS] V.Caselles,C.Sbert,What is the best causal space for three dimentional
images,SIAM J.Appl.Math.,vol.56,n.4,(1996) 11991246.
[CHL] M.G.Crandal,H.Hishii,P.L.Lions,Users guide to viscosity solutions
of second order partial dierential equations,Bull.A.M.S.27,1,(1992),
167.
[ES] L.C.Evans,J.Spruck,Motion of level sets by mean curvature,J.Di.
Geom,33,(1991),635681.
[GGIS] Y.Giga,S.Goto,H.Ishii,M.H.Sato,Comparison principle and con
vexity preserving properties for signal degenerate parabolic equations on
unbounded domains,Ind.Univ.Math.J.,40,(1990),443470.
[G1] F.Guichard,Axiomatisation des analysis multiechelles d'images et des
lms,PhD.Thesis University Paris IX Dauphine (1994).
17
[G] F.Guichard,Multiscale analysis of movies,Proc.Eioght Workshop
on Images and multidimensional Signal processing (1993,Cannes),IEEE,
NewYork,236237.
[KM1] J.Kacur,K.Mikula,Solution of nonlinear diusion appearing in image
smoothing and edge detection,Appl.Num.Anal.,17,(1995),4753.
[KM2] J.Kacur,K.Mikula,Slowed anisotropic diusion in scalespace theory
in computer vision,Lecture notes in Computer Science,1252,Springer,
(1997),357360.
[IS]H.Ischii,P.Souganidis,Generalized motion of noncompact hypersurfaces
with velocity having arbitrary growth on the curvature tensor,Tokokn
Math.J.,47,(1995),227250.
[LS]C.Lamberti,F.Sgallari,Edge detection and velocity eld for the analysis
of heat motion,Digital Signal Processing,91,Elsevier,(1991),603608.
[LSU]??Linear and quasilinear equations of parabolic type.Translations of
Mathematical Monographs.AMS,vol 23,1968.
[MSL] K.Mikula,A.Sarti,C.Lamberti,Geometric diusiuon in 3D
echocardiography,Proceedings of Algoritmy 1997,Conference on Scientic
Computing,West Trater MountainsZuberec,(1997).167181.
[OST1] P.Olver,G.Sapiro,A.Tannenbaum,Invariant geometric evolutions
of surfaces and volumetric smoothing,SIAMJ.Appl.Math,57,1,(1997),
176194.
[OST2] P.Olver,G.Sapiro,A.Tannenbaum,Classication and Uniqueness
of invariant geometric ows,C.R.A.S.Sci.Paris,t.319,Serie I,(1994),
339344.
[PM] P.Perona,J.Malik,Scale space and edge detection using anisotropic
diusion,Proc.IEEE Comp.Soc.Workshop on Computer Vision,(1987).
[S] J.Sethian,Level set methods evolviong interfaces geometry, uid mechan
ics,computer vision and material science.,Cambridge University Press
(1996).
[So] H.M.Soner,Motion of a set by curvature of its boundary,J.Di.Eqs,
101,2,(1993),313372.
[SMS] A.Sarti,K.Mikula,F.Sgallari,Nonlinear Multiscale Analysis of
3D Echocardiographic Sequences,IEEE,Trans,Medical Imaging,18,9,
453466.
[ST1] G.Sapiro,A.Tannenbaum,On invariant curve evolution and image
analysis,Ind.Univ.Math.J.,42,3,(1993),9851003.
18
[ST2] G.Sapiro,A.Tannenbaum,Ane invariant scale space,J.Funct.Anal.,
19,(1994),79120.
[ZSL] G.Zini,A.Sarti,C.Lamberti,Application of continuum theory and
multigried methods to mothin evaluation from3D echocardiography.,IEEE
Trans.,Ultr.Ferr.Freq.Control.44,2,(1997),297308.

Perona Malik
[KK] B.Kawohl,N.KutevMaximumand comparison principle for onedimensional
anisotropic diusion.Math.Ann.311,No.1,107123 (1998).
[K] S.Kichenassamy,The PeronaMalik paradox.SIAM J.Appl.Math.57,
No.5,13281342 (1997).
19
Enter the password to open this PDF file:
File name:

File size:

Title:

Author:

Subject:

Keywords:

Creation Date:

Modification Date:

Creator:

PDF Producer:

PDF Version:

Page Count:

Preparing document for printing…
0%
Comments 0
Log in to post a comment