79
Acta Math. Univ. Comenianae
Vol. LXXIV, 1(2005), pp. 79–94
ERROR ESTIMATES FOR FINITE VOLUME SCHEME FOR
PERONA-MALIK EQUATION
A. HANDLOVIČOVÁ and Z. KRIVÁ
Abstract. We present Perona-Malik nonlinear image selective smoothing equation
(modified in the sense of Catté, Lions, Morel and Coll) which is investigated esspecially from numerical point of wiev. Error estimates in L2 norms for fully discrete
numerical finite volume scheme are derived and proved. Some numerical examples
are presented.
1. Introduction
1.1. Mathematical model of the problem
We are dealing with Perona-Malik type problem discussed in [4] in the following
form
(1)
(2)
(3)
∂t u − ∇.(g(|∇Gσ ∗ u|)∇u) = 0 in QT ≡ I × Ω,
∂ν u = 0 on I × ∂Ω,
u(0, ·) = u0 in Ω,
where Ω ⊂ IRd is a rectangular domain, I = [0, T ] is a scaling interval, and
(4)
g(s) is a Lipschitz continuous decreasing function,
(5)
with Lipschitz constant Lg
g(0) = 1, 0 < g(s) → 0 for s → ∞,
(6)
Gσ ∈ C ∞ (IRd ) is a smoothing kernel with compact support Kσ
Z
with
Gσ (x)dx = 1
IRd
(7)
and Gσ (x) → δx for σ → 0, δx – Dirac function at point x,
initial condition u0 is such that regularity below is fulfiled.
Received June 17, 2003.
2000 Mathematics Subject Classification. Primary 35K55, 65M12.
Key words and phrases. Perona Malik equation, image processing, finite volume method,
error estimates.
The work was supported by NATO Collaborative Linkage Grant PST.CLG.979123 The
Efficient and Robust Comuptational methods for Biomedical Image Analysis, 2002-2004 and it
was partly done during the stay at Stefan Banach Center of Excellence Warszawa in Poland.
80
A. HANDLOVIČOVÁ and Z. KRIVÁ
We can rewrite the partial differential equation (1) in the form
(8)
∂t u − ∇.(g(|J(u)(x)|)∇u) = 0 in QT ≡ I × Ω,
where J(u) : L2 (Ω) → (C ∞ (Ω))d . In our case we use J(u)(t, ·) = ∇Gσ ∗ u(t, ·) for
t fixed, but we can choose any smoothing operator with these properties.
Let us define a weak solution to the problem (8),(2),(3). Equation (8) is multiplied
by a test function ϕ ∈ Ψ, where Ψ is the space of smooth test functions
Ψ = {ϕ ∈ C 1,2 ([0, T ] × Ω), ∇ϕ.~n = 0 on (0, T ) × ∂Ω , ϕ(T, .) = 0}.
After integrating over [0, T ] and Ω and after applying integration by parts and
properties of a test function, we come to a definition of the weak solution.
Definition 1.1. The weak solution of the regularized Perona-Malik problem
(1)–(3) is a function u ∈ L2 (I, W 1,2 (Ω)) satisfying the identity
(9)
ZT Z
u
∂ϕ
(t, x) dx dt +
∂t
0 Ω
Z
u0 (x)ϕ(0, x) dx
Ω
−
ZT Z
g(|J(u(t, x))|)∇u(t, x)∇ϕ(t, x) dx dt = 0
0 Ω
for all ϕ ∈ Ψ.
It is well known from the regularity theory of such a solution [10] that, because
of the properties of the operator J(u), the weak solution of our problem is a
function u ∈ L2 (I, W 2,2 (Ω)) for initial condition u0 ∈ L∞ (Ω). Moreover from the
embedding theory for dimension d = 2, or d = 3 follows that u ∈ C(Q̄T ).
For our error estimates we need futher regularity of the solution, more precise
u ∈ L2 (I, W 2,2 (Ω)) ∩ L∞ (I, W 1,2 (Ω)) and utt ∈ L1 (I, L1 (Ω)).
1.2. Formulation of the finite volume scheme
Let τh be a uniform mesh of Ω with cells p of measure m(p) (we assume rectangular
cells here). For every cell p we consider a set of neighbours N (p) consisting of all
cells q ∈ τh for which the common interface of p and q, denoted by epq , is of nonzero measure m(epq ). We denote the set of all these edges for all volumes p ∈ τh
by E and by epq I we denote the edge which connects the volumes p and q. (Clearly
epq = eqp = epq I). It is assumed that for every p, there exists a representative
x −x
point xp ∈ p, such that for every pair p, q, q ∈ N (p), the vector |xqq −xpp | is equal
to a unit vector npq which is normal to epq and oriented from p to q. We denote
dpq := |xp − xq |. In a simple case of a uniform grid xp is just the center of the
pixel. Then, let xpq be the point of epq intersecting the segment xp xq . We define
(10)
Tpq :=
m(epq )
.
dpq
Discrete approximation of a solution of partial differential equation is considered
to be piecewise constant on control volumes [5].
ERROR ESTIMATES FOR PERONA-MALIK EQUATION
81
Let unp be the value of the numerical solution in the n-th scale step on a volume
p. The finite volume semi-implicit scheme on a uniform grid is then written as
follows:
Let 0 = t0 ≤ t1 ≤ . . . ≤ tn . . . ≤ tNmax , Nmax ·k = T denote the scale discretization
steps with tl = tl−1 + k, where k is the discrete scale step, l = 1, 2, . . . , Nmax .
For n = 0, . . . , Nmax we look for un+1
, p ∈ τh , satisfying the identities
p
P σ,n
,
gpq Tpq un+1
− un+1
un+1
− unp m (p) = k
q
p
p
(11)
q∈N (p)
u0p
1
=
m(p)
Z
u0 (x)dx,
p
σ,n
gpq
:= g (|J(ũ (tn , xpq )) |) ,
(12)
where ũ is a periodic extension of the discrete image computed in the n-th scale
step. Its L2 norm can be estimated with constant B by L2 norm of function u.
unp is a value of the numerical solution on the volume p in the n-th scale step.
2. Stability and convergence results
We briefly mention results of Mikula and Ramarosy, see [12], obtained for the semiimplicit finite volume scheme concerning the stability and convergence properties.
Explicit time discretization are discussed also in [7] and [8]. Stability estimates
are of the following type [12]:
Lemma 2.1 (A priori estimates in L2 (QT )). It holds, that there exist positive
constants C1 , C2 such that
X
2
ulp m(p) ≤ C1 ,
(i) max
0≤l≤Nmax
(ii)
N
max
X
l=0
k
p∈τh
X
(p,q)∈E
ulp − ulq
dpq
2
m (epq ) ≤ C2
and the constants C1 , C2 do not depend on the h, k.
Let us denote by uh,k the finite volume numerical solution for some fixed space
and scale mesh h and k. This solution is piecewise constant on each finite volume
and in each scale step as it is usual for finite volume numerical schemes of a
parabolic type. By ūl we denote the function piecewise constant on each finite
volume in the l-th scale step. Then we have:
Lemma 2.2 (Convergence of uh,k ). There exists u ∈ L2 (QT ) which is the weak
solution of (9) such that
uh,k → u in L2 (QT )
as h, k → 0. Furthermore, the convergence is pointwise.
82
A. HANDLOVIČOVÁ and Z. KRIVÁ
3. Error estimates
3.1. L∞ stability for a discrete solution
We rewrite the original discrete equation (11) in the following way:
un+1
+
p
(13)
X
k
σ,n
gpq
Tpq un+1
− un+1
= unp .
p
q
m (p)
q∈N (p)
un+1
p
Now let
be the maximal value for the fixed scale step n + 1 and p ∈ τh . Then
the second term on the left hand side of (13) is nonnegative and:
un+1
≤ unp .
p
Recursively we have
||un+1 ||L∞ ≤ |un+1
| ≤ |u0p | ≤ ||u0 ||L∞ ≤ C.
p
(14)
3.2. Error estimates
Let now tn < t ≤ tn+1 . Multiplying PDE (8) by vpn+1 and then integrating over
volume p and using integration by parts, we have:
Z
Z
(15)
∂t u(t, x) vpn+1 dx − g(|J(u)|)∇u · np vpn+1 dx = 0,
p
∂p
where ∂p is the boundary of the volume p and np is the outward unit normal vector
to the boundary of volume p and further analogously npq will be the outward unit
normal vector to the edge epq . We can write
[
∂p =
epq .
q∈N (p)
For the discrete scheme we have
X
n+1
un+1
− unp vpn+1 m (p)
p
σ,n
(16)
−
gpq
Tpq un+1
− un+1
vp = 0.
q
p
k
q∈N (p)
Now we denote
enp = u(tn , xp ) − unp ,
where xp is a representative point of a volume p, p ∈ τh .
Then posing vpn = enp and subtracting (16) from (15) we obtain:
!
Z
X Z
un+1 − un+1
en+1
− enp n+1
p
p
σ,n q
gpq
ep
+
− g(|J(u)|)∇u · npq en+1
p
k
dpq
q∈N (p)epq
p
=
Z p
u(tn+1 , xp ) − u(tn , xp )
− ∂t u en+1
.
p
k
83
ERROR ESTIMATES FOR PERONA-MALIK EQUATION
Now after summation over all volumes p ∈ τh and integration over In =
htn , tn+1 ) for all n = 0, 1, . . . , m − 1 and rearranging the equation we obtain:
(17)
Z
m−1
XZ
|en+1 − en |2
|em |2 +
n=0 Ω
m−1
XZ
Ω
X X Z
+2
n=0 I
=
Z
p∈τh q∈N (p)e
pq
n
0 2
|e | + 2
Ω
m−1
XZ
n=0 I
un+1 − un+1
p
σ,n q
gpq
− g(|J(u)|)∇u · npq
dpq
!
en+1
p
X Z u(tn+1 , xp ) − u(tn , xp )
− ∂t u en+1
.
p
k
p∈τ
h
n
p
The third term on the left hand side of the last equation can be expressed as it
is usual in the finite volume theory, see [5]:
”Third” = 2
m−1
XZ
n=0 I
=2
n
σ,n
gpq
p∈τh q∈N (p)e
pq
n
m−1
XZ
n=0 I
X X Z
XZ
Ee
σ,n
gpq
un+1
− un+1
q
p
− g(|J(u)|)∇u · npq
dpq
un+1
− un+1
q
p
− g(|J(u)|)∇u · npq
dpq
pq I
!
!
en+1
p
(en+1
− en+1
).
p
q
After rearranging we get:
”Third“ = 2
m−1
XZ
n=0 I
+2
m−1
XZ
X Z
m−1
XZ
X Z
n=0 I
+2
n=0 I
n
n
E e I
pq
n
X Z
E e I
pq
en+1
− en+1
p
q
g(|J(u)|)
dpq
σ,n
gpq
− g(|J(u)|)
g(|J(u)|)
E e I
pq
2
un+1
− un+1
q
p
(en+1
− en+1
)
p
q
dpq
u(tn+1 , xq ) − u(tn+1 , xp )
− en+1
).
− ∇u · npq (en+1
p
q
dpq
Involving these terms to the (17) equation we obtain:
Z
Ω
m 2
|e | +
m−1
XZ
n=0 Ω
|e
n+1
n 2
−e | +2
m−1
XZ
n=0 I
n
X Z
E e I
pq
en+1
− en+1
p
q
g(|J(u)|)
dpq
2
84
A. HANDLOVIČOVÁ and Z. KRIVÁ
=
Z
|e0 |2 + 2
Ω
+2
m−1
XZ
n=0 I
+2
n
m−1
XZ
n=0I
n
m−1
XZ
n=0 I
h
n
X Z
p
σ,n
gpq
− g(|J(u)|)
E e I
pq
XZ
X Z u(tn+1 , xp ) − u(tn , xp )
− ∂t u en+1
p
k
p∈τ
g(|J(u)|
Ee I
pq
un+1
− un+1
q
p
(en+1
− en+1
)
p
q
dpq
u(tn+1 , xq ) − u(tn+1 , xp )
− en+1
),
− ∇u · npq (en+1
p
q
dpq
or briefly
2
Z
m−1
m−1
XZ
XZ X Z
en+1
− en+1
p
q
m 2
n+1
n 2
|e | +
|e
−e | +2
g(|J(u)|)
dpq
n=0
n=0
Ω
=
Z
Ω
In
E e I
pq
|e0 |2 + I + II + III.
Ω
Remark. To obtain an appropriate error estimate we must take into account
the regularity of the solution which plays an important role in error analysis. Error
estimates could be done also better, but further regularity for time derivative is
needed. If we suppose u0 ∈ L∞ (Ω) only, no further regularities are available.
Now we must estimate each of the last three terms on the right hand side.
m−1
X Z X Z u(tn+1 , xp ) − u(tn , xp )
− ∂t u en+1
I =2
p
k
p∈τ
n=0
h
In
=2
m−1
X
XZ
n=0 p∈τh p
=2
XZ
p∈τh p
+ 2
p
(u(tn+1 , xp ) − u(tn+1 , x) + u(tn , x) − u(tn , xp )) en+1
p
m−1
X
n=0
XZ
p∈τh p
n+1
(u(tn , xp ) − u(tn , x)) enp − ep
!
0
(u(tm , xp ) − u(tm , x)) em
p − (u(0, xp ) − u(0, x)) ep
XZ
√ m−1
√ Z
≤ 2
h|∇u(tn , ·)||en+1 − en | + 2 h|∇u(tm , ·)||em |
n=0
Ω
√ Z
+ 2 h|∇u(0, ·)||e0 |.
Ω
Ω
ERROR ESTIMATES FOR PERONA-MALIK EQUATION
85
After using Young’s inequality we get
Z
m−1 Z
m−1
XZ
1 X
2
n+1
n 2
2
2
|e
I≤h
−e | + h
|∇u(tn , ·)| +
|∇u(tm , ·)|2
2
n=0 Ω
n=0 Ω
Ω
Z
Z
Z
1
1
m 2
2
0 2
2
|e | + h
|e | .
|∇u(0, ·)| +
+
2
2
Ω
Ω
Ω
Finally
m−1 Z
1X m2
1 X
|en+1 − en |2 +
|e |
I≤
2 n=0
2
Ω
Ω
2
Z
1
h T
2
0 2
+ 2h ||∇u||L∞ (I,L2 (Ω))
+
|e | +
2
k
Ω
We estimate the second term in the following way:
m−1
XZ X Z
un+1
− un+1
q
p
σ,n
II = 2
gpq
− g(|J(u(x))|)
(en+1
− en+1
)dx.
p
q
d
pq
n=0
E e I
pq
In
First we estimate
σ,n
|gpq
− g(|J(u)|)| = |g(|∇Gσ ∗ ũ(tn , xpq )|) − g(|∇Gσ ∗ ũ(t, x)|)|
Z
Z
≤ Lg |
∇Gσ (xpq − η)ũh,k (tn , η)dη − ∇Gσ (s − η)ũ(t, η)dη|
d
IRd
Z IR
≤ Lg
|∇Gσ (xpq − η) − ∇Gσ (s − η)||ũh,k (tn , η)|dη
IRd
Z
+ Lg
|∇Gσ (s − η)||ũh,k (tn , η) − ũ(t, η)|dη.
IRd
We obtain
σ,n
|gpq
− g(|J(u)|)|
Lg B
≤ √ · h||D2 Gσ ||L∞ (Ω) ||uh,k ||L∞ (QT ) m(Kσ ) + Lg B||∇Gσ ||L∞ (Ω)
2


 21
x
Z Ztn
Z
Z
Z
X
∂u(t, y)


|dydx ,
|∂t u(s, x)|dsdx +
|
·  |en |2 dx +
∂n
p∈τ
Ω
Ω
t
h
p xp
where m(Kσ ) is measure of the compact support Kσ , σ is fixed, B is the estimation
for mirror reflexion function. We denote
C3 = 2Lg B||D2 Gσ ||L∞ (Ω) ||uh,k ||L∞ (QT ) m(Kσ ),
C4 = 2Lg B||∇Gσ ||L∞ (Ω) ,
Cg is such that g(|J(u)|) ≥ Cg .
86
A. HANDLOVIČOVÁ and Z. KRIVÁ
The last estimate can be established due to the properties of the solution u. Hence
the whole term II can be estimated as follows:
 12 
 12
Z
Z
m−1
X
|en+1
− en+1
|2 
|un+1
− un+1
|2   X X

p
q
q
p
II ≤ C3 h · 
k
k
 

dpq
dpq
n=0
n=0

m−1
X
E e I
pq
E e I
pq
 12
 12

 21
Z
Z
Z
n+1
n+1 2
n+1
n+1 2
X
X
|ep − eq | 
|uq − up |  

+ C4 ·
k

  |en |2 
d
d
pq
pq
n=0
m−1
X
+ C4 ·
m−1
XZ
n=0 I
Ee I
pq
Ee I
pq

|un+1
q
Z
X

E e I
pq
n
Ω
 12 
un+1
|2 
p
−
dpq
|en+1
p
Z
X
 
E e I
pq
 12
en+1
|2 
q
−
dpq




Z Ztn
Z Zx
X
∂u(t, y)



|dydx
·
|∂t u(s, x)|dsdx + 
|
∂n
p∈τ
Ω
h
t
Z
m−1 Z
1 X X
4C2 C32 h2
+
II ≤
Cg2
2
4C 2
+ 24 ·
Cg
+
4C42
·
Cg2
m−1
X
k
n=0 I
E e I
pq
n=0
m−1
XZ
n=0 I
n
X Z
E e I
pq
=
E e I
pq
en+1
− en+1
p
q
g(|J(u)|)
dpq
|un+1
− un+1
|2
q
p
dpq
Z
2
|en |2
Ω
|un+1
− un+1
|2
q
p
dpq


x
Z Ztn
Z
Z
X
∂u(t, y)


·
|dydx
|∇ · (g(|J(u)|∇u) |dsdx +
|
∂n
p∈τ
Ω
4C2 C32 h2
Cg2
n
X Z
p xp
t
h
p xp
+ II1 + II2 + II3 ,
where the inequalities (14), (ii) and the equation (1) has been used. The last term
can be estimated using the properties of the exact solution:
II3 ≤
4C42 C2
8C42 Lg C2
||D2 Gσ ||L∞ (Ω) ||∇u||L∞ (I,L2 (Ω)) +
||∆u||L2 (I,L2 (Ω))
2
Cg
Cg2
2
8C4 Lg C2
||DGσ ||L∞ (Ω) ||∇u||L∞ (I,L2 (Ω)) · h.
+
Cg2
·k
87
ERROR ESTIMATES FOR PERONA-MALIK EQUATION
Finally the third term can be estimated:
III = 2
m−1
XZ
n=0 I
n
X Z
g(|J(u)|)
E e I
pq
u(tn+1 , xq ) − u(tn+1 , xp )
− ∇u · npq
dpq
· (en+1
− en+1
)
p
q
We denote
Rpq =

1

−
m(epq )
Z
∇u · npq +
13454568epq I

u(tn+1 , xq ) − u(tn+1 , xp )

m(epq ) .
dpq
Applying the properties of function g, this term can be estimated as
|III| ≤ 2
m−1
XZ
n=0 I
n
X Z
E e I
pq
|Rpq ||en+1
− en+1
|.
p
q
Now using the regularity of a weak solution and the estimates well known in the
finite volume method see, [5, Chapter 3.1.6], we get
m−1 Z Z
C 2X
(H(u)(z))2
|III| ≤ h
Cg
n=0
In Ω
+
Here |H(u)(z)|2 =
d
P
1
4
i,j=1
m−1
XZ
n=0 I
n
X Z
g(|J(u)|)
E e I
pq
en+1
− en+1
p
q
dpq
2
.
|Di Dj u(z)|2 and Di denote the weak derivatives with
respect to the component zi . Since u ∈ L2 (I, W 2,2 (Ω)) we can denote
C5 =
C
||H(u)||2L2 (QT )
Cg
and we have
2
Z
m−1 Z
en+1
− en+1
1 X X
p
q
g(|J(u)|)
III ≤ C5 h +
.
4 n=0
dpq
2
In
E e I
pq
88
A. HANDLOVIČOVÁ and Z. KRIVÁ
Putting all these estimates together, we obtain:
2
Z
m−1
m−1
XZ
XZ X Z
en+1
− en+1
p
q
m 2
n+1
n 2
|e | +
|e
−e | +
g(|J(u)|)
dpq
n=0
n=0
Ω
Ω
≤4
Z
In
E e I
pq
h2 T
|e | + 2(
+ 2h2 )||∇u||L∞ (I,L2 (Ω)) +
k
0 2
Ω
4C2 C3
+ 2C5 h2
Cg
8C42 Lg
4C42 C2
2
||D
G
||
||∇u||
+
||∆u||
σ L∞ (Ω)
L∞ (I,L2 (Ω))
L2 (I,l2 (Ω)) k
Cg2
Cg2
2
8C4 Lg C2
||DGσ ||L∞ (Ω) ||∇u||L∞ (I,L2 (Ω)) · h
+
Cg2


Z
Z
m−1
n+1
n+1
2
|uq − up |
4C4  X X

+
|en |2  .
·
k
Cg
d
pq
n=0
+
E e I
pq
Ω
If we realize, that the first term on the right hand side with e0 is less than Ch
because of the properties of the exact solution and the definition of u0p , we are
prepared to use Gronwall’s lemma in the form:
Lemma 3.1. If u(t) and v(t) are non-negative measurable functions for t ≤ 0
Rt
and u0 is a non-negative constant, then the inequality u(t) ≤ u0 + v(s) u(s)ds
0
t
R
implies that u(t) ≤ u0 exp
v(s)ds .
0
To estimate the last term of the previous inequality let us denote for t ∈ In =
htn−1 , tn )
Z
X Z |un+1
− un+1
|2
q
p
v(t) =
,
u(t) = |en |2 dx.
dpq
E e I
pq
Ω
If we use the properties of function v then we can obtain


Z
Z
m−1
n+1
n+1 2
2
X
X
|u
−
u
|
h


q
p
+ k) · exp(
|em |2 ≤ C(h2 + h +
k

k
d
pq
n=0
E e I
pq
Ω
h2
),
k
where C is a generic constant depending only on domain Ω, time T and some
norms of the exact solution. To obtain convenient error estimate result we can
choose
≤ C · exp(C2 )(h2 + k + h +
(18)
k = Ch,
ERROR ESTIMATES FOR PERONA-MALIK EQUATION
89
Theorem 3.1. Let the relation between scale and space discretization be chosen
as in (18). Then for the error estimates for Perona-Malik weak solution and
numerical solution obtained via finite volume method it holds
NX
max Z Z
|u(tn+1 , x) − uh,k (tn+1 , x)|2 ≤ Ch
n=0 I Ω
n
and
m−1
XZ
n=0 I
n
X
epq I

n+1
 uq
−
dpq
m(epq )dpq 
un+1
p
−
1
m(epq )
Z
epq
Proof. It is easy to see that
NX
max Z Z
|u(tn+1 , x) − uh,k (tn+1 , x)|2
2

∇u · npq  ≤ Ch.
n=0 I Ω
n
≤ 2h2 k∇ukL2 (I,L2 (Ω) + 2
NX
max Z
Z
n=0 I Ω
n
|en+1 |2 ≤ Ch
and the first inequality is proved. Now
2

Z
Z
m−1
n+1
n+1
X
X
1

 uq − u p
−
∇u · npq 
m(epq )dpq 
dpq
m(epq )
n=0
In epq I
epq
≤C
m−1
XZ
n=0 I
+C
n
XZ
epq Ie
m−1
XZ
n=0 I
≤ Ch,
n
en+1
− en+1
p
q
g(J(u))
dpq
pq
dpq
2
2

Z
n+1
X  un+1
− up
1

q
−
∇u · npq 

dpq
m(epq )
epq I
epq
where we have again used the estimate of the finite volume method for the second
term.
4. Numerical experiments
In this section we present experiments with some artificial images perturbed by
various types of noise. We want to confirm the relation between scale step, mesh
size and the data coefficients obtained in the previous theorem. In simulations, we
use the function
1
g(s) =
1 + Ks2
90
A. HANDLOVIČOVÁ and Z. KRIVÁ
Figure 1. The first column shows the work for parameter K = 10 for different scale steps
k = 0.1, 0.5, 1, 2.5., the second column shows the work for parameter K = 100 for scale steps
k = 0.5, 1, 4, 10. for Example 1.
ERROR ESTIMATES FOR PERONA-MALIK EQUATION
91
Figure 2. Pictures shows the work for parameter K = 10 for different scale steps k = 0.2, 1, 5, 10
(from the top to bottom, from left to right) for Example 2.
92
A. HANDLOVIČOVÁ and Z. KRIVÁ
Figure 3. Pictures shows the work for parameter K = 100 (top) for different scale steps k =
1, 2, 5, 10, and for parameter K = 1000 for scale parameter k = 10, 20 for Example 2.
ERROR ESTIMATES FOR PERONA-MALIK EQUATION
93
and the convolution is realized with the kernel
Gσ (x) =
2
1 |x||x|
e 2 −σ2 ,
Z
where the constant Z is chosen so that Gσ has unit mass.
Example 1. To every position of the initial image we apply a 10% salt and
pepper noise.
Example 2. We have chosen another type of picture with a noise function f
defined as follows: if ψ(x) is a functiongenerating random values in [0, 2C], then
for every position x
f (u0 (x)) = MIN(255, MAX(0, u0 (x) − C + ψ).
C = 100 and the difference in intensity between the two values of the initial image
is 200.
In both examples the size of one finite volume corresponds to the size of one
pixel. We computed the same example for different scale steps. In both figures we
choose the best visual result for every parameter K in function g which plays an
important role in smoothing effect. For the best cases it seems that the relation
between grid mesh, scale step and parameter K remains is constant.
References
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of Image Processing, Arch. Rat. Mech. Anal. 123 (1993), 200–257
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Acta Numerica (1994), 1–59
3. Alvarez L., Lions P. L and Morel J. M., Image selective smoothing and edge detection by
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equation, Proceedings of ALGORITMY 2002, Conf. on Scientific Computing, 276–28.
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94
A. HANDLOVIČOVÁ and Z. KRIVÁ
A. Handlovičová, Department of Mathematics, Slovak University of Technology, Radlinského 11,
81368 Bratislava, Slovakia, e-mail : angela@vox.svf.stuba.sk
Z. Krivá, Department of Mathematics, Slovak University of Technology, Radlinského 11, 81368
Bratislava, Slovakia, e-mail : kriva@vox.svf.stuba.sk