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.
1.1.
Introduction
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,
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.
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where Ω ⊂ IRd is a rectangular domain, I = [0, T ] is a scaling interval, and
(4)
(5)
(6)
g(s) is a Lipschitz continuous decreasing function,
with Lipschitz constant Lg
g(0) = 1, 0 < g(s) → 0 for s → ∞,
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.
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.
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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
ZT Z
(9)
0 Ω
∂ϕ
u (t, x) dx dt +
∂t
ZT Z
Z
u0 (x)ϕ(0, x) dx −
g(|J(u(t, x))|)∇u(t, x)∇ϕ(t, x) dx dt = 0
0 Ω
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 non-zero 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 point xp ∈ p, such that for every pair p, q, q ∈ N (p), the vector
xq −xp
|xq −xp | 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
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Discrete approximation of a solution of partial differential equation is considered to be piecewise constant on
control volumes [5].
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
un+1
− unp m (p) = k
gpq Tpq un+1
− upn+1 ,
p
q
(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 semi-implicit 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
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(i)
(ii)
max
X
0≤l≤Nmax
N
max
X
l=0
k
ulp
2
m(p) ≤ C1 ,
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.
3.
3.1.
Error estimates
L∞ stability for a discrete solution
We rewrite the original discrete equation (11) in the following way:
(13)
un+1
+
p
X
k
σ,n
gpq
Tpq un+1
− un+1
= unp .
p
q
m (p)
q∈N (p)
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Now let un+1
be the maximal value for the fixed scale step n + 1 and p ∈ τh . Then the second term on the left
p
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
n+1
(15)
∂t u(t, x) vp 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
(16)
X
n+1
un+1
− unp vpn+1 m (p)
p
σ,n
−
gpq
Tpq un+1
− un+1
vp = 0.
q
p
k
q∈N (p)
Now we denote
enp = u(tn , xp ) − unp ,
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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
en+1
− enp n+1
p
ep
+
k
q∈N (p)epq
p
Z =
σ,n
gpq
un+1
− un+1
q
p
− g(|J(u)|)∇u · npq
dpq
!
en+1
p
u(tn+1 , xp ) − u(tn , xp )
− ∂t u en+1
.
p
k
p
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:
Z
m 2
|e | +
(17)
m−1
XZ
n+1
|e
n 2
−e | +2
n=0 Ω
Ω
Z
=
Ω
|e0 |2 + 2
m−1
XZ
n=0 I
n
m−1
XZ
X X Z
n=0 I
p∈τh q∈N (p)e
pq
n
σ,n
gpq
un+1
− un+1
q
p
− 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
p
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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
=2
m−1
XZ
X X Z
n=0 I
p∈τh q∈N (p)e
pq
n
m−1
X Z XZ
n=0 I
Ee
n
σ,n
gpq
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
!
en+1
p
!
(en+1
− en+1
).
p
q
pq I
After rearranging we get:
”Third“ = 2
m−1
XZ
n=0 I
+2
m−1
XZ
n=0 I
+2
n
m−1
XZ
n=0 I
n
X Z
E e I
pq
X Z
E e I
pq
n
X Z
E e I
pq
en+1
− en+1
p
q
g(|J(u)|)
dpq
2
un+1
− un+1
q
p
σ,n
gpq
− g(|J(u)|)
(en+1
− en+1
)
p
q
dpq
g(|J(u)|)
u(tn+1 , xq ) − u(tn+1 , xp )
− ∇u · npq (en+1
− en+1
).
p
q
dpq
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Involving these terms to the (17) equation 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 | +2
g(|J(u)|)
dpq
n=0
n=0
Ω
Ω
Z
0 2
|e | + 2
=
+2
m−1
XZ
n=0 I
+2
m−1
XZ
n=0 I
Ω
X Z
E e I
pq
n
m−1
XZ
n=0I
n
XZ
E e I
pq
In
n
X Z u(tn+1 , xp ) − u(tn , xp )
− ∂t u en+1
p
k
p∈τ
h
p
un+1
− un+1
q
p
σ,n
gpq
− g(|J(u)|)
(en+1
− en+1
)
p
q
dpq
g(|J(u)|
E e I
pq
u(tn+1 , xq ) − u(tn+1 , xp )
− ∇u · npq (en+1
− en+1
),
p
q
dpq
or briefly
Z
m 2
|e | +
Ω
Z
=
m−1
XZ
n+1
|e
n=0 Ω
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
|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.
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Now we must estimate each of the last three terms on the right hand side.
I =2
m−1
XZ
n=0 I
=2
m−1
X
n
X Z u(tn+1 , xp ) − u(tn , xp )
− ∂t u en+1
p
k
p∈τ
h
XZ
p
(u(tn+1 , xp ) − u(tn+1 , x) + u(tn , x) − u(tn , xp )) en+1
p
n=0 p∈τh p
=2
XZ
m−1
X
p∈τh p
n=0
+ 2
XZ
!
(u(tn , xp ) − u(tn , x))
enp
−
en+1
p
0
(u(tm , xp ) − u(tm , x)) em
p − (u(0, xp ) − u(0, x)) ep
p∈τh p
XZ
√ Z
√ Z
√ m−1
≤ 2
h|∇u(tn , ·)||en+1 − en | + 2 h|∇u(tm , ·)||em | + 2 h|∇u(0, ·)||e0 |.
n=0 Ω
Ω
Ω
After using Young’s inequality we get
m−1
XZ
Z
m−1 Z
1 X
|en+1 − en |2 + h2 |∇u(tm , ·)|2
2 n=0
n=0 Ω
Ω
Ω
Z
Z
Z
1
1
m 2
2
2
0 2
+
|e | + h
|∇u(0, ·)| +
|e | .
2
2
I ≤ h2
Ω
|∇u(tn , ·)|2 +
Ω
Ω
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Finally
2
Z
m−1 Z
1 X
1X m2 1
h T
n+1
n 2
0 2
2
I≤
|e
−e | +
|e | +
|e | +
+ 2h ||∇u||L∞ (I,L2 (Ω))
2 n=0
2
2
k
Ω
Ω
Ω
We estimate the second term in the following way:
m−1
XZ X Z
un+1
− un+1
q
p
σ,n
(en+1
− en+1
)dx.
II = 2
gpq
− g(|J(u(x))|)
p
q
d
pq
n=0
In
E e I
pq
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


 12
Z
Z Ztn
Z Zx
X
∂u(t, y)


·  |en |2 dx +
|∂t u(s, x)|dsdx +
|
|dydx ,
∂n
p∈τ
Ω
Ω
t
h
p xp
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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 .
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 

m−1
X

II ≤ C3 h · 
k
|un+1
q
X Z
E e I
pq
n=0
un+1
|2 
p
−
dpq

 
m−1
X
 12
k
X Z
n=0
|en+1
p
E e I
pq
en+1
|2 
q
−
dpq

 12
 21

 12
Z
Z
Z
n+1
n+1
2
n+1
n+1
2
X
X
|u
−
u
|
|e
−
e
|



p
q
q
p
|en |2 
+ C4 ·
k


d
d
pq
pq
n=0
m−1
X
E e I
pq
E e I
pq

m−1
X Z X Z
+ C4 ·

n=0 I
E e I
pq
n

Z Ztn

·
Ω
t
Ω
1
2
 
Z
n+1 2
|un+1
−
u
|
 X
q
p
 
dpq
E e I
pq
 12
|en+1
p
en+1
|2 
q
−
dpq



Z Zx
X
∂u(t, y)


|∂t u(s, x)|dsdx + 
|
|dydx
∂n
p∈τ
h
p xp
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2
Z
m−1 Z
en+1
− en+1
4C2 C32 h2
1 X X
p
q
II ≤
+
g(|J(u)|)
Cg2
2 n=0
dpq
In
4C 2
+ 24 ·
Cg
+
4C42
·
Cg2
m−1
X
k
X Z
n=0
E e I
pq
m−1
XZ
X Z
n=0 I
n
E e I
pq
E e I
pq
|un+1
− un+1
|2
q
p
dpq
Z
|en |2
Ω
|un+1
− un+1
|2
q
p
dpq


Z Ztn
Z Zx
X
∂u(t, y)


·
|∇ · (g(|J(u)|∇u) |dsdx +
|
|dydx
∂n
p∈τ
Ω
=
t
h
p xp
4C2 C32 h2
+ II1 + II2 + II3 ,
Cg2
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:
8C42 Lg C2
4C42 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
II3 ≤
·k
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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 +
u(tn+1 , xq ) − u(tn+1 , xp )

m(epq ) .
dpq
13454568epq I
Applying the properties of function g, this term can be estimated as
|III| ≤ 2
m−1
XZ
n=0 I
X Z
|Rpq ||en+1
− en+1
|.
p
q
E e I
pq
n
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
XZ Z
C
(H(u)(z))2
|III| ≤ h2
Cg n=0
In Ω
1
+
4
m−1
XZ
n=0 I
n
X Z
E e I
pq
en+1
− en+1
p
q
g(|J(u)|)
dpq
2
.
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Here |H(u)(z)|2 =
d
P
|Di Dj u(z)|2 and Di denote the weak derivatives with respect to the component zi . Since
i,j=1
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
III ≤ C5 h +
g(|J(u)|)
.
4 n=0
dpq
2
In
E e I
pq
Putting all these estimates together, we obtain:
Z
m 2
|e | +
m−1
XZ
n+1
|e
n 2
−e | +
n=0 Ω
Ω
Z
≤4
|e0 |2 + 2(
m−1
XZ
n=0 I
n
X Z
E e I
pq
en+1
− en+1
p
q
g(|J(u)|)
dpq
h2 T
+ 2h2 )||∇u||L∞ (I,L2 (Ω)) +
k
2
4C2 C3
+ 2C5 h2
Cg
Ω
8C42 Lg
4C42 C2
2
+
||D Gσ ||L∞ (Ω) ||∇u||L∞ (I,L2 (Ω)) +
||∆u||L2 (I,l2 (Ω)) k
Cg2
Cg2
2
8C4 Lg C2
+
||DG
||
||∇u||
σ L∞ (Ω)
L∞ (I,L2 (Ω)) · h
Cg2


Z
m−1
n+1
n+1 2 Z
X
X
|uq − up |
4C4 

+
·
k
|en |2  .
Cg
d
pq
n=0
E e I
pq
Ω
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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 and u0 is a non-negative
Rt
Rt
constant, then the inequality u(t) ≤ u0 + v(s) u(s)ds implies that u(t) ≤ u0 exp
v(s)ds .
0
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
Ω
m−1
X
2
|em |2 ≤ C(h2 + h +
h

+ k) · exp(
k
k
n=0
≤ C · exp(C2 )(h2 + k + h +

X Z
E e I
pq
|un+1
q
un+1
|2 
p
−
dpq

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
(18)
k = Ch,
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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
X
n=0 I
epq I
n
2

m−1
XZ
n+1
 uq
m(epq )dpq 
un+1
p
−
dpq
−
Z
1
m(epq )

∇u · npq  ≤ Ch.
epq
Proof. It is easy to see that
NX
NX
max Z Z
max Z Z
|u(tn+1 , x) − uh,k (tn+1 , x)|2 ≤ 2h2 k∇ukL2 (I,L2 (Ω) + 2
|en+1 |2 ≤ Ch
n=0 I Ω
n
n=0 I Ω
n
and the first inequality is proved. Now

2
Z
Z
m−1
n+1
n+1
X
X
1
 uq − up

m(epq )dpq 
−
∇u · npq 
d
m(e
)
pq
pq
n=0
In epq I
≤C
m−1
XZ
n=0 I
n
epq
XZ
epq Ie
pq
g(J(u))
en+1
p
2
en+1
q
−
dpq
+C
m−1
XZ
n=0 I
n
2

dpq
X
n+1
 uq

epq I
un+1
p
−
dpq
−
1
m(epq )
Z

∇u · npq 
epq
≤ Ch,
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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
and the convolution is realized with the kernel
|x|2
1
Gσ (x) = e |x|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.
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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.
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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.
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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.
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2. Alvarez L. and Morel J. M., Formalization and computational aspects of image analysis, Acta Numerica (1994), 1–59
3. Alvarez L., Lions P. L and Morel J. M., Image selective smoothing and edge detection by nonlinear diffusion II, SIAM J. Numer.
Anal. 29 (1992), 845–866
4. Catté F., Lions P. L., Morel J. M. and Coll T., Image selective smoothing and edge detection by nonlinear diffusion, SIAM J.
Numer. Anal. 29 (1992), 182–193.
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(Eds.), North Holland.
6. Kichenassamy S., The Perona-Malik paradox, SIAM J. Appl. Math. 57(5) (1997), 1328–1342.
7. Krivá Z., Adaptive Computational Methods in Image processing, Edition of scientific works STU Bratislava, 15 (2004).
8. Krivá Z. and Handlovičová A., Modified explicit finite volume scheme for Perona-Malik equation, Proceedings of ALGORITMY
2002, Conf. on Scientific Computing, 276–28.
9. Kačur J. and Mikula K., Solution of nonlinear diffusion appearing in image smoothing and edge detection, Applied Numerical
Mathematics 17 (1995), 47–59.
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(in Russian).
11. Lions P. L., Axiomatic derivation of image processing models, Math. Models Methods in Appl. Sci. 4 (1994), 467–475.
12. Mikula K. and Ramarosy N., Semi-implicit finite volume scheme for solving nonlinear diffusion equations in image processing,
Numerische Mathematik,
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on Computer Vision 1987.
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
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