Perona-Malik equation error estimates for
numerical finite volume schemeAngela Handlovičová
Zuzana Krivá
Department of Mathematics Slovak University of Technology, Bratislava, Slovakia
Initial noisy image
Smooth initial noisy image and preserve edges
Perona-Malik Equation
)()0,( 0 xuxu
,0)),(|)),(*(|(),(
xtuxtuGg
txtu
Itxxtu
,,0),(
Itx ,
Perona-Malik equation - properties of datag(s) is Lipschitz continuous decreasing function,g(0)=1, 0< g(s) 0 for s
)( dCG
is a smoothing kernel with compact support
u0(x) is initial condition
Kwith
d
dxxG 1)( andxxG )( for 0
Dirac function at point x
Results obtained for finite volume numerical scheme• Existence of weak solutions and regularity-Catté,
Lions, Morel, Coll (1992)
• Convergence of semi-implicit finite volume numerical scheme – Mikula Ramarossy (2001)
• Convergence of explicit finite volume scheme Krivá (2003)
• Error estimates for semi-implicit FV scheme Handlovičová, Krivá (2005)
• Error estimates for explicit scheme H,K (2005)
Finite volume
p…volume of measure m(p), with representative point xpN(p) …neighbors of p
epq…common edge between volumes p and q of measure m(epq)
dpq:=|xp-xq|, Tpq :=m(epq)/d pq
upn …numerical solution
maxNT
k
• Explicit scheme
)()()( 11
)(
1,1
np
nqpq
pNq
npq
np
np uuTgkpmuu
p
p dxxupm
u )()(
10
0
|))(*(|: 1,1pq
nnpq xuGgg
• Semi - implicit scheme
)()()()(
1,1 np
nqpq
pNq
npq
np
np uuTgkpmuu
p
p dxxupm
u )()(
10
0
|))(*(|: 1,1pq
nnpq xuGgg
Regularity of the solution
))(,()(
)),(,(
)),(,(
)),(,(
11
11
2,1
2,22
LILu
LILu
WILu
WILu
t
tt
Error estimates
2Chk
max
0
21,1 |),(),(|
N
n I
nkhn
n
Chxtuxtu
max
0
211
))(
1()(
N
n I Ie e
pqpqpq
np
nq
pqpq
n pq pq
Chnuemd
uudem
For explicit scheme
For semi -implicit scheme Chk
Thank you
Angela HandlovičováZuzana Krivá
Department of Mathematics Slovak University of Technology, Bratislava, Slovakia
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