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