Preconditioned Level Set Flows. Martin Burger Institute for Computational and Applied Mathematics European Institute for Molecular Imaging (EIMI) Center for Nonlinear Science (CeNoS) Westfälische WilhelmsUniversität Münster. Introduction.
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
Martin Burger
Institute for Computational and Applied Mathematics
European Institute for Molecular Imaging (EIMI)
Center for Nonlinear Science (CeNoS)
Westfälische WilhelmsUniversität Münster
Level Set Flows AIP 2007, Vancouver, June 07
Level Set Flows AIP 2007, Vancouver, June 07
Level Set Flows AIP 2007, Vancouver, June 07
where K is a class of admissible shapes (eventually including additional constraints).
Level Set Flows AIP 2007, Vancouver, June 07
Level Set Flows AIP 2007, Vancouver, June 07
d
0
(
)
V
J
V
0
(
(
)
)
(
(
)
)
J
J
V
7
!
t
t
n
n
=
n
d
t
Level Set Flows AIP 2007, Vancouver, June 07
Level Set Flows AIP 2007, Vancouver, June 07
Level Set Flows AIP 2007, Vancouver, June 07
 tangent spaces of such a manifold can be identified with normal velocities
 Riemannian structure induces scalar product on tangent spaces
Level Set Flows AIP 2007, Vancouver, June 07
1
2
(
(
)
)
(
(
)
)
d
V
J
V
i
+
¿
¿
m
n
!
n
n
;
;
;
2
V
¿
n
DeGiorgi 1974, AmbrosioGigliSavare 2005
Level Set Flows AIP 2007, Vancouver, June 07
(
)
k
k
d
b
b
¡
=
@
@
1
2
;
1
2
Level Set Flows AIP 2007, Vancouver, June 07
Z
Z
Z
Z
Z
0
(
(
)
)
(
(
)
)
d
d
d
d
d
J
J
V
W
V
V
W
+

¾
c
·
x

c
c

·
¾
¾
¾
=
=
=
n
n
n
n
n
@
@
@
@
Level Set Flows AIP 2007, Vancouver, June 07
Z
¢
V
h
i
d
·

c
V
W
r
V
r
W
=
¢
¾
¾
n
=
n
n
¾
n
¾
n
;
@
Level Set Flows AIP 2007, Vancouver, June 07
Level Set Flows AIP 2007, Vancouver, June 07
Level Set Flows AIP 2007, Vancouver, June 07
Allaire, Jouve et al 06/07
Level Set Flows AIP 2007, Vancouver, June 07
Level Set Flows AIP 2007, Vancouver, June 07
Level Set Flows AIP 2007, Vancouver, June 07
@
u
§
V
¡
u
V
=
§
¡
n
=
n
@
n
V
r
¡
u
=
§
Level Set Flows AIP 2007, Vancouver, June 07
V
r
¡
u
=
§
Level Set Flows AIP 2007, Vancouver, June 07
Level Set Flows AIP 2007, Vancouver, June 07
Level Set Flows AIP 2007, Vancouver, June 07
(
)
f
(
)
d
d
@
i
t
m
a
x
s
u
p
s
x
=
1
2
2
@
2
;
;
;
x
1
(
)
g
d
@
i
t
s
u
p
s
x
1
@
2
;
x
2
Level Set Flows AIP 2007, Vancouver, June 07
0
0
h
i
(
)
(
)
V
W
J
V
W
¼
(
)
N
H
n
n
n
;
;
Level Set Flows AIP 2007, Vancouver, June 07
Residual and L1error, noisy data
Level Set Flows AIP 2007, Vancouver, June 07
0.1 % noise
Iterations
5,10,20,25
Level Set Flows AIP 2007, Vancouver, June 07
1% noise 2% noise
3% noise 4% noise
Level Set Flows AIP 2007, Vancouver, June 07
Papers and talks at
www.math.unimuenster.de/u/burger
or by email
martin.burger@unimuenster.de
Based on joint work with:
Norayr Matevosyan, Stan Osher
Thanks for input and suggestions to:
B.Hackl, W.Ring, M.Hintermüller, U.Ascher
Austrian Science Foundation FWF, SFB F 013
Level Set Flows AIP 2007, Vancouver, June 07