medical image processing with orientation scores erik franken*,...
TRANSCRIPT
11
Med
ical Im
ag
e P
rocessin
g
wit
h O
rien
tati
on
Sco
res
Erik
Fra
nken
*, R
emco
Dui
ts,
Mar
kus
van
Alm
sick
,B
art t
erH
aar
Rom
eny
*E-m
ail:
e.m
.fran
ken@
tue.
nl
Ein
dhov
en U
nive
rsity
of T
echn
olog
yD
epar
tmen
t of B
iom
edic
al E
ngin
eerin
g
Mat
hem
atic
a T
echn
olog
yC
onfe
renc
e 20
06O
ctob
er13
th 2
006,
Cha
mpa
ign
22
Ou
tlin
e
•A
bo
ut
ou
rR
es
ea
rch
Gro
up
•O
rient
atio
nS
core
s•
Diff
usio
nin
Orie
ntat
ion
Sco
res
•S
toch
astic
Com
plet
ion
Fie
lds
•U
sing
Mat
hem
atic
a
33
Th
e B
iom
ed
icalIm
ag
e A
naly
sis
gro
up
44
Sta
rrin
g:
Pet
rŠ
ered
a(P
ilsen
, Cze
chR
epub
lic)
Tim
Pee
ters
(E
cht,
The
Net
herla
nds)
Math
em
ati
ca in
th
e B
ioM
IMla
b
55
Ou
rM
ath
em
ati
ca In
frastr
uctu
re
•F
ullc
ampu
s lic
ense
for
Mat
hem
atic
a•
The
nee
dfo
r“b
igm
ath”
kern
else
rver
s–
Big
mat
h1: T
yan
TX
46, 4
x O
pter
on2.
2Ghz
, 32G
B–
Big
mat
h2: T
yan
VX
50, 4
x D
ualc
ore
Opt
eron
2.2G
hz, 6
4GB
–+
3 o
lder
serv
ers
•U
seof
Par
alle
lMat
hem
atic
a
66
Ou
rM
ath
em
ati
ca In
frastr
uctu
re
•F
ullc
ampu
s lic
ense
for
Mat
hem
atic
a•
The
nee
dfo
r“b
igm
ath”
kern
else
rver
s–
Big
mat
h1: T
yan
TX
46, 4
x O
pter
on2.
2Ghz
, 32G
B–
Big
mat
h2: T
yan
VX
50, 4
x D
ualc
ore
Opt
eron
2.2G
hz, 6
4GB
–+
3 o
lder
serv
ers
•U
seof
Par
alle
lMat
hem
atic
a
77
Math
Vis
ion
To
ols
Com
pute
r V
isio
nLi
brar
yfo
rM
athe
mat
ica:
•G
auss
ian
deriv
ativ
es•
Geo
met
rydr
iven
diffu
sion
•O
rient
atio
nsc
ore
func
tions
•Im
age
tran
sfor
mat
ions
•D
ICO
M im
port
/exp
ort
ww
w.m
ath
vis
ion
too
ls.n
et
88
1
2
3
4
5
6
7
8
9
10
11
Math
em
ati
ca in
Bach
elo
r E
du
cati
on
Imag
e A
naly
sis
for
Pat
holo
gy.
•G
roup
sof
8 2
nd y
ear
stud
ents
•“I
nven
t”im
age
anal
ysis
algo
rithm
sin
M
ath
em
atica
•C
ompe
titiv
eel
emen
t•
6 w
eeks
pro
ject
99
1stor
der
(edg
es)
2ndor
der
(rid
ges)
3rdor
der
(T-ju
nctio
ns)
For
exa
mpl
eR
otat
ion
inva
riant
T-ju
nctio
n de
tect
ion:
Exam
ple
: D
iffe
ren
tialin
vari
an
ts
1010
Ou
tlin
e
•A
bout
our
Res
earc
h gr
oup
•O
rie
nta
tio
nS
co
res
•D
iffus
ion
in O
rient
atio
nS
core
s•
Sto
chas
ticC
ompl
etio
nF
ield
s•
Usi
ngM
athe
mat
ica
1111
1.T
he r
etin
a co
ntai
nsre
cept
ive
field
sof
var
ying
size
s�
multi-sca
lesa
mpl
ing
devi
ce2.
Prim
ary
visu
alco
rtex
is m
ulti-ori
enta
tion
Bio
log
ical In
sp
irati
on
•C
ells
in th
e pr
imar
y vi
sual
cor
tex
are
orie
ntat
ion-
spec
ific
•S
tron
g co
nnec
tivity
be
twee
n ce
lls th
at
resp
ond
to (
near
ly)
the
sam
e or
ient
atio
n
Mea
sure
men
t in
Prim
ary
Vis
ual C
orte
x
Bos
king
et a
l., J
. Neu
rosc
ienc
e 17
:211
2-21
27, 1
997
1212
Ori
en
tati
on
Sco
res
Fro
m2D
imag
e f(
x,y
) to
orie
ntat
ion
scor
e U
f(x,y
,θ)
with
pos
ition
(x,y
) an
d or
ient
atio
n θ
An
orie
ntat
ion
scor
e is
a fu
nctio
non
the
Eu
clid
ea
nm
otio
ng
rou
p
1313
Ou
rap
pro
ach
: Im
ag
e P
rocessin
g v
ia
Ori
en
tati
on
Sco
res
“Enh
ance
men
t”op
erat
ion
Initi
alim
age
“Enh
ance
d”im
age
Orie
ntat
ion
scor
e tr
ansf
orm
atio
n
Inve
rse
orie
ntat
ion
scor
e tr
ansf
orm
atio
n
Seg
men
ted
stru
ctur
es
Seg
men
t str
uctu
res
of
inte
rest
1414
Invert
ible
Ori
en
tati
on
Sco
re T
ran
sfo
rmati
on
Des
ign
cons
ider
atio
ns: r
econ
stru
ctio
n,
dire
ctio
nal,
spat
iall
ocal
izat
ion,
qua
drat
ure
1515
Ou
tlin
e
•A
bout
our
Res
earc
h gr
oup
•O
rient
atio
nS
core
s•
Dif
fus
ion
in O
rie
nta
tio
nS
co
res
•S
toch
astic
Com
plet
ion
Fie
lds
•U
sing
Mat
hem
atic
a
1616
Th
e D
iffu
sio
nE
qu
ati
on
on
Imag
es
f=
imag
eu
= s
cale
spac
eof
imag
eD
= d
iffus
ion
tens
or
Line
ardi
ffusi
onP
eron
a&M
alik
Coh
eren
ce-e
nhan
cing
diff.
t=
0
1717
Th
e D
iffu
sio
nE
qu
ati
on
on
Imag
es
f=
imag
eu
= s
cale
spac
eof
imag
eD
= d
iffus
ion
tens
or
Line
ardi
ffusi
onP
eron
a&M
alik
Coh
eren
ce-e
nhan
cing
diff.
t=
10
1818
Dif
fusio
nin
ori
en
tati
on
sco
res
curv
atur
e
Diff
usio
nor
thog
onal
toor
ient
edst
ruct
ures
Diff
usio
nta
ngen
t to
orie
nted
stru
ctur
esD
iffus
ion
inor
ient
atio
n
Evo
lvin
gor
ient
atio
nsc
ore
Rot
atin
gta
ngen
t spa
ceco
ordi
nate
basi
s
Left-
inva
riant
deriv
ativ
es
are
left-
inva
riant
deriv
ativ
eson
Euc
lidea
nm
otio
ngr
oup,
i.e.
∂ξ
∂η
∂θ
x
θ
y
∂ξ
∂η
∂θ
∂ξ,∂η,and∂θ
Lg∂iU=∂iLgU,i∈{ξ,η,θ}
1919
Exam
ple
dif
fusio
nkern
els
2020
•O
rient
edre
gion
s: D
’ 11
and
D33
smal
l, D
22la
rge
and κ
acco
rdin
gto
estim
ate
•N
on-o
rient
edre
gion
s: D
’ 11
larg
e,
D22
=D
33la
rge,
κ=
0∂ξ
∂η
∂θ
x
θ
y
∂ξ
∂η
∂θ
Ho
wto
Ch
oo
se
Co
nd
ucti
vit
yC
oeff
icie
nts
2121
Diff
usio
nin
orie
ntat
ion
scor
eC
oher
ence
enha
ncin
gdi
ffusi
onR
esu
lts
Siz
e: 1
28 x
128
x 6
4
t=
0
2222
Diff
usio
nin
orie
ntat
ion
scor
eC
oher
ence
enha
ncin
gdi
ffusi
onR
esu
lts
Siz
e: 1
28 x
128
x 6
4
t=
10
2323
Co
llag
en
imag
e
Diff
usio
nin
orie
ntat
ion
scor
eC
oher
ence
enha
ncin
gdi
ffusi
on
Siz
e: 2
00 x
200
x 6
4
t=
0
2424
Co
llag
en
imag
e
Diff
usio
nin
orie
ntat
ion
scor
eC
oher
ence
enha
ncin
gdi
ffusi
on
Siz
e: 2
00 x
200
x 6
4
t=
30
2525
Ou
tlin
e
•A
bout
our
Res
earc
h gr
oup
•O
rient
atio
nS
core
s•
Diff
usio
nin
Orie
ntat
ion
Sco
res
•S
toc
ha
sti
cC
om
ple
tio
nF
ield
s
•U
sing
Mat
hem
atic
a
2626
Oth
er
PD
E:
the s
toch
asti
cco
mp
leti
on
field
Res
olve
ntof
line
arP
DE
Itre
nder
spr
obab
ility
dens
ityfie
ld fo
rlin
eco
ntin
uatio
nba
sed
onra
ndom
wal
ker
prio
r
2727
Co
nvo
luti
on
on
Ori
en
tati
on
Sco
res
Nor
mal
conv
olut
ion
(on
tran
slat
ion
grou
p)
G-c
onvo
lutio
n, w
here
G is
the
Euc
lidea
nm
otio
ngr
oup
An
imag
e is
a fu
nctio
non
the
tran
slat
ion
grou
p
An
orie
ntat
ion
scor
e is
a fu
nctio
non
the
Euc
lidea
nm
otio
ngr
oup
2828
�
(Fro
mM
.Sc.
thes
is b
yR
ensk
e de
Boe
r)
Fillin
gG
ap
sin
Cu
rves
2929
Sou
rce
imag
eLo
cali
nfor
mat
ion
�R
esul
t�
En
han
cin
g e
dg
es in
Med
ical im
ag
es
3030
Ou
tlin
e
•A
bout
our
Res
earc
h gr
oup
•O
rient
atio
nS
core
s•
Diff
usio
nin
Orie
ntat
ion
Sco
res
•S
toch
astic
Com
plet
ion
Fie
lds
•U
sin
gM
ath
em
ati
ca
3131
•M
athe
mat
ica
is h
elpf
ulin
sol
ving
the
mat
h(e
.g. n
on-c
omm
utin
gop
erat
ors)
•N
DS
olve
in m
ath
em
atica
is n
otus
able
for
our
type
of P
DE
sas
far
as I
know
•P
DE
sol
ver
is w
ritte
nin
C+
+, l
inke
dw
ithM
athl
ink
•T
ypic
alpr
oble
ms
of o
urP
DE
–H
ighl
yan
isot
ropi
c, n
otal
igne
dw
ithgr
id–
Non
-com
mut
ing
oper
ator
s–
Con
vect
ion
+ d
iffus
ion
Usin
gM
ath
em
ati
ca
3232
Ackn
ow
led
gem
en
ts
•R
emco
Dui
ts•
Mar
kus
van
Alm
sick
•B
art t
er H
aar
Rom
eny
•B
art J
anss
en•
Arje
n R
icks
en•
Ren
ske
de B
oer
For
ques
tions
/ mor
e re
fere
nces
abou
tthi
sw
ork,
co
ntac
t e.m
.fran
ken@
tue.
nl