Gene alized Ellipsoids and Aniso opic il e ing o
Segmen a ion Imp o emen in 3D Medical Imaging1
R. Dosil, X. M. Pa do
Dep . Elec ónica e Compu ación
Uni e sidade de San iago de Compos ela, Spain.
[email p o ec ed], [email p o ec ed]
De o mable models ha e demons a ed o be e y use ul echniques o image
segmen a ion. Howe e , hey p esen se e al weak poin s. Two o he main p oblems
wi h de o mable models a e he ollowing: (1) esul s a e o en dependen on he ini ial
model loca ion, and (2) he gene a ion o image po en ials is e y sensi i e o noise.
Modeling and p ep ocessing me hods p esen ed in his pape con ibu e o sol e hese
p oblems. We p opose an ini ializa ion ool o ob ain a good app oxima ion o global
shape and loca ion o a gi en objec in o a 3D image. We also in oduce a no el
echnique o co ne p ese ing aniso opic di usion il e ing o imp o e con as and
co ne measu es. This is use ul o bo h guiding ini ializa ion (global shape) and
subsequen de o ma ion o ine uning (local shape).
Keywo ds: egis a ion, de o mable models, segmen a ion, aniso opic di usion,
su ace pa ch saliency, 3D medical images.
Co esponding au ho :
Raquel Dosil Lago
Dep . Elec ónica e Compu ación
Uni e sidade de San iago de Compos ela
Mon e da Condesa, Campus Su
15782, San iago de Compos ela, SPAIN
e-mail: [email p o ec ed]
Fax: +34 981 599412
1 This wo k was suppo ed by Spanish Go e nmen and Xun a de Galicia by p ojec s
TIC2000-0399-C02-02 and PGIDT99PXI20606B espec i ely
Gene alized Ellipsoids and Aniso opic il e ing o
Segmen a ion Imp o emen in 3D Medical Imaging1
R. Dosil, X. M. Pa do
Dep . Elec ónica e Compu ación
Uni e sidade de San iago de Compos ela, Spain.
[email p o ec ed], [email p o ec ed]
Abs ac . De o mable models ha e demons a ed o be e y use ul echniques o
image segmen a ion. Howe e , hey p esen se e al weak poin s. Two o he main
p oblems wi h de o mable models a e he ollowing: (1) esul s a e o en dependen on
he ini ial model loca ion, and (2) he gene a ion o image po en ials is e y sensi i e o
noise. Modeling and p ep ocessing me hods p esen ed in his pape con ibu e o sol e
hese p oblems. We p opose an ini ializa ion ool o ob ain a good app oxima ion o
global shape and loca ion o a gi en objec in o a 3D image. We also in oduce a no el
echnique o co ne p ese ing aniso opic di usion il e ing o imp o e con as and
co ne measu es. This is use ul o bo h guiding ini ializa ion (global shape) and
subsequen de o ma ion o ine uning (local shape).
Keywo ds: egis a ion, de o mable models, segmen a ion, aniso opic di usion,
su ace pa ch saliency, 3D medical images.
1. In oduc ion
A de o mable model [13] is an ene gy minimiza ion me hod, whe e he ene gy
unc ional is de ined in e ms o in insic shape a ibu es (in e nal ene gy) and desi ed
1 This wo k was suppo ed by Spanish Go e nmen and Xun a de Galicia by p ojec s
TIC2000-0399-C02-02 and PGIDT99PXI20606B espec i ely
image ea u es (ex e nal po en ial), such as g adien and cu a u e. The ex e nal
po en ial o igina es o ces ha a ac he model o speci ic image ea u es while he
in e nal ene gy causes s ess o ces ha y o main ain model con inui y and
smoo hness. When o ces a e balanced, he model eaches equilib ium, and he
geome ic de o ma ion inishes. The e o e, he model de o ms i sel om i s ini ial
loca ion o app oach he nea es ene gy minimum, which maybe does no co espond o
he a ge objec su ace. De o mable ma ching can e icien ly deal wi h small and local
shape changes, bu ails i global misalignmen is oo la ge. I is e y impo an ha he
s a ing model loca ion and he objec bounda y a e nea enough, so a good ini ializa ion
me hod should be aluable.
When de o mable models a e applied o 3D medical da a, wo kinds o segmen a ion a e
possible: 2 ½ D (slice by slice) and 3D. Ini ializa ion is simple o 2 ½ D segmen a ion,
since i is applied jus o he i s slice. Each slice is ini ialized wi h he esul o he
p e ious one. Manual edi ion o ini ial 3D su aces is e y labo ious, and he au oma ic
o semiau oma ic ini ializa ion is usually mo e complex han he 2D coun e pa . As a
coun e weigh o he low obus ness due o he usual low accu acy in ini ializa ion,
de o mable su ace models ha e he powe o ensu e smoo hness and cohe ence in 3D
shapes.
In de o mable model li e a u e, we can ind di e en app oaches o cope wi h he
ini ializa ion p oblem in 3D. Some app oaches a e based on he inco po a ion o balloon
o ces o o e come po en ial minima. McIne ney and Te zopoulos [17], o example,
used a balloon model o ini ialize he en icle acking in MRI da a. The main di icul y
wi h his scheme is conce ned wi h he inhe en ade-o in he choice o he
in la ion/de la ion o ce.
O he app oaches a e based on he manual o semiau oma ic selec ion o ancho poin s.
Among hem is he imposing o in e ac i e cons ain s in he o m o sp ings and
olcanos [13], and mo e ecen ly he me hod o Neuenschwande e al. [21], ha allows
he use o ix a se o seed poin s and hei no mal ec o s, which canno be changed
du ing de o ma ion.
Some au ho s p opose mul iple ini ializa ion h ough se e al seed models. Du ing he
de o ma ion p ocess, se e al ini ial models will me ge and he supe luous ones should
be emo ed. This solu ion was used, among o he s, by Tek and Kimia [31], and
Leona dis e al. [15]. Impo an decisions ha e o do wi h: (1) he numbe and loca ion
o ini ial seed models, (2) he s opping c i e ia o he seed g owing p ocess, and (3)
choosing one esul among he inal models.
Fully au oma ic ini ializa ion can be achie ed by ma ching he objec in he image wi h
a p o o ype, as done by Bajcsy and Ko acic [1] who used b ain a lases in hei speci ic-
pu pose ini ializa ion echniques. The e a e se e al p oblems wi h he de o mable a las
app oach. The echnique is sensi i e o ini ial posi ioning o he a las and he p esence
o neighbo ing ea u es may cause ma ching p oblems. One solu ion is o use image
p ep ocessing in conjunc ion wi h he de o mable a las; Sando and Leahy [26] used
his app oach.
Se e al esea che s, as Coo es e al. [6], and S aib e al. [30], augmen ed snake-like
models wi h p io in o ma ion abou ypical mean shapes and no mal a ia ions. A
numbe o esea che s ha e inco po a ed knowledge o objec shape using de o mable
shape empla es. Among hem, supe quad ics ha e gained popula i y in medical image
esea ch [12, 18, 32].
The segmen a ion o human o gans om CT o MR images is a good example whe e
model-based econs uc ion can be applied. The model-based econs uc ion p oblem
could be s a ed in wo (no necessa ily disjoin ) phases [7]: egis a ion and ee o m
de o ma ion. On he one hand, egis a ion desc ibes a ans o ma ion wi h a less
deg ees o eedom han he ee o m de o ma ions. The e o e, hei abili y o ep esen
shape a ia ions is less impo an han he ee o m de o ma ion. On he o he hand,
because o hei es ic ed deg ees o eedom, hey end o be mo e obus han ee
o m de o ma ions. The e o e, he i s is be e o desc ibing global shape and loca ion
and he second is be e in de ec ing ine de ails.
In his wo k, we p opose a ully au oma ic ini ializa ion me hod ( egis a ion) based on
ma ching 3D da a wi h models ha comp ise global shape and high le el ea u e
in o ma ion. The main idea is o pe o m he ini ializa ion phase no aking in o accoun
he ele ance o indi idual poin ea u es, bu he p ope ies and global saliency o
connec ed poin ea u es (su ace pa ches in 3D). The me hod includes he cons uc ion
o a p io i models om es images. Bo h shape model cons uc ion om es images
and ma ching wi h a new image a e achie ed by means o i ing a pa ame ic su ace o
a cloud o poin s belonging o he objec bounda y. This pa ame ic su ace is
ep esen ed by a gene alized ellipsoid scheme. Bounda y poin s a e ex ac ed om he
image, g ouped in pa ches and inally selec ed acco ding o high le el p ope ies. In his
way, we y o ex end o 3D he ideas de eloped in a p e ious pape [22] whe e a p io i
knowledge on con ou segmen s was used in o de o ob ain an imp o ed ini ializa ion
o 2D de o mable models.
Ex ac ion o bounda y poin s is no a c i ical s ep because he ele ance o he poin
ea u e depends on all he neighbo s in he same su ace pa ch, and ma ching wi h
global shape models is e y obus . Howe e , accu acy in bounda y de ec ion has
ce ain in luence in he size o de ec ed su ace pa ches and con ibu es o elimina e
noise in luence.
Noise elimina ion and bounda y de ec ion can be achie ed simul aneously using he
de i a i e o Gaussian il e , bu his p esen s se e al d awbacks, such as alse nega i es
and disloca ion o g adien maxima a g ea scales, and alse posi i es a low scales.
Nonlinea di usion me hods [33] pe mi noise elimina ion while p ese e meaning ul
s uc u es. They simula e a hea -sp eading phenomenon in which di usi i y depends on
local p ope ies o he image. Pe ona and Malik [23] in oduced a g adien dependen
di usion coe icien o s op di usion a bounda y poin s, elimina ing bounda y blu ing
bu also main aining noise a bounda y poin s. Weicke [34] p oposed a nonlinea
aniso opic di usion me hod o smoo h su aces jus along he angen plane a
bounda y poin s, educing noise also a bounda ies wi hou blu ing. The sho coming
o Weicke ’s app oach o aniso opic di usion is ha i causes a ounding e ec on
co ne s.
Aniso opic di usion has been equen ly used in medical image p ocessing [11, 14]
[27]. In pa icula , K issian e al. [14] ha e al eady applied i o su ace ex ac ion
success ully. They de eloped a di ec ional aniso opic di usion echnique o enhance
su ace ex ac ion on essel images, ob aining be e esul s han he ones p o ided by
Gaussian il e ing. In hei app oach, di usi i y in he maximum cu a u e di ec ion is
annula ed o educe he ounding p oblem. This is made a he expense o noise
educ ion in he a o emen ioned di ec ion.
The second con ibu ion o he p esen wo k is o in oduce a co ne -p ese ing
aniso opic di usion me hod, as a p ep ocessing ool o imp o e g adien and co ne
measu e and de ec ion. This is achie ed by de ining cu a u e dependen di usion
coe icien s. The inco po a ion o he p ep ocessing s ep ep esen s an incoming o
bo h ini ializa ion, imp o ing su ace de ec ion, and de o ma ion oo, since alse minima
elimina ion and co ec placemen o bounda ies and co ne s imp o es he ex e nal
po en ial de ini ion. Mo eo e , he cu a u e based app oach p oposed in his pape
leads o a be e loca ion o g adien po en ial minima and enhancemen o he cu a u e
based po en ial, as ounding p oblem is a oided.
The pape is o ganized as ollows. In nex sec ion, an o e iew o he comple e
ini ializa ion me hodology is desc ibed. In sec ion 3, he gene alized ellipsoid model is
de ined. Sec ion 4 p esen s he op imiza ion echnique used o model cons uc ion and
ma ching. In sec ion 5 all p ep ocessing s eps a e desc ibed. These a e, image denoising,
bounda y su ace pa ches de ec ion and pa ch classi ica ion. Finally, in sec ion 6
e icacy o he whole p ocess is s udied, paying special a en ion o i s obus ness in he
p esence o noise and loss o in o ma ion caused by con as a ia ions o e su aces.
2. Ini ializa ion me hodology
The goal o 3D econs uc ion me hods is o ob ain a de ailed desc ip ion o objec s
p esen in a olume da a. To his end, de o mable models a e a good choice, because
hey possess g ea lexibili y and gua an ee ce ain smoo hness and con inui y
p ope ies. The p oblem wi h de o mable models is ha hey in e ac locally wi h image
ea u es. I is necessa y o de ine a good s a ing geome ic con igu a ion o ensu e he
success o he de o ma ion p ocess. The ini ial su ace mus be close o he bounda y o
he objec o in e es , which implies o de e mine posi ion, o ien a ion and global
s uc u e o ha objec in he 3D domain. Figu e 1 illus a es he ini ializa ion p ocess
p oposed in his pape .
The a ainmen o o a high le el desc ip ion o ce ain objec is sepa a ed in wo
s ages: 1) modeling o global shape o he objec class and 2) ma ching he class
model wi h he ins ance objec . The i s s age is pe o med o -line o e a se o
sample shapes o he same objec class ( igu e 1, s ep 1). The esul ing class model
is called he
a p io i
model. The sample images a e segmen ed manually o a oid
ypical au oma ic segmen a ion p oblems and hen a su ace p o o ype is ex ac ed
om hem. A e wa ds, global shape o he objec su ace p o o ype is de e mined
by i ing a ma hema ical model o i . To ep esen global shape we ha e chosen
he gene alized ellipsoid model, also called supe quad ic.
Once an
a p io i
model is a ailable, posi ion, o ien a ion and scale o an ins ance
objec in a new image is de e mined by a ma ching echnique. To pe o m
ma ching i is necessa y o ex ac some ea u e om he image ha desc ibes he
objec su ace p ope ly and o ind he ans o ma ions ha lead o a
co espondence be ween image ea u es and model su ace. In his wo k su ace
poin s a e used as image ea u es ( igu e 1, s ep 2). The ex ac ion o such low le el
ea u es om g adien in o ma ion is no obus . Again, he use o p io
knowledge is needed o dis inguish he a ge objec su ace poin s om o he s
p esen in he image as, o example, poin s belonging o s uc u es o he ha he
one unde s udy o noise a i ac s. To his end, poin s a e no conside ed
indi idually, bu hey a e g ouped in pa ches. Resul ing pa ches a e cha ac e ized
by some o hei a e age p ope ies as, o example, a ea, con as o shape
desc ip o s, and hese a e used in a selec ion p ocess o exclude undesi ed pa ches
( igu e 1, s ep 3). The s a ing su ace o de o ma ion is ob ained a e ma ching
be ween model and objec ( igu e 1, s ep 4). This is done by inding he pa ame e s
o he igid ans o ma ion ha minimizes an e o unc ion ela ed o he
dis ances om selec ed image poin s o he pa ame ic su ace.
In sho , wha is p esen ed he e is a me hod ha akes ad an age om bo h
bo om-up and op-down p ocesses. Ini ializa ion is accomplished by ob aining
highe and highe le el desc ip ions o image con en s: om olume poin s wi h
associa ed g ay le els, o bounda y poin ep esen a ion, om bounda y poin s o
su ace pa ches, ea u ed by local and global desc ip o s ha allow disc imina ing
be ween desi ed and spu ious pa ches, and om su ace pa ches o a global
su ace model wi h he help o p io knowledge. The nex s age, no desc ibed
he e, would walk he in e se way. S a ing om he coa se ep esen a ion o he
su ace esul ing om ini ializa ion, he de o ma ion p ocess in oduces local
deg ees o eedom o each a de ailed desc ip ion o he su ace objec . Thus, he
con lic ing goals o high obus ness and high esolu ion can be achie ed.
3. Global shape models
Bo h bounda y de ec ion and calculus o shape desc ip o s, based on cu a u e
measu es, equi e he compu a ion o di ec ional de i a i es o g ay le el alues. The
de i a i e o Gaussian ope a o pe o ms di e en ia ion and smoo hing simul aneously
[19], bu i modi ies g adien maxima posi ion, disloca ing su aces. Mo eo e , small
s uc u es can be elimina ed. Nonlinea aniso opic il e s o e be e esul s.
In his wo k, a co ne p ese ing aniso opic il e has been de eloped o smoo h 3D
images wi hou al e g adien and cu a u e alues nei he misplacing bounda ies no
ounding co ne s. A e image denoising, de i a i es a e app oxima ed by cen al
di e ences. Nex subsec ions desc ibe he main s eps in ea u e ex ac ion: aniso opic
smoo hing, bounda y de ec ion, and su ace pa ch selec ion.
5.1 Aniso opic di usion
Le us in oduce he gene al me hod o di usion il e ing in 3D p oposed by Weicke
[34]. Le Ω be a 3D image domain and ∂Ω i s bounda y. Gi en an image I(x, y, z), i s
il e ed e sion u(x, y, z, ) is ob ained by he nex exp ession, wi h e lec ing bounda y
condi ions
(
)
(
)
(
)
()
()()
()
(
)
()
∞×Ω∂
Ω
∞×Ω
=∇
==
∇∇∇=∂
,0
,0
on
on
on
0,,,,
,,0,,,
,,,,,,
n zyxuC
zyxI zyxu
zyxuuC zyxu
(15)
whe e n is he ou e no mal, 〈·,·〉 is he inne p oduc and subsc ip s s and o pa ial
de i a i es. In he iso opic case, di usion coe icien C is a scala magni ude. Usually,
i is a dec easing unc ion o ||∇u|| wi h alues belonging o he in e al [0, 1]. In his
way, di usion is s opped in he p esence o bounda ies.
P e ious exp ession is o en ela ed o he ene gy minimiza ion o mula ion. The ene gy
unc ional E(u) is de ined as he in eg al o e he image o a po en ial Φ(||∇u||).
Equa ion (15) is ob ained by applying he g adien descen me hod o minimize he
ene gy unc ional. Bo h app oaches a e ela ed by
()
(
)
uuuC ∇∇Φ=∇ ' (16)
Using his ela ion and ope a ing wi h equa ion (15), nex exp ession o nonlinea
iso opic di usion can be ob ained
(
)
(
)
ξξξξ
uuuuu −∆∇Φ+Φ= ''' (17)
whe e uξξ s ands o he second de i a i e o u in he no mal di ec ion ξ. A possible
choice o he po en ial unc ion is he one p oposed by G een [10]
Φ(s)=(
α
2/2)log cosh(s) (18)
whe e
α
ep esen s he g adien h eshold a which di usi i y s ops g owing. O he
app oaches we e s a ed by Pe ona and Malik [23] and Cha bonnie e al. [5] among
o he s.
A di usion p ocess is called aniso opic when di usi i y akes di e en alues
{λ1, λ 2
, λ3} in di e en di ec ions {e1, e2, e3} o space. As a esul , di usi i y is a
enso ial magni ude, and hen, he lux ec o C∇u is no pa allel o he g adien
di ec ion. In he e e ence ame de ined by he basis {e1, e2, e3}, di usi i y u ns in o a
diagonal enso D = diag(λ1, λ2, λ3). The exp ession o he di usion enso C in a
gene al ame is C = TDT T whe e T is he ma ix o med by he column basis ec o s.
Ano he way o cons uc ing an aniso opic il e is gene alizing equa ion (17), allowing
independen di usion coe icien s o each e m, as done by K issian e al. [14]. The
al e na i e exp ession is a ained by aking he enso ial app oach and se ing he
di usi i y eigen alues such ha
λ
i = Φi (||∇u||)/||∇u||, wi h i = 1, 2, 3, and applying
some ma hema ical ela ions, ob aining
(
)
(
)
2211 321 ''''
ηηηηξξ
uuuuuu ∇Φ+∇Φ+Φ= (19)
Using his app oach o il e an image o a gi en ime is as e han using he enso ial
app oach, because he second de i a i es o u in he ex eme cu a u e di ec ions can be
compu ed di ec ly wi hou calcula ing he Hessian eigen ec o s, using he nex ela ion
21 2211 kuukuu ∇−=∇−=
ηηηη
(20)
whe e k1 and k2 a e he maximum and minimum cu a u es espec i ely.
A ypical scheme o aniso opic di usion is cons uc ed by se ing di usi i ies
associa ed o angen di ec ions o cons an alues, while aking a dec easing unc ion o
||∇u|| in he g adien di ec ion. Thus, di usion is s opped in he p esence o bounda ies
only in he no mal di ec ion bu no in he angen plane, elimina ing noise also a
su aces.
5.1.1 Co ne p ese ing di usion
Smoo hing in he angen di ec ions lessens cu a u e alues as he sys em e ol es. This
e ec elimina es noise by la ening su aces, bu also al e s he shape o objec s,
elimina ing small de ails and ounding co ne s. One way o p e en ounding is
a oiding di usion in he maximum cu a u e di ec ion. In his way, he highes
cu a u e alue is no modi ied bu , consequen ly, noise is main ained in he
co esponden di ec ion oo.
A be e choice would be educing di usion only in he p esence o co ne s, while
keeping i a la egions o su aces whe e he angen ec o s a ia ion is smoo h. In
his wo k, we p opose such a di usion me hod. To his aim, di usi i y in he maximum
cu a u e di ec ion has been modi ied in ela ion o he iso opic app oach o in oduce a
dependency on a ce ain co ne measu e c. This di usion coe icien is expec ed o be
high in he absence o co ne s and dec ease as co ne measu e g ows. This beha io can
be modeled wi h he G een unc ion p esen ed o con as p ese ing il e ing, jus by
changing he bounda y de ec o by a co ne de ec o and he g adien h eshold by a
co ne h eshold
β
. The e o e, di usion coe icien s in equa ion (19) a e
()
(
)
1' anh'cosh'' 32
2
1=∇Φ=∇Φ∇=Φ −uccuu
ββα
(21)
The co ne de ec o is ela ed o he local p incipal cu a u es o he image. Cu a u e
measu es he a ia ion o he angen ec o wi h he a c leng h in some di ec ion, bu i
canno be conside ed as a co ne de ec o i sel , because high cu a u e alues can be
o igina ed by noise. To dis inguish be ween eal ea u es and noise a i ac s, cu a u e is
usually mul iplied by some powe o he g adien magni ude. Di e en app oaches o a
co ne de ec o wi h hese cha ac e is ics a e s udied and compa ed by Spo ing e al.
[29]. Among he a ious possibili ies discussed he e, he e i has been selec ed nex
max
kuc ⋅∇= (22)
As a esul , an image poin is conside ed a co ne when bo h i s g adien modulus and
i s maximum cu a u e ha e high alues. Highe o de powe s o he g adien modulus
can ejec co ne s om s uc u es wi h low con as .
The selec ion o he h eshold alues
α
and
β
is c ucial in he accu acy o he ob ained
esul s, since hey es ablish whe he a ea u e mus be smoo hed o p ese ed. To ob ain
au oma ic pa ame e es ima ion, a ool om obus s a is ics [3, 24] is employed. The
median absolu e de ia ion abou he median, MAD, is aken as a measu e o he obus
scale
σ
e o some magni ude. I medianI is he median o some magni ude compu ed
om all poin s belonging o image I, hen he obus scale o g adien is
()
(
)
[
]
6745.0medianmedian6745.0MAD III IIe ∇−∇=∇=
σ
(23)
Cons an 0.6745 is he MAD o a ze o-mean no mal dis ibu ion wi h uni a iance.
Scale
σ
e is he con as alue a which lux mus s op g owing. I he s opping c i e ion
is o each a ce ain ac ion x o he asymp o ic limi o he iso opic lux unc ion ∞,
pa ame e
α
can be ela ed o
σ
e by ∞
⋅
=
=
∇
xI e),||(||
α
σ
. Fo he G een unc ion
∞ =
α
, so he h eshold pa ame e is
(
)
x
ea anh
σ
α
=
(24)
He e, i has been aken x = anh(1) = 0.7619, so ha
α
=
σ
e. The same es ima ion can be
done o
β
, compu ing MADI (c).
5.2 Bounda y de ec ion
Once he image is smoo hed, di ec ional de i a i es can be compu ed using a cen al
ini e di e ences scheme. An image poin is conside ed a su ace poin i i is a local
maximum o he g adien modulus. Monga and Benayoun [19] accomplish g adien
maxima de ec ion by compa ing he modulus magni ude o each poin only wi h
alues co esponden o p e ious and nex poin s in he g adien di ec ion ∇u( ),
ep esen ed by + and – and calcula ed by
(
)
(
)
uu ∇∇±=
±/ (25)
When
(
)
(
)
(
)
{
}
_
, uuu ∇∇>∇ + (26)
is a g adien maximum. I + o – do no coincide wi h an image posi ion, g adien
modulus is app oxima ed by ilinea in e pola ion in a icini y o .
5.3 Su ace pa ches selec ion
A e g adien maxima de ec ion, image con en s a e desc ibed by a se o candida e
bounda y poin s. Now, i is necessa y o de e mine wha poin s a e o be used in he
i ing p ocess. To his end, a collec ion o su ace poin s is no an app op ia e
desc ip ion o image objec s, since poin s do no ca y in o ma ion abou which objec
hey belong o and wha is he global aspec o ha objec . Simple h esholding o
indi idual poin a ibu es, as g adien modulus o cu a u es, can cause discon inui ies
on ele an su aces due o local luc ua ions. Fu he mo e, no only s uc u es unde
s udy a e eco e ed, bu also undesi ed su aces o noise a i ac s can appea .
Fo hose easons, bounda y poin ep esen a ion o image objec s is eplaced by a
highe le el desc ip ion. Ex ending he idea poin ed by Pa do and Cabello [22]
om 2D o 3D, g adien maxima a e g ouped in connec ed componen s o ob ain
su ace pa ches. Global in o ma ion abou objec s can be ex ac ed om his new
desc ip ion as, o example, a ea, con as o shape. These global ea u es a e used
o de e mine he alue one unique label o each poin ha indica es whe he i
belongs o he objec su ace o no . In a mo e gene al case his label may
ep esen he deg ee in which ha poin can be said o belong o he su ace objec .
This is, each poin
i
belonging o ce ain su ace pa ch
P
j
is cha ac e ized by a
label alue de e mined by a labeling unc ion
L
such ha
L
(
i
) =
L
(
P
j
), ∀
i
∈
P
j
, o
wha is he same, all poin s in a pa ch ha e he same label alue and his alue is
de e mined om he con ibu ions o all indi idual poin s. Labeling unc ion
dependency on su ace pa ch a ibu es is de e mined acco ding o p io
knowledge abou image con en s.
L
mus show g ea alues o pa ches wi h global
desc ip o alues simila o he ones expec ed
a p io i
o ha objec class and low
alues o any o he pa ch. The e o e, an exp ession o
L
mus be cons uc ed
om hose
a p io i
desc ip o alues o each objec class.
To accomplish g ouping, a 26-connec i i y c i e ion is used. Many o he poin s
de ec ed as g adien maxima do no co espond o su aces o he desi ed objec s in he
image, bu hey a e noise a i ac s. When g ouping bounda y poin s o cons uc su ace
pa ches, hose poin s mus no be conside ed. Simple h esholding is no a good
echnique o dis inguish noise a i ac s om eal su ace poin s, as con as , in gene al,
is no uni o m o e he objec su ace. He e, hys e esis h esholding is used. Hys e esis
in ol es de ining wo h eshold le els. The lowes h eshold le el de e mines whe he a
poin belongs o a su ace. I his le el is chosen app op ia ely, su ace agmen a ion is
educed. In addi ion, each connec ed componen mus ha e, a leas , a numbe n o
poin s wi h g adien g ea e o equal o he o he h eshold le el. This is supposed o
exclude su ace pa ches o igina ed by noise. He e, minimum numbe o poin s o e he
highes h eshold le el has been aken n = 1.
Ou goal is o cons uc a ep esen a ion o he image ha emphasizes salien loca ions.
We seek o associa e a measu e o saliency, deno ed by he labeling unc ion, o each
su ace pa ch. A p ope y ha seems o play an impo an ole in bounda y saliency is
he combina ion o size, global and/o local (smoo hness) shape, and con as . A
labeling unc ion ha would accoun o ou wo king examples is one ha a o s long,
smoo h shape and high g adien su ace pa ches. In ou p oposal, smoo hness is ela ed
o cu a u e ype o cu a u e a ia ions.
The exac o mula ion o L can be adap ed o he speci ic applica ion domain. He e,
wo king hypo heses a e ha ue su aces ha e highe a ea and highe a e age g adien
le el han noise a i ac s. In addi ion, hey can be use ul o disc imina e among a ious
ana omical s uc u es wi h known p ope ies. Fo example, co ical bone issue in CT
images is cha ac e ized by i s high con as in ela ion o muscle and abecula bone
issues. Rela i e sizes o objec s a e also known in gene al. Shape desc ip o s a e used
o disc imina e ana omical s uc u es wi h di e en mo phologies. In his wo k
Gaussian cu a u e K and mean cu a u e H a e used as shape desc ip o s.
()
−+−+
⋅
=
+
=kkKkkH 2 (27)
whe e k+ and k– a e he ex eme cu a u e alues a each poin . Using hei alues, nex
classi ica ion o poin s can be made:
H
> 0
H
= 0
H
< 0
K
> 0 conca e ellip ic − con ex ellip ic
K
= 0 conca e cylind ical plane con ex cylind ical
K
< 0 conca e hype bolic saddle poin con ex hype bolic
Su ace pa ch desc ip o s can be ob ained om poin desc ip o s by a e aging hei
alues. Howe e , he e o commi ed in he calculus o K is he p oduc o he e o s
co esponden o k+ and k–. As cu a u e alues a e e y sensi i e o noise, esul s
ob ained o K a e no e y eliable. Be e esul s should be ob ained by a e aging
ex eme cu a u es and hen using esul ing alues in equa ion (27), a leas o he
Gaussian cu a u e sign, despi e mean(k+ · k–) ≠ mean(k+)·mean(k–).
A e age Gaussian and mean cu a u e signs classi y su aces in a e y ough manne .
Thei u ili y is limi ed o simple objec s. I maximum and minimum cu a u e signs
a y s ongly o e he su ace, a e age measu es a e no longe desc ip i e o su ace
shape. Le us hink, o example, in he shape o a e eb a. In hese cases, he cu a u e
a ia ions along he su ace pa ch can be used as a measu e o smoo hness.
Labeling unc ion mus depend on hose pa ch a ibu es in such a manne ha a ge
su aces ha e high label alues while he emainde pa ches ha e low alues. Labels can
be used in se e al ways o decide how he i ing p ocess is o be done. A gene al
me hod in ol es conside ing labels as weigh ac o s in he calculus o he e o
unc ion, modula ing he con ibu ion o each su ace poin o he o al e o . Then, he
e o unc ion can be ede ined in he nex way
() () ( )
∑
=
=N
i
ii DLE
1
22 ,q q (28)
Then, i a poin belongs o a salien pa ch, i s con ibu ion o he accumula ed dis ance is
decisi e in he esul . As label alue dec eases, he in luence o all poin s on he pa ch is
educed.
6. Resul s
Pe o mance o he whole me hodology depends on h ee aspec s: he capabili y o he
chosen model o ep esen objec s in an app op ia e manne in ce ain ield o
applica ion, he e iciency o he op imiza ion me hod and he e ec i eness o he
p ep ocessing echnique in su ace pa ches ex ac ion. In his sec ion, he ini ializa ion
me hod is es ed aking all his poin s in o accoun .
6.1 Modeling wi h supe quad ics
In sec ion 4.1, se e al e o me ics we e p esen ed. Each o hem ep esen s a i ness
measu e in ce ain me ic. To make a compa ison be ween esul s ob ained wi h each
one, i is necessa y o es ablish a me ic-independen quali y c i e ion. The p ocedu e is
as ollows. A syn he ic supe quad ic su ace, illus a ed in igu e 4, is designed wi h
ec o pa ame e q, ep esen ing i as a poin cloud. A e wa ds, pa ame e s a e
es ima ed op imizing h ee di e en e o unc ions: D2, D3 and D4 co esponden o
equa ions (11), (12) and (13) espec i ely. Then, pa ame e s q’ es ima ed wi h each
e o unc ion a e compa ed wi h eal alues o de e mine which me ic o use.
The AG scheme employed o i he su ace has he ollowing cha ac e is ics. Pa ame e
ec o has been ep esen ed wi h G ay code o 16 bi s. Popula ion size has been se o
80 indi idual o modeling and 40 o ma ching. The e o unc ion is mapped by linea
ank be o e selec ion o candida es o be used by gene ic ope a o s. Selec ion me hod is
p obabilis ic ou namen . Gene ic ope a o s applied a e wo-poin c osso e and
mu a ion wi h p obabili ies pc = 0.8 and pm = 0.1 espec i ely. The bes 20% indi iduals
a e ep oduced in nex i e a ion o he algo i hm.
To compa e eal and es ima ed pa ame e s, he disc epancy d be ween he wo su aces
has been de ined. I ep esen s he Euclidean dis ance be ween he wo pa ame e
ec o s. To assign he same weigh o each pa ame e in he dis ance measu e, each
componen qi is scaled wi h a ac o
α
i. Scale ac o s a e de e mined heu is ically and
a e ela ed o he ange o a ia ion o each pa ame e . Then, d is
() ( )
∑−=
i
iii qqd 22 ',
α
q'q (29)
Pa ame e s o he syn he ic supe quad ic a e shown in he i s column o able 1. The
second column con ains he weigh ac o s o each pa ame e . The emainde columns
show pa ame e ec o s es ima ed wi h h ee di e en e o unc ions. They ha e been
() ()
(
)
(
)()
<><>
=
o he wise0
and,0,i 1 minmax
2/12
hj hjjj
j
kPkkPkPHlaPg
PL (30)
whe e a is he pa ch a ea, g(Pj) is he g adien modulus a e aged o e all pa ch poin s
and l is a h eshold alue es ima ed om ypical noise, muscle and bone g adien and
a ea alues. He e i has been aken l = 1e5. The use o his labeling unc ion as a weigh
ac o in he i ing p ocess is equi alen o he elimina ion o less salien pa ches.
Selec ed poin s, p esen ed in o h ow o igu e 19, con ibu e equally o he e o
unc ion.
The su ace is no comple e due o a enua ion o he ibia in ensi y le el in he knee
egion. The ma ching p ocess has been ealized using he geome ic model ob ained in
sec ion 6.1. Resul s in i h ow o igu e 19 show ha ini ial model is nea he su ace
o be modeled. As his model has been ob ained om he same image by manual
segmen a ion, his p o ides a ool o measu e he co ec ness o he igid ans o ma ion
pa ame e es ima ion. Resul s in able 5 show pa ame e s q ob ained in he model
cons uc ion phase and pa ame e s q’ ob ained by ma ching he model o he p ocessed
image. I he dispa i y measu e be ween bo h pa ame e ec o s is compu ed, he esul
is d = 0.06839, which a good esul o a ela i ely complex su ace.
MRI Images a e cha ac e ized by hei high signal o noise a io (SNR). Fis ow o
igu e 20 shows an example o an MRI image o he ao a a e y. I can be seen ha i is
e y noisy and he a e age con as is low. In addi ion, o he s uc u es a e p esen in
he image besides he ao a. The applica ion o aniso opic di usion o scale
σ
= 5
enhances he MRI image ( igu e 20, 2nd ow), gi en ha noise is almos comple ely
elimina ed bu bounda ies a e no blu ed. Th eshold pa ame e s a e
α
= 2.7 and
β
= 0.42.
Because o he low con as o he image, high h eshold mus be also low. He e, i has
been aken 1 = 40. Low h eshold mus be ela i ely high, 2 = 25, o a oid connec ion
o dis inc su aces. Connec ed componen s a e shown in hi d ow o igu e 20. Again,
h esholding is now enough o ob ain desi ed su ace and pa ches selec ion is equi ed.
The same c i e ion used in he p e ious example is used he e o ex ac ao a su ace,
which is also cylind ical, bu conca e now.
() ()
(
)
(
)()
<>>>
=
o he wise0
and,0,i 1 minmax
2/12
hj hjjj
j
kPkkPkPHlaPg
PL (31)
Fo his kind o MRI images, i is aken l = 1e4. A e selec ion o pa ches, esul s in
o h ow o igu e 20 a e ob ained. Pa ches ha do no belong o he ao a ha e been
elimina ed, bu pa o he su ace has been los du ing p ep ocessing. A p io i model
has no been ex ac ed om es images his ime. A simple cylinde model is enough o
ep esen oughly he a e y shape. Resul s o he ma ching p ocess a e shown in i h
ow o igu e 20.
7. Conclusions
De o mable models a e e y eliable modeling echniques. They o ce con inui y and
smoo hness in segmen a ion bu hey ha e a local ield o ac i i y, so s a ing
con igu a ion de e mines he success o he p ocedu e. In his pape , an au oma ic
ini ializa ion ool o guide de o ma ion is p esen ed. To his end, a comple e
me hodology has been de eloped o ob ain a desc ip ion o global shape, loca ion,
o ien a ion and size o an objec om a 3D image.
Objec iden i ica ion is accomplished by combining high le el in o ma ion om images
and in oducing a p io i knowledge. A me hodology, based on a p io i knowledge
abou su ace ea u es, has been designed o isola e su ace pa ches belonging o he
a ge objec . This me hod pe mi s o elimina e s uc u es ha co espond o noise o
o he objec s p esen in he image. Modeling wi h supe quad ics does he emaining
wo k. Su ace pa ches a e used only o de e mine igid ans o ma ion pa ame e s, and
ine uning o lea ned geome ic ea u es.
The p incipal ad an age o his echnique is ha ini ializa ion is highly au oma ed and i
p o ides a good app oxima ion be ween su aces o he desi ed objec and i s model,
ensu ing p oximi y o he co ec ene gy minimum in a de o mable model scheme.
Gene ali y is also impo an . A p io i models can be easily cons uc ed in any
applica ion domain, so ha a de ailed ana omical a las is no necessa y. Desc ip ion
wi h implici su aces simpli ies he es ima ion o ma ching e o , since i is no
necessa y o de e mine co espondences be ween model poin s and image poin s. The
disc e iza ion and bounding o he solu ion space allows using a GA o ind he global
op imum wi hou an excessi e ime cos .
In u u e wo ks we expec o supply he echnique wi h mechanisms o de ec di e en
componen s o mul ipa o b anched objec s. Thus, each pa can be ini ialized wi h a
di e en model in a hie a chical scheme.
The o he con ibu ion in his wo k is he co ne p ese ing aniso opic il e ing.
Cu a u e dependen di usion coe icien s ha e been designed, so ha di usion is
s opped a co ne poin s in he angen di ec ion co esponden o he maximum
cu a u e le el. The e o e, noise is elimina ed a e e y image egion, in e - egion poin s
included, wi hou disloca e bounda ies. This ac implies an impo an imp o ing o
su ace de ec ion, since shape is now mo e eliable. Su ace me ging also is a oided.
Fu he mo e, his p ocessing echnique is expec ed o imp o e he measu es o he
ex e nal ene gy o he de o mable model.
Resul s p esen ed show ha bo h g adien and co ne de ec o s o e be e esponses
a e aniso opic il e ing in compa ison wi h Gaussian blu ing. The e ec o he
de ini ion o he h eshold pa ame e s is e y ele an o his applica ion. The
au oma ic se ing o hese pa ame e s p o ides a use ul ool o ind a comp omise
be ween smoo hing and bounda y o co ne p ese ing. Howe e , he e a e cases in
which he de ini ion o global h esholds is insu icien . Va ia ions on he backg ound
in ensi y le el, a enua ion o he ea u es in ensi y o he p esence o s uc u es o
di e en in ensi ies can be he eason o impo an loss o in o ma ion. To sol e he
p oblem, he h eshold pa ame e s should be es ima ed locally on a icini y o each
poin . This is a p oposal o u u e wo ks, whe e iabili y o he app oach mus be
s udied in e ms o compu a ional e iciency.
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Pa e n Analysis and Machine In elligence, 13(10) (1991) 1038-1049.
Cap ions
Figu e 1. Sequence o s eps o achie e he s a ing con igu a ion o he de o mable
model.
Figu e 2. Supe quad ic su aces o di e en ε1 and ε2 alues.
Figu e 3. Global de o ma ions.
Figu e 4. Syn he ic supe quad ic used o es i ing me hod.
Figu e 5. Tibia p o o ype o be modeled.
Figu e 6. Tibia model ob ained wi h e o unc ion D4.
Figu e 7. Tibia model ob ained wi h e o unc ion D3.
Figu e 8. (a) O iginal es image, consis ing o a cube (1) a cylinde (2) and a sphe e
(3). I has been smoo hed wi h (b) Gaussian il e (c) co ne p ese ing
aniso opic di usion, bo h wi h scale
σ
= 6. Slices ep esen plane z = 30.
Figu e 9. G adien a slice z = 30 o (a) Gaussian il e ing (b) aniso opic il e ing.
Figu e 10. Bounda ies de ec ed a slice z = 30 wi h (a) Gaussian il e (b) aniso opic
il e .
Figu e 11. Co ne measu e a slice z = 30 o (a) Gaussian il e ing (b) aniso opic
il e ing.
Figu e 12. Displacemen o co ne s wi h il e ing o scale
σ
= 6. Co ne s a e labeled
by quad an .
Figu e 13. Disloca ion o sphe e bounda y h ough di usion ime.
Figu e 14. Ex eme cu a u es es ima ion a di e en scales o he Gaussian il e .
Figu e 15. Ex eme cu a u es es ima ion a di e en scales o aniso opic di usion.
Figu e 16. Di e en sec ions o a syn he ic image o es pa ch classi ica ion.
Figu e 17. Di e en selec ions o su ace pa ches.
(a) Non-plana | kmax | > k h
(b) Plana | kmax | < k h
(c) Conca e H > k h
(d) Con ex H < −k h
(e) Ellip ical | kmax | > k h & | kmin | > k h & K > 0
( ) Hype bolic | kmax | > k h & | kmin | > k h & K < 0
(g) Cylind ical | kmax | > k h & | kmin | < k h
(h) Selec ion o indi idual cylind ical poin s
Resul s a e p esen ed o slice z = 25 excep in (h), which shows z = 40.
Figu e 18. 1s ow: es image wi h Gaussian noise o a iance
σ
n = 5 and a ia ion o
he backg ound colo along y axis. 2nd ow: Image smoo hed wi h
aniso opic di usion wi h scale
σ
= 5. 3 d ow: Selec ion o cylind ical
pa ches. 4 h ow: cylinde model – esul ing om ma ching wi h ex ac ed
pa ch –supe imposed o o iginal image.
Figu e 19. 1s ow: Slices o he o iginal ibia CT image. 2nd ow: image il e ed wi h
aniso opic di usion o scale
σ
= 5. 3 d ow: con ex cylind ical pa ches
selec ed wi h u1 = 20 and u2 = 50. 4 h ow: ini ial model supe imposed o
o iginal image.
Figu e 20. 1s ow: Slices o he o iginal ao a MRI image. 2nd ow: image il e ed wi h
aniso opic di usion o scale
σ
= 5. 3 d ow: conca e cylind ical pa ches
selec ed wi h u1 = 25 and u2 = 40. 4 h ow: ini ial model supe imposed o
o iginal image.
Table 1. Columns om le o igh : syn he ic supe quad ic pa ame e s,
co esponden weigh ac o s, a e age es ima ed pa ame e s qi’ ob ained
wi h di e en e o unc ions.
Figu e 14.
Figu e 15.
y = 50
z = 25
x = 70
x = 150
x = 213
Figu e 16.
Table 3.
Pa ch k1 k2 H K
Plane 0.001219 −0.006986 −0.002883 −0.000009
Plane 0.001622 −0.007492 −0.002935 −0.000012
Con ex hype bola 0.010841 −0.036092 −0.012626 −0.000391
Conca e hype bola 0.040352 −0.015881 0.012236 −0.000641
Con ex cylinde −0.000721 −0.038698 −0.019710 0.000028
Conca e cylinde 0.052256 −0.009572 0.021342 −0.000500
Con ex sphe e −0.042745 −0.047356 −0.045050 0.002024
Conca e sphe e 0.074956 0.069251 0.072103 0.005191
(a)
(b)
(c)
(d)
(e)
( )
(g)
(h)
Figu e 17.
z = 50 x = 50 y = 50
Figu e 18.
q
i
α
i
q
i’ gauss
q
i’ aniso opic
a
0
1 1
9.758e−1 1.069e+1
β
0 0.159
−1.203e−1 1.452e−2
1
50 0.02 5.011e+1 5.032e+1
2
50 0.02 5.013e+1 5.203e+1
3
50 0.02 4.924e+1 5.122e+1
Table 4.
z = 50
z = 140
x = 98
y = 112
Figu e 19.
Table 5
q
i
q
i’
a
0
1.00e+0 9.54e−1
α
5.31e−1 3.88e−1
β
−9.10e−2 −1.08e−1
γ
−1.27e−1 1.03e−1
1
9.65e+1 9.77e+1
2
1.04e+2 1.04e+2
3
8.73e+1 8.78e+1
z = 3
x = 1
y = 2
Figu e 20.
