HDM: AN HETEROGENEOUS STRUCTURES
DEFORMATION MODEL
Mon se Bigo da Dani Tos
27 h Ma ch 1998
Keywo ds:
Volume de o ma ions - FFD - Medical Applica ions- Volume Da a
1 In o duc ion
In he las decade Compu e G aphics applica ions in medicine ha eg own due
o he ad ances o inpu echnologies such asMRAandCTwhich enable he
cons uc ion o h ee-dimensional ep esen a ion o ana omical s uc u es and
hei isualiza ion. Volume da a ep esen a ion schemes ha e b een s udied, pa -
icula ly he oxel mo del o egula inpu da a Kau90] and, o non- egula
da a se s, he e ahed al cell mo del CFM
+
94] along wi h mo e compac ep e-
sen a ions such as o c ees WG92], mul i e as, and equency domain ep esen-
a ions, as wa ele es Mu 93]. Ma jo a en ion has b een paid o he p oblems
o he accu acy o he ep esen a ions, hei memo y equi emen s and hei
adequacy o he isualiza ion. Howe e , hese mo dels a e gene ally s a ic: he
ep esen a ion o he emp o al e olu ion o olume da a, pa icula ly hei de-
o ma ion has b een less add essed.
Die en medical applica ions need he simula ion o de o ma ions o he shap e
o ana omical s uc u es. As an example, in oncological s udies, i is o en
necessa y o simula e o o p edic he g ow h o a umo whichmay p o duce
he de o ma ion o he su ounding s uc u es and he e o e p o oke seconda y
pa hologies. A pa icula case o his, a e he emb olisms caused by he s enosis
o a ce eb al essel due o he p essu e o a b ain umo . O he examples o
de o ma ions in medicine a e hose p o duced by ex e nal o ces, such as he
bis u i p essu e.
In his pap e , he p oblem o he de o ma ion o he e ogeneous olume da a
se s is analyzed. A gene al me ho d is p op osed, enabling he simul aneous de-
o ma ion o b o h he in e io and he shap e o a ious imb ica ed s uc u es.
The me ho d supp o s die en de o ma ion mo dels acco ding o he pa icu-
la hyp o hesis o each s uc u e elas ic b eha io . The pap e is s uc u ed in o
h ee sec ions:  s he p e ious wo k is e iewed, nex he p op osed me ho d
is desc ib ed and nally some simula ion examples a e discussed b e o e he con-
clusions.
1
2 Backg ound
2.1 P e ious wo k
Shap e de o ma ion consis s in he mo dica ion o ei he he geome y o he
op ology o a su ace. The e a e wo main de o ma ion b eha io s: elas ic ones,
which a e asso cia ed o geome ical mo dica ions, and non-elas ic ones, which
co esp ond o s uc u al changes such as ac u es. Mos o he de o ma ion li -
e a u e has o cused a shap es ep esen ed explici ly ei he as p olygonal meshes
o smo o h sculp u ed su aces, al hough, ecen ly, some pap e s add ess he de-
o ma ion o disc e e, oxel-based, su aces.
Su ace de o ma ion has b een mainly used o he ollowing pu p oses:

Simula ion o physically ealis ic de o ma ions such as hose p o duced by
ob jec s collisions and ma e ial hea ing and usion. TW88], KT91].

Smo o h su aces in e ac i e design by successi e de o ma ions o an ini ial
ough shap e SP86], Co q90], HHK92] and sculp u ing o disc e e su aces
in bina y oxel mo dels (GH91],WK95])

Anima ion o non- igid s uc u es, such as a icula ed o legged gu es
GVP91] and blobby mo dels (WMW86]).

Mo phing b e ween woshapes KCP92], BN92], CSB95] and mo phing
be ween wo olume da a se s Hug92], LGL95].

Segmen a ion o egions o in e es in 2D images and 3D olume da a
by successi e de o ma ions o a emp a i e cu e (snake) KWT88]o a
p olyhed al app oxima ion MBL
+
91](GW92]) o he shap e o he egion.

Ma ching b e ween wo geome ical o olume mo dels in o de , o ins ance,
o compa e he ana omy o a pa ien wi h a p e iously c ea ed ep esen-
a ion o an ana omical a las mo del (BK89]) o o ma ch wo die en
mo dali ies o egis a ion Da 97]

Recons uc ion o a bina y disc e e 3D mo del om a se o semi- anspa en
image p o jec ions o i (Mu 91]).
Shap e de o ma ion mo dels all in o h ee main ca ego ies: kinema ic, dynamic
and mo dula mo dels. The kinema ic mo dels enable o compu e he de o med
mo del on he basis o geome ical in o ma ion only, ypically byin e p ola ing
new p osi ions using a use -sp ecied subse o displacemen s. Two main die -
en kinema ic app oaches ha e b een published: he applica ion o non-linea
geome ical ans o ma ions such as b ending, ap e ing and wis ing (Ba 84],
CR94]and F ee-Fo m-De o ma ions (SP86]) along wi h hei nume ous ex-
ensions (Co q90], CJ91], HHK92]). Dynamical mo dels simula e physically
ealis ic b eha io s and mo del he de o ma ions p o duced by o ces and o ques
acco ding o physical laws o elas ici y and ma e ials esis ance TF88]. Mo du-
la mo dels, also called
laye ed
mo dels, a e based on a mul i-le el ep esen a ion
o he ob jec s: a  s simplied laye , o which dynamical de o ma ions can b e
applied, and a second laye , comp osed by he su ace o skin o he s uc u e
ied o he ke nel laye acco ding o kinema ic cons ain s GVP91].
2
2.2 S a emen o he p oblem
Mos o he p e iously ci ed de o ma ion mo dels a e
su ace-based
, i.e. hey
assume ha he ob jec s a e emp y and hey do no add ess he impac o he
su ace de o ma ion on hei in e nal s uc u e CEO
+
93]). This hyp o hesis,
alid in nume ous cases, is no longe accep able in many medical applica ions
whe e he eal b eha io o he ob jec s should b e simula ed. Se e al a emp s
ha e b een done o mo del he 3-D olume ic na u e o he ob jec s suchas
CEO
+
93] and BNC96] which gene alize o 3D he elas ic de o ma ion mo del,
using a 3D ni e elemen mesh, in o de o b e e simula e su gical cu s. How-
e e , hese mo dels assume ha he in e io o he ob jec s is homogeneous o ,
a leas , ha he p op e y alues in hei in e io emains unmo died unde
de o ma ion. In addi ion, he de o ma ion is applied o a single 3D ana omical
s uc u e mo del ex ac ed om medical images.
Ne e heless, ana omical egions a e comp osed o a ious homogeneous and he -
e ogeneous imb ica ed s uc u es. The su ace o an ana omical s uc u e canno
always b e de o med aside om he su ounding egions, b ecause i s de o ma ion
mayp o oke he de o ma ion o he ex e nal and he in e nal s uc u es as well.
Figu e 1 illus a es his idea. In gu e 1.a a mo del comp osed o wo- egions
is ep esen ed schema ically wi h wo die en ll-a ea pa e ns. The su ace
o he in e nal egion has b een iden ied. In gu e 1.b, his su ace has b een
de o med indep enden ly om he p op e y alues o he su ounding olume.
The eec p o duced is ha he b ounda y no longe encloses he ci cle-pa e n
egion. Figu e 1.c illus a es he desi able esul , in which he olume p op e ies
ha e b een mo died in acco dance o he su ace displacemen .
Figu e 1: In e ela ionship b e ween olume and su ace de o ma ions
He ein a gene al amewo k o he de o ma ion o olume mo dels is p op osed.
I s main ea u e is ha i is
hyb id
: i enables de o ma ions o b o h he su aces
and he olume. This is accomplished by p opaga ing he su aces displacemen s
o hei in e nal and o hei su ounding olume and he e o e, o he o he
emb edded s uc u e b ounda ies. This gene al amewo k is sui able o any
de o ma ion mo del, kinema ic as well as dynamic. He ein howe e , a sp ecic
de elopmen is analyzed, based on a kinema ic mo del wi h cons ain s. This
mo del will b e e e ed as HDM
(Hyb id De o ma ion Model)
in he es o he
pap e .
3
3 The Hyb id De o ma ion Mo del
3.1 Gene al amewo k
The pip eline o he p op osed amewo k is illus a ed in gu e 2. The ep e-
sen a ion mo del is a g ey-le el, he e ogeneous oxel mo del emb edding die en
egions co esp onding o a ious ana omical s uc u es: o gans, b ones e c. This
mo del is ob ained by egis a ion o da a using medical de ices suchasCTsand
MRs and by applying l e ing and segmen a ion p o cesses whiche en ually e-
mo e he noise o he inpu slices and enable he iden ica ion o he ana omical
egions. The b ounda y su aces o he s uc u es a e no ep esen ed explici ly
bu hey can b e ex ac ed om he oxel mo del ei he using a ma ching cub es
algo i hm o bycon ou ex ac ion and iling b e ween successi econ ou s. The
applica ion ecei es as an inpu a se o displacemen s o p oin s b elonging o he
su ace o one o mo e ana omical s uc u es in e io o he oxel mo del. These
inpu p oin s can b e ob ained in die en ways: hey can b e, o ins ance, he
co o dina es o an elec onical scalp el p essing he su ace o an o gan, o hey
can b e in e ac i ely sampled on he su ace o a umo con iguous o ana omical
s uc u es, whose de o ma ion is b eing s udied. Wi h hese inpu da a and he
o iginal mo del, he applica ion compu es he shap e de o ma ion and he esul -
ing mo dica ion o he p op e ies o he oxels inside and ou side he s uc u e.
The ou pu o he applica ion is a new g ey-le el oxel mo del emb edding he
de o med egions. This new mo del can b e manipula ed as he o iginal one, i.e.,
i can b e isualized, he de o med su aces o he inne egions can b e ex ac ed
om i , e c.
model oxel model De o ma ion
Voxel
displacemen s
Su ace poin s
De o med
oxel model
s uc u es
iden i ica ion
S uc u es Segmen ed
classi ied
Figu e 2: Pip eline o he p op osed me ho d
3.2 Desc ip ion o he me ho d
3.2.1 The da a
As men ioned ab o e, he disc e e ep esen a ion o he olume ic mo del b e o e
he de o ma ion,
V
,is a oxel mo del such ha :
V
=
ij k
j
p op
(
ij k
)=

ij k
g
(1)
whe e he oxel
ij k
is cha ac e ized by i s p op e y alues, o ins ance, i s
densi y

ij k
.
4
The die en egions, o ana omical s uc u es, inside he oxel mo del ha e
b een p e iously segmen ed in suchaway ha each egion has an unambiguous
p op e y alue ange.
1
. The egion su ace pass h ough he b ounda y oxels
cha ac e ized by a non-homogeneous neighb o ho o d. In addi ion o he classical
ans e unc ions which asso cia e o he die en p op e y anges opaci yand
colo alues, new ans e unc ions ha e b een designed which p o ide elas ical
p op e ies o each ange. These unc ions a e empi ical, based on he physicians
knowledge.
The inpu da a o he de o ma ion,
D
, a e pai s o homologous p oin s o he
egion su aces b e o e and a e he de o ma ion:
D
=
(
p
1
p
0
1
)
 :::
(
p
n
p
0
n
)
j8
i
=1
::n p
i
2
Sp
0
i
2
S
0
p
0
i
=
De o m
(
p
i
)
g
(2)
whe e
S
is he ini ial su ace s uc u e and
S
0
is he de o med su ace s uc u e.
3.2.2 The de o ma ion
The p op osed me ho d is comp osed o h ee consecu i e s eps, called
Iden ica-
ion
,
De o ma ion
and
Res uc u ing
.
A he  s s age, he su ace o in e es is iden ied inside he oxel mo del
V
.F om he inpu da a
D
o he desi ed de o ma ions and, applying a su ace
de o ma ion echnique, he de o ma ion s age de o ms he whole oxel mo del,
ob aining a non- egula la ice mo del. The es uc u ing s ep compu es a eg-
ula oxel mo del equi alen o he non- egula la ice mo del.
Eachs epmay b e p e o med acco ding die en s a egies. Nex a pa icu-
la implemen a ion o his pip eline is desc ib ed. Howe e , any o he su ace
iden ica ion and su ace de o ma ion could ha e b een applied as well.
Iden ica ion
As men ioned ab o e, he HDM de o ms b o h he su ace and
he olume. The e o e, b o h in o ma ion mus b e ep esen ed simul aneously
and p oin one o each o he . Thus, he equi emen o he su ace iden ica ion
is o c ea e a p olygonal mo del o he su ace, o which a su ace de o ma ion
mo del could b e applied, while p ese ing in o ma ion o he oxels o which he
su ace aces b elong.
The echnique mos used o ex ac a su ace om a oxel mo del, is he
Ma ching
Cubes
echnique LC87], which gi es an app oxima ion o he su ace as a se
o up o h ee p olygons inside each oxel, using ilinea in e p ola ions o he
p op e y alues o compu e he su ace e ices. The ma ching-cub es su ace
mo del is complex, made o many small aces, and i lo oses he in o ma ion o he
oxels o which he e ices b elong, unless sp ecic da a s uc u es a e designed
o keep his in o ma ion.
He ein, a simple app oxima ion o he su ace is used: he cub e ille mo del
UG93] which is comp osed o he b ounda y aces o he b ounda y oxels. Be o e
1
In MRA da a, ep esen ing ascula in o ma ion, his ange segmen a ion is no always
easible, as ascula s uc u es do no co esp ond o sp ecic anges bu o lo cal maxima. In
his case, he olume mo del is cons uc ed a e he segmen a ion using lab elling p op e y
alues ins ead o he o iginal da a.
5

he de o ma ion, his su ace is comp osed by iso he ic aces which a e pa allel
o he co o dina e planes. A e he de o ma ion, hese aces a e ob iously no
longe iso he ic. Al hough he su ace ep esen a ion is ough, i has a lowe
memo y equi emen han he ma ching cub es mo del, b ecause i needs o s o e
only one e ex p e oxel. Bounda y oxels and hei b ounda y aces can b e
compu ed on he y.
De o ma ion
The de o ma ion s age i sel p e o ms wo ela ed p o cesses: he compu a ion
o he displacemen s o he e ices o he oxels unde de o ma ion
su ace-
de o ma ion
and he compu a ion o he new p op e y alues o he de o med
oxels
olume de o ma ion
.
The su ace-de o ma ion implemen ed he ein is an ex ended kinema ic mo del
based on he
F ee-Fo m de o ma ion (FFD)
SP86] me ho d. I consis s o h ee
s eps:
1.- C ea ion o pa allelepip ed egula la ice enclosing he cub e ille su ace o
he whole ana omical s uc u e. A lo cal co o dina e sys em is asso cia ed o
his la ice, ha ing a e ex o he la ice
P
0
as he new o igin and b eing
he main di ec ions
S
,
T
,
U
pa allel o he la ice.
The e ices
V
ij k
o he la ice cells (
con ol poin s
) b e o e he de o ma ion
a e:
V
ij k
=
P
0
+
i
l
S
+
j
m
T
+
k
n
U
(3)
whe e l, m, n a e he numb e o sub di isions o he mesh in each di ec ion.
2.- Sp ecica ion o he de o ma ion in e ms o displacemen s o he con ol
p oin s o he la ice. As he inpu de o ma ions
D
a e p oin s o one o
mo e cub e ille su aces b eing de o med, he displacemen s o he con ol
p oin s should  s b e compu ed. This p oblem has b een add essed in
HHK92]. I equi es he compu a ion o he pseudo-in e se o a ma ix
o de i e he displacemen s o he con ol p oin s ha minimize he e o
be ween he sp ecied displacemen s and he ac ual de o ma ion using a
squa ed die ence e o me ics. As p oin ed ou in LWCS96], he pseudo-
in e se ma ix compu a ional cos is e y high when he numbe o poin s
ha should b e mo ed is la ge. The e o e, he p op osed me ho d uses he
app oacho LWCS96] which p o duces simila esul s o HHK92] wi hou
calcula ing he pseudo-in e se ma ix.
The in e ace o he p oin s de o ma ion sp ecica ion es ic s he ange o
he allowable displacemen s and gua an ees ha he FFD la ice is s ill
s uc u ed and ha i s cells do no au o-in e sec .
In addi ion, some simple dynamic cons ain s can b e aken in o accoun ,
p e en ing he e ices o some s uc u es in e io o he de o med cu-
b e ille su aces om b eing mo died. This enables, o ins ance, o de o m
s uc u es such as he skin while keeping he b one unmo died. In o de o
keep igid s uc u es, he closes la ice e ices enclosing hem a e xed
and again he in e ace p e en s he sp ecied de o ma ions o b eak he
egula s uc u e o he la ice.
6
3.- Compu a ion o he displacemen s o he e ices o he cub e ille su -
ace and o he e ices o he inne s uc u e oxels. All he e ices
a e de o med excep hose b elonging o igid s uc u es. The egula de-
o ma ion o a e ex is compu ed acco ding o he o mula p op osed in
SP86]:
P
d
=
l
X
i
=0

l
i

(1
;
s
)
l
;
i
s
i
2
4
m
X
j
=0

m
j

(1
;
)
m
;
j
j
"
n
X
k
=0

n
k

(1
;
u
)
n
;
k
u
k
V
ij k
#
3
5
(4)
The olume de o ma ion consis s o compu ing he p op e y alues o he de-
o med oxels. The olume is conside ed as an he e ogeneous se o a ious
s uc u es which a e hemsel es homogeneous b e o e and a e he de o ma ion.
Rigid s uc u e oxels emain unmo died, whe eas oxels in e io o de o med
shap es ha eacons an mo died alue. In he de o ma ion, he olume o he
s uc u es ei he inc eases, dec eases o emains cons an . The densi y compu-
a ion is based on he hyp o hesis ha he ma e quan i y emains cons an
h ough de o ma ion and hus, he changes in he densi y alues a e p op o -
ional o he mo dica ion o he olumes. Finally, as a esul o he de o ma ion
s age a non- egula de o med la ice mo del is ob ained.
Res uc u ing
A his p oin o he pip eline, he o iginal oxel mo del has b een de o med and i is
now non- egula al hough i is s ill s uc u ed and op ologically equi alen o i s
ini ial shap e. The es uc u ing s age allows o egula ize i , while keeping he
de o ma ions o he inne s uc u es. This s age is hus essen ially equi alen
o a e- oxeliza ion Han90], al hough, by opp osi e o o he e- oxeliza ions
p o duced by ane ans o ma ions such as o a ions o he olume i has o
deal wi h a non-ane ans o ma ion.
The equi alen egula oxel mo del is compu ed as he same esolu ion as he
o iginal one. Howe e i may b e compu ed a any esolu ion as well. The
es uc u ing s age consis s hus in a scanning he egula oxel mo del and
compu ing o each oxel which oxels o he de o med mo del ha e a non-ze o
in e sec ion wi h i .
I should b e no iced ha a oxel o he la ice mo del can co esp ond o one o
mo e oxels in he oxel mo del. I he co esp ondence is unique, he egula
oxel p op e y is simply se o he de o med oxel p op e y.Howe e i mo e
han one de o med oxel in e sec s he egula one, hei esp ec i e p op e y
alue a e weigh ed and accumula ed (see gu e 3).
4 Simula ions and esul s
Colo Pla es 1 o 8 show he esul s o a 2D p o o yp e simula ion o he de o -
ma ion mo del. Being a 2D, i would b e mo e p op e o alk in e ms o pixels
7
Figu e 3: Res uc u ing
a he han oxels, howe e , o cohe ence wi h he es o he pap e , he e m
oxel has b een p e e ed.
In Colo Pla e 1 he o iginal image da a o a CT scan o a head a e isualized.
Two cub e ille su aces a e iden ied in he image: he ex e nal su ace o he
b ain and he su ace o a umo , depic ed in blue and ed esp ec i ely in Colo
Pla e 2. The elas ic b eha io o b o h su aces is conside ed iden ical.
The FFD ne compu ed as he b ounding b ox o he whole ob jec is shown in
Colo Pla e 3. F om sp ecied alues o displacemen s o se e al p oin s o he
b ain su ace, he de o ma ion o he con ol e ices o he FFD a e compu ed
and ep esen ed in Colo Pla e 4.
Colo Pla e 5 shows he esul s o he de o ma ion on he olume. In o de o
allow a b e e unde s anding o he image,
mac o-pixels
o 10x10 a e ep esen ed.
I can b e obse ed ha he o iginal oxel mo del is no longe iso he ic. The
de o med cells b eha e as closed compa men s which
d ag
he ma e inside
hem in hei de o ma ion. The colo o he cells changes acco ding o he
mo dica ion o he densi y alue inside he cell. The new densi y is compu ed
as he p e ious alue o densi ymul iplied by he a ion b e ween he p e ious
oxel a ea and he new oxel a ea. The de o ma ion has b een applied only a
he oxels which a e inside and on he cub e ille su ace o he b ain. Being
inside he b ain, he umo is also de o med. Colo Pla e 6 shows he olume
once he es uc u ing s ep has b een p e o med. The gene al asp ec o he image
is qui e simila o Colo Pla e 5. Howe e he oxels a e now pa allel o he
co o dina e axis. The oxels which all comple ely inside a de o med s uc u e
a e conside ed homogeneous and he e o e hey a e no mo died. The oxels
which exhibi die ences wi h Colo Pla e 5 a e hose ha in e sec he su ace,
b ecause hei alue is compu ed as a weigh ed a e age o he de o med oxels
whichco e hem.
Finally Colo Pla es 7 and 8 show a mo e complex de o ma ion.
5 Conclusions and u u e ends
A gene al amewo k o he 3D de o ma ion o mul iple s uc u es ep esen ed
implici ly in a olume ic oxel mo del has b een p op osed. The mo del is hyb id
in he sense ha i enables he de o ma ion o a su ace inside he olume
8
mo del and he mo dica ion o he in e nal p op e y alues as a esul o he
comp ession o expansion o he de o med su ace. The su ounding olume is
also mo died in ela ion o he de o ma ion.
A  s kinema ic p o o yp e implemen a ion o he mo del on 2D images, based
on he use o FFD has b een desc ib ed. The esul s o he simula ions a e
encou aging and make i necessa y o implemen he h ee-dimensional e sion
o he mo del. This implemen a ion is cu en ly b eing done.
The in eg a ion o dynamic cons ain s o he mo del is ano he esea ch line
unde p og ess. Up o know only homogeneous and igid b eha io a e allowed.
The use o he ue elas ic p op e ies should b e enabled. Non-homogeneous
p opaga ion o he de o ma ion hough he olumes p op e ies should also b e
s udied.
Finally, a u u e ex ension o he me ho d is i s applica ion a die en le els o
esolu ion o he oxel s uc u e, enabling highe p ecision in zones o in e es s
and coa se de o ma ion in a eas o less ele ance.
Re e ences
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ACM
Compu e G aphics
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pu e Vision, G aphics and Image P ocessing
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BN92] T. Beie and S. Neely.Fea u e-based image me amo phosis.
ACM
Compu e G aphics
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BNC96] M. B o-Nielsen and S. Co in. Real- ime olume ic de o mable mo d-
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Eu og aphics96
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93] S.A. Co e , N.F. Ezque a, J. O'B ien, R.Rowe, T.Gadacz, and
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