A Nurbs-Based Technique For The Segmentation Of Medical Images

Abstract

Extracting the human brain from magnetic resonance head scans is difficult because of its highly convoluted and nonuniform geometry. A technique based on Non-Uniform Rational B-Splines (NURBS) and energy minimising deformable models to extract the brain surface accurately from MR head scans is presented. The weighting parameter that comes with the NURBS definition is explored to attract the surface into the regions showing high curvature. The weight at each control point is adjusted automatically according to the curvature properties of the evolving surface. This process facilitates a deformable surface with increased local flexibility that adapts to complex geometrical features of the brain. The results show that the proposed model is capable of capturing the correct brain surface with a higher accuracy than the existing techniques. 1.

Full text

Page 1

A NURBS-BASED TECHNIQUE FOR THE SEGMENTATION OF
MEDICAL IMAGES
Ravinda G.N. Meegama and Jagath C. Rajapakse
School of Computer Engineering, Nanyang Technological University
Nanyang Avenue, Singapore 639798
e-mail: gayan@sentosa.sas.ntu.edu.sg, asjagath@ntu.edu.sg
Abstract
Extracting the human brain from magnetic resonance head
scans is difficult because of its highly convoluted and
nonuniform geometry. A technique based on Non-Uniform
Rational B-Splines (NURBS) and energy minimising de-
formable models to extract the brain surface accurately
from MR head scans is presented. The weighting parameter
that comes with the NURBS definition is explored to attract
the surface into the regions showing high curvature. The
weight at each control point is adjusted automatically ac-
cording to the curvature properties of the evolving surface.
This process facilitates a deformable surface with increased
local flexibility that adapts to complex geometrical features
ofthebrain. Theresults showthat theproposed modelis ca-
pable of capturing the correct brain surface with a higher
accuracy than the existing techniques.
1. Introduction
Magnetic Resonance (MR) head scans indicate the skull,
subarachnoid fluid and other anatomical structures that
hamper the visibility of the brain surface. Therefore, in or-
der to carry out an accurate morphometry in neurological
studies, the brain has to be separated from these external
structures. Such a segmentation routine is of immense as-
sistance for various clinical studies because the surface pat-
terns of the brain are considered to be natural pathways on
the subarachnoid system that can be used to access patho-
logical structures within the brain [10].
Several researchers have investigated the problem of
modeling the brain surface using deformable models [1,
7, 14, 15, 17]. Our research, an extension to “Dynamic
NURBS” [13], is motivated by the limitations found in the
previous techniques that require manual initialisation, user
interactionduringdeformation, poorlocalshapecontroland
redundant points to capture the brain surface accurately.
The approach presented in this paper emphasises on de-
veloping a surface model to segment the brain accurately
from MR head scans with minimum user intervention. The
main reason for the selection of a NURBS surface for our
algorithm is its inherent locally adaptable features [12]. In
other words, a movement of a particular control point (or a
change in the weight of a control point) alters the shape of
the surface segment lying immediately close to that point.
This is in sharp contrast to other deformable models based
on irregular mesh configurations where such changes tend
to propagate globally on the surface [1, 8, 9, 15].
The proposed method consists of two stages: First, we
obtain the brain matter using an approximate segmenta-
tion routine. Manual initialisation is avoided by taking the
output image from this procedure to construct the initial
NURBS surfacethat is positioned close to the actual bound-
ary of the brain matter. The second stage involves deform-
ing the above NURBS surface according to an energy min-
imisation algorithm and also, modifying the weight associ-
ated with each control point automatically depending on the
curvature properties of the evolving surface. This weight
modification forces parts of the surface near sulci (grooves
on the brain surface) to attract towards the cavities with-
out having to duplicate the control points near such regions.
By using a NURBS-based deformable model, we have suc-
ceeded in; (a) increasing the local flexibility of the surface
by automatically adjusting the weights of the control points,
(b) controlling the local shape with B-Spline basis functions
and (c) automating surface initialisation. Experiments to
determine the reliability of the proposed technique are car-
ried out on MR head scans of 16 healthy subjects.
2. Initial Brain Segmentation
The purpose of this stage is to obtain an initial surface lo-
cated close to the actual brain surface. This surface may not
necessarily be the true brain surface but an approximation
that will be deformed to conform to the actual boundary of
the brain matter. At the beginning, the voxelsof the original
MR head scans (Fig. 1(a)) are converted to have isotropic
dimensions of
mm
mations. The scans are then filtered using an edge preserv-
ing 3D neighbourhood averaging routine [5]. The filtered
?????????
? to facilitate distance transfor-

Page 2

(a) (b)(c)(d)
(e)(f) (g)
Figure 1: Approximate initial segmentation of MR head scans: (a) original MR head scan, (b) intensity clustered image, (c)
binary masking of gray and white matter, (d) boundary points of gray and white matter, (e) binary opening (f) binary closing
of the largest connected component and (g) masking with the source image.
imageis then clusteredinto fiveclasses; cerebro-spinal fluid
(CSF), gray matter (GM), white matter (WM), skull and
background (Fig. 1(b)). Next, a binary mask containing
GM and WM is generated (Fig. 1(c)). To open the narrow
paths that links the brain with other external structures, bor-
der points of brain matter are extracted (Fig. 1(d)) and a 3D
distance transformation [2] is applied to yield all the con-
nected regions located outside 2 pixels from these boundary
points resulting in a binary image (Fig. 1(e)). This distance
is chosen based on the percentage of CSF found between
the skull and the brain.
Other than the background region, the largest connected
component in the distance-transformed binary image is the
brain matter. A 3D binary closing [6] is applied to this
component and this morphed image (Fig. 1(f)) is masked
with the filtered MR source to obtain an approximate brain
matter (Fig. 1(g)) that will be used to construct the initial
NURBS surface.
2.1. Initial NURBS surface
The border points of the extracted brain, as shown in Fig. 2,
form a continuous stream of voxels around the brain mat-
ter. From this representation, we sample a set of coordinate
positions by rotating an axis about the centre of gravity of
the boundary in each slice (the
x-y plane and in the
secting points of this axis and the boundary give the control
points of the NURBS surface. The angle of rotation of the
axis at each sample determines the number of control points
needed.
? and
? directions are on the
? direction, respectively). The inter-
2.2. Energy Minimising Surface
The brain surface is a two-parameter vector-valued math-
ematical function
arbitrary point on the surface is parametrically expressed as
????? ??????????? ???????????? such that an
??? ?!??"$# ,
?%?&?'??#)(*? ??????? . The internal energy
+?,.-0/ at a point
??? ?!??"$# on the brain surface
? is defined as
13254?6'798:7<;>=@?BACA!DFE
G
H!I@JLK3M
M
M
M9N
8O7<;P=@?BA
N
;
M
M
M
M Q:R
ISKTJUM
M
M
M9N
8O7<;P=@?VA
N
?
M
M
M
MQ
RXW
JCJUM
M
M
M
N
Q
8O7<;P=@?BA
N$Y?N$Z
M
M
M
M
Q
R?W
[3]. The external energy
Q
K?M
M
M
M
N
Q
8O7<;>=[?BA
N$Y
Q
M
M
M
M
Q
R\W
K
Q
M
M
M
M
N
Q
8O7<;P=@?]A
N$Z
Q
M
M
M
M
QT^
(1)
where
_?,a`?(b?+dcSe?/ at a point
??? ?!??"$# on the brain surface
? is defined as

Page 3

(a)(b)
u
v
x
y
z
(c)(d)
Figure 2: Constructing the initial NURBS surface of the
brain: (a) approximately segmented brain slice, (b) bound-
ary of the brain, (c) extraction of the control points and (d)
tessellated NURBS surface.
f?g h?i[jlk3jSm:n?o?p p3qsr)tvu wyxzj<k?j m!n?o$p p?u{
(2)
where
image
at time
given by
t?|~}
and

j<k?j m!n?o$p p
gives the intensity of the
 at a point
k?j m!n?o$p. If
k3jSm!n?o€n‚ pdenotes a surface
k
ƒ
|~„ …?n?†*p, an active brain surface at time
ƒU‡‰ˆ is
k?j m!n?oPn?
‡Šˆ
p?qŒ‹V'Ž’‘”“
•?–V—
jl˜’jlk3jSm!n?o€n?Tp p'p
(3)
where
™ represents all the possible surfaces at time
ƒ and
f?j<k?j m!n?oPn? p'pšq››
œž]Ÿ  [¡
–]•
j9f?¢.£0i?jlk3jSm!n?o€n‚ p p
‡
f
gSh?i
j<k?j m!n?oPn? p'p p?¤]¥z¤V¦
gives the total energy of
k
at time
ƒ .
2.3. NURBS-based Deformable Surface Model
Let
points, having weights
surface
, at time
§©¨
ª
Ÿ«,
¬
qŒ…€n
ˆ
n?­ž­.­.n'®¯r
ˆ ,
°
qŒ…€n
ˆ
n?­ž­.­.n'±*r
ˆ be thecontrol
²
i
¢
Ÿ³
|F}?´
, of an active NURBS
k
ƒ , given by
k3j%¥&n'¦>n
ƒ
p?q
£$µ:¶
·
¢ž¸:¹©º
µ!¶
·
³
¸!¹¼»
¢
Ÿ³
j%¥&n'¦>n
ƒ
p
§
i
¢
Ÿ³
(4)
where
»
¢
Ÿ³
jL¥OnT¦Pn
ƒ
p?q
²
i
¢
Ÿ ³?½
¢
Ÿ¾
jL¥zp
½
³'Ÿ¿
j%¦?p
£?µ!¶
·À
¸!¹
º
µ:¶
·
Á
¸!¹
²
i
À
Ÿ
Á
½
À
Ÿ¾
j%¥!p
½
Á
Ÿ¿
jL¦$p
are the rational basis functions in which
B-Spline basis function of degree
½ÃÂ%ÄÅ gives the
such that
Æ
iLÇ
È
|ÊÉ
´
½ÃÂ
Ÿ
¹Ëj9̂p?qÎÍ
ˆ
n
if
otherwise
ÌÃ|τÌ
Â
n@Ì
Â
´
¶
p
…€n
and
½
Â
Ÿ
Å
jl̂p3q
jlÌ)rÐÌ
Â
p
½
Â
Ÿ
Å
µ!¶
j9̂p
Ì
Â
´
Å
rÐÌ
Â
‡
j9Ì
Â
´
Å
´
¶
rÐ̂p
½
Â
´
¶
Ÿ
Å
µ!¶
j9̂p
Ì
Â
´
Å
´
¶¯rÐÌ
Â
´
¶
where
[12].
Ì
Âare positive real numbers referred to as knots
2.4. Deformable NURBS Surface
The brain image in Fig.
3D patterns of the brain surface because it contains low
intensity CSF that fills sulci cavities.
stage of our technique, the approximate NURBS sur-
face obtained previously is deformed so that it wraps
itself onto the cavities and the hills on the brain sur-
face.The NURBS surface given in Eq.
be expressed in matrix form as
1(g) does not exhibit the true
In the second
(4) can also
where
kÑq
§&Ò&Ó§
q
Ô
§©¨
¹
Ÿ
¹
n
§©¨
¹
Ÿ
¶
n?­.­.­žn
is given by
§3¨
¹
ŸÕ
µ!¶
n
§©¨
¶
Ÿ
¹
n
§©¨
¶
Ÿ
¶
n?­.­ž­.n
§©¨
Ö
µ:¶
ŸÕ
µ!¶[×
Òand
Ó
qØj<ÙX¹
Ÿ
¹Vn[ÙX¹
Ÿ
¶?n?­ž­.­žn?ÙX¹
ŸÕ
µ!¶?n?Ù?¶
Ÿ
¹Bn?­.­.­žn?Ù
Ö
µ:¶
ŸÕ
µ!¶[p
Ò
Using Eq. (1), the total internal energy of the NURBS
surface
k
f?¢.£0i?j<kOp3qŒ…?­aÚ
§
ÒOÛ
§
(5)
where
Û
q››
œ.VŸ [¡
–]•
Ô9Ü
¶C¹
Ó
ÒÝ
Ó
Ý
‡
Ü
¹?¶
Ó
ÒÞ
Ó
Þ
‡
Ü
¶T¶
Ó
ÒÝ?Þ
Ó
Ý?Þ
‡
Ü
{
¹
Ó
ÒÝ?Ý
Ó
Ý?Ý
‡?ß
¹
{
Ó
ÒÞ'Þ
Ó
Þ'Þ
×
¤]¥à¤]¦
To compute the internal energy function in Eq. (5), the
partial integrations of the rational basis functions
tobe derived. Note thatfor thesakeof simplicity,thedegree
»
¢
Ÿ³have

Page 4

term
derive the first partial integrations of
á of the B-Spline basis functions will be omitted. We
âäãLåæ as:
â¼ç
ãLåæ]è%é&ê'ë>êTìSí?îðï?ñ
ã%åæ?ò
æ
è%ë?íSóõô@è ö>íC÷øó?ùXèSúËí
ûýü
and
â¼þ
ã%åæËèLéOêTëPê@ìSí?î
ï?ñ
ãLå æ?ò
ã
è%é!íSóÐè%é!íC÷øóÏÿ
?
è úËí
ûýü
where
óÐè??
íŒîè
???
è??aí@ê
???
è??aí@ê??????žê
and
???
ÿ
?
è??
í'í ,
óõôTèSö>íŒî
???
ô
è ö>íSóÐè ö>í??
ther, the second partial integrations are derived as:
?
ô@èSöàíSó
?
èSö>í ,
ó
ÿ
?
è úËí~î
???
?
è úËí ó?ù©èSú$í??
?
?
èSúËí ó?ù
?
èSúËí ,
ò
?
?
è??‚íî
?
ò
?
è??‚í????????????
è ?Tê"!$í@ê??#?
è%é&ê'ë?í,
֔%$
ï?ñ
ã%åæ'&)(
ÿ
?+*-,
ÿ
?
û
î
û
è%é&ê'ë>êTìSí?î
./.
ï?ñ
ãLåæ
ò
ã
è%é!í
ò
æ
è%ë?íî óÐèLézíS÷øó?ù3èSú$í0? Proceeding fur-
â
ç‚ç
ãLåæ
è%é&ê'ë>êTìSí?îðï?ñ
ã%åæ
ò
æ
è%ë?í!è
û
ó
?
ã
è%é!í?12?436587Ëó
ô
è ö>í?1Êí
û
?
and
â¼þ[þ
ã%åæ]èLéOêTëPê@ìSí?î
ï?ñ
ãLåæ
ò
ã
è%é!í:9
û<;
óÏÿ
?
?
èSú$í=?>3?5:?
;
óÏÿ
?
èSúËíA@
û
?
where
1
îŒ÷øó?ù3èSú$í and
;
îŒóÐè ö>íC÷
. Further,
â ç‚þ
ãLåæ
è%é&ê'ë>êTìSí?î
ï?ñ
ã%åæ
û
ó
ã
è%é!íC÷øóCB
?
è úËí=?D3
?
?
è úËíE5
?
ó
ô
èSö>íA1
û
?
where
óCB
æ
èLë$íÃî
ò
æ
è%ë?íSó?ù
?
èLë$í8F
ò
?
æ
è%ë?íSó?ùXè%ë?í ,
û
ç
î
?
û
?E?àé and
û
þ
î2?
where
û
?E?àë . The above partial integrations
are used to asses
The total external energy of
G
7 ,
G
? ,
G
7?7 ,
G
??? ,
G
7
? from
G .
H , using Eq. (2), is given by
IKJ?L
ñ
è
H
í©îM?KN
N
îPOQO
R
ç
å
þ?S?T)UWV
XZY
7
X [
ù
G]\
F
Y
?
X?[
ù
G^\_\a`
ö
`
ú
with
b?c
X?[
ù
G^\
îed
d
d
d
?
???gfih
b
X [
ù
G]\
d
d
d
d
ü
?Q???ÏèLéOêTë$í
in which
Therefore, the total energy of the NURBS surface
fjh
denotes convolution with a Gaussian.
H is
I
è
H
í3î
X?k
?ml
[
ùjn
[
?po
\
(6)
p
NURBS surface
brain surface
control
points
i,j
t
q
Figure 3: The NURBS surface is attracted toward the sulci
cavity when the weight
creased (dashed line indicates the surface location before
weight adjustment).
ï?ñ
ã%åæ of the control point
[:q
ô
å
? is in-
The energy minimising process of a deformable surface,
given in Eq. (3), is solved iteratively by considering the
Euler-Lagrange equation
equilibrium to Eq. (6) from which the new locations of
the control points inare computed. Subsequently, a new
NURBS surface is fitted over
adjusted by modifying the weights of the control points as
described next.
r
I
è
H
íÊî
k
. It gives a state of
[
[
whose local shape is further
2.5. Weight Adjustment
When the deformable NURBS surface evolves through the
energy minimising states, some segments may not be sit-
uated close to the actual brain matter. In such a case, the
onlyway tomovethesurfacetowards theseregions,without
translating the control points, is by increasing the weight of
therelevantcontrol points [11]. Moreover,thisweight mod-
ification must not effect parts of the surface located outside
sulci cavities.
As shown in Fig. 3, let
a sulci towards which the NURBS surface
attracted by adjusting the weight of the control point
(note that
the NURBS form of the evolving surface becomes
s
be a point at the bottom of
H
needs to be
[
ñ
ã%åæ
s is not a control point). If
H is at
s at time
ìtF
?,
u>u
ï
ñ
B
?
ã%åæ
ò
ã
è%é!í
ò
æ
èLë$í
[
ñ
ãLåæ
u>u
ï
ñ
B
?
ã%åæ
ò
ã
è%é!í
ò
æ
è%ë?í
?
upu
ï
ñ
ãLåæ
ò
ã
è%é!í
ò
æ
è%ë?í
[
ñ
ã%åæ
u>u
ï
ñ
ãLåæ
ò
ã
è%é!í
ò
æ
è%ë?í
î
s
?
H
è ö!ê?úPê_vSí
(7)
Note that during weight adjustments, the position of a
control point remains unchanged; i.e.
weight change is denoted by
can be written as
[
ñ
B
?
ãLåæ
î
[
ñ
ãLåæ . If the
w
ïdñ
ãLåæ
î
ï
ñ
B
?
ã%åæ
?
ï?ñ
ãLåæ , Eq. (7)

Page 5

û
X
./.
ïdñ
ãLåæ?ò
ã
è%é!í
w
ò
æ
èLë$í
[
ñ
ãLåæ
F
./.
w
ïdñ
ãLå æ?ò
ã
è%é!í
ò
æ
è%ë?í
[
ñ
ã%åæ
\
?
./.
ï?ñ
ã%åæ
ò
ã
è%é!í
ò
æ
è%ë?í
[
ñ
ã%åæ
X
û
F
./.
w
ï?ñ
ãLåæ
ò
ã
è%é!í
ò
æ
èLë$í
\
î
û
X
û
F
./.
ï?ñ
ã%å æ‚ò
ã
èLézí
ò
æ
èLë$í
\
è
s
?
H
èSö!ê?ú€ê_v'í í
(8)
Multiplying Eq. (8) by
?
?
û
yields
(
ÿ
?
x
ã?y
?
,
ÿ
?
x
æ?y
?
w
ï
ñ
ã%åæ
ò
ã
èLézí
ò
æ
è%ë?í
X
[
q
ô
å
?
?
s
\
îè
s
?
H
èLéOêTëPê@ìSí í
û
èLéOêTëPê@ìSí
(9)
Now, Eq. (9) gives the weight changes
the deformable NURBS surface
because it causes a perspective functional translation of the
points on the effected NURBS segment toward
Due to the continuous nature of the NURBS surface,
an analytical solution for the curvature of the surface near
the control point
between and the average vector
connected neighbouring control points of
w
ï?ñ
ãLåæ to expand
H towards the sulci cavity
[aq
ô
å
?.
[:q
ô
å
? is computed by taking the distance
[aq
ô
å
?
z
è
[aq
ô
å
?
í of all the
[
q
ô
å
?, denoted by
{
è
[aq
ô
å
?
í, which approximates the surface curvature at
The approximate surface curvature at
where
[aq
ô
å
?.
[aq
ô
å
?,
|
ñ
ã%åæ , is given by
|
ñ
ãLåæ
î
d
d
[aq
ô
å
?
?
z
è
[aq
ô
å
?
í
d
d
ü
z
è
[
q
ô
å
?
í©î
?
}
}
{
è
[
q
ô
å
?
í
}
}
x
~€
?ƒ‚ „??
T+…
R
~€

‚„
S
[
q
ô
?
å
?0?
in which
}
}
{
è
[aq
ô
å
?
í
}
}gives the number of control points
in
(say, the average surface curvature
weight
{
è
[aq
ô
å
?
í . If the surfacecurvature
|
ñ
ã%åæ exceeds a threshold
î
?
(
*-,
./.
|
ñ
ãLåæ ), the
ï?ñ
ã%åæ is updated such that
ï
ñ
B
?
ãLåæ
î
ï
ñ
ãLåæ?FD†
|
ñ
ãLåæ
‡‰ˆ?Š
ãLåæ
$
|
ñ
ãLåæ
&
where
the surface towards the control point.
control of the surface, due to weight adjustments, comes
from the B-Spline basis functions where
forThe knots for the two B-Spline
basis functionsand
vectors
tively. Each knot pair
†P?Œ‹
controls the amount of attraction of
This local shape
ò?
å
Ž
è??‚íÎî
k
???‘?

ê??

B
Ž
B
?
í .
ò
ãLå
’
è%é!í
ò
æ'å
“
èLë$í
are given by the
è%é
?
êTé
?
ê+?”???.ê'é
(
B
’
íCù and
è%ë
?
ê'ë
?
ê????”?.ê'ë
,
B
“
íCù , respec-
èLé
ã
ê'ë
æ
í relates the parameters
é and
ë to a weight
only the surface segment
ï?ñ
ãLåæ such that an adjustment of
ï?ñ
ãLåæ affects
H
è%é&ê'ë>êTìSí ,
èLéOêTë$í??4
é
ã
ê'é
ã
B
’
B
?
í
?

ë
æ
êTë
æ
B
“
B
?
í [12]. Finally, the brain matter is extracted af-
ter removing all the voxels situated outside the deformed
NURBS surface.
Figure 4: MR image slices of the brain: (top) original MR
images, (middle) NURBS surface initialised close to the
brain and (bottom) brain slices segmented with NURBS-
based deformable surface.
3. Results
To validate the preciseness of the proposed technique, 16
MR head scans of healthy volunteers, between the ages
of 6 and 27, were processed. The actual boundary of the
brain surface was extracted by manually outlining the outer
perimeter of the brain matter of each slice.
Fig. 4 depicts segmented results of an MR head scan.
Although our algorithm is implemented in 3D, the images
are shown as 2D slices for enhanced clarity. The accuracy
and the shape difference of the NURBS-based and manual
segmentationwasmeasuredusingtheaveragedistanceerror
between the two surfaces [16]. It is the average distance
between each point on the deformed NURBS surface
the closest point on the actual brain surface
distance error ,
• and
–• . The average
— , is given by
—€˜"•=™
–
•jš8›œ
ž Ÿ8Ÿ
¡”¢)£¤?¥"¦)§%¨i©«ª
¡?¢_¬ £ ¤?¬­¥?¦¯®§±°
°
°
•:˜³²´™?µ¶š=·
–
•8˜ ²¹¸ ™?µ'¸?š
°
°
°?º
²
º
µ