3D BENDING OF SURFACES AND VOLUMES WITH AN
APPLICATION TO BRAIN TORQUE MODELING
Antonietta Pepe and Jussi Tohka
Department of Signal Processing, Tampere University of Technology
P.O. Box 553, FIN-33101 Tampere, Finland
Keywords:
Space deformation, 3D surfaces and volumes bending, Brain asymmetry, Magnetic resonance imaging.
Abstract:
In this work, we propose a novel space deformation model for local bending of 3D volumes and surfaces.
The model can be easily controlled through accommodation of a few intuitive parameters. Experiments on
volumes, parametric surfaces, and polygonal surfaces show that our method has increased modeling capabil-
ities when compared to the previous space deformation methods for local bending. We apply this new, more
flexible model for space bending to model human brain asymmetry. In particular, we develop an image pro-
cessing pipeline for automatic generation of a set of realistic 3D brain magnetic resonance (MR) images for
which the asymmetry is known. This dataset can be used for the quantitative validation of voxel and surface
based methods for studying brain shape asymmetry. The pipeline encompasses a realistic modeling of the
anatomical rightward bending of the inter-hemispheric fissure in human brain.
1 INTRODUCTION
Space Deformation (SD) methods are efficient and in-
tuitive class of methods for deforming multidimen-
sional shapes. In SD methods, the object’s shape
is modified by deforming the underlying space in
which the object is embedded regardless of the par-
ticular shape representation employed. Besides the
scaling, translation, rotation and affine transforma-
tions, more sophisticated parametric remappings of
the space (e.g. the global linear bending) have been
proposed (Barr, 1984).
Free Form Deformation (FFD) methods (Sederberg
and Parry, 1990) are another class of shape morphing
techniques that enable the deformation of geometric
models through deformations of the underlying space.
In FFD techniques, the object to be deformed is first
embedded in a lattice. Control points of the lattice are
then moved to control the space deformation.
Even though FFD methods can express a larger num-
ber of deformations than SD methods, there are still
some advantages in using SD. First, the control lattice
used to deform the space in FFD is not directly related
to the coordinates of the object being deformed. Due
to this indirect framework, controlling the deforma-
tion requires expertise. Second, SD provides a more
compact way to encode the deformations if compared
to the high number of control points which are re-
quired in FFD. Related to this, moving the control
points can be time consuming (Xiaogang et al., 2001;
Hsu, 1992).
In this work, we used SD based methods for modeling
of the bending deformation. The main contributions
of this work can be summarized as follows:
1) We extended the linear bending deformation
proposed in (Barr, 1984) to allow more flexible trans-
formations of the space, as well as to include con-
straints on the deformation while maintaining the sim-
plicity of the model. The usefulness of the proposed
bending deformations,in terms of increased modeling
power, is demonstrated in this work.
2) The two most consistently reported patterns of
normal asymmetry in the human brain anatomy are
the protrusion of one hemisphere over the other at
the frontal and occipital lobes (petalia), and the right-
ward bending of the inter-hemispheric fissure (known
as brain torque or Yakovlevian anti-clockwise torque)
due to the protrusion of the left hemisphere over the
right one in the occipital region (Toga and Thompson,
2003). In this work, we develop a realistic model of
the left occipital right frontal petalia and of the brain
torque deformations. The proposed model is suitable
for any brain shape representation.
3) There exist automatic methods for the statical
shape analysis of the anatomical asymmetries in the
human brain based on 3D magnetic resonance (MR)
images. In such methods, the brain shape asymmetry
can be either analyzed through morphometric voxel
411
Pepe A. and Tohka J. (2012).
3D BENDING OF SURFACES AND VOLUMES WITH AN APPLICATION TO BRAIN TORQUE MODELING.
In Proceedings of the 1st International Conference on Pattern Recognition Applications and Methods, pages 411-418
DOI: 10.5220/0003773804110418
Copyright
c
 SciTePress

level measures computed from flipped and unflipped
MR images (Voxel Based Morphometry) (Ashburner
and Friston, 2000; Good et al., 2001), or through mea-
sures computed along a surface mesh representation
of the cerebral hemispheres at corresponding loca-
tions on the two sides of the brain (Surface Based
Morphometry) (Thompson et al., 1997; Pepe et al.,
2011). In this work, we have developed an automatic
image processing pipeline for the generation of vali-
dation databases for voxel and surface based methods
for the analysis of brain shape asymmetry from 3D
MR images.
2 BACKGROUND
Space Deformations. SD techniques modify the
shape of objects by deforming the space in which they
are embedded. The space deformation can either be
applied to the whole space, S = F (s), or defined lo-
cally, J
i
(s) = ∂F(s)/∂s
i
(Barr, 1984; Sumner, 2005).
In the above expressions F : R
n
:→ R
n
denotes the
global transformation function, J denotes the Jaco-
bian matrix of the transformation F , s denotes the
object embedded in the space operated by F , and S
denotes the resulting deformed object.
Often, it is useful to calculate the normal and tan-
gent vectors of the deformed object S. These can
be used to obtain the surface’s orientation, to derive
the reflectivity of a light source onto a surface for
lighting, to add details to a surface, e.g., roughness
to a flat surface, and for surface interpolation (shad-
ing). The computation of normal and tangent vec-
tors can be a computationally intensive task
1
, espe-
cially for polygonal meshes having thousands of ver-
texes. However, the tangent t
S
of the deformed ob-
ject S can be computed directly from the tangent nor-
mal of the undeformed object (t
s
) through a simple
matrix multiplication by the Jacobian of the deforma-
tion: t
S
= Jt
s
(Barr, 1984). Similarly, the normal vec-
tor n
S
of the deformed object S can be computed as
proportional to the inverse transpose of the Jacobian
matrix times the normal vector of the undeformed ob-
ject: n
S
= det(J)J
−1T
n
s
(Barr, 1984). Such tangent
1
For a parametric surface s = s(u,v), the tangent vector
t
s
can be computed as the linear combination of the partial
derivatives of s with respect to the two variables u and v;
whereas the normal vector n
s
can be computed as the cross
product of two linearly independent tangent vectors. Simi-
larly, the normal vector at a face on a surface mesh s can be
calculated as the cross product of two linearly independent
tangent vectors at that face, while the normal vector at a ver-
tex on the surface mesh is equal to the normalized sum of
the normals in each face connected to the vertex (3 in case
of triangulated surface meshes).
and normal transformation rules allow a fast calcu-
lation of the tangent and normal vectors, increasing
the practical usefulness of SD methods for geometry
modeling and surface morphing applications.
Global Linear Bending. A model for the isotropic
Global Linear Bending (GLB) deformation was first
introduced in (Barr, 1984). In GLB, the space is
bended along a line parallel to one of the axis by a
piece-wise constant deformation. More in detail, in
the bending region, the bending deformation is ap-
proximated trough simultaneous rotations and trans-
lations of two components of each point around the
third one. In the non bending regions, the space is
rigidly rotated and translated. In the following sec-
tions, we propose modifications to the GLB model
named as modified GLB (mGLB), adaptive modified
GLB (amGLB), and rescaled adaptive modified GLB
(ramGLB).
3 DEFORMATION MODELS
3.1 Modified Global Linear Bending
Let (x,y,z) and (X
m
,Y
m
,Z
m
) denote the original
(undeformed) and deformed x, y and z coordinates,
respectively. The modified Global Linear Bending
(mGLB) along a centerline parallel to the y axis is
defined as follows (bending along lines parallel to the
x and z axes are obtained in the same way as for the
y axis):
X
m
=















C
θ
(x−
1
k
1
) +
1
k
1
+ S
θ
(y− y
A
) if y < y
A
C
θ
(x−
1
k
1
) +
1
k
1
if y
A
≤ y < y
1
C
θ
(x−
1
k
1
) +
1
k
1
+ S
θ
(y− y
1
) if y
1
≤ y ≤ y
2
C
θ
(x−
1
k
2
) +
1
k
2
if y
2
< y ≤ y
C
C
θ
(x−
1
k
2
) +
1
k
2
+ S
θ
(y− y
C
) if y > y
C
(1)
Y
m
=
˜
Y
m
+ ˜y
m
=
=















−S
θ
(x−
1
k
1
) +C
θ
(y− y
A
) + y
1
if y < y
A
−S
θ
(x−
1
k
1
) + y
1
if y
A
≤ y < y
1
−S
θ
(x−
1
k
1
) +C
θ
(y− y
1
) + y
1
if y
1
≤ y ≤ y
2
−S
θ
(x−
1
k
2
) + y
2
if y
2
< y ≤ y
C
−S
θ
(x−
1
k
2
) +C
θ
(y− y
C
) + y
2
if y > y
C
(2)
Z
m
= z (3)
where:
ICPRAM 2012 - International Conference on Pattern Recognition Applications and Methods
412

(a) (b) k
1
= 0.1, k
2
=
0.1, n
1
= 1, n
2
= 1
(c) k
1
= 0.1, k
2
=
−0.1, n
1
= 1, n
2
= 1
(d) k
1
= 0.2,k
2
= −0.2,
n
1
= 2, n
2
= 1
(e) k
1
= 0.1, k
2
= −0.2,
n
1
= 2, n
2
= 1, y
1
= y
2
= 0
(f) k
1
= 0.1, k
2
= 0.1, n
1
=
1, n
2
= 1, y
1
= y
2
= 0
Figure 1: The modified Global Linear Bending (mGLB) deformation. A space (x,y) ∈ [−5,5] × [−5,5] with bending region
defined by y
A
= −3, y
1
= −1.5, y
2
= 1.5 , and y
C
= 3(a) is remapped in (b)-(d) using the mGLB with varying values for k
1
,
k
2
, n
1
, and n
2
. Undeformed and deformed spaces are depicted in 2D as the mGLB deformation around the y axis (as defined
by Eq. 1-4) is constant along the z axis. Panels (b)-(e) show that the space deformations in the two bending regions (depicted
in black color) can have different directions and intensities. Panel (f) shows one of the 3 possible cases when the mGLB
reduces to the GLB space deformation.
(a) k
1
= 0.2, k
2
= 0.2
n
1
= 1, n
2
= 1
(b) k
1
= 0.27, k
2
= 0.27
n
1
= 2, n
2
= 1
(c) k
1
= 0.4, k
2
= 0.4
n
1
= 2.1, n
2
= 2.1
(d) k
1
= 0.1, k
2
= −0.1
n
1
= 1.5, n
2
= 1.5
(e) k
1
= 0.2, k
2
= 0.2
n
1
= 2.1, n
2
= 2.1
(f) k
1
= 0.2, k
2
=
−0.2 n
1
= 2, n
2
= 5
Figure 2: The effects of the factors of bending amplification on mGLB space deformation. Panels (a)-(e) show a space
(x,y) ∈ [−5,5] × [−5,5] deformed by mGLB with y
A
= −3, y
1
= y
2
= 0, y
C
= 3, and with varying values for k
1
= k
2
and
n
1
= n
2
. The mGLB remappings depicted in panels (a)-(c) reduce to the original GLB deformation. A crimp effect can be
seen in the top right most part of (b) and (c). The mGLB space deformations depicted in panels (d) and (e) produce similar
amount of bending as in (b)-(c), but with no crimp effect. In (f), a more general example of mGLB space deformation with a
relatively high level of bending and no crimp effect is depicted.
θ =













n
1
k
1
(y
A
− y
1
) if y < y
A
n
1
k
1
(y− y
1
) if y
A
≤ y < y
1
0, if y
1
≤ y ≤ y
2
n
2
k
2
(y− y
2
) if y
2
< y ≤ y
C
n
2
k
2
(y
C
− y
2
) if y > y
C
(4)
In above, θ is the bending angle; y
1
and y
2
are the
centers of the deformations; k
1
and k
2
are the constant
bending rates measured in radiants per unit length; n
1
and n
2
are the constants factors of bending amplifica-
tion; and C
θ
and S
θ
are defined as cos(θ) and sin(θ),
respectively. Parameters y
A
,y
1
,y
2
and y
C
define the
bending regions.
In the non bending regions y ∈ (∞,y
A
) and y ∈ (y
C
,∞),
the mGLB deformation consists of rigid body transla-
tions and rotations with a constant angle θ. The non
bending region y ∈ [y
1
,y
2
] remains unaffected by the
mGLB transformation as θ = 0. In the bending re-
gions y ∈ [y
A
,y
1
) and y ∈ (y
2
,y
C
], θ changes linearly
with y, and the bending deformations are approxi-
mated through simultaneous rotations and translations
of two components of each point around the third one.
The mGLB model is C
0
continuous as the deformed
X
m
, Y
m
, and Z
m
functions have continuous values
along y but their derivatives with respect to y are not
continuous at regions’ boundaries.
If y
1
= y
C
and n
1
= 1; or if y
2
= y
A
and n
2
= 1; or
if y
1
= y
2
and n
1
= n
2
= 1 and k
1
= k
2
, then the
mGLB reduces to the original GLB deformation in
(Barr, 1984). The mGLB deformation is a generaliza-
tion of the GLB deformation in following two aspects.
1) Whereas in GLB there were two non-bending
and one bending regions, the mGLB deformation in
Eq. 1-4 is composed of three non bending regions:
y ∈ (∞,y
A
), y ∈ (y
C
,∞), and y ∈ [y
1
,y
2
]; and two bend-
ing regions: y ∈ [y
A
,y
1
) and y ∈ (y
2
,y
C
]. Moreover,
the bending rates k
1
and k
2
and the factors of bending
amplification n
1
and n
2
in the mGLB deformation can
have different signs and values, and thus the direction
and intensity of the deformation can be different in
the two bending regions (see Fig. 1.(b)-(d)). Due to
this, the mGLB deformation can model a number of
phenomena not otherwise possible through GLB, e.g,
the brain torque.
2) The second novelty of the mGLB is the addition
of the multiplicative factors n
i
,i = 1,2 to the defini-
tion of the bending angle θ. If |n
i
| > 1, the degree of
bending increases by a factor |n
i
| through amplifica-
tion of the angles spanned by the mGLB in the bend-
ing regions ([y
A
,y
1
) if i = 1, (y
2
,y
C
] if i = 2). Nega-
tive values of n
i
result in a swap of the direction of the
bending with respect to the one expressed by the sign
3D BENDING OF SURFACES AND VOLUMES WITH AN APPLICATION TO BRAIN TORQUE MODELING
413

(a) (b) k
1
= k
2
= 0.1, n
1
= n
2
= 1,
y
1
= y
2
= 0, y
A
= −3, y
C
= 3
(c) k
1
= k
2
= 0.5, n
1
= n
2
= 1, y
1
=
y
2
= 0, y
A
= −3, y
C
= 3
(d) k
1
= k
2
= 0.7, n
1
= n
2
= 1,
y
1
= y
2
= 0, y
A
= −3, y
C
= 3
(e) k
1
= k
2
= 0.6, n
1
= n
2
= 11,
y
1
= y
2
= 0, y
A
= −2.5, y
C
= 2.5
(f) k
1
= k
2
= 0.6, n
1
= 14, n
2
= 1,
y
1
= y
2
= 0, y
A
= −2.5, y
C
= 2.5
(g) k
1
= k
2
= 0.1, n
1
= 1, n
2
= 5,
y
1
= −1.5, y
2
= 1.5, y
A
= −3, y
C
= 3
(h) k
1
= 0.2, k
2
= −0.2, n
1
= 2,
n
2
= 1, y
1
= y
2
= 0,y
A
= −3, y
C
= 3
(i) k
1
= 0.2, k
2
= −0.2, n
1
= 2, n
2
= 5,
y
1
= −1.5, y
2
= 1.5, y
A
= −3, y
C
= 3
Figure 3: Examples of shapes deformed by the modified Global Linear Bending (mGLB) deformation. (a) A parametric
surface (a sphere - |x|
2
+ |y|
2
+ |z|
2
= 1 - with a cross inside) is embedded in a 3D space (x,y,z) ∈ [−5,5] × [−5,5] × [−5,5],
and then deformed (b)-(i) by the mGLB with varying values of the model parameters. Panels (b)-(d) show examples of objects
deformed by the original GLB with increasing bending rates. A crimp effect can be observed for high degrees of the bending
in panels (c) and (d) but not in the case of mGLB in panels (e) and (f). Panels (g)-(i) show surface deformed by the mGLB
space deformation with k
1
n
1
6= k
2
n
2
: different directions and/or intensities of the deformations in the two bending regions.
Table 1: The normal and tangent rules, and the rates of local volumetric change for the mGLB space deformation.
J J
−1T
det(J)
ˆ
k ˆn



C
θ
S
θ
(1−
ˆ
kx) ˆn 0
−S
θ
C
θ
(1−
ˆ
kx) ˆn 0
0 0 1






C
θ
(1−
ˆ
kx) ˆn S
θ
0
−S
θ
(1−
ˆ
kx) ˆn C
θ
0
0 0 (1−
ˆ
kx)n



(1−
ˆ
kx)n







k
1
if y
A
≤ y < y
1
k
2
if y
2
≤ y < y
C
0 otherwise



n
1
if y < y
2
n
2
if y ≥ y
2
of k
i
. This bending angle amplification helps to avoid
the crimp effect as is illustrated in Fig. 2 in the sim-
plified case of y
1
= y
A
, y
2
= y
C
and k
1
= k
2
. Partic-
ularly, in Fig. 2.(a)-(c), three examples of the mGLB
deformations that reduce to GLB deformations with
increasing values of k
1
= k
2
(and constant values of
y
A
, y
C
and n
1
= n
2
= 1) are shown. A crimp effect,
consisting of an overlap in the deformed portions of
the grid in the bending and non bending regions, can
be seen in the top right most parts of Fig. 2.(b) and
(c). This crimp effect, also described in (Barr, 1984),
increases as |k
i
| increases. However, in many prac-
tical situations, it is necessary to model a relatively
high degree of bending and still avoid the crimp ef-
fect as that could result in unrealistic deformations
(Fig. 3.(c) and (d)). This is especially true if the
objects occupy a relatively large volume of the space
being deformed. In Fig. 2.(d)-(e), we show how a
proper tuning of the pair (k
i
, n
i
) can be used to pro-
duce high degree of bending and still avoid the crimp
effect (see also Fig. 3.(e) and (i)). The rule of thumb
is to fix n
i
= 1 and choose the maximum value of k
i
(
˜
k) not producing the crimp effect; then, for a fixed
k
i
=
˜
k
1
, to increase n
i
until the desired level of bend-
ing is achieved. In principle, any degree of bending
with no crimp effect can be achieved using the mGLB
deformation.
The mGLB space transformations were used to
deform a few parametric surfaces in Fig. 3. Note
how the addition of a few parameters to the model
resulted in remarkably more ductile and realistic de-
formations.
The expressions for the Jacobian matrix (J), its in-
verse transpose (J
−1T
), and its determinant (det(J))
for the mGLB deformations are given in 1 . The
above quantities are needed, respectively, for the tan-
gent and normal transformation rules, and to calculate
the rate of local volumetric change introduced by the
proposed deformation.
3.2 Adaptive Modified Global Linear
Bending
Even though the mGLB deformation would in prin-
ciple enable the simultaneous leftward and rightward
bending of the inter-hemispheric fissure at the frontal
and occipital lobes, respectively, fine adjustments of
the mGLB are needed to better model this pattern
of structural asymmetry of the human brain. More
specifically, the bending of the cerebral hemispheres
ICPRAM 2012 - International Conference on Pattern Recognition Applications and Methods
414

(a) (b)
Figure 4: The adaptive modified Global Linear Bending (amGLB). (a) Different space deformations applied at different
z ∈ [z
min
,z
max
]; and (b) definition of the model parameters n
1
(z), n
2
(z), y
1
(z), and y
2
(z). In the transition area between
the cerebellum and the lower portions of the cerebrum (when z > z
min
), the bending amplification n
1
(z),n
2
(z) is increased
gradually while the bending regions ([y
A
,y
1
(z)) and (y
2
(z),y
C
]) become gradually narrower. Viceversa in the transition area
between the lower and higher portions of the cerebrum.
is localized in the most inferior part (i.e. the part with
a small z-coordinate) of the cerebrum. Thus, we apply
a mGLB style deformation only when z
min
≤ z ≤ z
max
.
This way, it is possible to deform the bottom part of
the cerebrum while leaving the cerebellum (z < z
min
)
and the upper portions of the cerebrum (z > z
max
) un-
deformed. To obtain smooth transitions between the
deformed (z
min
≤ z ≤ z
max
) and undeformed (z < z
min
,
z > z
max
) regions along the z axis, the adaptive mod-
ified Global Linear Bending (amGLB) model is de-
signed to enable scalable levels of the mGLB style
space deformations.
The amGLB deformation along a line parallel to the
y axis is defined as the mGLB space deformation in
Eqs. (1)-(4), but the factors of bending amplifications,
n
1
and n
2
, and the parameters y
1
and y
2
, are the fol-
lowing functions of z-coordinate:
ˆn
i
(z) =









n
max
z
1
−z
min
(z− z
min
) if z
min
< z < z
1
n
max
z
2
−z
max
(z− z
max
) if z
2
< z < z
max
n
max
if z
1
≤ z ≤ z
2
0 otherwise
(5)
ˆy
1
(z) =









y
1
−y
A
z
1
−z
min
(z− z
min
) + y
A
if z
min
< z < z
1
y
1
−y
A
z
2
−z
max
(z− z
max
) + y
A
if z
2
< z < z
max
y
1
if z
1
≤ z ≤ z
2
y
A
otherwise
(6)
ˆy
2
(z) =









y
2
−y
A
z
1
−z
min
(z− z
min
) + y
C
if z
min
< z < z
1
y
2
−y
A
z
2
−z
max
(z− z
max
) + y
C
if z
2
< z < z
max
y
2
if z
1
≤ z ≤ z
2
y
C
otherwise
(7)
where i = 1,2; y
1
and y
2
are parameters for the centers
of the deformation; z
min
≤ z
1
≤ z
2
≤ z
max
and n
max
are
model parameters. See Fig. 4 for an illustration.
The amGLB deformation introduces local
changes on the size of the deformed space. In fact,
while the protrusions of the left occipital and the
right frontal lobes, thus the local volumetric changes,
at the interhemispheric region are desirable, the
total volume of the brain should not change. In the
same way, the shape and position of the skull and
scalp regions should not be modified by the space
deformation. To prevent this, we added to the model a
rescaling factor in the x (r
x
(z)) the (r
y
(z)) y direction,
and define the rescaled adaptive deformation Global
Linear Bending (ramGLB) deformation along a line
parallel to the y axis as follows:
X
ram
= r
x
(z)X
am
(8)
Y
ram
= r
y
(z)
˜
Y
am
+ ˜y
am
(9)
Z
ram
= Z
am
(10)
where the X
am
,Y
am
, and Z
am
denote the amGLB trans-
formed x, y, and z coordinates. To define the rescal-
ing factors, we define points A(z) = (x
A
,y
A
,z),B(z) =
(x
B
,y
A
,z), and C(z) = (x
A
,y
C
,z), where x
A
,x
B
are pa-
rameters whose meaning will be explained shortly
and y
A
,y
C
are as in Eqs. (1) - (7), as functions of z-
coordinate. Let X
am
(I, z), Y
am
(I, z), I = A,B,C denote
the x and y coordinates of these points after amGLB
deformation. Then,
r
x
(z) =





















x
B
−x
A
X
am
(A,z))−X
am
(z,−θ)
+
x
A
+X
am
(A,z)
2
if x ≥
x
B
−x
A
2
, y ≤ y
1
(z), z
min
≤ z ≤ z
max
x
B
−x
A
X
am
(B,z))−max(X
am
(z,−θ)
+
x
A
+X
am
(B,z)
2
if x <
x
B
−x
A
2
, y ≥ y
2
(z), z
min
≤ z ≤ z
max
1 otherwise
(11)
3D BENDING OF SURFACES AND VOLUMES WITH AN APPLICATION TO BRAIN TORQUE MODELING
415

(a) (b) (c) (d)
Figure 5: The rescaling factors of the rescaled adaptive modified Global Linear Bending (ramGLB). In (a) and (b) is shown the
rescaling field factor along the x axis to be used in the ramGLB space remapping and in (d) the resulting ramGLB deformed
space. We found experimentally that better results were achieved by adding the mean terms in the expression of the rescaling
factor along x than in the case this additive mean terms were not used (c).
r
y
(z) =





y
A
−y
1
(z)
Y
am
(A,z))−y
1
(z)
if y ≤ y
1
(z) z
min
≤ z ≤ z
max
y
C
−y
2
(z)
Y
am
(C,z))−y
2
(z)
if y ≥ y
2
(z) z
min
≤ z ≤ z
max
1 otherwise
(12)
where X
ma
(A,z,−θ)) denotes the value of X
ma
(A,z) at
point A in case of the flipping the sign of θ in Eq. (4).
The rescaling in the x direction is designed to pro-
duce a rightwards bending of the portion of the space
corresponding to the left occipital lobe (y ≤ y
1
(z),
x ≤ (x
B
− x
A
)/2, here the rescaling factor along the
x axis is 1), while at the same time resizing the por-
tion of the space corresponding to the right occipital
(y ≤ y
1
(z), x ≥ (x
B
− x
A
)/2) lobe (see Fig. 5). With
these rescalings, we can synthesize the protrusion of
the left occipital lobe overthe right, while not deform-
ing the exterior regions (e.g. skull) to the right occipi-
tal lobe. The rescaling along the y direction follows a
similar idea for the image space corresponding to the
right temporal lobe.
4 APPLICATION TO BRAIN
IMAGING
Our motivation for studying the global bending de-
formations originated from a need to develop a sim-
ple, global model for the brain torque. Based on this
model given in previous section, we can introduce a
known amount of realistic asymmetry to any given
brain MR image, and therefore, construct a set of im-
ages with a known and easily parametrized asymme-
try pattern. This dataset can be used for the quantita-
tive validation of voxel level and surface based mor-
phometric methods of brain shape asymmetry. This
section describes an automatic image pipeline to gen-
erate such a dataset.
First, two artificially symmetrized images are ob-
tained from each MR image. Next, the synthetic
images are deformed by a customized model of the
brain torque. In order to make sure that the paramet-
ric global linear bending deformation would produce
a consistent level of bending deformations among
images (despite of the eventually different image
and voxel sizes), the deformation is applied to each
synthetic image in stereotactic space and the de-
formed volumes are then registered back to the na-
tive space. All the image registration stages of the
pipeline are performed using the FLIRT tool (Jenkin-
son and Smith, 2001) of the FMRIB FSL Software Li-
brary (http://www.fmrib.ox.ac.uk/fsl/). The pipeline
workflow is depicted in Fig. 6 and described in the
following 6 consecutive steps.
STEP 1: Image Pre-processing. Each T1-
weighted MR image is corrected for intensity
non-uniformity (also referred as bias field) using a
multi-resolution approach with a minimum intensity-
gradient entropy criteria (Manj´on et al., 2007). The
bias field corrected images are denoised using a non-
local means denoising algorithm which automatically
adapts to the spatially varying noise levels of the MR
images (Manj´on et al., 2010).
STEP 2: Symmetrized Images Generation. The
Left-Left (LL) and the Right-Right (RR) artificially
symmetrized images are extracted from the pre-
procssed images. Each LL image is generated by re-
placing the right hemisphere (voxels on the right hand
side of the mid-sagital plane) of the pre-processed im-
age with the flipped version of the left hemisphere.
The synthetic RR images are created in a similar
way. If the original image exhibits an appreciable
leftward (rightwards) asymmetry, than the LL (RR)
image is discarded as a non realistic double inter-
hemispheric fissure would appear along the inter-
hemispheric plane in the regions of higher degree of
ICPRAM 2012 - International Conference on Pattern Recognition Applications and Methods
416

Figure 6: The proposed automatic image processing pipeline for the generation of a dataset for quantitative validation of
morphometric methods for studying brain asymmetry from T1- weighted 3D MR images.
right (left) hemisphere protrusion.
STEP 3: Addition of Rician Noise. In MRI, mea-
surement noise can be considered to be Rician dis-
tributed (Gudbjartsson and Patz, 1995). In order to
create images with non-symmetric noise realizations,
we add Rician noise to the denoised, symmetrized im-
ages.
STEP 4: Spatial Normalization. The synthetic
symmetrical LL and RR images are spatially normal-
ized to symmetrical subject- and image- specific tem-
plate images using affine registration. Affine registra-
tion parameters are estimated from the skull and scalp
stripped synthetic symmetrical images and applied
then to the corresponding non skull stripped synthetic
symmetrical images. The skull stripping is performed
by the Brain Extraction Tool ( BET) (Smith, 2002)
with default parameters. The above mentioned sym-
metrized subject- and image- specific template im-
ages are obtained for each synthetic image by first av-
eraging it with its flipped version around the x axis.
Next, the resulting average image is registered to the
stereotaxic space (181 x 218 x 181 voxels of size 1.0
x 1.0 x 1.0 mm
3
) using a 7 parameter affine transfor-
mation. By construction, these template images are
symmetrical and registered to the stereotaxic space.
STEP 5: Volume Deformations. The non skull
stripped phantom images in stereotaxic space are de-
formed by the ramGLB deformation with the follow-
ing model parameters: k
1
= 0.00003, k
2
= −0.0001,
n
max
= 10, y
A
= −90, y
1
= −22, y
2
= 62, y
C
= 91,
z
1
= 74, z
2
= 76 z
min
= 45 and z
max
= 105. In-
deed, for MR images in the stereotaxic space, we
found experimentally that a realistic modeling of the
inter-hemispheric fissure bending can be obtained by
remapping the sub-space with z ∈ [45,105] while the
remaining space was kept unchanged.
The modeling of more (less) intense patterns of the
inter-hemispheric bending at the occipital and frontal
lobes could be easily achieved e.g. by increasing (de-
creasing) the |k
1
| and |k
2
| values. A realistic modeling
of a typical pattern of invertedbrain torque can also be
obtained by the ramGLB space deformation with the
following model parameters: k
1
= −0.000015, k
2
=
−0.0001, n
max
= 10, y
A
= −90, y
1
= −22, y
2
= 62,
y
C
= 91, z
min
= 45 and z
max
= 105.
STEP 6: Back to the Native Space. The 7-
parameter affine registration matrix used in step 4 is
inverted and applied to the deformed synthetic images
computed in step 5. As a result, the deformed syn-
thetic images are mapped back to their native spaces.
The ramGLB-like deformation was applied to a
dataset previously described in (Laakso et al., 2001)
consisting of 19 T1-weighted MR images of right-
handed healthy controls (7 females, 12 males). Re-
sults were visually inspected to verify that the de-
formed synthetic images were modeling a realistic
rightward bending of the interhemispheric fissure as
well as the hemispheric protrusions of the right oc-
cipital and left frontal lobes. A few examples of de-
formed synthetic images are depicted in Fig. 7. Par-
ticularly, Fig. 7 shows how well the images processed
by the proposed image processing pipeline resemble
the brain torque in the corresponding original MR im-
ages.
5 CONCLUSIONS
In this work, we have proposed space deformation
methods to approximate the bending deformation in
3D volumes and surfaces, which add flexibility to
the current SD methods for global bending. Partic-
ularly, the ramGLB was developed for the model-
ing of the inter-hemispheric bending of the human
brain. The ramGLB space deformation can introduce
a known amount of realistic asymmetry to any given
3D T1-weighted brain MR image. Due to the sim-
plicity of the model and to the automatism of the
whole image processing pipeline, the latter can be
used for the quantitative validation of voxel and sur-
face based morphometric methods for the study of the
brain asymmetry in large databases. The proposed
image processing pipeline was designed and tested for
3D T1-weighted MR images although it can be easily
adapted for other structural imaging modalities.
ACKNOWLEDGEMENTS
This work was supported by the Academy of Finland
grants 130275 and 129657, Finnish Programme for
Centres of Excellence in Research 2006-2011).
3D BENDING OF SURFACES AND VOLUMES WITH AN APPLICATION TO BRAIN TORQUE MODELING
417

(a) (b) (c)
Figure 7: The original (left) T1-weighted MR images of three subjects (rows) are modified with the developed image pro-
cessing pipeline (right). For each subject, three transversal views are shown (a)-(c). Synthesized images on right appear
remarkably similar to the original images, but their asymmetry pattern is known exactly.
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