DEFORMABLE STRUCTURES LOCALIZATION AND
RECONSTRUCTION IN 3D IMAGES
Davide Moroni, Sara Colantonio, Ovidio Salvetti
Institute of Information Science and Technologies (ISTI), Italian National Research Council (CNR), Pisa, Italy
Mario Salvetti
Department of Mathematics, University of Pisa, Pisa, Italy
Keywords:
Deformable structures, Image segmentation, Artificial neural networks, Cardiac MRI.
Abstract:
Accurate reconstruction of deformable structures in image sequences is a fundamental task in many appli-
cations ranging from forecasting by remote sensing to sophisticated medical imaging applications. In this
paper we report a novel automatic two-stage method for deformable structure reconstruction from 3D image
sequences. The first stage of the proposed method is focused on the automatic identification and localization
of the deformable structures of interest, by means of fuzzy clustering and temporal regions tracking. The final
segmentation is accomplished by a second processing stage, devoted to identify finer details using a Multi-
level Artificial Neural Network. Application to the segmentation of heart left ventricle from MRI sequences
are discussed.
1 INTRODUCTION
In the field of computer vision, analyzing the de-
formation pattern of non-rigid structures may con-
vey useful information in a variety of settings. For
example satellite image sequences display tempo-
ral evolution of complex structures like clouds and
vortices, whose analysis is essential for meteoro-
logical forecast (Papin et al., 2000); reinforcement
of speech recognition by visual data may also be
based on the analysis of lips deformation (Bregler and
Konig, 1994). More interestingly for our purposes,
deformable structures show up in human anatomy
(lungs and heart being key examples) and their defor-
mation modes are of key importance in understanding
the functional properties of the related organs and as-
sessing their health-state (Moroni et al., 2006).
Imaging modalities provide an invaluable aid in
analyzing such complex structures. However image
sequences contain a huge amount of high dimensional
data (2 or 3 spatial dimensions plus time) which can-
not be fully exploited unless with the help of suitable
tools for image processing and pattern recognition.
The main goal is to compactly but faithfully describe
deformable structure in such a way to allow for defor-
mation pattern characterization and assessment. Such
an encoding would be useful to build up a reference
database for similarity searches or data mining proce-
dures.
Of course, an essential step in characterizing de-
formable structures is first of all their localization and
reconstruction from an image sequence.
In this paper we address this preliminary problem,
assuming that structures are periodically deformable
3D structures. The developed method consists in a
two-stage procedure, based on fuzzy clustering and
Artificial Neural Networks (ANN), for the identifica-
tion and reconstruction of the deformable structures
of interest in an image sequence. As an application of
this procedure we describe the segmentation of heart
Left Ventricle (LV) from Magnetic Resonance (MR)
image sequences, extending and refining (Colantonio
et al., 2005). Actually the LV, pumping oxygenated
blood to the body, is of great importance in assess-
ing the health or pathological states of the heart since
it has been demonstrated that different pathologies of
the heart are deeply correlated to its dynamics.
The paper is organized as follows. In Section 2
we define the class of structures we are interested
in, making explicit the necessary assumptions. Then
in Section 3 the proposed approach is outlined and
its basic modules are described in detail. More pre-
215
Moroni D., Colantonio S., Salvetti O. and Salvetti M. (2007).
DEFORMABLE STRUCTURES LOCALIZATION AND RECONSTRUCTION IN 3D IMAGES.
In Proceedings of the Second International Conference on Computer Vision Theory and Applications, pages 215-222
DOI: 10.5220/0002071202150222
Copyright
c
 SciTePress

cisely in Section 3.1 we address the problem of au-
tomatic localization of periodically deforming struc-
tures, while in Section 3.2 we propose a method for
the actual reconstruction. In Section 4 we discuss LV
segmentation and show some experimental results,
whereas conclusions and directions for further work
are briefly collected in Section 5.
2 PERIODICALLY
DEFORMABLE STRUCTURES
A structure O embedded in the background space Ω ⊂
R
3
is a collection
O =
{
V
α
}
α=1,2,...,k
of smooth surfaces V
α
embedded in Ω. The smooth-
ness assumption is a quite common hypothesis in
computational anatomy (see e.g. (Grenander and
Miller, 1998)) and it is satisfied in practice to a large
extent. We use, moreover, collection of surfaces -
instead of a single one- to be able to describe and
characterize also subparts of the structures.
A deformable structure
C = (O
t
)
t=1,2,...
is a tem-
poral sequence of structures satisfying some smooth-
ness constraint. Each O
t
=
{
V
α
t
}
1≤α≤k
should be re-
garded as the snapshot of the deformable structure at
time t.
We require that each surface V
α
t
appearing in the
snapshot at time t can be smoothly deformed into
V
α
t+1
in the subsequent snapshot. Tears or crack of
any structure subpart are, therefore, ruled out; more-
over, in such a way, we avoid dealing with changes
in topology, that would require to model shape tran-
sitions. Such a task would be essential for example
in meteorological applications, but is far beyond our
present scopes in biomedical problems.
Finally, a periodically deformable structure is a
deformable structure for which there exists an inte-
ger T such that ∀t : O
t
= O
t+T
. In other words, the
deformable structure depicts a periodic motion; thus,
a periodically deformable structure is characterized
by a finite list of snapshots (O
0
,O
2
,...,O
T−1
), which
will be referred to as its deformation cycle.
It is assumed that a 3D image sequence
(S
t
)
0≤t≤T−1
has been acquired from which morphol-
ogy of the structure can be inferred so as to represent
faithfully a physical body or phenomenon of inter-
est. Without loss of generality, considering biomedi-
cal applications, we assume that each scan S
t
consists
of a set of h parallel 2D slices. The pixels of each slice
are identified by their position (x,y) in the slice plane
and by a third coordinate, z (z = 1,...,h), which refers
to the index of the slice itself in the stack. In the fol-
lowing, the three coordinates (x, y,z) will be referred
to as a voxel and S
t
(x,y,z) will denote its intensity
value.
3 TWO-STAGE DEFORMABLE
STRUCTURE SEGMENTATION
We address the problem of deformable structures seg-
mentation with a two-stage method, which, firstly, au-
tomatically localize the deformable structure and then
extracts its finer details, looking for precise contours
of the whole structure and of its subparts.
To each scan S
t
, the following two-stage proce-
dure is applied:
1. Structure automatic localization: a cluster analy-
sis, based on the fuzzy c-means algorithm, is ap-
plied to identify and label homogeneous regions
in each scan. Through a region tracking proce-
dure, the behavior of these regions is analyzed
over an entire cycle, in order to extract a rough ap-
proximation
C
′
= {O
′
t
}
0≤t≤T−1
of the deformable
structure
C .
2. Segmentation refinement: C
′
is used to com-
pute the approximate orientation of the real struc-
ture
C , which, in turn, is used to extract three-
dimensional features processed by a dedicated
ANN, in order to complete the segmentation, by
identifying accurate contours of
C .
3.1 Automatic Localization of
Deformable Structures
We assume that shape descriptors of the deformable
structure tracked on time exhibit a periodical behav-
ior, with main frequency concentrated in the motion
frequency. Further we assume that the subparts of
the deformable structures appear as homogeneous re-
gions at some scale. However the latter assumption
is dictated by our implementations and can be substi-
tuted without altering the spirit of this contribution.
3.1.1 Clustering
Homogeneous image regions are first labelled us-
ing an unsupervised clustering method, based on
the fuzzy c-means algorithm (FCM) (Bezdek, 1981).
This algorithm groups a set of data in a predefined
number of regions so as to iteratively minimize a cri-
terion function, namely the sum-of-squared-distance
from region centroids, weighted by a cluster member-
ship function. A membership grade p ∈ [0, 1] is asso-
VISAPP 2007 - International Conference on Computer Vision Theory and Applications
216

ciated to each element of the data set, describing its
probability to be in a particular cluster.
The FCM algorithm is applied to each 3D scan S
t
to produce a number of clusters: for any voxel x, a
features vector (I
0
(x),I
1
(x),I
2
(x),...,I
r
(x)) is com-
puted so that I
0
(x) = S
t
(x), and for d = 1,...,r, we
set I
d
(x) =
G
d
∗S
t
(x), where
G
d
is a Gaussian kernel
with standard deviation σ ∝ d.
This, in turn, induces a partition of the image domain
into a set P
t
= {R
1
t
,R
2
t
,...} of disjoint connected re-
gions, where the upper indices 1,2,... are region la-
bels. In the following, ρ
t
will denote the generic re-
gion in P
t
.
3.1.2 Region Tracking
Once eliminated regions of negligible volume (island
removal), an intra-cycle tracking procedure is per-
formed. A simple centroid-based tracking algorithm
associates, to any region ρ
t
∈ P
t
in the phase t, its
correspondent region T(ρ
t
) ∈ P
t+1
in the subsequent
phase t + 1. The procedure can be iterated, thus pro-
ducing a region sequence
ρ
t
= T
0
(ρ
t
) → T
1
(ρ
t
) → T
2
(ρ
t
) → . . .
which may be thought as the evolution of the start-
ing region ρ
t
in the different phases. Considering
t = 0 as reference phase, for each ρ
t
∈ P
t
the re-
gions appearing in its evolution are collected in a list
Ev(ρ
0
) = (T
t
(ρ
t
))
0≤t≤T−1
.
3.1.3 Features Extraction
For any region ρ
0
∈ P
0
, the behavior in time of a shape
descriptor G (such as elementary geometric proper-
ties: volume, inertia moments etc. ) can be estimated
by evaluating G for every element in the list Ev(ρ
0
),
thus obtaining a vector f
G
(ρ
0
) = (G.T
t
(ρ
0
))
0≤t≤T−1
.
To detect the oscillatory behavior of f
G
(ρ
0
), it is
effective and convenient to switch to frequency do-
main and consider its power spectrum density (PSD)
function.
In more detail, first the vector f
G
(ρ
0
) is normal-
ized:
µ
G
(ρ
0
) =
T−1
∑
t=0
G.T
t
(ρ
0
)
f
′
G
(ρ
0
) =
1
µ
G
(ρ
0
)
f
G
(ρ
0
)
to obtain a scale invariant vector and, thus, getting rid
of the relative size of G. Then the Fourier transform
in non negative frequencies is computed:
F
′
G
(ρ
0
) =
F ( f
′
G
(ρ
0
))
and the PSD is estimated using the periodogram
method:
PSD
G
(ρ
0
) = |F
′
G
(ρ
0
)|
2
The first harmonic coefficient ν
G
(ρ
0
) in PSD
G
(ρ
0
) is
then selected as a salient feature. Indeed, for fixed re-
gions the variations in G during time are essentially
due to noise; instead for regions in periodic motion
the spectrum power is concentrated in the motion fre-
quency.
Finally, for a predetermined list {G, H,...}
of shape descriptors, a features vector
I (ρ
0
) =
(µ
G
(ρ
0
),ν
G
(ρ
0
),µ
H
(ρ
0
),ν
H
(ρ
0
),...) is associated to
each ρ
0
∈ P
0
.
3.1.4 Region Classification
Let O
′
t
denote the region corresponding to the de-
formable structure
C at the phase t. At first, the ref-
erence phase is considered and O
′
0
is searched among
regions ρ
0
∈ P
0
, taking into account their features vec-
tors
I (ρ
0
). More precisely, a set of learning examples
is used to introduce a Mahalanobis distance in the fea-
ture space. Let I
1
,
I
2
,...
I
s
be a set of observed fea-
ture vectors relative to a training set of regions
C
′
with
mean m and covariance matrix Σ. The associated Ma-
halanobis distance, defined by
D(
I ) =

(I − m)
t
Σ
−1
(I − m)

1/2
,
measures the dissimilarity of a feature vector w.r.t. to
the expected region feature vector. Thus, for any new
case, O
′
0
is selected among candidate regions ρ
0
∈ P
0
according to the criterion:
O
′
0
= arg min
ρ
0
∈P
0
D(
I (ρ
0
))
In subsequent phases, the region O
′
t
is singled out by
means of the tracking algorithm, namely O
′
t
is defined
as T
t
(O
′
0
).
3.2 Segmentation Refinement
The localization of the deformable structure in the
previous section supplies as a byproduct a rough ap-
proximation of its boundary surface, which may suf-
fer from poor intensity contrast or the presence of spu-
rious structures.
The aim of this section is to refine the segmenta-
tion found in the previous section and to identify as
well the contours of the structure subparts.
The set up is as follows. Let Ω ⊂ R
3
be the image
domain of the scan S
t
. First we define in 3.2.1 a fea-
tures function
F
t
: Ω → R
s
, that assigns to each point
x ∈ Ω a vector
F (x) of local features extracted from
the image data S
t
. Then we use an approach based
DEFORMABLE STRUCTURES LOCALIZATION AND RECONSTRUCTION IN 3D IMAGES
217

on Multi-Level Artificial neural networks (MANN) to
find functions Φ
α
: Ω × R
s
→ R (α = 1, 2, . . . , k) s.t.
the level sets:
V
α
= {x ∈ Ω| Φ
α
(x,
F (x)) = 0} α = 1,2,...k
correspond to the surface V
α
respectively.
The functions Φ
α
are learned using a training set of
segmented images and they can be used subsequently
to segment new instances.
3.2.1 Features Extraction
Given a scan S
t
: Ω → R we briefly describe how a
features function
F
t
: Ω → R
s
may be constructed.
Since the neural network will eventually use this func-
tion for the identification of image edges, it is clear
that the function
F
t
should include “edge detector”-
like clues.
The involved features can be divided into two classes.
First, low-level features are considered: they are
context-independent and do not require any knowl-
edge and/or pre-processing. Some examples are voxel
position, gray level value, gradients and other differ-
entials, texture, and so forth. Middle-level features
are also selected, since voxel classification can benefit
from more accurate clues, specific of the problem at
hand. In particular, the knowledge of the deformable
structure orientation, obtained as a byproduct in Sec-
tion 3.1, can be used to individuate an Intrinsic Ref-
erence System (IRS) suitable to describe the structure
shape. If, in addition, a priori information about the
structure shape is available, a reliable clue for detect-
ing edges in the images is given by the gradient along
the normal direction to the expected edge orientation.
Moreover, a multiscale approach is adopted: the fea-
tures are computed on blurred images, supplying in-
formation about the behavior of the voxel neighbor-
hood, which results in a more robust classification.
3.2.2 MANN-based Voxel Classification
The set of selected features are processed to accom-
plish the voxel classification by means of a Multilevel
Artificial Neural Network (MANN), which assures
several computational advantages (Di Bona et al.,
2003).
For each voxel x, its computed features vector
F
t
(x) is divided into vectors
F
i
t
(x), each one contain-
ing features of the same typology and/or correlated.
Then each
F
i
t
(x) is processed by a dedicated classi-
fier based on an unsupervised Self Organizing Maps
(SOM) architecture. The set of parallel SOM modules
constitutes the first level of the MANN which aims at
clustering each portion of the feature vector into crisp
classes, thus reducing the computational complexity.
Cluster indexes, in turn, are the input of the final de-
cisional level, operated by a single EBP network. The
output of this last module consists in a vector of mem-
bership grade of the voxel x to the various surfaces
V
α
(1 ≤ α ≤ k). The SOM modules are trained ac-
cording to Kohonen algorithm (Kohonen, 1997). For
the EBP module, a set of 3D scans should be pre-
classified by an expert observer and used for super-
vised training, performed according to the Resilient
Back-Propagation algorithm (Riedmiller and Braun,
1993).
4 STUDY CASE: LEFT
VENTRICLE SEGMENTATION
Accurate segmentation of cardiac cavities is funda-
mental in assessing cardiac function and determin-
ing quantitative parameters. Magnetic Resonance
Imaging (MRI) is a high quality and well-established
imaging modality in analyzing heart diseases and has
proved to be more reliable than other techniques,
both in supplying accurate and reproducible morpho-
logical information and in assessing heart functions
(ACC\AHA Task Force, 2005). However, due to
noise or acquisition artifacts, visual information can
be corrupted or ill defined: in a usual edge map
of a cardiac MR slice, boundaries belonging to the
LV appear broken or, even worse, close to stronger
edges of other structures. In such cases, only expert
knowledge may help: the exact location of the con-
tours cannot be based only on image evidence, but
should be learned from examples provided by expert
observers. Usually, researchers have tried to design
ad hoc algorithms able to incorporate a priori infor-
mation about the LV shape. Model based surface
detector have been widely used: for example, (De-
clerck et al., 1997) employed a Canny-Deriche edge
detector in a 3D polar map to segment endocardial
and epicardial surfaces, while (Faber et al., 1991) de-
fined a hybrid spherical-cylindrical coordinate sys-
tem. Snakes, since their introduction in the semi-
nal paper by (Kass et al., 1988), have been a pow-
erful tool in cardiac images analysis for segmenta-
tion and motion tracking. Recent improvements in
this field include works by (Jolly et al., 2001), who
reduced sensitivity to initial contour through Dijk-
stra algorithm, and by (Paragios, 2002) and (Huang
et al., 2004) who introduced deformable models influ-
enced by forces derived from image region informa-
tion. (Mitchell et al., 2002;
¨
Uz
¨
umc
¨
u et al., 2003) used
the concept of active appearance model (AAM). An
AAM is a technique of analysis by synthesis, which,
in principle, could describe any heart through a set
VISAPP 2007 - International Conference on Computer Vision Theory and Applications
218

of learned 3D shapes and a set of allowed variations.
Although AAMs mitigate the segmentation problem
exploiting a shape prior, they are however very sensi-
tive to initialization and intensity variations occurring
at random locations. Aiming at solving this two is-
sues, in (de Bruijne and Nielsen, 2004) a particle fil-
tering scheme is introduced in the AAM framework to
get relatively initialization-independent results, while
a shape model inference on the basis of pixel classifi-
cation is used to cope with local intensity variations.
Finally, also neural networks approaches have been
proposed. In (Stalidis et al., 1999) a Generating-
Shrinking Neural Classifier is used to distinguish
among lung, blood and myocardium points. This
classification allows to extract a set of points on my-
ocardial surfaces and, then, to assess parameters for
a wavelets-based model. Two dedicated neural net-
works are presented instead in (Coppini et al., 1995).
The first is used to select from an edge map bound-
aries belonging to the ventricle. To get a meaning-
ful and unbroken surface, these edges are further pro-
cessed: a thin plate model for the left ventricle is in-
troduced and a stable configuration of minimum po-
tential is found by means of an analog neural net-
work implementation. The methods reported above
depend tightly from the choice of model parameters
and initial conditions. The last problem has tradition-
ally been solved with a manual intervention by an ex-
pert observer, but this contrasts the need of a fully
automatic segmentation.
4.1 Data
The two-stage method described in the previous sec-
tion has been applied to short axis gradient echo
MR images, acquired with the FIESTA, GENESIS
SIGNA MRI device (GE medical system), 1.5 Tesla,
TR = 4.9 ms, TE = 2.1 ms, flip angle 45
◦
and reso-
lution r
x
× r
y
× r
z
= (1.48 × 1.48 × 8) mm. Sets of
T = 30 3D scans, consisting of h = 11 2D slices,
were acquired at the rate of 30 ms for cardiac cy-
cles [diastole-systole-diastole]. Various clinical cases
were considered, for a total of 360 scans, correspond-
ing to 12 cardiac cycles.
4.2 Experimental Results
The LV segmentation can be viewed as a bi-modal
problem (Paragios et al., 2002): the structure of in-
terest is the myocardium, which can be identified
and extracted seeking the separation among the en-
docardium and the epicardium surfaces.
frame 0 frame 1 frame 2 frame 3 frame 4
frame 5 frame 6 frame 7 frame 8 frame 9
frame 10 frame 11 frame 12 frame 13 frame 14
frame 15 frame 16 frame 17 frame 18 frame 19
frame 20 frame 21 frame 22 frame 23 frame 24
frame 25 frame 26 frame 27 frame 28 frame 29
Figure 1: Visualization of the results of clusterization and
tracking algorithm (slice number 6).
4.2.1 LV Localization
FCM was applied separately to each scan to produce
two cluster using 2 as fuzziness parameter; we con-
sidered as a feature vector (I
0
,I
1
,...,I
r
) where I
d
=
G
d
∗S
t
and
G
d
is a gaussian kernel of standard devia-
tion d times the inslice resolution r
x
:
G
d
(x) =
1
(2π)
3/2
(d · r
x
)
1/2
exp

−||x||
2
(d · r
x
)
2

Experimental testing showed that setting r = 2 is
sufficient to get a good partition of the image domain.
The result of the tracking procedure on a middle slice
is shown in Figure 1. The convex-hull volume and
the inertia moments were considered as geometric
properties. The use of convex volume (instead of
the simpler volume) reduces the effect of papillary
muscles that sometimes move towards the boundary
of the region corresponding to the LV. Processing was
performed only on middle slices, thus eliminating
the apical cap and the basal segments of the LV.
Analysis of various clinical cases has been used to
introduce the Mahalanobis distance D; for simplicity,
the covariance matrix Σ has been assumed to be
diagonal.
4.2.2 LV Boundary Extraction
The previously found region corresponds roughly to
the LV cavity (LVC) and may be used to introduce
an IRS. Since LV is essentially bullet shaped, a hy-
brid sphrical/cylindrical reference system is suitable
to characterize its geometry and extract salient edge
information. To describe more in detail the IRS, sup-
pose, without loss of generality, that the z-axis of
DEFORMABLE STRUCTURES LOCALIZATION AND RECONSTRUCTION IN 3D IMAGES
219

Ω ⊂ R
3
coincides with the long axis of the LV com-
puted in the previous section and that it is oriented
from the apex to the base of the LV. A point O =
(0,0,z
0
) on the long axis is selected as the switching
point between cylindrical and spherical coordinates.
Cylindrical coordinates (r,θ,h) are assigned to points
x = (x,y, z) ∈ R
3
satisfying z− z
0
≥ 0:
r(x, y,z) =
p
x
2
+ y
2
(1)
θ(x,y,z) = tan
−1
(
y
x
) (2)
h(x,y,z) = z− z
0
(3)
whereas spherical coordinates (r, θ, φ) are given to
points satisfying z− z
0
≤ 0:
r(x, y,z) =
q
x
2
+ y
2
+ (z− z
0
)
2
(4)
θ(x,y,z) = tan
−1
(
y
x
) (5)
φ(x,y,z) = cos
−1
(
z− z
0
r(x, y,z)
) (6)
(Inverse tangent must be suitable defined to take the
correct quadrant into account)
Note that r and θ are defined consistently everywhere.
The unit vector field ˆr(x) =
∂
∂r
x/||
∂
∂r
x|| (pointing in
direction of increasing radial coordinate r) is then
given in cartesian coordinates by:
ˆr(x) =

(cosθ,sinθ,0) if z− z
0
≥ 0
(cosθsinφ,sinθsinφ,cosφ) if z− z
0
≤ 0
(7)
Note that the field ˆr is almost orthogonal to cardiac
surfaces and, therefore, the derivative
∂S
t
∂r
along the
radial direction may be used as a clue for edge detec-
tion. Indeed for a point on a cardiac surface, the mod-
ulus of radial derivative is likely to be a high fraction
of total gradient magnitude (see Figure 2). Moreover
the degree of freedom in the choice of the switching
point O may be used to tune the IRS to the peculiar
cardiac geometry under exam. An automatic proce-
dure for the selection of the switching point is de-
scribed in the Appendix. The hybrid reference system
is used to associate to each point x = (x,y, z) ∈ Ω, a
vector consisting of the following features extracted
from the data :
• Position: The position of a point x w.r.t. the
IRS is expressed as a quadruple (r,θ,φ,h). If
z − z
0
≤ 0 the entries r,θ,φ represent its spheri-
cal coordinates, whereas h is set to 0. Similarly,
for z−z
0
≥ 0, the entries r, θ, h represent its cylin-
drical coordinates whereas φ is set to π/2. No-
tice that with this choice both definitions agree for
points in the plane z = z
0
.
• Intensity: The intensity value S
t
(x) as well as its
smoothed analogues
G
d
∗S
t
(x).
(a) Slice 3 (b) Slice 5
(c) Slice 7 (d) Slice 9
Figure 2: Example of computed features: radial derivative.
• Gradient norm: Gradient norm
||∇(
G
d
∗S
t
(x))(x)|| of the smoothed images
G
d
∗S
t
.
• Radial derivative: The radial derivative of the
smoothed images
∂G
d
∗S
t
(x)
∂r
(x) = ∇(
G
d
∗S
t
(x)) · ˆr.
Using the 2-level ANN, voxels are classified on
the basis of their features vector as belonging or not to
epi- and endocardial surfaces. More in detail, the set
of extracted features is divided into two vectors
F
1
,
F
2
containing respectively 1) position and intensity
and 2) position, gradient norm and radial derivative.
The position w.r.t. IRS is replicated in both vectors
because it reveals salient for clustering both features
subsets. Then, the first level of the MANN consists of
two SOM modules, which have been defined as 2D
lattice of neurons and dimensioned experimentally,
controlling the asymptotic behavior of the number of
excited neurons versus the non-excited ones, when
increasing the number of total neurons (Di Bono
et al., 2004).
A 8× 8 lattice SOM was then trained for cluster-
ing the features vector
F
1
, while
F
2
was processed by
a 10× 10 lattice SOM.
A single EBP module has been trained to com-
bine the results of the first level and supply the final
response of the MANN. The output layer of this final
module consists in two nodes, which are used sepa-
rately for reconstructing the epicardium and the endo-
cardium. Since each cardiac surface divides the space
into two connected regions (one of which is bounded),
each output node can be trained using the signed dis-
tance function with respect to the relative cardiac sur-
face. In this way, points inside the surface are given
negative values, whereas positive values are given to
points in the outside. Henceforth the surface of in-
terest correspond to the zero-level set of the output
VISAPP 2007 - International Conference on Computer Vision Theory and Applications
220

function.
Different architectures have been tested, finding
the best performance for a network with only one
hidden layer of 15 units. A manual segmentation
was performed with expert assistance on the available
data. A set of 240 scans was used for network train-
ing, while the remaining ones were used for network
performance test.
The voxel classification, supplied by the MANN,
may be directly used for visualization purposes by us-
ing an isosurface extraction method, as shown in Fig-
ure 3. Figure 4 shows the intersection of the two car-
diac surfaces with a slice plane.
Figure 3: Different views of the rendered left ventricle at
end diastole. The surfaces are obtained applying march-
ing cubes on the two output functions of the network. To
eliminate satellites, a standard island removing procedure
is applied.
5 CONCLUSIONS
In this paper we presented a general approach to
the localization and reconstruction of periodically de-
formable structures, based on fuzzy-clustering and
multilevel artificial neural networks.
The elective case studies are represented by the anal-
ysis of heart deformable anatomical structures. Ac-
tually, for demonstrating the effectiveness of the pro-
posed framework, we have shown the preliminary re-
sults in the reconstruction of the heart left ventricle.
(a) Endocardium (b) Epicardium
Figure 4: Intersection of cardiac surfaces with a slice plane.
The next step will be to employ the obtained results
for defining a method to characterize the state of the
deformable structure (a task already started in (Colan-
tonio et al., 2006; Moroni et al., 2006)) with the
goal of building up a reference database for similar-
ity searches or data mining procedures.
ACKNOWLEDGMENTS
This work was partially supported by European
Project Network of Excellence MUSCLE - FP6-
507752 (Multimedia Understanding through Seman-
tics, Computation and Learning) and by European
Project HEARTFAID “A knowledge based platform
of services for supporting medical-clinical man-
agement of the heart failure within the elderly
population”(IST-2005-027107).
REFERENCES
ACC\AHA Task Force (2005). ACC\AHA 2005 guideline
update for the diagnosis and management of chronic
heart failure in the adult. Technical report, American
College of Cardiology Foundation and the American
Heart Association.
Bezdek, L. (1981). Pattern Recognition with Fuzzy Objec-
tive Function Algorithm. Plenum Press, New York.
Bregler, C. and Konig, Y. (1994). “Eigenlip” for robust
speech recognition. In Acoustics, Speech, and Signal
Processing ICASSP-94., volume II, pages 669–672.
Colantonio, S., Moroni, D., and Salvetti, O. (2005). MRI
left ventricle segmentation and reconstruction for the
study of the heart dynamics. In IEEE ISSPIT, pages
213–218, Athens, Greece.
Colantonio, S., Moroni, D., and Salvetti, O. (2006). Shape
comparison and deformation analysis in biomedical
applications. In Eurographics Italian Chapter Con-
ference, pages 37–43.
Coppini, G., Poli, R., and Valli, G. (1995). Recovery of the
3-D shape of the left ventricle from echocardiographic
images. IEEE TMI, 14(2):301–317.
de Bruijne, M. and Nielsen, M. (2004). Image segmentation
by shape particle filtering. ICPR, 03:722–725.
Declerck, J., Feldmar, J., Goris, M., and Betting, F. (1997).
Automatic registration and alignment on a template
of cardiac stress and rest reoriented SPECT images.
IEEE Trans. Med. Imag., 16:727–737.
Di Bona, S., Niemann, H., Pieri, G., and Salvetti, O.
(2003). Brain volumes characterisation using hier-
archical neural networks. Artificial Intelligence in
Medicine, 28(3):307–322.
Di Bono, M., Pieri, G., and Salvetti, O. (2004). A tool for
system monitoring based on artificial neural networks.
WSEAS Transactions on Systems, 3(2):746–751.
DEFORMABLE STRUCTURES LOCALIZATION AND RECONSTRUCTION IN 3D IMAGES
221

Faber, T., Stokely, E., Peshock, R., and Corbett, J. (1991). A
model-based four-dimensional left ventricular surface
detector. IEEE Tran. Med. Imag., 10:321–329.
Grenander, U. and Miller, M. I. (1998). Computational
anatomy: an emerging discipline. Q. Appl. Math.,
LVI(4):617–694.
Huang, X., Metaxas, D., and Chen, T. (2004). Meta-
Morphs: Deformable shape and texture models.
CVPR, 01:496–503.
Jolly, M., Duta, N., and Funka-Lea, G. (2001). Segmen-
tation of the left ventricle in cardiac MR images. In
ICCV, pages 501–508.
Kass, M., Witkin, A., and Terzopulos, D. (1988). Snakes:
Active contour models. IJCV, 1:321–331.
Kohonen, T. (1997). Self-Organizing Maps, volume 30 of
Springer Series in Information Sciences. 2nd edition.
Mitchell, S., Bosch, J., Lelieveldt, B., van der Geest, R.,
Reiber, J., and Sonka, M. (2002). 3D active appear-
ance models: Segmentation of cardiac MR and ultra-
sound images. IEEE TMI, 21(9):11671178.
Moroni, D., Perner, P., and Salvetti, O. (2006). A gen-
eral approach to shape characterization for biomedical
problems. In Perner, P., editor, Industrial Conference
on Data Mining ICDM - Workshop on Mass-Data
Analysis of Images and Signals, IBAI CD-Report,
pages 56–65, Leipzig.
Papin, C., Bouthemy, P., M
`
emin, E., and Rochard, G.
(2000). Tracking and characterization of highly de-
formable cloud structures. In Computer Vision -
ECCV 2000, volume 1843 of LNCS, pages 428–442.
Springer Verlag.
Paragios, N. (2002). A variational approach for the segman-
tation of left ventricle in cardiac image analysis. IJCV,
50(3):345–362.
Paragios, N., Rousson, M., and Ramesh, V. (2002).
Knowledge-based registration segmentation of the left
ventricle: A level set approach. In Proc. Sixth IEEE
Workshop on Application of Computer Vision, pages
37–42.
Riedmiller, M. and Braun, H. (1993). A direct adap-
tive method for faster backpropagation learning: The
RPROP algorithm. In Proc. of the IEEE Intl. Conf. on
Neural Networks, pages 586–591, San Francisco, CA.
Stalidis, G., Maglaveras, N., Dimitriadis, A., and Pappas,
C. (1999). Using learning classification and four-
dimensional parametric modeling for the analysis of
myocardial thickening. In Computers in Cardiology.
¨
Uz
¨
umc
¨
u, M., Frangi, A., Sonka, M., Reiber, J., and
Lelieveldt, B. (2003). ICA vs. PCA active appearance
models: Application to cardiac MR segmentation. In
LNCS, volume 2878, pages 451–458.
APPENDIX: IRS SWITCHING
POINT SELECTION
An automatic procedure that tune the IRS to the pe-
culiar cardiac geometry under exams may be devised
Figure 5: The mean z-profile of the LVC is plotted, together
with outer normals to the curve (green) and radial vector ˆr
O
(red) in selected points.
exploiting our knowledge of LVC.
Let O = (0, 0, z
0
) be a point on the LVC long axis and
let ˆr
O
be the unit vector field given by eq. 7. A natural
objective function that estimates orthogonality of the
field ˆr
0
w.r.t. endocardial surface is given by:
J(O) =
Endocardium
(1− ˆr
O
· ˆn)
2
(8)
where ˆn is the outer normal to the endocardial surface.
In the previous equation, it is possible to approximate
the (unknown) endocardial surface with the boundary
of the LVC:
J(O) =
∂(LVC)
(1− ˆr
O
· ˆn)
2
(9)
Note that, in the spirit of Section 3.2.1, suitability of
IRS to the cardiac geometry means in particular that
ˆr
O
should be orthogonal to cardiac surfaces. There-
fore selection of the switching point may be translated
into the optimization problem:
ˆ
O = argmin
O
J(O) (10)
However, for our purposes, it is not necessary to solve
the optimization problem in eq. 10; indeed it is not
meaningful to compute a so fine estimation of the
switching point on the base of the rough data pro-
vided by LVC boundary. Instead, we prefer to con-
vert this 2D fitting problem (integration on a surface)
to a 1D problem (integration on a curve), by taking
into account the z-axis mean profile of the LVC. This
consists in the curve obtained considering the mean
radius of the sections of LVC with a pencil of parallel
planes {z = cost} (see figure 5).
In this new setting the orthogonality criterion 10 may
be restated with obvious modification.
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