APPLICATION OF MODAL ANALYSIS FOR EXTRACTION OF
GEOMETRICAL FEATURES OF BIOLOGICAL OBJECTS SET
Mchal Rychlik, Witold Stankiewicz
Division of Methods of Machine Design, Poznan University of Technology, ul. Piotrowo 3, Poznan, Poland
Marek Morzyński
Division of Methods of Machine Design, Poznan University of Technology, ul. Piotrowo 3, Poznan, Poland
Keywords: 3D geometry reconstruction, anthropometric measurements, PCA (Principal Component Analysis),
registration, reverse engineering.
Abstract: This article presents application of modal analysis for the computation of data base of biological objects set
and extraction of three dimensional geometrical features. Authors apply two types of modal analysis:
physical (vibration modes) and empirical (PCA – Principal Component Analysis) for human bones. In this
work as the biological objects the fifteen human femur bones were used. The geometry of each bone was
obtained by using of 3D structural light scanner. In this paper the results of vibration modal analysis (modes
and frequencies) and PCA (mean shape and features – modes) were presented and discussed. Further the
possibilities of application of empirical modes for creation three dimensional anthropometric data base were
presented.
1 INTRODUCTION
Nowadays, many engineering CAD technologies
have an application not only in mechanics but also in
different disciplines like biomechanics,
bioengineering, etc. This interdisciplinary research
takes advantage of reverse engineering, 3D
modelling and simulation, PCA analysis and other
techniques. The 3D virtual models have a numerous
applications such as visualisation, medical
diagnostics (e.g. virtual endoscopies), pre-surgical
planning, FEM analysis, CNC machining, Rapid
Prototyping, etc. Several engineering technologies
can be used for analysis of biological objects.
Usually the populations of the biological objects
like bones, are used to be described only in two
dimensional space, by the set of the dimensions (e.g.
distance). Thereby traditional anthropometric data
base contains information only about some
characteristic points, while other parameters are not
collected. Generally data acquisition process is made
with usage of the conventional measurements
equipment (e.g. calliper). For any new research work
(when not existing parameter is needed) completely
new study and measurements process must be done.
The new methods of statystical analysis and
storage of complete parametric data for each of all
elements from population are researched. The
methods which can be used to describe a
geometrical parameters of 3D objects are modal
analysis.
2 MODAL ANALYSIS METHODS
In this chapter authors present modal analysis
methods which are used for geometry description of
three dimensional objects and data base creation.
These methods are used to simplify and minimize
the number of parameters which describe 3D
objects.
One of the methods that is based on modal
decomposition is PCA (Principal Component
Analysis, known also as POD – Proper Orthogonal
Decomposition). While empirical modes (PCA) are
optimal in the sense of information included inside
each of the modes (Holmes, Lumley and Berkooz,
1998), often other decompositions, based on
mathematical (e.g. spherical harmonics) or physical
modes (vibration modes) are used. The kind of
227
Rychlik M., Stankiewicz W. and Morzy
´
nski M. (2008).
APPLICATION OF MODAL ANALYSIS FOR EXTRACTION OF GEOMETRICAL FEATURES OF BIOLOGICAL OBJECTS SET.
In Proceedings of the First International Conference on Biomedical Electronics and Devices, pages 227-232
DOI: 10.5220/0001048902270232
Copyright
c
SciTePress
modal method (mathematical, physical or empirical)
which is applied to analysis has a fundamental
importance for results.
The goal of using mathematical modes is
conversion of physical features onto mathematical
features (synthetic form). In the case of the
mathematical modes the features which describe
geometry of 3D object are usually saved as the
vectors. Each vector is obtained through splitting of
the 3D model onto several classes (different
diameter spheres) and calculation of common areas
between 3D object and surfaces of individual
spheres. All areas are described by a set of vectors
(spherical functions). For spherical functions Fourier
transformation is used, resulting in easier
multidimensional description of feature vectors. For
representation of feature vectors spherical harmonics
are used (figure 1.).
Figure 1: Example of spherical harmonics of 3D model of
aeroplane and application of spherical harmonic in
reconstructtion of geometry of the cube (Vranic and
Saupe, 2002).
Application of spherical modes is not optimal
solution and sometimes causes increased
computation costs, because all objects are
approximated by deformed sphere. Reconstruction
of geometry of the cube requires very large number
of spherical harmonics. This problem is analogous to
Fourier decomposition of rectangular signal.
The second group of modal decompositions of
3D objects is represented by physical (mechanical)
modes. These modes – also known as the vibration
modes – are obtained by solution of eigenproblem
for elastic model of analyzing object. Vibration
modal decomposition provides alternative
parameterisation of degrees of freedom of the
structure (translations of the nodes in x, y, z
directions only) based on eigenmodes of the objects
and correlated frequencies (eigenvalues). Usually
eigenmodes related with low frequencies, describing
deformation vectors for individual nodes of FEM
grid, are used. This way the deformation of
geometry of base object and its fitting into searched
object is possible. Vibration modes computed for
rigid body represent translations and rotations of 3D
model and vibration modes of elastic body describe
different variations of the base model’s shape
(figure 2.).
Figure 2: Graphical representation of seven low frequency
vibration modes for surface model of ellipsoid (Syn and
Prager, 1994).
PCA transformation gives orthogonal directions
of principal variation of input data. Principal
component which is related with the largest
eigenvalue, represent direction in data space of the
largest variation. This variation is described by
eigenvalue of largest magnitude. The second
principal component describes the next in order,
orthogonal direction in the space with the next
largest variation of data. Usually only a few first
principal components are responsible for a majority
of the data variations. The data projected onto other
principal components often have small amplitude
and can be treated as measurement noise. Therefore,
without the loss of accuracy, components related to
smallest eigenvalues can be ignored.
3 PHYSICAL MODES –
VIBRATION MODES
Decomposition basis on vibration modes uses
similar procedure like in analysis of dynamical
problems. For describing of complicated moving
they used set of simple functions (1):
() ()()
=
=
N
n
nn
zyxtqtzyxu
1
,,,,,
φ
(1)
where N is the number of used functions,
n
φ
is the
vector (mode) of the object’s vibration, and
n
q is
coefficient for
n -th mode in time t .
Linear elastic structures can be described by
surface or volume finite elements. After
discretization in FEM software the eigenanalysis is
done, using the mathematical oscillation model (2):
()
tfKuuCuM =
+
+
(2)
where CM , and
K
are adequately: mass,
damping and stiffness matrix, and
u
is vector of
grid node displacements.
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3.1 FEM Model
Computations have been done using NASTRAN
software. Model geometry is based on surfaces
resulting from 3D-scanning of real human femoral
bone. Finite element mesh consists of approximately
4800 nodes and 5500 elements of two types.
External layer of 1940 triangular plate elements
represents compact (cortical) bone and 2560
tetrahedral elements represent internal, trabecular
bone. Model was fixed in condyles part.
In both cases, orthotropic material, based on
measurement data (Ogurkowska et al, 2002), was
used.
3.2 Eigenmodes
The result of eigenproblem solution is a set of
eigenvalues (representing the vibration frequencies)
and eigenmodes. The eigenmodes related with
lowest frequencies, added to mean shape of femur,
are presented at figure 3.
Mode 1: freq=28.73Hz
Mode 1: freq=30.56Hz
Mode 3: freq=189.28Hz
Mode 4: freq=206.35Hz
Mode 5: freq=261.36Hz
Mode 6: freq=507.37Hz
Figure 3: Eigenmodes related with lowest frequencies.
Gray scale levels represent total translation and
dark bone is a mean, undeformed shape. Each bone
is presented in two views: posteriori and anterior
view.
Practical application of vibrational modes is
strongly limited due to high number of modes
required to reconstruct different geometries.
Additionally eigenmodes don’t represent any
biophysical features of human femoral bones.
4 EMPIRICAL MODES – PCA
Despite the fact that the method is called differently
in various application areas, the used algorithm is
generally the same and is based on statistical
representation of the random variables.
The shape of the every object is represented in
the data base as the 3D FEM grid and described by
the vector (3)
[
]
,,,,
21
T
iNiii
sssS = ,,,2,1 Mi =
(3)
where
(
)
zyxs
ij
,,
=
describes coordinates of each
of the nodes of FEM grid in Cartesian coordinates
system.
M
is the number of the objects which are in
database,
N is the number of the FEM nodes of
every single object. The decomposition is based on
computation of the mean shape
S
and covariance
matrix
C (4):
=
=
M
i
i
S
M
S
1
1
,
=
=
M
i
T
ii
SS
M
C
1
~~
1
(4)
The difference between mean shape and current
object from data base is described by the
deformation vector
SSS
ii
=
~
. The statistical
analysis of the deformation vectors gives us the
information about the empirical modes. Modes
represent the features: geometrical (shape), physical
(density) and others like displacement and rotation
of the object. Only few first modes carry most of the
information, therefore each original object
i
S can
be reconstructed by using some
principal
components (5):
=
Ψ+=
K
k
kkii
aSS
1
,
,,,2,1 Mi =
(5)
where
k
Ψ
is an eigenvector representing the
orthogonal mode (the feature computed from data
base),
ki
a is coefficient of that eigenvector and
i
-th
data base model. The example of low dimensional
reconstruction for three different values of the
coefficient of the first mode is presented on the
figure 4. For
0
=
ki
a we obtain mean value, for
different values we get new variants of object’s
shape.
APPLICATION OF MODAL ANALYSIS FOR EXTRACTION OF GEOMETRICAL FEATURES OF BIOLOGICAL
OBJECTS SET
229
Figure 4: The visualisation of the reconstruction for
different coefficient values.
4.1 Data Acquisition – 3D Scanning
As the input data (data base) 15 femur bones were
measured (6 female, 9 male). For 3D scanning
(Rychlik Morzyński and Mostowski, 2001) the
structural light 3D scanner – accuracy 0,05mm –
was used (figure 5). Each bone was described by
individual point cloud (1.5mln points) and triangle
surface grid (14000nodes, 30000 elements).
Figure 5: Data acquisition: a) input femur bones,
b) measurement process, c) final triangle surface grid.
4.2 Data Registration
The Principal Component Analysis requires the
same topology of the FEM mesh for all objects (the
same number of nodes, connectivity matrix, etc.). To
achieve this, every new object added to data base,
must be registered. The goal of registration is to
apply the base grid onto geometry of the new
objects. The registration is made in two steps. First
step (preliminary registration) is the rigid
registration - a simple geometrical transformation of
solid object in three-dimensional space (rotation and
translation). The second step is the viscous fluid
registration. For this registration the modified
Navier-Stokes equation in penalty function
formulation (existing numerical code: Morzynski,
Afanasiev and Thiele, 1999; source segment:
F Bro-
Nielsen and Gramkow, 1996) is used (6):
()
0
Re
1
,,,,
=+
++


segmentsource
i
codenumericalexisting
jijjjijjii
fgfVVVVV
ρ
λ
ε
(6)
where
ρ
is fluid density,
i
V velocity component,
Re Reynolds number,
λ
bulk viscosity. In this
application parameters
ε
and
λ
are used to control
the fluid compressibility,
f is the base object,
g
is
the target object (input model). The object is
described by the FEM grid. The displacements of the
nodes are computed from integration of the velocity
field. Computed flow field provides information
about translations of the nodes (FEM grid) in both
sections. After computation we obtain dislocation of
nodes of the base grid onto new geometry
(figure 6.).
Figure 6: FEM grid deformation (from the left): base
object, new object, base FEM grid on geometry of the new
objects.
4.3 Empirical Modes – PCA
For that prepared database of 15 femur bones the
Principal Component Analysis was done. The result
of this operation is the mean object, fifteen modes
and coefficients (figure 7).
Table 1: Participation of the modes in reconstruction.
Number of
the mode
Participation of
the mode [%]
Total participation
of the modes [%]
1 74.9212416
74.9212416
2 10.5438352
85.4650767
3 4.2699519
89.7350286
4 3.3128685
93.0478971
5 1.6659793
94.7138765
6 1.4234329
96.1373093
7 1.0359034
97.1732127
8 0.6781645
97.8513772
9 0.5866122
98.4379894
10 0.4796167
98.9176061
11 0.3301463
99.2477523
12 0.3080968
99.5558492
13 0.2516839
99.8075330
14 0.1924670
100.0000000
15 0.0000000
100.0000000
The first fourteen modes include one hundred
percent of information about decomposed geometry
(table 1.). Fifteenth mode contains only a numerical
noise and it is not used for further reconstruction.
Modes describe the features of the femur bones.
First mode describes the change of the length of the
femur bone, second mode – the change of the
position of the head of the bone, third - change of
the arc of the shaft (body). Further modes describe
more complex deformations. For example fourth
mode describes the change of position of the greater
trochanter and lesser trochanter and also the thick-
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230
Figure 7: Visualisation of the mean value and first nine
empirical modes of femur bones (anterior and posterior
view).
ness of the shaft (body). Fifth mode describes
deformation of the greater trochanter in other
directions. Sixth describes deformation of the greater
trochanter and lesser trochanter, and position of the
shaft (body) in other directions.
Results of the statistical analysis (empirical
modes) can be used for reconstruction of the
geometry (in CAD systems) of individual features of
the object. Empirical modes give as information
about 3D mean shape of population of objects and a
set of the geometrical features that describes
principal deformations in analyzed population of the
objects.
This method can be used for creation of complete
3D anthropometric database and gives us possibility
to measure any dimension on the surface of the
bone.
Real 3D anthropometric database is also
necessary in practical application of the method of
reconstruction of 3D biological objects basing on the
few RTG images (Rychlik, Morzynski and
Stankiewicz, 2005).
5 CONCLUSIONS
Although we can use several modal methods to
describe the geometry of 3D objects, only empirical
modes give us an optimal statistical data base.
Graphical representation of spherical harmonics
is very specific and it is impossible to find similarity
with input model, with exception of algebraic
relations.
There are several differences between methods
producing physical (vibration) modes and empirical
modes (PCA, POD, Karhunen-Loeve), that are used
in modeling of 3D objects.
A large limitation of usage of physical modes
(vibration modes) is the impossibility to obtain the
modes that describe resizing (scaling) of whole
object or it’s parts. These features are skipped out
and they cannot be used in decomposition. The
problem is also the large number of the modes that
must be used in description of the shape of the
object. Sometimes for reconstruction of a very
simply geometry (e.g. cube) we must use a lot of
modes (even up to 200 modes).
In case of physical modes, the only available
determinant of mode’s suitability is the vibration
frequency (eigenvalue). One can assume that modes
with high frequencies will represent numerical noise
only, but the number of modes related with low
eigenvalues that have to be used in reconstruction of
another 3D object of the population is unknown.
While the total number or eigenmodes is equal to the
number of degrees of freedom of the model (in our
case: 3x4800 DOF), the modal description of the
population using physical modes might require
larger data storage than input data (separate grids for
each of the objects), and might still be incomplete
(the scaling mentioned before).
Empirical modes describe features of the object
that are dependent on frequent occurrences in
population. The largest eigenvalues are related with
Mean
value
Coefficient value
min
Coefficient value
max
Mode
1
Mode
2
Mode
3
Mode
4
Mode
5
Mode
6
Mode
7
Mode
8
Mode
9
APPLICATION OF MODAL ANALYSIS FOR EXTRACTION OF GEOMETRICAL FEATURES OF BIOLOGICAL
OBJECTS SET
231
modes describing the most important features, what
makes the reduction of data storage quite simple.
For Karhunen-Loeve analysis of data base which
consists of several similar, but not the same objects,
differing from each other only in the scale, this
feature (size of the object) will be the most dominant
empirical mode. Additionally, the number of modes
required to reconstruct the whole population of
objects without quality losses is assured to be
smaller or equal to the number of objects in that
population. In practice, a number of empirical modes
can be used to describe the population with accuracy
higher than in case of any other modes (optimality of
PCA mentioned before).
For empirical modes (in data base) it is possible
to keep the additional information’s, e.g. data from
diagnostic systems, density, Young’s modulus, and
other material properties.
PCA can be used for creation of complete three
dimensional anthropometric data base.
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