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

K

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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