EVALUATION OF QUALITY RESULTS CONCERNING IMAGE

PROCESSING BASED STRUCTURAL MONITORING

Rharã A. Cardosoa, Nicholas A Mota

d, Flavio S. Barbosa

a,b, Patricia H. Hallak

a

Fernando M. A. Nogueirac, Jair G.O. Borges

e

aDepartment of Applied and Computational Mechanics, Federal University of Juiz de Fora, Juiz de

Fora – MG, Brazil

http://www.ufjf.br/mac

bGraduation Program in Computational Modeling, Federal University of Juiz de Fora, Juiz de Fora –

MG, Brazil

http://www.ufjf.br/mmc

cDepartment of Production and Mechanics Engineering, Federal University of Juiz de Fora, Juiz de

Fora – MG, Brazil

http://www.ufjf.br/ep

dTutorial Educational Program of Civil Engineering, Federal University of Juiz de Fora – MG, Brazil

http://www.ufjf.br/petcivil

eCivil Engineering Laboratory, State University of North Fluminense, Campos dos Gotacazes

- RJ, Brazil

http://www.uenf.br

Keywords: Structural Identification; Image Processing; Time and Frequency Domain

Identification Methods

Abstract. The use of image processing based structural monitoring system has increased over

the last years. Professional integrated monitoring systems are been developed and, in a short

time period, it is expected an augmentation of the application of this non-contact

instrumentation method. In the present work, qualitative and quantitative evaluations of

structural instrumentation results obtained by means of an image processing monitoring

system are carried out. To this end, traditional instrumentation sensors as strain-gages and

accelerometers are used in order to define benchmark solutions which are compared to image

processing based results counterpart. Time and frequency domain modal identification

methods, such as Ibrahim Method, Random Decrement Method, Exponential Decrement

Method, Peak Picking Method and Frequency Decomposition Method, are applied to the

experimental data obtained from the performed laboratory tests allowing comparisons

between traditional and image processing based instrumentation.

1 INTRODUCTION

The use of digital images to analyze dynamical behavior of structures is present in some

works in the literature (Poudel, 2005 and Ram and Caldwell ,1996). Specifically, in the

structural dynamic domain, researches analyze structural damages (Ewins, 2000) and, in the

most cases, performed a modal identification (Asmussen, 1998). In all these cases, it is

important to verify if the experimental data obtained with cameras allow accurate results. This

is the main objective of this paper.

Traditionally, time histories of an analyzed structure are extracted by using, for example,

accelerometers, or strain-gages. In this paper, these kinds of sensors are adopted in order to

evaluate the quality of an image processing based experimental data.

To this end, a cantilever beam is instrumented with an accelerometer, a strain-gage and a

non-contact sensor (image processing based technique) and dynamically tested.

Firstly, modal characteristics extracted from strain-gage and non-contact experimental data

are compared. In this case, a computer model is also used to support comparisons.

Secondly, acceleration time histories captured by the accelerometer are directly compared

to the ones obtained with the non-contact sensor. In this case, the second derivative of the

obtained non-contact sensor displacements is numerically calculated in order to allow

acceleration comparisons.

The organization of the text includes, besides the present introduction, section 2, where the

applied modal identification methods are briefly presented; section 3, where the image

processing based technique is detailed; section 4, where the experimental tests and numerical

model are shown; section 5 with the evaluation of the non-contact instrumentation and

discussions; and the conclusion section.

2 IDENTIFICATION METHODS

2.1 Methods in Time Domain

a) Random Decrement Method - RD (Asmussen, 1998)

The random decrement method is a temporal modal method which was used at the first

time by H. Cole in the late 60s. Since that time, this method has not stopped growing. It is

particular used in the aerospace industry for the analysis of experimental vibration data. The

success of this method is its simplicity of use and ability to analyze data in real time.

This method transforms a stochastic processes y(t) into a random decrement function

(Dy()) defined as:

( )( ) ( )y y tD E y t T

(1)

where Ty(t) is called as trigger condition.

An estimation for the random decrement function may be written using mean values

described in following equation:

( )

1

1( ) ( ( ) )

i

N

y i y t

i

D y t TN

(2)

Equation (2) is usually interpreted as an approximation of the structural free vibration

response. In that way, methods that use this kind of time history for modal identification, as

Exponential Decrement Method (Clough and Penzien, 1993) or Ibrahim Method (Ewins, 2000), may be applied.

b) Exponential Decrement Method - ED (Clough and Penzien, 1993)

This is one of the most simple identification method. For this reason, this method is widely

used. By curve fitting modal parameters to structural free vibration time history responses

filtered around a specific natural frequency, it is possible to identify damping ratio and

natural frequencies. Natural frequencies are expressed by Eq. (3) and damping ratio by means

of Eq. (4).

ii T

2 (3)

q

p

iD

Dln

2

1

(4)

being i the i-th natural circular frequency of vibration period Ti; i the i-th damping ratio; Dp

and Dq two consecutive peaks of the free vibration response D.

c) Ibrahim Time Domain Method – ITD (Ewins, 2000)

By solving the eigenvalue problem described by:

(A - iI) = 0 (5)

where A is the state matrix of the system (directly extracted from the free vibration system

response), natural frequencies and damping ratios may be achieved from the eigenvalues i,

and modal amplitudes from the eigenvectors

2.2 Methods in Frequency Domain

a) Peak Picking Method - PP (Felber, 1993)

The peak picking method (PP), also called as basic method, assumes that the modal

parameters may be identified if the structure is excited by a Gaussian white noise. With this

assumption, it is possible to identify the modal parameters using only its response over time

The natural frequencies are recognized through the resonant peak of the Average

Normalized Power Spectral Density (ANPSD):

GLn

ini

GL

n NPSDn

ANPSD1

)(1

)( (6)

where:

N

in

nni

PSD

PSDNPSD

1

)(

)()(

(7)

In Eq. (6) GLn is the number of the analyzed degrees of freedom. In Eq. (7), )( nPSD

denotes the Power Spectrum Density for n , being N the among of discrete frequencies of the

analyze.

In order to verify if identified natural frequencies are not noises, a coherence function

described in Eq. (8) may be used. In this equation, i and j are the degree of freedom, ijS~

are

the elements of the spectral density matrix and ijγ is the coherence function, which can range

from 0 to 1. Values of ijS~

close to unity indicate a higher degree of linearity between the

measured signals, pointing that the identified frequencies are actual natural frequencies of the

analyzed structure.

)(ωS)(ωS

)(ωS)(ωγ

mijmij

mij

mij ~~

~ 2

2

(8)

The modal configuration is attained by using the transfer function, which relates the PSD

of a degree of freedom with a reference degree of freedom.

The damping ration is evaluated using the traditional half-power method. However, many

researchers report some uncertainty in this evaluation (Borges, 2010). This method requires

for a reasonable estimation of damping that natural frequencies are not close from each other.

b) Frequency Domain Decomposition Method – FDD (Brincker, 2000)

The Frequency Domain Decomposition method was developed in order to solve the

problems of PP method. It is based on the singular value decomposition of the PSD matrix,

known at discrete frequency i . It was firstly proposed by Brincker et al (2000) and

Brincker et al (2001) . This decomposition is shown in Eq. (9), where the matrix

imiii uuuU ,...,, 21 is an unitary matrix having the singular vectors iju and iS is a diagonal

matrix with the scalar singular values ijs .

H

iiiiyy USUjG )(

(9)

Near to a peak, corresponding to the k-th mode in the spectrum, the correspondent mode

will be dominating. Thus, in this case, the first singular vector is an estimation of the

mode shape:

1iu

(10)

This PSD function is identified around the peak by comparing the estimated mode shape

with singular vectors for the frequency near this peak. Modal Assurance Criterion (MAC)

value is used to compare these vectors.

Using the Single Degree of Freedom (SDOF) PSD obtained around a peak, the natural

frequency and the damping can be achieved. In this paper the PSD’s piece related to a SDOF

is taken back to the time domain by an Inverse Fast Fourier Transform (IFFT). Frequency and

the damping are estimated from the crossing times and the logarithmic decrement of the

corresponding SDOF autocorrelation function.

3 IMAGE PROCESSOR ALGORITHM (Gonzales and Woods, 2002)

The methodology used in this work consists on recording fixed targets placed on a

structure under vibration using a digital camera. The obtained video is transferred to a

computer and the developed software is applied in order to automatically identify the central

target coordinates for each video frame. This process, after simple algebraic operations,

results in displacement time histories. Modal parameters may be extracted by using

identification methods (section 2).

The targets must allow a robust identification through automatic methods and it should

present geometric characteristics compatible to the application. In this way, a black circle on a

white background is used as target.

A simple thresholding is capable to transform a grayscale (8 bits) or true color (24 bits)

image into a binary (1 bit) image as presented in Fig. 1. In this case a grayscale image is

transformed into a binary image. The thresholding defines the limit between the light and dark

image pixel grayscale, transforming light ones into white pixels, and dark ones into black

pixels.

Using a binary image, the pixels attached to the black color (inside the circle) have the

label b(x,y)=1, and the pixels attached to the white color (outside the circle) have the label

b(x,y)=0, as shown in Figure 1.

Figure 1 - The threshold operation

The thresholding limit may be considered constant if the illumination conditions are stable

during all operation.

After the target pixel identification, it is possible to determine the black circle image

centroid yx,

coordinates x and y , shown in Figure 1, with sub-pixel precision using:

xyxbMN

xN

x

M

y

.,1

1 1

(11)

and:

yyxbMN

yN

x

M

y

.,1

1 1

(12)

where: N and M are the number of columns and rows of the image, respectively.

By taking the coordinates yx,

obtained for each frame, it is possible to generate time history

series tx and

ty , where t represents the frame number. These time series allow structural

natural frequency identification that, in this work, is obtained by using algorithms described

in section 2.

This methodology is quite simple and relatively not expensive but it may be not generally

applied due to some limitations. A modal identification using the proposed method may present errors in some particular situations.

The first limitation is attached to the aliasing problem. The standard digital camera used in

the analysis has acquisition rate equals to 30 frames per second, and this situation demands a

dynamic behavior with maximal frequency component of 15 Hz (Nyquist frequency). This

low frequency limit may be increased using a higher frame rate camera, which is obviously

more expensive than the used camera.

Secondly, it must be observed that images are bi-dimensional (2D) projections of the three-

dimensional (3D) space. In that case, the perspective effects of this 3D to 2D transformation

affect natural frequency identification when the camera sensor plan is not parallel to the

analyzed structure oscillation plan. If this parallelism does not occur, magnitude of modal

components are affected and ghost frequencies may appear in the time seriesty

(Nogueira et al,

2005).

Finally, illumination and reverberation problems may cause inaccurate results. This kind of

problem is inherent to image processing methods.

4 EXPERIMENTAL TESTS AND NUMERIAL MODEL

4.1 Description of the tested structure:

Figure 2a shows a picture of the cantilever beam in which four masses with four circular

black targets were added and a strain gage was bounded. Figure 2b presents a detail of the

accelerometer connected under the beam at the same position of a target. Table 1 presents

physical and geometrical properties of this beam.

a) The cantilever beam with the targets and a bounded strain-

gage b) A detail of the connected accelerometer

Figure 2 – Pictures of the experimental tests.

Table 1. Physical and geometrical properties of the beam.

Material aluminum

Elasticity module 70.3 GPa

Specific mass 2600 kg/m3

Cross section area 39 mm2

Moment of inertia 29.25mm4

This beam was dynamically excited with an impulsive load and the three kinds of sensors

recorded time histories in terms of accelerations, strains and displacements. For accelerometer

and strain-gage results, frequencies of acquisition were set in 300 Hz and a low-pass filter in

20Hz was used. The camera frame rate was 30 Hz.

Figures 3, 4 and 5 present typical time histories obtained with the accelerometer, the strain-

gage and the camera, respectively.

Figure 3 – An accelerometer signal.

Figure 4 – A strain gage signal.

Figure 5 – Non-contact sensor signals. Series node 1 refers to the first target, from the left to the right (see Fig. 2)

and series node 3 refers to the third target, also from the left to the right.

4.2 Description of the numerical model:

A numerical simulation of the dynamic behavior of the tested beam was applied in order to

aid the evaluation of the non-contact sensor. A Finite Element Model (FEM) with the

following consideration was constructed:

i. The vibrations occur around an undeformable configuration, it means, the static

deformation imposed by adding masses were disregarded;

ii. The structure were discretized using frame elements with nodes located in the

geometrical middle of each mass;

iii. The four masses added in the system were introduced in the model as

concentrated masses.

The numerical model reached with these assumptions takes the form of the scheme

presented in Fig. 6, with 5 nodes and 4 elements. Table 2 resumes the dynamic properties

obtained in the numerical simulation for the first two mode shapes (and respective natural

frequencies ( and ). Note that the amplitude of each mode i were normalized using the

amplitude of free extreme (node 1).

Figure 6 – Numerical model.

-15

-10

-5

0

5

10

15

0 2 4 6 8 10t(s)

dis

pla

cem

ent

(mm

) node 1 node 3

0.20m 0.20m 0.20m 0.22m

78g 82g 72g 72g

0.025m

1

1

2 3 4 5

2 3 4

0.20m 0.20m 0.20m 0.22m

78g 82g 72g 72g

0.025m

1

1

2 3 4 5

2 3 4

Table 2 – Dynamic properties of the numerical model

Node

Mode 1

f1=1.41 Hz

Mode 2

f2=9.10 Hz

1 1.00 1.00

2 0.66 -0.60

3 0.34 -1.24

4 0.10 -0.70

5 0.00 0.00

5 EVALUATION OF NON-CONTACT SENSOR RESULTS AND DISCUSSIONS

5.1 Evaluation of modal identification

The signal of Fig. 4 was used to determine modal reference results. By curve fitting

theoretical free vibration solution for the model presented in Eq. (13), it was possible to

obtain natural frequencies and damping.

2

1)exp()cos()(

i iiiii tttx (13)

being, x(t) the strain amplitude; i the i-th modal amplitude; i and i the i-th natural circular

frequency and damping ratio, respectively; and i the i-th phase angle.

In Table 3 are presented frequencies, damping ratios obtained and magnitudes of and

for each mode1.

Table 3 – Frequencies and damping ratio of the simulated signal.

Mode fi(Hz) i(%) i i

1 1.49 0.30 0.34 0

2 10.00 0.50 0.70 0

After the curve fitting process, x(t) must have to approximate the original signal obtained

with the strain gage. In order to assure the fidelity between both signals, MAC (modal

assurance criterion) coefficient was used. In Fig. 7 is plotted x(t) and the original signal from

the strain gage, for a short period of time. The MAC coefficient obtained was approximately

96%, indicating a very good correlation between both signals.

Figure 7 – Comparison between strain gage signal and theoretical curve fitted model.

Modal values presented in Table 3 were considered as a benchmark, since they allow an

almost perfect correlation between the strain gage response and the theoretical curve fitted

model x(t). 1 Note that damping ratios are very small (lower than 1%). Consequently, damped and undamped natural

frequencies are considered identical.

-200

-150

-100

-50

0

50

100

150

200

2 2.5 3 3.5 4 4.5 5t (s)

ms

theoretical curve fitted model

strain gage signal

In Table 4, the frequencies and damping ratio obtained using the ITD (after applying RD),

PP and FDD methods applied to the non-contact experimental data are given. Achieved

modes shapes are presented in Table 5.

Table 4 – Frequency and damping ratio results (Hz).

Mode Frequency (Hz)

Damping ratio

(%)

ITD PP FDD ITD PP FDD

1 1.50 1.56 1.56 1.30 6.00 3.50

2 9.80 9.96 9.96 7.30 1.20 0.60

Table 5 – Modes shapes results.

Mode 1

Node ITD PP FDD

1 1.00 1.00 1.00

2 0.67 0.67 0.67

3 0.37 0.37 0.37

4 0.13 0.14 0.13

5 0.00 0.00 0.00

Mode 2

Node ITD PP FDD

1 1.00 1.00 1.00

2 -0.55 -0.50 -0.52

3 -1.20 -1.16 -1.19

4 -0.80 -0.79 -0.81

5 0.00 0.00 0.00

Reference natural frequencies – Ref. (obtained from the strain gage instrumentation –

Table 3), and camera results, are shown in Table 6. Values of frequency are presented with

percentage differences to Ref. values.

Table 6 – Frequency comparisons (Hz)

fi Ref.

Camera

ITD

(-%)

PP

(-%)

FDD

(-%)

1 1.49 1.50

(0.67%)

1.56

(4.70%)

1.56

(4.70%)

2 10.0 9.80

(-2,00%)

9.96

(-0.40%)

9.96

(-0.40%)

Regarding Table 6, it is possible to observe that natural frequencies were correctly

determined with non-contact data.

Considering, damping ratios, strain-gage data and non-contact data allow an erratic

performance as it may be verified in Table 7. Discrepancies in damping ratio identification are

not rare in modal analysis. Lozano (2003) and Mendes and Oliveira (2008) face the same

behavior for damping ratio identification.

Table 7 –Damping ratios comparison

Mode benchmark Camera

ITD PP FDD

1 0.30 1.30 6.00 3.50

2 0.50 7.30 1.20 0.60

Finally, first and second mode shapes and FEM’s results are compared in Fig. 8 and 9,

respectively. FEM’s mode shapes were defined as Reference, since strain-gage and

accelerometer instrumentations have only a single measure point.

Figure 8 – First identified mode shapes and FEM’s result

Figure 9 – Second identified mode shape and FEM’s result

The identified mode shapes have practically coincident amplitudes and FEM’s results are

slightly different from then, in both figures. Table 8 presents MAC comparisons for mode

shapes #1 and #2, respectively. For all cross mode shape comparison, MAC values are

superior than 99%, indicating an excellent correlation.

Table 8 –Mode shapes comparisons via MAC

Mode 1

FEM ITD PP FDD

FEM 100 99.93 99.90 99.93

ITD 99.93 100 99.99 100

PP 99.90 99.99 100 99.99

FDD 99.93 100 99.99 100

FEM 100 99.93 99.90 99.93

Mode 2

FEM ITD PP FDD

FEM 100 99.58 99.34 99.36

ITD 99.58 100 99.94 99.96

PP 99.34 99.93 100 99.99

FDD 99.36 99.96 99.99 100

FEM 100 99.58 99.34 99.36

Table 8 shows that image processing data allow correct identification of mode shapes.

5.2 Acceleration time history comparisons

This last evaluation is made by comparing acceleration time histories obtained with the

accelerometer and non-contact sensor results counterpart. As the camera results are recorded

in terms of displacements, numerical derivative is required. Equation (14) presents an

0

1

1 2 3 4 5Node

FEM ITD PP FDD

-2

-1

0

1

1 2 3 4 5Node

FEM ITD PP FDD

approximation for the second derivative used in this work.

( ) ( ) ( ) ( )

(14)

where, ( ) and ( ) are, respectively, the acceleration and the displacement for , and

is the discrete time step. In this case, three beam lengths were tested, providing different fundamentals frequencies.

This strategy was adopted in order to evaluate camera results in different frequency levels.

In that way, three fundamental frequencies levels were approximately set: 4, 8 and 12Hz.

Figures 10, 11 and 12 present, respectively, comparisons for these different levels of

fundamental frequencies

Concerning the first and second fundamental frequencies, it is possible to verify that as the

fundamental frequency increases, the results with non-sensor acquisition get worse, but it can

be observed that differences between both acquisition techniques are not enormous. Of course

numerical errors due to derivative applied to the via image processing results also contribute

to increase the differences.

For the higher fundamental frequency (12Hz), camera results are completely erroneous.

Two major factors contribute to this weak performance: firstly, it is important to observe that

12Hz is very close to the Nyquist frequency of the camera acquisition system: 15Hz.

Secondly, the numerical errors due to the application of Eq. (14) are more pronounced for

higher frequencies.

Figure 10 – Acceleration time histories for frequency close to 4Hz

Figure 11 – Acceleration time histories for frequency close to 8Hz

Figure 12 – Acceleration time histories for frequency close to 12Hz

6 CONCLUSIONS

In this paper, time and frequency domain identification methods were applied to time

histories obtained by means of an image processing based procedure used in a structural

experimental test.

A strain gage classical instrumentation and a numerical model were used to evaluate the

quality of the obtained image processing time histories. Comparisons concerning mode shapes

and natural frequencies presented good agreement. On the other hand, damping ratios were

not correctly identified.

Time domain methods presented better performance for the first natural frequency. It is

due to limitations regarding camera’s frame rate, leading to a Nyquist frequency limit of 12.5

Hz. Near to this frequency the process tends to be more inaccurate. This effect is highly

increased because a second derivative is taken by a numerical method (Eq. (14)) to lead

displacement (primary data obtained via image processing) to acceleration so as to make the

comparison with accelerometer data possible.

Frequency domain methods were specially limited by frequency resolution. For low

frequencies, this limitation is more pronounced.

It is important to notice that all conclusions extracted from this paper are not inherent to

time histories obtained by image processing based algorithms. Any kind of instrumentation

method having 0.1Hz of frequency resolution and 30Hz of acquisition frequency tend to

present the same facilities and difficulties. In the face of that, it can be concluded that an

image processing based instrumentation can allows equivalent results in terms of modal

identification, if compared to classical instrumentation methods.

Acknowledgments

The authors would like to thank UFJF (Universidade Federal de Juiz de Fora), FAPEMIG

(Fundação de Amparo à Pesquisa do Estado de Minas Gerais), CNPq (Conselho Nacional de

Desenvolvimento Científico e tecnológico) and CAPES (Coordenação de Aperfeiçoamento de

Pessoal de Nível Superior) for the financial support.

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