H.V

High voltage probe

R=50Ω

Pressuregauge

Digital oscilloscop

Gas outlet

Gas inlet

Camera

Analysis of Time Evolution of Patterns Based on Various Image Processing Techinques

Junying Chen, Lifang Dong*, Han Yue, Hong Xiao, Yuanyuan Li College of Physics Science and Technology

Hebei University Baoding 071002, China

* corresponding author: donglfhbu@163.com

Abstract—Images of patterns are processed with some image processing technology, such as spatial intensity distribution, spatial correlation function and Fourier transformation, thus some important intrinsic elements of active plasma and pattern formation can be presented more clearly. Images of disordered grids, spiral pattern and big spot hexagonal pattern are obtained in dielectric barrier discharge system at different times under the same experiment condition. Using spatial intensity distribution and spatial correlation function, it is found that the average distance becomes unique and the structure turns more regular with time relaxed, and most hexagonal cells in big spot hexagonal pattern are relatively perfect. For further investigation, these images are transformed to frequency domain and Fourier spectra are studied. It is found that the mode changes with time relaxed. Grids pattern presents disordered structure. The spiral pattern is formed with single-wavelength but orientation unselected. The stable big spot hexagonal pattern is formed with single-wavelength three-wave resonance. With time relaxed, separation of system modes leads to increase of system periodicity by means of self organization.

Keywords- image processing; spatial correlation function; Fourier spectra

I. INTRODUCTION Image processing has become a critical component in

contemporary science and technology and has many important applications. For example, it has been used to analyze the spatiotemporal patterns in nonlinear science. Patterns are generally regularly periodic structures resulting from self-organization, which can be found either in a natural or a manmade system. They have been investigated experimentally and numerically. The formation of pattern has been observed in various physical, chemical, and biological systems [2-5], and has been simulated by various models [6]. The spreading of research to these subjects is based not only on the intrinsic of development of science but also on advanced information processing, from which principles governing the behavior of self-organized media can be investigated.

In this work, three methods of image processing, spatial intensity distribution, spatial correlation function of pattern image and angular spectral distribution in Fourier space, are employed to analyze patterns obtained in dielectric barrier discharge, which is a kind of non-equilibrium ac gas discharge. It is shown that image processing is helpful to study the development of patterns.

II. EXPERIMENTAL SETUP

Figure 1. Schematic diagram of the experimental setup

Figure 2. Time evolution of patterns: (a) disordered grids, 0s; (b) spiral wave, 16s; (c) big spot hexagonal, 57s. The other discharge parameters: P=1atm,

d=2.0 mm, U=8kV, f=55 kHz.

The schematic diagram of experimental device is shown in Fig. 1. Two cylindrical containers with inner diameter of 70

This work is supported by the National Natural Science Foundation of China under Grants No. 10775037 and No.10975043, and the Natural Science Foundation of Hebei Province, China, under Grants No. A2008000564

(a)

(c)

(b)

2010 International Conference on Electrical and Control Engineering

978-0-7695-4031-3/10 $26.00 © 2010 IEEE

DOI 10.1109/iCECE.2010.114

441

2010 International Conference on Electrical and Control Engineering

978-0-7695-4031-3/10 $26.00 © 2010 IEEE

DOI 10.1109/iCECE.2010.114

441

mm are filled with tap water. There is a metallic ring immersed in each of the containers and connected to an ac power supply. Thus, the water acts as a liquid electrode. The parallel quartz with thickness 1.5 mm serves as dielectric layers. All of the apparatus are enclosed in a big container filled with argon/air mixture. The ac power supply has a maximum peak voltage of 30 kV with a fixed frequency of 55 kHz. A CCD camera (Konica Minolta Dimage Z2) is used to record the images. The discharge voltage and current are recorded by Digital oscilloscop ( Tektronix DPO 4054，500MHz ).

In this experiment, the discharge parameters are: pure argon, gas pressure P=1atm, applied voltage U=8kV, gas gap d=2.0 mm. A development of patterns occurs as following at different times: disordered grids, spiral pattern and stable big spot hexagonal pattern. After breakdown of the gas, the voltage is increased to 8 kV, and grids pattern occurs. As time relaxed, spiral pattern is selected. A few seconds later, the spiral pattern begins to split. About 57s later, it changes to stable big spot hexagonal pattern which shown in Fig. 2(c) and keeps fixed.

III. METHODOLOGY In Fig. 2 (a)-(c), it can be seen that the characteristics of the

patterns are different from each other. In order to analyze the change of their spatial characteristics and selective mechanism of the system with time, we employ some image processing technology, compiling codes based on the MATLAB software. For different pattern we take different approach. The program includes four parts: spatial distribution of the brightness calculation, primary image processing, spatial correlation function calculation, Fourier spectra and corresponding angular spectral distribution computing.

A. Spatial distribution of the brightness calculation The recorded images of patterns take much information about the patterns such as the spatial distribution of the discharge cells. The information could be obtained by analyzing the size and brightness of cells and the spatial intensity distribution by image processing. In the image processing, the gray value represents brightness. Each peak in the distribution of the gray value of the pattern image represents a discharge cell in dielectric barrier discharge.

B. Primary image processing Inevitably, noises often appear in the obtained images and

videos. By decreasing background noises, grey-scale images of patterns are obtained at first. They demonstrate that the patterns are composed of individual spots, and each spot denotes a three-dimension filament. Since each spot in images takes up certain pixels area, some methods in morphology can be used, such as removing pixels on the boundaries of spot but not allowing spot to break apart and removing pixels so that spot without holes shrink to a point. As a result, the exactitude binary images are obtained. This image format also stores an image as a matrix but can only color a pixel black or white (nothing in between). The processing makes the filament location exact, which serve as a prelude for the further space investigation.

C. Spatial correlation function To analyze the spatial distance and structure in filaments

patterns, the spatial correlation of filaments has been calculated. Through primary processing, the positions of filaments have been determined. For a given filament marked with i, the number Ni(r, Δr) of filaments between r and r+Δr from the primary one is counted, where Δr is determined by the resolution of the camera that has been used. This procedure is repeated for all filaments in the analyzed area and finally summed. Now Ni(r, Δr) is determined for each filament in the discharge area. This gives nr=pr(2πrΔr ), from which it is easy to calculate pr .From pr we obtain the normalized probability ρr by division with the average number of filaments within the discharge area. ρr is the probability density to find a couple of filaments within this distance. We refer to ρr as the spatial correlation function.

∑ ΔΔ==

02

),(1

00 N

irr rr

rrNNN

pπ

ρ

In the definition of spatial correlation function, the boundary of the image has been ignored. In the factual algorithm, some tips considering the finite image are employed. In term of this definition, it is found through computer simulation that spatial correlation functions of perfect structures, such as squares and hexagons, present certain narrow peaks at different certain distances r. The first peak can indicate the distance between the neighbor spots. For hexagonal structure the distance ratio of the first peak to the

second one is 3 .

D. Fourier spectrum and angular spectral distribution Fast Fourier transformation is a powerful image processing

program based on the FFT algorithm given by Frigo et al. (1998), which can transforms images of patterns from the spatial domain into the frequency domain. Signals presented by different domain can give us more information of the intrinsic mechanism. In Fourier spectrum, every spot represents a frequency or a wave vector or a mode exited from the system and its magnitude tells “how much” of a certain frequency component is present. Based on it, the associated angular spectral distributions can be computed for getting a sum of energy in each direction. As a result, the spatial modes distribution can be seen clearly, and the mechanism of patterns can be studied according to this angular spectral distributions.

IV. RESULTS AND DISCUSSIONS

A. Spatial distribution of the brightness For the symmetric character of spiral pattern, the spatial

intensity distribution intensity along the line in Fig.1 are measured and shown in Fig. 3. There are seven peaks with approximately the same gray and diameter along the given straight line shown in Fig. 2(b) except the center one. It can be deduced that the rolls along this direction are ordered arrangement. The results of other directions are the same as that of this line direction. It indicates that the system wavelength has been selected 16s after the disordered grids.

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Angle (deg.)

I(ar

b.un

its)

(a)

(b)

(c)

Figure 3. Intensity distribution corresponding to patters in figure 2 (b)

B. Spatial correlation function analysis Corresponding to the pattern in Fig. 2(c), the spatial

correlation function is shown in Fig. 4, there are some order sharp peaks. The first peak presents at r1 =5.14, and the second one at r2 =8.92, the ratio of r2 to r1is 1.735, approximating

to 3 , which turns out that the structure of the hexagonal pattern is perfect.

Figure 4. Spatial correlation function corresponding to patters in figure2 (c).

Above all, with time changing, the distance between neighboring filaments becomes unique and the system selection experiences disorder to order by self organization.

C. Fourier spectrum and angular spectral distribution The Fourier spectra and angular spectral distributions for

the patterns shown in Fig. 2 are presented in figure 5 and Fig. 6 respectively, where they are denoted by the respective letters.

In Fig. 5(a) the spectra distribution is so disordered, which indicates that the wavenumber of pattern in Fig. 2(a) is random. Moreover, the corresponding angular spectral distributions, shown in Fig. 6(a), present several small broad peaks, which illustrates that the wavelength selective mechanism of system is not sure and the orientation is stochastic.

About 16 seconds later, there is a single apparent ring around the central dot, shown in Fig. 5(b), and its radius presents the average wavenumber of pattern in Fig. 2(b). From its angular spectral distribution, it can be seen that the angular spectral distributions is almost flat with some random broad peaks. That is to say, the wavelength is selected but the orientation is isotropic.

Figure 5. From top to bottom, Fourier spectra of patterns corresponding to patterns in figure 2(a)-(c).

Figure 6. From top to bottom, angular spectra distributions of Fourier spectra corresponding to patterns in figure 4(a)-(c).

Distance(pixels)

P(ar

b.un

its)

Distance(pixels)

I(ar

b.un

its)

(a)

(b)

(c)

q1

q2 q3

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For the hexagonal pattern, as shown in Fig.5 (c), the radius of the ring is the same to that of spiral pattern in Fig.5 (b). However, the ring shrinks into six dots. From its angular spectral distribution, shown in Fig. 6(c), it can be seen that the phase interval between the neighboring dots in the same ring is

about π/3. And their presented wave vectors, denoted by 1

→q ,

→

2q , →

3q , satisfy the resonance relationship 0321 =++→→→qqq .

For the radius can denote the mean wavenumber of the pattern, it can be deduced that the mean distance between the neighboring ring in Fig. 2(b) is approximately equal to the neighboring filaments in Fig. 2(c), which is in agreement with the fact illustrated by the corresponding images.

Above all, with time relaxed, the modes selective mechanism of system self organization undergoes a sequence of distribution transformation in both wavelength and orientation. This illustration keeps accordance with the scenario of periodicity of the pattern evolvement.

V. CONCLUSIONS In conclusion, image processing is employed to investigate

the pattern evolution, disordered grids, spiral pattern and big spot hexagonal pattern with time relaxed effectively. Using spatial intensity distribution and spatial correlation function, it is found that the average distance becomes unique and the structure turns more regular with time relaxed, and most hexagonal cells in big spot hexagonal pattern are relatively perfect. For further investigation, these images are transformed

to frequency domain and Fourier spectra are studied. It is found that the mode changes with time relaxed. Grids pattern presents disordered structure. The spiral pattern is formed with single-wavelength but orientation unselected. The stable big spot hexagonal pattern is formed with single-wavelength three-wave resonance. With time relaxed, separation of system modes leads to increase of system periodicity by means of self organization. Image processing plays an important role in the developments of plasma investigation and pattern formation in nonlinear science.

REFERENCES

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[2] L. Yang, M. Dolnik, A. M. Zhabotinsky, and I. R. Epstein, “Turing patterns beyond hexagon and stripes”, Chaos, 16, 037114, 2006.

[3] G. D’Alessandro and W. J. Firth, “Spontaneous hexagonal formation in a nonlinear optical medium with feedback mirror”, Phys. Rev. Lett, 66, pp.2597-2600, 1991.

[4] A. Kudrolli, B. Pier and J. P. Gollub, “Superlattice pattern in surface waves”, Phys. D, 123, pp.99-111, 1998.

[5] John R. Royer and Guenter Ahlers. “Wave-number selection by target patterns and sidewalls in Rayleigh-Bénard convection”. Phys. Rev. E. 70, 036313, 2004.

[6] M. Bachir, P. Borckmans and G. Dewel, “Formation of rhombic and superlattice patterns in bistable systems”, Europhys. Lett., 54 (5), pp. 612–618 ,2001.

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