An Image Analyzing Method for the Vaguely Grain Boundary Detection by a Reaction Diffusion System

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Procedia Materials Science 12 ( 2016 ) 72 – 76

Selection and peer-review under responsibility of the VŠB - Technical University of Ostrava,Faculty of Metallurgy and Materials Engineering doi: 10.1016/j.mspro.2016.03.013

ScienceDirect

6th New Methods of Damage and Failure Analysis of Structural Parts [MDFA]

An Image Analyzing Method for the Vaguely Grain Boundary

Detection by a Reaction Diffusion System

K. Nakane

1*

, H. Mahara

2

, K. Kida

3

1

Osaka University, Faculty of medicine, 1-7, Yamadaoka, Suita, Osaka, Japan 2

Chiba University Hospital, 1-8-1, Inohana, Chuo-ku, Chiba-shi, 260-8677, Japan 3

University of Toyama, Gofuku 3190, Toyama1-8-1, Inohana, Chuo-ku, Chiba-shi, 260-8677, Japan

Abstract

To analyze the images of structures, we often detect the edges of grains. Since the grain boundaries are not so clear, it takes a lot of time to detect them, manually. Here, we present a support method to detect the edge from structure images. We make blurry images clear by solving non-linear partial differential equations numerically. Combining ordinary image analysis methods, we can detect edges of grains easily. In this note, we introduce the results of our method and discuss the possibility of this method. © 2014 The Authors. Published by Elsevier Ltd.

Selection and peer-review under responsibility of the VŠB - Technical University of Ostrava, Faculty of Metallurgy and Materials Engineering.

Keywords: Reaction diffusion, Edge detection, Homology.

1.Introduction

To detect the edges of grains in structure images, many methods have been developed. Mainly, filtering methods have been studied. For example: Gradient, Roberts, Sobel and so on. If there are variations in brightness in the image, they do not work effectively.

Here, we introduce the edge detection method by solving the reaction-diffusion equation. This method has been developed for the edge detection and the figure-ground separation [1].

* Corresponding author.

E-mail address: [email protected] m

Selection and peer-review under responsibility of the VŠB - Technical University of Ostrava,Faculty of Metallurgy and Materials Engineering

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It can achieve a high quality processing of noisy images as compared with the Median filter [2]. Nomura et al. [3] extended this algorithm in order to detect edges in a gray-scale image. Several researchers have observed that a reaction diffusion mechanism works as an image processing system. Kuhnert [4, 5] found that a reaction diffusion mechanism built into a real chemical reaction system worked for visual functions such as the edge enhancement [6, 7].

2.The model equation

The two-variable reaction diffusion system is used as a base of the presented method that is called the FitzHugh-Nagumo (FHN) equations. The FHN equations are a model of nerve membrane [8, 9]. This model with two variables

u(t, x) and v(t, x) is expressed by:

(1) (2) where De and Diare diffusion coefficients for the variables u and v, the parameter is a positive small constant (0< <<1) and f(u, v) and g(u, v) are reaction terms. The two variables u and v are excitatory and inhibitory variables, respectively. The reaction terms of the FHN model are expressed by:

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(4) where the parameters a and b are positive constants.

We used the following condition for figure-ground separation: b>4/(1-a)2_{. The parameter}_a_{is the threshold value} of u. The value of u goes to the fixed point 0 if the initial value of u is under the threshold value and it goes to another stable fixed point _{if the initial value exceeds the threshold value. By}

utilizing this property, the ambiguous parts of the image are emphasized.

Fig.1. The transverse axis is the initial value of u at the ground. The vertical axis is the u value of the figure. The value n means the pixel number of the figure width. Curves determine the threshold value for the figure-ground separation.

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The threshold value for the figure-ground separation depends on the width of the figure and the value of the ground. Fig. 1 shows these dependencies in one-dimensional reaction diffusion system. The threshold value increases as the value of the ground decreases when the width is sufficiently small (1~4 pixels). The threshold is a constant and does not depend on the value of the ground when the width of the figure is larger than 5 pixels.

In this note, we use the diffusion equation whose variable is a. It was introduced by Nomura et al. [3] for averaging of threshold. Precise equations and the initial conditions for u, v, and a are described in the reference [3].

3. Analytical Results

Fig. 2(a) shows the microstructure of the material SUJ2 after triple quenching and tempering. The prior austenite grain size refinement was achieved through repeated quenching as shown in Fig. 3. The polished surfaces of SUJ2 samples were etched in a modified Picral solution with two drops of HCl.

We applied the reaction diffusion system to this image. We useda computer: Dell Vostro (Intel Core i7-3632QM 2.20GHz 4.00 GB). The initial values are taken from Fig. 2(a) and the numerical result is shown in Fig. 2(b). It took 4-5 seconds to process one image.

We put the red color on Fig. 2(b) and superimposed it on 2(a). As we can see in Fig. 2(c) the edges can be taken exactly.

a b

c

Fig. 2: (a) The original image of material SUJ2Q3T1); (b) The numerical result of the reaction diffusion equation; (c) The superimposed images of Fig. 2(a) and Fig. 2(b).

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Fig. 3. Heat treatment scheme, triple quenching and tempering (Q3T1).

a b

Fig. 4. (a) The result of noise reduction in Fig. 2(c). (b) The image Fig 4(a) is superimposed on Fig. 2(b).

The result of noise reduction is shown in Fig. 4(d). If we add a manual correction (noise reduction and so on), we detect the edges easily.

Fig. 5 is the result of the completion of Fig. 4(d) by using of the ordinary image analyzing method. At a glance, this result seems to be good. However, by taking a closer look, there are several edges that are out of the correct positions. If we can complete the edges, we can calculate the grain numbers automatically [10]. We need to check effectiveness of this completion by applying a number of images.

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4. Conclusions

The imaging methods for detecting of grain boundaries from the structure images have been studied for a long time. The methods use filters mainly. It does not work effectively, because we have to adjust the parameters for each image, if we apply these methods to images where the brightness is not uniform.

Here, we have applied the reaction diffusion method to micrographs of etched metal surfaces for the edge detection (Fig. 4). We can get clear edges from vaguely images. As a result, this method can be applied for many different condition images at the same time. Because we only solve the differential equation system numerically, this method can be applied to many types of images.

Moreover, we completed the edges automatically (Fig. 5). Several edges were out of their exact positions. In near future, we should show these shifts to be not a big problem to get the grain numbers. If this completion has enough accuracy to get the grain numbers, we can apply the homology method [10]. This implies that we could calculate the grain numbers from images automatically. We are going to check this in near future.

Acknowledgement

This work was supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (B) Grant Number 26310209. References

[1] Nomura, A., Ichikawa M., Miike H., Ebihara M., Mahara H., Sakurai T.: Ralizing Visual Functions with Reaction Diffusion Mechanism: Journal of the Physical Society of Japan 72, 2003, pp.2385-2395.

[2] Ebihara M., Mahara H., Sakurai T., Nomura A., Miike H., Image processing by a discrete Reaction-Diffusion System: Proceeding of

Visualization, Imaging and Image Processing 396, 2003, pp.145-150.

[3] Nomura, A., Ichikawa M., Sianipar R. H., Miike H.: Edge Detection with Reaction-Diffusion Equations Having a Local Average Threshold: Pattern Recognision and Image Analysis 10, 2008, pp.289-299.

[4] Kuhnert. L, Agladze, K. I. & Krinsky, V. I.: Image processing using light-sensitive chemical wavesNature 319 (1986) 393. [5] Kuhnert. L, Agladze, K. I. and Krinsky, V. I.: Image processing using light-sensitive chemical waves, Nature 337 (1989) 244. [6] Krinsky, V. I., Biktashev, V. N. and Efimov, I. R.: Autowave principles for parallel image processing Physica D 49 (1991) 247.

[7] Aliev, R. R.: J. Phys. Oscillation Phase Dynamics in the Belousov-Zhabotinsky Reaction. Implementation to Image Processing Chem. 98 (1994) 3999

[8] FitzHugh R.: Impulses and Physiological States in Theoretical Models of Nerve Menbrene: Biophysical Journal 1, 1961, pp.445-466. [9] Nagumo J., Arimoto S., Yoshizawa S.: An Active Pulse Transmission Line Simulationg Nerve Axon: Proceeding of the IRE 50, 1962,

pp.2061 - 2070.

[10] Nakane K., Mizobe K. and Kida K.: Homology estimate of grain size measurement based on the JIS samples: Advanced Materials Research 813, 2013, pp. 120-123, DOI:10.4028/www.scientific.net/AMR.813.120.