Inte rnatio nal Jo urnal o f Sc ie ntific & Eng inee ring Re se arc h, Vo lume 3, Issue 3, Marc h-2012 1

ISS N 2229-5518

The Segmentation of FMI Image Layers Based on FCM Clustering and Otsu thresholding

J. Gholampour, A.A. Pouyan

—————————— ——————————

He difficulty in carbonate reservoir evaluation come from the complexity of reservoir storage spaces (such as pores, vugs, fractures and their combination) and thir hetero- geneity. Because of fractures, the validity of the reservoir is not easily identified by conventional log. The imaging log provides a new means for such a complex reservoir evalua- tion. FMI (Full Micro-resistivity Imager) provides borehole wall resistivity imaging. The FMI tool is made of eight elec- trode pads with 24 button electrodes on each pad. The FMI tool sample interval in depth and well circumference is 0.1 inch. The tool covers approximately 80% of the borehole wall in an 8.5 inch hole. The FMI image clearly shows the geologi-

cal phenomena around the borehole wall [1].

In order to extract qualitative information from FMI, a basic

step is used to segment the FMI image to get sub-image of layers. Then, the segmented images are analyzed and processed to extract relevant information. The segmentation result directly affects the accuracy of parameter calculation [2].

A number of image segmental algorithms have been devel-

oped in the literature. They can be rouly grouped in two cate-

gories: area description-based algoriths and edge detection- based methods [3].

In this paper, we present a segmentation method based on

Fuzzy C-mean algorithm and Otsu thresholding method. The

The complete block diagram representation of the proposed method is shown in Figure 1. After inputting a FMI image, we select intensity values of red, green, blue and equivalent gray level of each pixel as features. Then we normalize these fea- tures for each row of image. At section 2.1 we illustrate the normalization method. After this step, we cluster pixel of FMI image using fuzzy c-mean (FCM). Feature selection, feature normalization and FCM have been performed to last row of FMI image.

Then k–nearest neighbor algorithm (KNN) has been called

on each row of image. KNN has been explained in section 2.3.

Then uncovered columns have been removed (section 2.4). We illustrate removing image inequality method in section 2.5. In section 2.6 we explain Otsu thresholding method.

As leave-one-out scheme is used, for each feature in the train- ing samples the feature normalization is adapted based on the following scheme:

f

first step classify pixel of each row of FMI image using fuzzy C-mean algorithm. At the second step, we use k-nearst neigh- bor for classifying. Then we remove uncovered columns and

f i i

i

(1)

remove image inequalities. At the end we use Otsu threshold-

ing method.

———— ——— ——— ——— ———

Where *f*i, *µ*i and *σ*i are respectively i-th feature, mean and

standard deviation of i-th feature [4].

Feature selection and feature normalization run on each row of FMI image, then fuzzy c-means (FCM) executed.This process runs until arrive to last row of FMI image (Fig 1).

FCM is a method of clustering which allows one piece of

data to belong to two or more clusters. This method is fre- quently used in pattern recognition [5]. It is based on minimi- zation of the objective function:

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Inte rnatio nal Jo urnal o f Sc ie ntific & Eng inee ring Re se arc h Vo lume 3, Issue 3 , Marc h-2012 2

ISSN 2229-5518

n c each member of the training set is to the test class that is being

J q (U ,V )

k 1 i 1

(*u*ik ) d

( *x*k , *v*i )

(2)

examined. Euclidean Distance measuring:

N

2 2 (8)

where *X *={x1, x2,…, xn}, n is the number of data items, *c *is the number of clusters with *2 ≤ c ≤ n*, *u*ik is the degree of mem- bership of *x*k in the i-th cluster, *q *is a weighting exponent on

d E (x , y )

i 1

x i y i

each fuzzy membership, vi is the prototype of the center of cluster i, d2 (xk, vi) is a distance measure between xk object and

cluster centre *v*i. A solution of the object function can be ob- tained via an iterative process, which is carried as follows:

1) Set value for *c, q *and ε.

2) Initialize the fuzzy partition matrix *U.*

3) Set the loop counter *b *= 0.

4) Calculate the *c *cluster centers {*v*i(b)} with *U*(b):

From this k-NN category, class label of the test pixel image is

determined by applying majority voting [8].

Start

n

(*u *ik ) *x *k

Last row of FMI image?

(*b *) * k *__1 __

i n

(3)

Yes

k 1

(*b *) *q ik*

NO

5) Ca lculate the me mbers hi *U*(b+1). For k = 1 to n,calcu late

the following

Feature selection

For the *k-th *column of the matrix, compute new membership values :

If *I*k = ᵠ , then

Feature

Normalization

Fuzzy C-Mean Clustering

(*b *1) 1

ik c d

(6)

( ik )2 (q 1)

j 1

d jk

KNN

(Number of rows of FMI Image times)

Else *u*ik(b-1) = 0 for all I *I*k and

u (*b *1) 1

(7)

Remove uncovered columns from

ik

i I k

next k

FMI Image

6) If U (b ) U (b 1)

and go to s tep 4 [6].

, s top; otherwis e, s et b=b+1

Removing image inequality

K-nearest neighbor (k-NN) is a supervised learning algo- rithm by classifying the new instances query based on majori- ty of k-nearest neighbor category. Minimum distance between query instance and the training samples is calculated to d e- termine the k-NN category. The k-NN prediction of the query instance is determined based on majority voting of the nearest neighbor category. Since query instance (test image pixel) will compare against all cluster [7].

In this works, for each test pixel image (to be predicted),

minimum distance from the test pixel image to the training set is calculated to locate the k-NN category of the training data set. Euclidean Distance measure is used to calculate how close

Otsu thresholding

Finish

Fig. 1. Block diagram of proposed system

At this step, we assume the columns that all pixels categorized to same class, are uncovered columns of FMI image (20% of FMI image). Therefore these columns remove.

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Inte rnatio nal Jo urnal o f Sc ie ntific & Eng inee ring Re se arc h Vo lume 3, Issue 3 , Marc h-2012 3

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In proposed system, a nonlinear filter is used that remove non-edge surface roughness image and keep edges FMI images. First, for each pixel we calculated eight derivatives in all direction. Then the combination of the eight derivatives added to pixels value. Figure 2 shows the pixel (n, m) and its eight neighbors. Equation (9) computed the derivative of pixel

neighbors. In this equation, I i represents the gray level of i-th neighbors of pixel (n, m). After calculating derivatives in all direction, these values multiplied by the coefficient δ and added to pixel value (equation (10)). As shown in equation (9), this filter has three parameters (K, α and δ). Their value in this study is considered 0.02, 2.5 and 1.25 respectively.

The means of class C1 and C2 are

The total mean of gray levels is denoted by *u*T

Figure 2: pixel (n, m) and 3 ×3 neighborhood

uT w 1u1 w 2u 2

(16)

i *I *i *I *(*n*, *m*)

(9)

The class variances are

*I *(*n*, *m*) *I *(*n*, *m*) { 1

}

1 1

1 8

(10)

2 2 *i *2

K K

The within -class variance is

M

As noted above, this filter removes non-edge surface rough-

ness and keeps edges in FMI images. Based on above relation- ships, if derived value in the edge pixel be larger than parame-

2 2

k k k 1

(19)

ter K then the absolute operand is greater than 1. After ^, ob-

The between-class variance is

tained a large value and changed a little on the intensity of edge pixel. Therefore the filter has no effect on the edge pixels.

2 w

1 u1

*u*T

2 w

2 u 2

*u*T

(20)

Furthermore, for non-edge pixels, the absolute value is smaller

than 1 and can skip it. Thus equation (10) in limit case is Lap-

The total variance of gray levels is

2 2 2

(21)

lace for non-edge pixels and acts as a smoothing filter. In addi- tion, we can use this nonlinear filter much time [8]. Figure 7 shows multiple repeated of this filter and in each stage, the image obtained is displayed [9].

In this section, we use Otsu method [10] as follows. Assuming

T w B

Otsu method chooses the optimal threshold *t *by maximizing the between-class variance, which is equivalent to minimizing the within-class variance, since the total variance (the sum of the within-class variance and the between-class variance) is constant for different partitions [11].

an image is represented in L gray levels [0, 1, …L-1]. The

*t * arg{ max { 2 (*t *)}} arg{ min { 2 (*t *)}}

(22)

number of pixels at level i is denoted by *n*i, and the total num-

B

0*t **L *1

w

0*t **L *1

ber of pixels is denoted by *N *= *n*1 + *n*2 +...+ *n*L . The probability of gray level i was denoted by

L 1

Otsu method can be extended to multilevel thresholding me- thod. Assuming that there are M-1 thresholds [t1,t2,…,tM-1] that divide the pixels in the image to M classes{C1, C2,…, CM}

pi ni / N , pi 0, 0

pi 1

(11)

t , t ,, t

arg{ max { 2 (*t *,*t*

,...,*t*

)}}

1 2 M 1

2

0*t **L *1

B 1 2

M 1

(23)

In bi-level thresholding method, pixels divided into two classes C1 with gray levels [0, 1, …, t] and C2 with gray levels [t+1, …, L-1] by the threshold *t *. The gray level p robability dis- tributions for two clas s es are

arg{ min {W (*t*1 ,*t *2 ,...,*t *M 1 )}}

0*t **L *1

Where

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Inte rnatio nal Jo urnal o f Sc ie ntific & Eng inee ring Re se arc h Vo lume 3, Issue 3 , Marc h-2012 4

ISSN 2229-5518

KNN Classifier, International Conference on Intelligent and Ad- vanced Systems(ICIAS), IEEE, 2010

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IEEE Transactions on System Man Cybernetics, Vol. SMC-9, No. 1:

62-66, 1979.

[11] S.S. Reddi, S.F. Rudin, and H.R. Keshavan, ―An optimal multiple

threshold scheme for image segmentation,‖ IEEE Trans. System Man

Cybernet. 14(4): 661 –665, 1984.

Original FMI image is represented in Fig. 3. The output of all steps of proposed system has been shown in Fig. 4 to Fig. 8. Fig. 4 is pixels clustering after FCM algorithm. The result of KNN algorithm is shown in Fig. 5. Removeing of uncovered columns of FMI image is determind in Fig. 6. Fig. 7 shows Output image after Removing Image Inequality. Finally, Fig. 8 shows the resulting segmented image after Otsu thresholding method.

We have proposed a segmentation method based on FCM clustering and Otsu thresholding. The field data processing examples show that sub image of layers can be accurately s e- prated from original FMI images.

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Fig. 3. Original FMI image.

Fig. 4. Pixels af ter FCM clustering algorithm.

Fig. 5. Pixels af ter KNN algorith m.

Fig. 6. The uncovered columns of FMI image are determind.

Fig. 7. Output image af ter Removing Image Inequality.

Fig. 8. Final output af ter Otsu thresholding method.

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