In this section, we will discuss how to determine which pixelsare belonged to the mineral. So far, sample was taken from the mineral which we would like to segment and membership functions were defined. Now, these membership functions are used to calculate all pixels’ memberships to the mineral from which a sample was taken.
Pixels’ memberships to the mineral are calculated for once but two inferences are done for pixels which have strong membership and pixels which have weak membership. It is assumed that the rest part of pixels has no membership with the mineral.
Rules take RGB components of the pixels as crisp inputs, and thenfuzzifythem according to the fuzzy sets that we already defined. After the fuzzification, fuzzy AND operator is applied to the membership values. Fuzzy AND operator simply selects the minimum of the membership values. The minimum of membership value informs us the pixel color’s membership to the mineral’s color. Then, inference is made by pre-defined thresholds to determine if pixel has strong or weak membership to the mineral’s color.
Let image’s row and column of the pixels’ red, green and blue components are represented as , , , and , . And membership functions of the sample image are represented as , and . So that the first fuzzy-rule is,
“IF , 0.8AND , 0.8AND , 0.8THEN pixel has
strong membership”
Pixels which have strong membership to the mineral are marked with red color in the Fig.4.
Fig.4.Pixels with Strong Membership The second fuzzy-rule is;
“IF , 0.2AND , 0.2AND , 0.2THEN pixel has
weak membership”
Pixels which have weak membership to the mineral are marked with red color in the Fig.5.
Fig.5.Pixels with Weak Membership
We accept that strong pixels definitely belong to the mineral from which sample was taken. But we cannot say the same thing for the weak pixels. So, we have to decide which of the weak pixels are going to be part of the mineral. This problem is solved by hysteresis threshold like used in Canny edge detector [12]. If any pixel has strong membership to a mineral, those pixels are become part of the output immediately, as the entire connected segment of the weak pixels. By using the hysteresis threshold, the probability of false positiveoutput pixels is greatly reduced because for a pixel to be segmented as a mineral it must be either have a strong membership or adjacent to a strong pixel. Also the probability of false negative mineral pixels is reduced because of the low threshold to determine the weak pixels. Segmentation result using both strong and weak pixels is shown in the Fig.6.
Fig.6.Segmentation by One Sample
However, this segmentation is not completed yet. Same mineral can have various RGB values because of the mineral weathering or plain light’s angle. So, users must take samples till all the different colored parts of the same mineral are segmented as one mineral. For the ongoing example of the segmentation of thin section image, the second sample is taken and shown in the Fig.7.
When another sample is taken from a thin section, again membership functions are defined. Pixels which were assigned to any mineral are not considered. And pixels which were not assigned to any mineral are used for calculations of the membership functions of the new sample, and then with hysteresis threshold segmentation is finalized. Figure 8 shows a final result of the segmentation by two samples.
Fig.7.Second Sample
Fig.8.Segmentation by Two Samples IV. EXPERIMENTAL RESULTS
Comparison is made between the segmentation results using proposed technique and knownFuzzy C-Means (FCM) applied to rock thin section images. The proposed technique and FCM technique were implemented using MATLAB R2011a. Many different rock thin section images are used for experiments. Segmentationresults of two different rock thin section images are discussed here. Those
images are segmented into 3 minerals because most of the igneous rock classification diagrams use 3 minerals for classification [11].
FCM took every pixel as 1 by 3 vector since each pixel has red, green and blue components. For FCM, number of cluster was chosen as 3 and initialization of the cluster centers were chosen randomly. The minimum amount of improvement was chosen as 0.00001 and maximum number of iteration was chosen as 100. For proposed image segmentation technique, two samples are taken for each mineral.
Fig.9 shows an input thin section image for the first experiment. The results (Fig.10 and Fig.11) show that the proposed image segmentation technique has significantly better results than FCM.
Fig. 9.Original Image
Fig.10.FCM Result
Fig.11.Segmentation by Two Samples
As the second experiment, Fig.12 is taken as input thin section image. Again in this experiment input image is segmented into 3 minerals. The results (Fig.13 and Fig.14) show that the proposed image segmentation technique has more complete mineral areas than FCM. This is because the proposed image segmentation technique considers spatial relationship of the pixels.
Fig.12.Original Image
Fig.13.FCM result
Fig.14.Segmentation by Two Samples V. CONCLUSIONS
In this paper, a fuzzy rule-based image segmentation technique for rock thin section images is proposed and implemented. Proposed segmentation technique uses pixel’s red, green and blue components as features and segments the rock thin section images into minerals.
The singularity of segmenting the rock thin section images is that thesame mineral can have various RGB values because of the mineral weathering or plain light’s angle. The known color segmentation techniques do not give sufficient results. So, in order to solve this problem the interactive technique was
proposed. Firstly, the users are allowed to take samples from minerals. Each sample is considered as new linguistic variable of mineral. As segmentation technique’s featuresthe red, green and blue components are used. Then, for each feature, membership functions are defined. These membership functions represent color components’ distributions in sample image. By fuzzy rules, membership degrees of the pixels to the mineral are calculated. According to their membership degrees strong and weak pixels are defined. Finally, we take strong pixels as seed points and include every adjacent weak pixel to the strong pixel into the output.
Proposed Fuzzy Rule-Based and known FCM techniques are applied to segment rock thin section images. Results showed that the proposed image segmentation technique has significantly better accuracy than FCM.
Obtained segmented thin section images are useful for classification of rocks by diagrams. For this purpose, in further works, output image will be used for calculations of parameters such as mineral grain’s area, major and minor axis lengths, angels, etc. Further works also include improvements on finding mineral edges by including new features such as texture.
ACKNOWLEDGMENT
This work is funded by the Scientific Research Project № 11B4343001 of Ankara University.
REFERENCES
[1] R. Prikryl,“Assessment of Rock Geomechanical Quality by Quantitative Rock Fabric Coefficients: Limitations and Possible Source of Misinterpretations”, in Engineering Geology, vol. 87, 2006, pp. 149-162
[2] R. C. Gonzalez and R. E. Woods,“Using Fuzzy Techniques for Intensity Transformation and Spatial Filtering”, in Digital Image Processing, 3rd ed. Prentice Hall, 2007, ch. 3, sec, 3.8, pp. 173-193
[3] R. Samet and M. Namazov,“Using Fuzzy Sets for Filtering Topographic Map Images”,in Applied Computational Mathematics,vol. 7, no 2, 2008, pp. 242-254
[4] G. C. Karmakar and L. Dooley, ”A Generic Fuzzy Rule Based Technique for Image Segmentation,”in Acoustic, Speech, and Signal Processing, vol. 3, Salt Lake City, 2001, pp. 1577-1580
[5] L. S. Dooley, G. C. Karmakar and M. Murshed, ”A Fuzzy Rule-Based Colour Image Segmentation Algorithm,”in Image Processing, vol. 1,2003, pp. 977-980
[6] M. Potrebic,“Iterative Fuzzy Rule Base Technique for Image Segmentation”, in Neural Network Applications in Electrical Engineering, 2004, pp. 221-224
[7] B-C Chien and M-C Cheng, “A Color Image Segmentation Approach Based on Fuzzy Similarity Measure”, in Fuzzy Systems, vol. 1, Honolulu, 2002, pp. 449-454
[8] L. B. Gonçalves and F. R. Leta, ”Macroscopic Rock Texture Image Classification Using a Hierarchical Neuro-Fuzzy Class Method,” in Mathematical Problems in Engineering, vol. 2010, Article ID 163635, 2010
[9] B. Aksoy and M. Ercanoğlu, “Landslide Identification and Classification by Object-Based Image Analysis and Fuzzy Logic: An Example from the Azdavayregion(Kastamonu, Turkey)”, in Computers &Geosciences,vol. 38, 2012, pp. 87-98 [10] L. A. Zadeh,“Fuzzy Sets”,in Information and Control,vol. 8,no. 3, 1965, pp. 338-353
[11] M. J. Le Bas and A. L. Streckeisen, “The IUGS systematics of igneous rocks”, in Journal of the Geological Society, vol.
148, London, 1991, pp. 825-833
[12] J. Canny,“A Computational Approach to Edge Detection”, inTransactions on Pattern Analysis and Machine Intelligence, vol. 8, no. 6, 1986, pp. 679-698