2011 IEEE International Conference on Fuzzy Systems June 27-30, 2011, Taipei, Taiwan 978-1-4244-7317-5/11/$26.00 ©2011 IEEE On the Type-1 and Type-2 Fuzziness Measures for Thresholding MRI Brain Images R. Rajesh Technology Development Centre Network Systems and Technologies Technopark, Thiruvananthapuram - 695581 Email: kollamrajeshr@ieee.org N. Senthilkumaran, J. Satheeshkumar, B. Shanmuga Priya, C. Thilagavathy, K. Priya Department of Computer Applications Bharathiar Universisty, Coimbatore - 641046, India Email: jsathee@rediffmail.com Abstract—The result of image thresholding is not always satisfactory due to the disturbing factors like vagueness, non- uniform illumination etc and to overcome these problems recently various researchers have proposed fuzzy image thresholding. The linear index of fuzziness for type-1 fuzzy sets by Zenzo et. al. and measure of ultrafuzziness for type-2 fuzzy sets by Tizhoosh has difficulties in handling MRI brain images with one level of gray value as background and other two levels of grayness as white matter and gray matter. Hence this paper proposes new modified thresholding measures for MRI brain images using type- 1 and type-2 fuzzy sets. The results show the effectiveness of the proposed modified thresholding measures. Index Terms—type-1 fuzzy sets, type-2 fuzzy sets, thresholding, ultrafuzziness I. I NTRODUCTION Image thresholding, considered as the simplest form of segmentation, is an important task in most of the image processing applications. Lot of research work has already been appeared on robust thresholding techniques [1], [2], [3]. Recently fuzzy set theory have been used extensively in image thresholding [4], [7], [8], [9], [10], [11], [12], [13] due to the ability of fuzzy logic in handling ambiguity/vagueness in the presence of disturbing factors like non-uniform illumination, vagueness etc. Fuzzy thresholding techniques can be classified into four categories [4] based on the way in which it is applied. They are namely (a) fuzzy clustering using fuzzy c-means, probabilistic c-means etc. (b) rule-based approaches (c) fuzzy geometrical approaches using spatial image information and geometrical measures(compactness, area coverage etc.) (d) information- theoretical approach by minimizing/maximizing measures of fuzziness (index of fuzziness, fuzzy entropy, fuzzy divergence etc.). Due to the simplicity and high speed of information- theoretical approach [7], [8], [9], [10], [11] it has been mostly used for image thresholding. The linear index of fuzziness for type-1 fuzzy sets by Zenzo et. al. [11] and measure of ultrafuzziness for type-2 fuzzy sets by Tizhoosh [4] has difficulties in handling (we mean thresholding) MRI brain images. Here the difficulties in thresholding are due to the presence of 3 levels of grayness in MRI images, one level of gray value as background and other two levels of grayness as white matter and gray matter. Hence this paper proposes new modified thresholding measures 0 5 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x u (a) Type−1 MF 0 5 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x u (b) Type−2 MF Fig. 1. Gaussian membership function with variance=0.9 of (a) type-1 with mean=5 (b) type-2 with uncertain means mean1=4.5, mean2=5.5 for MRI brain images using type-1 and type-2 fuzzy sets. The results show the effectiveness of the proposed modified thresholding measures. This paper is organized as follows. Section II deals with measure of fuzziness and section III introduces modified measures of fuzziness. Section IV presents the experimental results and section V concludes the paper. II. MEASURE OF FUZZINESS A. Type-1 fuzzy sets A type-1 fuzzy set A in X is defined by a type-1 member- ship function given by (1) Figure 1a shows the Gaussian membership function of type-1 fuzzy set with variance = 0.9 and mean = 5. The linear index of fuzziness [11] for an image subset with L gray levels can be given by (2) where h(g) is the histogram. is the membership function and can be defined by the standard S-function, Huang & Wang function, triangular membership function, LR type fuzzy number etc.. 992