Image thresholding is one of the most important approaches for image segmentation and it has been extensively used in many image processing or computer vision applications. In this paper, a new image thresholding method is presented using type-2 fuzzy sets based on GLSC histogram of human visual nonlinearity characteristics (HVNC).The traditional GLSC histogram takes the image spatial information into account in a different way from two-dimensional histogram. This work refines the GLSC histogram by embedding HVNC into GLSC histogram. To select threshold based on the redefined GLSC histogram, we employ the type-2 fuzzy set, whose membership function integrates the effect of pixel gray value and local spatial information to membership value. The type-2 fuzzy set is subsequently transformed into a type-1 fuzzy set for fuzziness measure computation via type reduction. Finally, the optimal threshold is obtained by minimizing the fuzziness of the type-1 fuzzy set after an exhaustive search. The experiment on different types of images demonstrates the effectiveness and the robustness of our proposed thresholding technique.
© 2011 OSA
Image thresholding, which extracts the object from the background in an input image, is often required for many image processing or computer vision applications, such as shape recognition, handwritten texts recognition and image enhancement [1–6]. Furthermore, the generation of binary image by image thresholding is employed for feature extraction and object recognition. Image thresholding could be regarded as the simplest form of image segmentation, or more generally, as a two-class clustering procedure [1–5].
Over the past decades, various approaches for automatic threshold selection have been reported, Weszka , Sahoo et.al  and Sankura et.al  surveyed on image thresholding at different times. In recent years, fuzzy set theory has been widely applied to thresholding [4–17]. By incorporating human perception and linguistic concepts such as similarity, fuzzy technique as a nonlinear knowledge-based method, can remove grayness ambiguities significantly. Most fuzzy thresholding methods [4,5,9–17] are based on the regular fuzzy sets (also referred as type-1 fuzzy sets ). In , Mendel et.al pointed out that there are at least four sources of uncertainties in fuzzy logic system. However, type-1 fuzzy sets are not able to directly model such uncertainties. In order to handle this problem, the concept of type-2 fuzzy sets, whose membership functions are fuzzy themselves, was proposed by Zadeh  in 1975. Unfortunately, type-2 fuzzy sets are not utilized widely because they are hard to understand compared with type-1 fuzzy sets. During the last decade, Mendel et.al [20–22], Wu et.al , Zhai et.al , Lucas et.al  and Aisbett et.al  discussed representation form and characteristics of type-2 fuzzy sets theoretically to motivate their applications. In 2005, Tizhoosh demonstrated the first use of type-2 fuzzy sets for thresholding  based on Mendel’s work . In Tizhoosh’s paper, he addressed a new membership function and utilized the entropy in interval valued fuzzy sets  as ultrafuzziness measure to capture the uncertainties within type-2 fuzzy systems. Vlachos et.al  and Bustince et.al  gave comments on Tizhoosh’s proposition recently to demonstrate the importance of type-2 fuzzy thresholding approach.
The existing thresholding methods that only depend on the first-order gray-level histogram (1-D histogram) have one common drawback that the spatial correlation between different gray levels is ignored. However, more information contained in the image could be used for better segmentation. Originating from the work of Kapur et.al  and that of Kirby and Rosenfeld , Abutaleb  extended the entropic thresholding technique by using two-dimensional histogram (2-D histogram) determined by the gray value and the local average gray value of the pixels. As following works, Brink , Pal et.al  and Sahoo et.al [33,34] refined Abutaleb’s method, and Liu et.al  suggested 2-D Otsu  technique. Moreover, Wang et.al was the first to employ 2-D histogram in fuzzy thresholding . However, 2-D thresholding methods suffer from the drawbacks of large time consumption and information loss during the optimal threshold search. Recently, a new entropic approach using gray level spatial correlation histogram (GLSC histogram) was proposed by Xiao et.al [37,38]. GLSC histogram, which takes image spatial correlation into account in a different way from 2-D histogram, is determined by using the gray value of the pixels and the number of neighboring pixels with similar gray value in the corresponding neighborhood. According to Xiao’s research [37,38], entropic thresholding method using GLSC histogram achieved equivalent or even better segmentation performance compared with 2-D histogram based ones, while saving time remarkably.
In this paper, we combine type-2 fuzzy sets with GLSC histogram to utilize their superiority addressed above for image thresholding. A new type-2 fuzzy thresholding approach based on GLSC histogram of human visual nonlinearity characteristics (HVNC)  is proposed. The original GLSC histogram is refined by embedding HVNC to enable it with human perception. Based on the redefined GLSC histogram, the type-2 fuzzy set for fuzzy thresholding is suggested. In order to obtain crisp fuzziness measure, the type-2 fuzzy set is subsequently transformed into a type-1 fuzzy set by vertical slice centroid type reduction (VSCTR) . Finally, the optimal threshold is obtained by minimizing the fuzziness of the type-1 fuzzy set after an exhaustive search. The overall flowchart of our thresholding method is shown in Fig. 1 . Meanwhile, the main difference between our work and Xiao’s proposition [37,38] is how to construct the criterion function for threshold selection. The criterion function in this paper is proposed based on fuzzy set theory, however Xiao addressed it by using the concept of Shannon entropy.
The remainder of this paper is organized as follows. Section 2 describes the GLSC histogram of HVNC. In section 3, type-2 fuzzy sets for thresholding is defined. Section 4 presents the fuzziness measure for type-2 fuzzy sets. Here, the optimal threshold is selected by minimizing the fuzziness. Section 5 shows comparative results to demonstrate the effectiveness and the robustness of the proposed approach and Section 6 draws the final conclusions.
2. GLSC histogram of HVNC
In this section, we first introduce the concept of 2-D histogram and GLSC histogram, while discussing their intrinsic drawbacks. Secondly, we illustrate in details how to refine GLSC histogram by embedding HVNC into it.
2.1 2-D histogram and its drawbacks
Figure 2 shows the 2-D histogram plane where T axis represents the gray value of pixels, S axis represents the local average gray value of the pixels and L is the number of gray levels.
2-D histogram is defined by assuming that pixels corresponding to object or background make more contributions to the diagonal quadrants, region A and D than edge pixels or the noise, as the gray level distribution is relatively more homogeneous in object or background. According to the optimal threshold vector selection in 2-D thresholding techniques [5,30–35], quadrants B and C are ignored. 2-D histogram provides an effective way for applying local neighborhood information to thresholding, which results in better segmentation. However, it still suffers from the drawbacks listed below:
- ● First, the bin - for object or background is occasionally farther away from region A and D than edge or noise, which conflicts with the assumption of 2-D histogram. For instance, as the two local neighborhoods of size shown in Fig. 3 , Fig. 3(a) of more homogenous gray level distribution corresponds to object or background with greater probability than Fig. 3(b) and it should locate closer to the diagonal line of 2-D histogram. However, the bins for Figs. 3(a) and 3(b) are and , obviously (a) is farther away from the diagonal quadrants.
- ● Secondly, as discussed above, some object or background bins may be discarded in 2-D thresholding for their location in region B and C, which is unreasonable. Additionally, in our opinion, ignorance of edge pixels would lead to serious information loss in 2-D thresholding.
- ● Lastly, the extension of 1-D approaches to 2-D histogram gives rise to the exponential increment of computation time.
2.2 GLSC histogram and its drawbacks
Object or background pixels have similar gray value as neighboring pixels more probably than edge or noise, for their gray level distribution is relatively more homogeneous. As a consequence, Xiao suggested that object or background make more contribution to the regions of high m value in GLSC histogram. Setting ζ to 4 as [37,38], the bins for Figs. 3(a) and 3(b) are and respectively, which is consistent with Xiao’s suggestion and the actual situation. In GLSC histogram based entropic thresholding approach [37,38], all the pixels in image are taken into consideration by assigning a weight correlated with m in entropy criterion function computation to avoid information loss. As proved by , entropic thresholding using GLSC histogram yielded equivalent or even better segmentation than 2-D histogram while saving time remarkably, for the computation complexity of Xiao’s method is while the original 2-D ones is .
As discussed above, GLSC histogram possesses some advantages over 2-D histogram. However, it also suffers from some drawbacks as referred in . An important affair is how to fix ζ to use GLSC histogram. In [37,38], ζ is an empirical constant for all pixels, which seems unreasonable. In the next subsection, we shall illustrate how to fix ζ in a statistical way by embedding HVNC into GLSC histogram.
2.3 The refined GLSC histogram
Research on vision physiology and psychology shows that under different background brightness - I, visual system does not yield the same perception response to absolute brightness difference - . Its discriminative ability decreases as background brightness is increased. Denote as a just noticeable brightness difference for the visual system, then will increase as I is enlarged. Apart from the scotopic region, the relationship between and I obeys the DeVries-Rose or Weber law . The relation curve in Fig. 4 serves to illustrate this nonlinearity principle. At low light intensities, is a constant and then converges asymptotically to DeVries-Rose or Weber behavior with I increasing as
Next, the human visual nonlinearity characteristics demonstrated above will be applied to GLSC histogram in this paper. In Fig. 5 , and are two square window centered at of size and () respectively, and is the central pixel. When we consider the relationship between and its neighboring pixels in for GLSC histogram computation, ζ is regarded as and the gray value mean of - is regarded as I in Eq. (6). According to HVNC, ζ could be calculated approximately asEq. (7) are set as , , , and respectively.
Figure 6 shows the cameraman image and its GLSC histogram of HVNC. Figure 7 shows the pixel mapping images corresponding to different m values ranging from 1 to 9 of the cameraman image, under the conditions that and . In each mapping image, pixels of corresponding m value remain their original gray value while the others are labeled as green color. From Fig. 7, we can see that the mapping images with high m value correspond to object or background more probably while the ones with low m value correspond to edge or noise oppositely.
In the next section, type-2 fuzzy sets based on the refined GLSC histogram are presented for thresholding.
3. Type-2 fuzzy sets based on GLSC histogram of HVNC
In this section, some basic concepts on type-2 fuzzy sets are introduced and then the proposed type-2 fuzzy sets based on GLSC histogram of HVNC are defined.
3.1 Basic concepts on type-2 fuzzy sets
In , Mendel et.al pointed out that there are at least four sources of uncertainty in type-1 fuzzy logic system : (1) the meaning of the used words is uncertain; (2) consequents of rules have a histogram of values, especially when knowledge is extracted from a group of experts who do not all agree; (3) measurements may be noisy; (4) the data used to tune the parameters of type-1 fuzzy logic system may also be noisy. All these factors listed above result in the assignment of a membership degree to an element in type-1 fuzzy sets being not certain. The main problem of type-1 fuzzy sets is that they are not able to handle such uncertainties because their membership functions are totally crisp. Oppositely, type-2 fuzzy sets  could model such uncertainties since their membership functions are fuzzy themselves .
As defined in Eq. (8), the membership value of x in is no longer a crisp number but the type-1 fuzzy set . Figure 8 depicts an example of type-2 fuzzy set in  and the shaded area is the footprint of uncertainty (FOU) used to verbalize the shape of type-2 fuzzy sets. In particular, , and the secondary membership function at is .
3.2 Type-2 fuzzy sets utilized in this paper
Tizhoosh  first applied type-2 fuzzy sets to image thresholding to eliminate the uncertainties in membership function selection. However, regardless of membership function shape, type-2 fuzzy sets are employed to remove the vagueness of image data in this paper. In our opinion, the membership value assigned to a pixel should not be only determined by its gray value as [4,6,9–17] but also the spatial correlation with neighboring pixels. As discussed in Section 2, the degree of a pixel belonging to object or background could be measured by its m value in GLSC histogram of HVNC, the greater the value of m, the more probably it belongs to object or background. Therefore, in fuzzy thresholding, pixels of greater m should be assigned greater membership value than the ones with the same gray value but of lower m. As a consequence, each gray level k would have a union of membership grades, which could be regarded as the primary membership of k denoted as . The secondary variable in is determined both by k and ... Additionally, the secondary grade of could be approximated as the normalized occurrence probability of in . Next, we shall illustrate how to define type-2 fuzzy sets based on GLSC histogram of HVNC in details.
For one image, let denote its GLSC histogram of HVNC and denote the type-2 fuzzy sets based on , where and . We haveEq. (9), , and are unknown and how to obtain them is the key point to construct . Next, their definition is illustrated in details.
- ● First, the definition of primary membership - is given as
- ● Secondly, the secondary variable - is obtained as
Motivated by what Huang  has addressed, is defined similarly to the membership function applied in the 1-D histogram based fuzzy method . Let denote the 1-D histogram. As shown in Fig. 9 , given a certain threshold t, the average gray value of background and object could be obtained as follows4] is given asEq. (14) is not easy to fix and we find that it affects the thresholding performance to some degree by experiment. To overcome this defect, is defined by making some changes on asEq. (15) and Eq. (16), it can be seen clearly that now no parameter needs to be set empirically in .
As discussed above, pixels of greater m should be assigned greater membership value than the ones with the same gray value but of lower m. So, should be obviously defined as a monotonically increasing function of m. In practical, is suggested as a nonlinear function belowFigure 10 shows an example of . It can be observed that monotonically increases with m. As a consequence, is also a monotonically increasing function of m, which agrees with the fact that object or background pixels should be assigned greater membership value than edge or noise of the same gray value. In practice, parameters in Eq. (17) are set as and respectively.
In image thresholding, for a given threshold t, any gray level in the input image should belong to either object or background. Hence, Huang  pointed out that the membership value of any gray level should be no less than 0.5. In , is the primary function and can be regarded as the tuning function for . Since is derived from , we expect that of any gray level should be also no less than 0.5. Here, a simple constraint is made to ensure being in the range .
- ● Finally, the secondary grade - is thought of as the possibility of to be as the membership grade of gray level k. Here, is approximated as the normalized occurrence probability of .. in , and we have
Using the type-2 fuzzy sets proposed in this section. How to choose the optimal threshold is presented next.
4. Optimal threshold selection
Being as a special case of type-2 fuzzy sets, interval type-2 fuzzy sets were employed in  and Tizhoosh selected the optimal threshold by maximizing the entropy of interval valued fuzzy sets addressed in , which seems a little unreasonable as what Bustince discussed in . However, in this paper is a general type-2 fuzzy set, and Tizhoosh’s suggestion on selection does not work under this condition. Here, is chosen in a different way illustrated below.
In order to obtain , the fuzziness measure of should be computed firstly as the criterion. Most recently, Zhai  has addressed how to compute the fuzziness for general type-2 fuzzy sets. While, the proposition in  is not suitable for practical image thresholding for two reasons: (i) the fuzziness measure suggested in  is a type-1 fuzzy set but not a crisp number and (ii) computational expense for fuzziness is very high.
For the sake of practical use and convenience in computation, is transformed into a type-1 fuzzy set via vertical slice centroid type reduction (VSCTR)  in our paper. And then is selected by minimizing the fuzziness proposed by Yager  of .
As illustrated in Section 3, the membership grade for gray level k is not a crisp number in but a type-1 fuzzy set also called the vertical slice defined as23], the centroid of is given as25] suggested.
Based on Yager’s proposition , the fuzziness of could be defined as
In order to demonstrate the performance of our proposed technique, 14 images of different histogram types (including unimodal, bimodal and multimodal) and sizes are chosen for test. The test images which are noisy or smooth contain small and large objects of clear and fuzzy boundaries, and those images have both simple and complex relationships between object and background. All the images are listed as ant (), bear (), block (), cameraman (), elephant (), field (), gear (), geometric1 (), geometric2 (), rhinoceros (), shadow1 (), shadow2 (), stele () and stone (). Most of the test images and the corresponding ground-truth images employed as gold standard are obtained form references [6,9,17], and the rest are downloaded from internet. Figure 12 shows the original test images, their gold standard and histograms.
Four well established algorithms are utilized to make comparison with the proposed approach (T2F2), they are Liu’s 2-D Otsu method (2DO) , Sahoo’s 2-D entropic technique (2DE) , Wang’s 2-D type-1 fuzzy means (2DT1F)  and Tizhoosh’s type-2 fuzzy approach (T2F1) . Among them, the former two are non-fuzzy ways while the others are fuzzy ones. The binary images yielded by the five algorithms are shown in Fig. 13 and Fig. 14 as PART I and II respectively.Table 1 . Mean - ϖ and standard deviation - σ employed to measure effectiveness and robustness of different approaches are also included. Figure 11 shows the performance comparison between our method and other techniques by graph.
As listed in Table 1, our method yields the highest performance of 94.41% with the lowest standard deviation of 5.96%, which is superior to other techniques apparently.
According to Fig. 11, a detailed comparison between T2F2 and other algorithms is made as follows.
- ● 2DO is famous for its good adaptability to different types of images. As seen from Fig. 13, Fig. 14 and Fig. 11(a), the segmentation yielded by 2DO for all the test images is acceptable, especially when the 1-D histogram is bimodal. However, for most cases 2DO is inferior to T2F2 significantly and its performance is not so satisfying when the image histogram is unimodal or multimodal, e.g., image 3, 5, 8, and 9.
- ● 2DE makes use of 2-D histogram to improve performance, but we find that it tends to yield over segmentation, e.g., image 2, 5, 7 and 12. This situation may be caused by the intrinsic characteristics of entropy. At the same time, there is also a great performance gap between 2DE and our method.
- ● 2DT1F integrates the concept of 2-D histogram and type-1 fuzzy sets together, generally its performance is close to T2F2 and sometimes even better as shown in Fig. 11(c). But under some condition, 2DT1F does not work, such as image 2 and 5. So, its robustness seems doubtful.
- ● As the first type-2 fuzzy thresholding algorithm, T2F1 gives the best segmentation to some images, which can demonstrate the usefulness of type-2 fuzzy sets in thresholding. Unfortunately, its robustness is still not good enough as its σ is the second greatest one of all the five approaches, while σ of T2F2 is the lowest.
From the experiment, it can be observed clearly that no method is the best choice for all the images. As a stable technique, our algorithm yields good segmentation for most test images with the lowest σ and averagely the greatest ϖ. But when the image content is simple, T2F2 seems not so outstanding, which needs to be improved.
Image thresholding is a fundamental and difficult task in image processing. Till now, no superior technique has been proposed for all types of images. So, it is necessary to develop new methods which are effective and robust. In this paper, a new thresholding approach using type-2 fuzzy sets based on GLSC histogram of HVNC is suggested. Our work focuses on how to make use of both gray level and local spatial information in fuzzy thresholding to improve segmentation. By embedding HVNC into GLSC histogram, we make it of human perception characteristics. On the basis of the refined GLSC histogram, type-2 fuzzy set is defined to model the vagueness in image data for thresholding, which type-1 fuzzy set is incapable of. Comparing with other four fuzzy and non-fuzzy algorithms, experiment on 14 images demonstrates the advantage of our method. As future work, more adaptive mathematical model for HVNC is desired. Additionally, we are also interested in how to define the ultrafuzziness of type-2 fuzzy set directly for thresholding instead of transforming it into type-1 fuzzy set.
This work was supported by the Project of the National Fundamental Research of China under Grant No. A1420061266, the Project of the National Natural Science Foundation of China under Grant No. 60736010 and the Project of Nonprofit Sector of China under Grant No. GYHY200906032.
References and links
1. J. S. Weszka, “A survey of threshold selection techniques,” Comput. Graph. Image Process. 7(2), 259–265 (1978).
2. P. K. Sahoo, S. Soltani, and A. K. C. Wong, “A survey of thresholding techniques,” Comput. Graph. Image Process. 41(2), 233–260 (1988).
3. M. Sezgin and B. Sankur, “Survey over image thresholding techniques and quantitative performance evaluation,” J. Electron. Imaging 13(1), 146–165 (2004).
4. L. K. Huang and M. J. Wang, “Image thresholding by minimizing the measure of fuzziness,” Pattern Recognit. 28(1), 41–51 (1995).
5. Q. Wang, Z. Chi, and R. Zhao, “Image thresholding by maximizing the index of nonfuzziness of the 2-D grayscale histogram,” Comput. Vis. Image Underst. 85(2), 100–116 (2002).
6. H. R. Tizhoosh, “Image thresholding using type II fuzzy sets,” Pattern Recognit. 38(12), 2363–2372 (2005).
7. I. K. Vlachos and G. D. Sergiadis, “Comment on: ‘Image thresholding using type II fuzzy sets’,” Pattern Recognit. 41(5), 1810–1811 (2008).
8. H. Bustince, E. Barrenechea, M. Pagola, J. Fernandez, and J. Sanz, “Comment on: ‘Image thresholding using type II fuzzy sets’: Importance of this method,” Pattern Recognit. 43(9), 3188–3192 (2010).
9. N. V. Lopes, P. A. Mogadouro do Couto, H. Bustince, and P. Melo-Pinto, “Automatic histogram threshold using fuzzy measures,” IEEE Trans. Image Process. 19(1), 199–204 (2010).
10. C. Murthy and S. Pal, “Fuzzy thresholding: mathematical framework, bound functions and weighted moving average technique,” Pattern Recognit. Lett. 11(3), 197–206 (1990).
11. O. J. Tobias and R. Seara, “Image segmentation by histogram thresholding using fuzzy sets,” IEEE Trans. Image Process. 11(12), 1457–1465 (2002).
12. C. V. Jawahar, P. K. Biswas, and A. K. Ray, “Analysis of fuzzy thresholding schemes,” Pattern Recognit. 33(8), 1339–1349 (2000).
13. S. K. Pal and A. Rosenfeld, “Image enhancement and thresholding by optimization of fuzzy compactness,” Pattern Recognit. Lett. 7(2), 77–86 (1988).
14. S. Di Zenzo, L. Cinque, and S. Levialdi, “Image thresholding using fuzzy entropies,” IEEE Trans. Syst. Man Cybern. B Cybern. 28(1), 15–23 (1998).
15. A. S. Pednekar and I. A. Kakadiaris, “Image segmentation based on fuzzy connectedness using dynamic weights,” IEEE Trans. Image Process. 15(6), 1555–1562 (2006). [PubMed]
16. C. V. Jawahar, P. K. Biswas, and A. K. Ray, “Investigations on fuzzy thresholding based on fuzzy clustering,” Pattern Recognit. 30(10), 1605–1613 (1997).
17. H. Bustince, E. Barrenechea, and M. Pagola, “Image thresholding using restricted equivalence functions and maximizing the measures of similarity,” Fuzzy Sets Syst. 158(5), 496–516 (2007).
18. L. A. Zadeh, “Fuzzy sets,” Inf. Control 8(3), 338–353 (1965).
19. L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reason I,” Inf. Sci. 8(3), 199–249 (1975).
20. J. Mendel and R. I. B. John, “Type-2 fuzzy sets made simple,” IEEE Trans. Fuzzy Syst. 10(2), 117–127 (2002).
21. J. Mendel, “Advances in type-2 fuzzy sets and systems,” Inf. Sci. 177(1), 84–110 (2007).
22. J. Mendel, “Type-2 fuzzy sets and systems: An overview,” IEEE Comput. Intell. Mag. 2, 20–29 (2007).
23. D. Wu and J. Mendel, “Uncertainty measures for interval type-2 fuzzy sets,” Inf. Sci. 177(23), 5378–5393 (2007).
24. D. Zhai and J. Mendel, “Uncertainty measures for general type-2 fuzzy sets,” Inf. Sci. 181(3), 503–518 (2011).
25. L. Lucas, T. Centeno, and M. Delgado, “General type-2 fuzzy inference system: analysis, design and computational aspects,” in Proceedings of IEEE International Conference on Fuzzy Systems (Imperial College, London, 2007), pp. 1–6.
26. J. Aisbett, J. T. Rickard, and D. G. Morgenthaler, “Type-2 fuzzy sets as functions on spaces,” IEEE Trans. Fuzzy Syst. 18(4), 841–844 (2010).
27. P. Burillo and H. Bustince, “Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets,” Fuzzy Sets Syst. 78(3), 305–316 (1996).
28. N. Kapur, P. K. Sahoo, and A. K. C. Wong, “A new method for gray-level picture thresholding using the entropy of the histogram,” Comput. Vis. Graph. Image Process. 29(3), 273–285 (1985).
29. R. Kirby and A. Rosenfeld, “A note on the use of (gray level, average gray level) space as an aid in the threshold selection,” IEEE Trans. Syst. Man Cybern. 9(12), 860–864 (1979).
30. A. S. Abutaleb, “Automatic thresholding of gray-level picture using two-dimensional entropy,” Comput. Vis. Graph. Image Process. 47(1), 22–32 (1989).
31. A. D. Brink, “Thresholding of digital images using two-dimensional entropies,” Pattern Recognit. 25(8), 803–808 (1992).
32. N. R. Pal and S. K. Pal, “Entropic thresholding,” Signal Process. 16(2), 97–108 (1989).
33. P. K. Sahoo and G. Arora, “A thresholding method based on two-dimensional Renyi’s entropy,” Pattern Recognit. 37(6), 1149–1161 (2004).
34. P. K. Sahoo and G. Arora, “Image thresholding using two-dimensional Tsallis-Havrda-Charvát entropy,” Pattern Recognit. Lett. 27(6), 520–528 (2006).
35. J. Z. Liu and W. Q. Li, “The automatic thresholding of gray-level pictures via two-dimensional Otsu method,” Acta Automatica Sin. 19, 101–105 (1993) (in Chinese).
36. N. Otsu, “A threshold selection method from gray-level histograms,” IEEE Trans. Syst. Man Cybern. 9(1), 62–66 (1979).
37. Y. Xiao, Z. G. Cao, and T. X. Zhang, “Entropic thresholding based on gray-level spatial correlation histogram,” in Proceedings of IEEE Conference on Pattern Recognition (Univ. of South Florida, Tampa, Florida, 2008), pp. 1–4.
38. Y. Xiao, Z. G. Cao, and S. Zhong, “New entropic thresholding approach using gray-level spatial correlation histogram,” Opt. Eng. 49(12), 127007 (2010).
39. G. Buchsbaum, “An analytical derivation of visual nonlinearity,” IEEE Trans. Biomed. Eng. BME-27(5), 237–242 (1980).
40. R. Yager, “On the measure of fuzziness and negation Part I: Membership in the unit interval,” Int. J. Gen. Syst. 5(4), 221–229 (1979).