Abstract

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

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  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).
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  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).

2011 (1)

D. Zhai and J. Mendel, “Uncertainty measures for general type-2 fuzzy sets,” Inf. Sci. 181(3), 503–518 (2011).

2010 (4)

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).

Y. Xiao, Z. G. Cao, and S. Zhong, “New entropic thresholding approach using gray-level spatial correlation histogram,” Opt. Eng. 49(12), 127007 (2010).

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).

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).

2008 (1)

I. K. Vlachos and G. D. Sergiadis, “Comment on: ‘Image thresholding using type II fuzzy sets’,” Pattern Recognit. 41(5), 1810–1811 (2008).

2007 (4)

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).

J. Mendel, “Advances in type-2 fuzzy sets and systems,” Inf. Sci. 177(1), 84–110 (2007).

J. Mendel, “Type-2 fuzzy sets and systems: An overview,” IEEE Comput. Intell. Mag. 2, 20–29 (2007).

D. Wu and J. Mendel, “Uncertainty measures for interval type-2 fuzzy sets,” Inf. Sci. 177(23), 5378–5393 (2007).

2006 (2)

P. K. Sahoo and G. Arora, “Image thresholding using two-dimensional Tsallis-Havrda-Charvát entropy,” Pattern Recognit. Lett. 27(6), 520–528 (2006).

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]

2005 (1)

H. R. Tizhoosh, “Image thresholding using type II fuzzy sets,” Pattern Recognit. 38(12), 2363–2372 (2005).

2004 (2)

M. Sezgin and B. Sankur, “Survey over image thresholding techniques and quantitative performance evaluation,” J. Electron. Imaging 13(1), 146–165 (2004).

P. K. Sahoo and G. Arora, “A thresholding method based on two-dimensional Renyi’s entropy,” Pattern Recognit. 37(6), 1149–1161 (2004).

2002 (3)

J. Mendel and R. I. B. John, “Type-2 fuzzy sets made simple,” IEEE Trans. Fuzzy Syst. 10(2), 117–127 (2002).

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).

O. J. Tobias and R. Seara, “Image segmentation by histogram thresholding using fuzzy sets,” IEEE Trans. Image Process. 11(12), 1457–1465 (2002).

2000 (1)

C. V. Jawahar, P. K. Biswas, and A. K. Ray, “Analysis of fuzzy thresholding schemes,” Pattern Recognit. 33(8), 1339–1349 (2000).

1998 (1)

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).

1997 (1)

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).

1996 (1)

P. Burillo and H. Bustince, “Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets,” Fuzzy Sets Syst. 78(3), 305–316 (1996).

1995 (1)

L. K. Huang and M. J. Wang, “Image thresholding by minimizing the measure of fuzziness,” Pattern Recognit. 28(1), 41–51 (1995).

1993 (1)

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).

1992 (1)

A. D. Brink, “Thresholding of digital images using two-dimensional entropies,” Pattern Recognit. 25(8), 803–808 (1992).

1990 (1)

C. Murthy and S. Pal, “Fuzzy thresholding: mathematical framework, bound functions and weighted moving average technique,” Pattern Recognit. Lett. 11(3), 197–206 (1990).

1989 (2)

N. R. Pal and S. K. Pal, “Entropic thresholding,” Signal Process. 16(2), 97–108 (1989).

A. S. Abutaleb, “Automatic thresholding of gray-level picture using two-dimensional entropy,” Comput. Vis. Graph. Image Process. 47(1), 22–32 (1989).

1988 (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).

S. K. Pal and A. Rosenfeld, “Image enhancement and thresholding by optimization of fuzzy compactness,” Pattern Recognit. Lett. 7(2), 77–86 (1988).

1985 (1)

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).

1980 (1)

G. Buchsbaum, “An analytical derivation of visual nonlinearity,” IEEE Trans. Biomed. Eng. BME-27(5), 237–242 (1980).

1979 (3)

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).

N. Otsu, “A threshold selection method from gray-level histograms,” IEEE Trans. Syst. Man Cybern. 9(1), 62–66 (1979).

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).

1978 (1)

J. S. Weszka, “A survey of threshold selection techniques,” Comput. Graph. Image Process. 7(2), 259–265 (1978).

1975 (1)

L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reason I,” Inf. Sci. 8(3), 199–249 (1975).

1965 (1)

L. A. Zadeh, “Fuzzy sets,” Inf. Control 8(3), 338–353 (1965).

Abutaleb, A. S.

A. S. Abutaleb, “Automatic thresholding of gray-level picture using two-dimensional entropy,” Comput. Vis. Graph. Image Process. 47(1), 22–32 (1989).

Aisbett, J.

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).

Arora, G.

P. K. Sahoo and G. Arora, “Image thresholding using two-dimensional Tsallis-Havrda-Charvát entropy,” Pattern Recognit. Lett. 27(6), 520–528 (2006).

P. K. Sahoo and G. Arora, “A thresholding method based on two-dimensional Renyi’s entropy,” Pattern Recognit. 37(6), 1149–1161 (2004).

Barrenechea, E.

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).

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).

Biswas, P. K.

C. V. Jawahar, P. K. Biswas, and A. K. Ray, “Analysis of fuzzy thresholding schemes,” Pattern Recognit. 33(8), 1339–1349 (2000).

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).

Brink, A. D.

A. D. Brink, “Thresholding of digital images using two-dimensional entropies,” Pattern Recognit. 25(8), 803–808 (1992).

Buchsbaum, G.

G. Buchsbaum, “An analytical derivation of visual nonlinearity,” IEEE Trans. Biomed. Eng. BME-27(5), 237–242 (1980).

Burillo, P.

P. Burillo and H. Bustince, “Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets,” Fuzzy Sets Syst. 78(3), 305–316 (1996).

Bustince, H.

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).

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).

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).

P. Burillo and H. Bustince, “Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets,” Fuzzy Sets Syst. 78(3), 305–316 (1996).

Cao, Z. G.

Y. Xiao, Z. G. Cao, and S. Zhong, “New entropic thresholding approach using gray-level spatial correlation histogram,” Opt. Eng. 49(12), 127007 (2010).

Chi, Z.

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).

Cinque, L.

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).

Di Zenzo, S.

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).

Fernandez, J.

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).

Huang, L. K.

L. K. Huang and M. J. Wang, “Image thresholding by minimizing the measure of fuzziness,” Pattern Recognit. 28(1), 41–51 (1995).

Jawahar, C. V.

C. V. Jawahar, P. K. Biswas, and A. K. Ray, “Analysis of fuzzy thresholding schemes,” Pattern Recognit. 33(8), 1339–1349 (2000).

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).

John, R. I. B.

J. Mendel and R. I. B. John, “Type-2 fuzzy sets made simple,” IEEE Trans. Fuzzy Syst. 10(2), 117–127 (2002).

Kakadiaris, I. A.

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]

Kapur, N.

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).

Kirby, R.

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).

Levialdi, S.

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).

Li, W. Q.

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).

Liu, J. Z.

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).

Lopes, N. V.

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).

Melo-Pinto, P.

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).

Mendel, J.

D. Zhai and J. Mendel, “Uncertainty measures for general type-2 fuzzy sets,” Inf. Sci. 181(3), 503–518 (2011).

J. Mendel, “Type-2 fuzzy sets and systems: An overview,” IEEE Comput. Intell. Mag. 2, 20–29 (2007).

J. Mendel, “Advances in type-2 fuzzy sets and systems,” Inf. Sci. 177(1), 84–110 (2007).

D. Wu and J. Mendel, “Uncertainty measures for interval type-2 fuzzy sets,” Inf. Sci. 177(23), 5378–5393 (2007).

J. Mendel and R. I. B. John, “Type-2 fuzzy sets made simple,” IEEE Trans. Fuzzy Syst. 10(2), 117–127 (2002).

Mogadouro do Couto, P. A.

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).

Morgenthaler, D. G.

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).

Murthy, C.

C. Murthy and S. Pal, “Fuzzy thresholding: mathematical framework, bound functions and weighted moving average technique,” Pattern Recognit. Lett. 11(3), 197–206 (1990).

Otsu, N.

N. Otsu, “A threshold selection method from gray-level histograms,” IEEE Trans. Syst. Man Cybern. 9(1), 62–66 (1979).

Pagola, M.

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).

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).

Pal, N. R.

N. R. Pal and S. K. Pal, “Entropic thresholding,” Signal Process. 16(2), 97–108 (1989).

Pal, S.

C. Murthy and S. Pal, “Fuzzy thresholding: mathematical framework, bound functions and weighted moving average technique,” Pattern Recognit. Lett. 11(3), 197–206 (1990).

Pal, S. K.

N. R. Pal and S. K. Pal, “Entropic thresholding,” Signal Process. 16(2), 97–108 (1989).

S. K. Pal and A. Rosenfeld, “Image enhancement and thresholding by optimization of fuzzy compactness,” Pattern Recognit. Lett. 7(2), 77–86 (1988).

Pednekar, A. S.

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]

Ray, A. K.

C. V. Jawahar, P. K. Biswas, and A. K. Ray, “Analysis of fuzzy thresholding schemes,” Pattern Recognit. 33(8), 1339–1349 (2000).

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).

Rickard, J. T.

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).

Rosenfeld, A.

S. K. Pal and A. Rosenfeld, “Image enhancement and thresholding by optimization of fuzzy compactness,” Pattern Recognit. Lett. 7(2), 77–86 (1988).

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).

Sahoo, P. K.

P. K. Sahoo and G. Arora, “Image thresholding using two-dimensional Tsallis-Havrda-Charvát entropy,” Pattern Recognit. Lett. 27(6), 520–528 (2006).

P. K. Sahoo and G. Arora, “A thresholding method based on two-dimensional Renyi’s entropy,” Pattern Recognit. 37(6), 1149–1161 (2004).

P. K. Sahoo, S. Soltani, and A. K. C. Wong, “A survey of thresholding techniques,” Comput. Graph. Image Process. 41(2), 233–260 (1988).

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).

Sankur, B.

M. Sezgin and B. Sankur, “Survey over image thresholding techniques and quantitative performance evaluation,” J. Electron. Imaging 13(1), 146–165 (2004).

Sanz, J.

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).

Seara, R.

O. J. Tobias and R. Seara, “Image segmentation by histogram thresholding using fuzzy sets,” IEEE Trans. Image Process. 11(12), 1457–1465 (2002).

Sergiadis, G. D.

I. K. Vlachos and G. D. Sergiadis, “Comment on: ‘Image thresholding using type II fuzzy sets’,” Pattern Recognit. 41(5), 1810–1811 (2008).

Sezgin, M.

M. Sezgin and B. Sankur, “Survey over image thresholding techniques and quantitative performance evaluation,” J. Electron. Imaging 13(1), 146–165 (2004).

Soltani, S.

P. K. Sahoo, S. Soltani, and A. K. C. Wong, “A survey of thresholding techniques,” Comput. Graph. Image Process. 41(2), 233–260 (1988).

Tizhoosh, H. R.

H. R. Tizhoosh, “Image thresholding using type II fuzzy sets,” Pattern Recognit. 38(12), 2363–2372 (2005).

Tobias, O. J.

O. J. Tobias and R. Seara, “Image segmentation by histogram thresholding using fuzzy sets,” IEEE Trans. Image Process. 11(12), 1457–1465 (2002).

Vlachos, I. K.

I. K. Vlachos and G. D. Sergiadis, “Comment on: ‘Image thresholding using type II fuzzy sets’,” Pattern Recognit. 41(5), 1810–1811 (2008).

Wang, M. J.

L. K. Huang and M. J. Wang, “Image thresholding by minimizing the measure of fuzziness,” Pattern Recognit. 28(1), 41–51 (1995).

Wang, Q.

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).

Weszka, J. S.

J. S. Weszka, “A survey of threshold selection techniques,” Comput. Graph. Image Process. 7(2), 259–265 (1978).

Wong, A. K. C.

P. K. Sahoo, S. Soltani, and A. K. C. Wong, “A survey of thresholding techniques,” Comput. Graph. Image Process. 41(2), 233–260 (1988).

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).

Wu, D.

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Figures (14)

Fig. 1
Fig. 1

Overall flowchart of the proposed algorithm.

Fig. 2
Fig. 2

2-D histogram plane.

Fig. 3
Fig. 3

Drawbacks of 2-D histograms. (a) Local neighborhood 1. (b) Local neighborhood 2.

Fig. 4
Fig. 4

Relationship between Δ I T and I.

Fig. 5
Fig. 5

Relationship between f ( x , y ) , N E 1 ( x , y ) and N E 2 ( x , y ) .

Fig. 6
Fig. 6

Cameraman image and its GLSC histogram of HVNC. (a) Cameraman image. (b) GLSC histogram of HVNC.

Fig. 7
Fig. 7

Mapping images corresponding to different m.

Fig. 8
Fig. 8

Example of a type-2 fuzzy set.

Fig. 9
Fig. 9

μ 0 ( t ) and μ 1 ( t ) computation.

Fig. 10
Fig. 10

Impact factor of local spatial information - Ψ ( m ) .

Fig. 12
Fig. 12

Test images, their corresponding ground-truth images and histograms.

Fig. 13
Fig. 13

Thresholding results of five algorithms - PART I. For each test image, from left to right: Liu’s 2-D Otsu method, Sahoo’s 2-D entropic technique, Wang’s 2-D fuzzy means, Tizhoosh’s type-2 fuzzy approach and our proposed algorithm.

Fig. 14
Fig. 14

Thresholding results of five algorithms - PART II. For each test image, from left to right: Liu’s 2-D Otsu method, Sahoo’s 2-D entropic technique, Wang’s 2-D fuzzy means, Tizhoosh’s type-2 fuzzy approach and our proposed algorithm.

Fig. 11
Fig. 11

Thresholding performance comparison between different algorithms. (a) 2DO and T2F2. (b) 2DE and T2F2. (c) 2DT1F and T2F2. (d) T2F1 and T2F2.

Tables (1)

Tables Icon

Table 1 Performance of Different Algorithms

Equations (25)

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H 2 d ( s , t ) = number of bin ( f ( x , y ) = s , g ( x , y ) = t ) Q × R , s [ 0 , L ] ; t [ 0 , L ] .
g ( x , y ) = 1 N 2 i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 f ( x + i , y + j ) , x [ 1 , Q ] ; y [ 1 , R ] .
H G L S C ( k , m ) = number of bin ( f ( x , y ) = k , d ( x , y ) = m ) Q × R , k [ 0 , L ] ; m [ 1 , N 2 ] .
d ( x , y ) = i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 T ( f ( x + i , y + j ) f ( x , y ) ) .
T ( f ( x + i , y + j ) f ( x , y ) ) = { 1 , if | f ( x + i , y + j ) f ( x , y ) | ζ 0 , if | f ( x + i , y + j ) f ( x , y ) | > ζ .  
Δ I T = { α 1 I , " D e V r i e s R o s e " r e g i o n α 2 I , " W e b e r " r e g i o n .
ζ = { η 1 M e a n 2 ( x , y ) + δ 1 , " D e V r i e s R o s e " r e g i o n η 2 M e a n 2 ( x , y ) + δ 2 , " W e b e r " r e g i o n       = { η 1 M e a n 2 ( x , y ) + δ 1 , 0 M e a n 2 ( x , y ) < D i v _ R η 2 M e a n 2 ( x , y ) + δ 2 , D i v _ R M e a n 2 ( x , y ) L .
A = x X [ u J x μ A ( x , u ) / u ] / x , J x [ 0 , 1 ] , μ A ( x , u ) [ 0 , 1 ] .
A G L S C _ H V N C = k [ 0 , L ] [ u ( k , m ) J k μ A G L S C _ H V N C ( k , u ( k , m ) ) / u ( k , m ) ] / k .
J k = m [ 1 , N 1 2 ] u ( k , m ) .
u ( k , m ) = ρ ( k ) ψ ( m ) .
μ 0 ( t ) = k = 0 t k H ( k ) .
μ 1 ( t ) = k = t + 1 L k H ( k ) .
m u ( k ) = { 1 1 + | k μ 0 ( t ) | / C , k [ 0 , t ] 1 1 + | k μ 1 ( t ) | / C , k [ t + 1 , L ] , m u ( k ) [ 0.5 , 1 ] .
ρ ( k ) = { 1 1 + sin ( ( | k μ 0 ( t ) | / D i f f _ m a x ) ( π / 2 ) ) , k [ 0 , t ] 1 1 + sin ( ( | k μ 1 ( t ) | / D i f f _ m a x ) ( π / 2 ) ) , k [ t + 1 , L ] , ρ ( k ) [ 0.5 , 1 ] .
D i f f _ m a x = the maximum value in { | k μ 0 ( t ) | , | k μ 1 ( t ) | | k [ 0 , L ] ; t [ 1 , L 1 ] } .
ψ ( m ) = ( ( 1 e ( 9 m / N 1 2 ϕ ) 1 + e ( 9 m / N 1 2 ϕ ) ) / ( 1 e ( 9 ϕ ) 1 + e ( 9 ϕ ) ) ) γ , ψ ( m ) ( 0 , 1 ] .
u ( k , m ) = 0.5 , i f u ( k , m ) < 0.5 .
μ A G L S C _ H V N C ( k , u ( k , m ) ) = H G L S C _ H V N C ( k , m ) / i = 1 N 1 2 H G L S C _ H V N C ( k , i ) .
A k = u ( k , m ) J k μ A G L S C _ H V N C ( k , u ( k , m ) ) / u ( k , m ) .
c ( A k ) = i = 1 N 1 2 u ( k , i ) μ A G L S C _ H V N C ( k , u ( k , i ) ) / i = 1 N 1 2 μ A G L S C _ H V N C ( k , u ( k , i ) ) .
A T 1 = k [ 0 , L ] c ( A k ) / k .
F p ( t ) = 1 [ k = 0 L | 2 c ( A k ) 1 | p H ( k ) ] 1 / p , p = 1 , 2 , 3 .
t * = A r g min { F p ( t ) | t [ 0 , L ] } .
λ = 100 × | B O B T | + | F O F T | | B O | + | F O | .

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