[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Image Edge Detection: A New Approach Based on Fuzzy Entropy and Fuzzy Divergence

  • Published:
International Journal of Fuzzy Systems Aims and scope Submit manuscript

Abstract

In image pre-processing, edge detection is a non-trivial task. Sometimes, images are affected by vagueness so that the edges of objects are difficult to distinguish. Hence, the usual edge-detecting operators can give unreliable results, thus necessitating the use of fuzzy procedures. In literature, Chaira and Ray approach is a popular technique for fuzzy edge detection in which fuzzy divergence formulation is exploited. However, this approach does not specify the threshold technique must be applied. Then, in this work, starting from Chairy and Ray procedure, we present a new fuzzy edge detector based on both fuzzy divergence (thought and proved to be a distance) and fuzzy entropy minimization for the thresholding sub-step in gray-scale images. Eddy currents, thermal infrared, and electrospinning images were used to test the proposed procedure after their fuzzification by a suitable adaptive S-shaped fuzzy membership function. Moreover, the fuzziness content of each image has been quantified by new specific indices proposed here and formulated in terms of fuzzy divergence. The results have been evaluated by suitable assessment metrics here formulated and are considered to be encouraging when qualitatively and quantitatively compared with those obtained by some well-known I- and II-order edge detectors.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Gonzales, R.C., Woods, R.F.: Digital Image Processing. Prentice-Hall, New York (2007)

    Google Scholar 

  2. Kaur, D., Kayr, Y.: Various image sementation techniques: a review. Int J Comput Sci Mobile Comput 3(5), 414–809 (2014)

    Google Scholar 

  3. Zhang, S., Ma, Z., Zhang, G., Lei, T., Zhang, R.: Semantic image segmantation with deep convolutional neural networks and quick shift. Symm. MDPI 12(427), 1–1 (2020)

    Google Scholar 

  4. Pont-Tuset, J., Arbelaez, P., Barron, J.T., Marquez, F., Malik, J.: Multiscale combinatorial grouping for image segmentation and object proposal generation. IEEE Trans. Pattern Recogn. (2016). https://doi.org/10.1109/TPAMI.2016.2537320

    Article  Google Scholar 

  5. Li, H.S., Qingxin, Z., Lan, S., Shen, C.Y., Zhou, R., Mo, J.: Image storagem retrieval, compression and segmentation in a quantum system. Quant. Inf. Process. 12(6), 2269–2290 (2013)

    Article  Google Scholar 

  6. Qasim, A.J., Din, R.E., Alyousuf, F.Q.A.: Review on techniques and file formats of image compression. Bull. Electr. Eng. Inf. 9(2), 602–610 (2020)

    Google Scholar 

  7. Combettes, P.L., Pesquet, J.C.: Proximal Splitting Methods in Signal Processing, Fixed-Point Algorithms for Inverse Problems in Science and Engineering, pp. 185–212 (2011)

  8. Peng, B., Zhang, L., Zhang, D.: A Survey of graph theoretical approaches to image segmentation. Pattern Recogn. 46(3), 1020–1038 (2013)

    Article  Google Scholar 

  9. Hay, G.J., Castilla, G., Wulder, M.A., Ruiz, J.R.: An automated object-based approach for the multiscale image segmentation of forest scense. Int. J. Appl. Earth Observ. Geoinform 7(4), 339–359 (2005)

    Article  Google Scholar 

  10. Sharon, E., Brandt, A., Basri, R.: Fast multiscale image segmentation. IEEE Comput. Soc. Conf. Comput. Vis. Pattern Recogn. 1, 70–77 (2020)

    Google Scholar 

  11. Russ, J.C., Brent Neal, F.: The Image Processing Handbook. CRC Press, Boca Raton (2018)

    Book  Google Scholar 

  12. Zhang, K., Zhang, Y., Wang, P., Tian, Y., Yang, J.: An improved sobel edge algorithm and FPGA implementation. In: Proceedings of the 8th International Congress of Information and Communication Technology (ICICT-2018), Procedia Computer Science, 131, pp. 243–248 (2018)

  13. Meltsov, V., Lapitsky, A.A, Rostovtsev, V.S.: FPGA-Implementation of a prediction module based on a generalized regression neural network. In: Proceedings of the IEEE Conference of Russian Young Researchers in Electrical and Electronic Engineering (EIConRus) (2020)

  14. Cao, J., Chen, L., Wang, M., Tian, Y.: Implementating a parallel image edge detection algorithm based on the Otsu–Canny operator on the hadoop platform. Comput. Intell. Neurosci. 3598284, 1–12 (2018)

    Article  Google Scholar 

  15. Verma, O.P., Parihar, A.S.: An optimal fuzzy system for edge detection in color image using bacterial foraging algorithm. IEEE Trans. Fuzzy Syst. 25(1), 114–127 (2017)

    Article  Google Scholar 

  16. Pattanaik, A., Mishra, S., Rana, D.: Comparative study of edge detection using Renyi entropy and differential evolution. Int. J. Eng. Res. Technol. 4(3), 1001–1005 (2015)

    Google Scholar 

  17. Chaira, T., Ray, A.K.: Fuzzy Image Processing and Application with MatLab. Taylor & Francis Group, CRC Press, Boca Raton, London, New York (2010)

    MATH  Google Scholar 

  18. Chaira, T.: Medical Image Processing, Advanced Fuzzy Set Theoretic Techniques. Taylor & Francis Group, CRC Press, Boca Raton, London, New York (2015)

    Book  Google Scholar 

  19. Pradeep Kumar Reddy, R., Nagaraju, C.: Improved canny edge detection technique using S-membership function. Int. J. Eng. Adv. Technol. (IJEAT) 8(6), 43–49 (2019)

    Article  Google Scholar 

  20. Mittal, M., et al.: An efficient edge detection approach to provide better edge connectivity for image analysis. IEEE ACCESS 7, 33240–33255 (2019)

    Article  Google Scholar 

  21. Ma, X., Liu, S., Hu, S., Geng, P., Liu, M., Zhao, J.: SAR image edge detection via sparse representation. Soft Comput. 22(8), 2507–2515 (2018)

    Article  Google Scholar 

  22. Chen, S.C., Cheng Chiu, C.C.: Texture construction edge detection algorithm. Appl. Sci. (MDPI) 9, 1–25 (2019)

    Google Scholar 

  23. Wang, X., et al.: Detection and localization of image forgeries using improved mask regional convolutional neural network. Math. Biosci. Eng. 16(5), 4581–4593 (2019)

    Article  Google Scholar 

  24. Sonka, M., et al.: Image Processsing. Analysis and Machine Vision. Brooks/Cole Publisher, London (2001)

    Google Scholar 

  25. Hagara, M., Kubinec, P.: About edge detection in digital images. Radioengineering 27(4), 1–11 (2018)

    Article  Google Scholar 

  26. Sekehravani, E.A., Babulak, E., Masoodi, M.: Implementing Canny edge detection algorithm for noisy image. Bull. Electr. Eng. Inf. 9(4), 1404–1410 (2020)

    Google Scholar 

  27. Nanda, A., et al.: Image edge detection using fractional calculus with features and contrast enhancement. Circ. Syst. Signal Process. 37, 3946–3972 (2018)

    Article  Google Scholar 

  28. Albuquerque, M.P., Esquef, I.A., Gesualdi Mello, A.R.: Image Thresholding using Tsallis entropy. Pattern Recogn. Lett. 25, 1059–1065 (2004)

    Article  Google Scholar 

  29. Sahoo, P.K., Arora, G.: A thresholding method based on two-dimensional Reny’s entropy. Pattern Recogn. 37, 1149–1161 (2004)

    Article  Google Scholar 

  30. Kenneth, H., Ohnishi, H.L., Ohnishi, N.: FEDGE-Fuzzy Edge Detection by Fuzzy Categorization and Classification of Edge, Fuzzy Logic in Artificial Intelligence, JCAI’95 Workshop, pp. 182–196 (1995)

  31. Silva, L.E.V., SenraFilho, A. C. S., Fazan, V.P.S., Felipe, J.C., MurtaJunior, L.O.: Two-dimensional sample entropy: assessing image texture through irregularity. Biomed. Phys. Eng. Express. (1976). https://doi.org/10.1088/2057-1976/2/4/045002

  32. Sadykova, D., James, A.P.: Quality assessment metrics for edge detection and edge-aware filtering: a tutorial review. https://doi.org/10.1109/ICACCI.2017.8126200

  33. Panetta, K., Gao, C., Agaian, S., Nercessian, S.: Nonreference medical image edge map measure. Int. J. Biomed. Imaging 2014, 1–8 (2014)

    Google Scholar 

  34. Bausys, R., Karakeviciute-Januskeviciene, G., Cavallaro, F., Usovaite, A.: Algorithm selection for edge detection in satellite images by neutrosophic WASPAS method. Sustain. MDPI 12, 1–12 (2020)

  35. Versaci, M., La Foresta, F., Morabito, F.C., Angiulli, G.: A fuzzy divergence approach for solving electrostatic identification problems for NDT applications. Int. J. Appl. Electromagn. Mech. 1, 1–14 (2018). https://doi.org/10.3233/JAE-170043

    Article  Google Scholar 

  36. Vollmer, M., Mollmann, K.P.: Infrared Thermal Imaging. WILEY-YCH Verlag GmbH & Co, New York (2018)

    Google Scholar 

  37. Ieracitano, C., Panto, F., Mammone, N., Paviglianiti, A., Frontera, P., Morabito, F.C.: Towards an Automatic Classification of SEM Images of Nanomaterial via a Deep Learning Approach. Multidisciplinary Approaches to Neural Computing, in press

  38. Versaci, M., Calcagno, S., Morabito, F.C: Image contrast enhancement by distances among points in fuzzy hyper-cubes. In: Lecture Notes in Computer Science, vol. 9257, pp. 494–505 (2015)

  39. Versaci, M., Calcagno, S., Morabito, F.C.: Fuzzy geometrical approach based on unit hyper-cubes for image contrast enhancement. In: Proceedings of the IEEE International Conference on Signal and Image Processing (ICSIPA 2015), Kuala Lumpur, Malaysia, pp. 488–493 (2015)

  40. Versaci, M., Morabito, F.C., Angiulli, G.: Adaptive image contrast enhancement by computing distances int a 4-dimensional fuzzy unit hypercube. IEEE Access 5, 26922–26931 (2017). https://doi.org/10.1109/ACCESS.2017.2776349

    Article  Google Scholar 

  41. Pavo, J., Gasparics, A., Sebestyen, I., Vertesy, G., Darczi, C.S., Miya, K.: Eddy Current Testing with Fluxset Probe. Applied Electromagnetics and Mechanics. JSAEM, Tokyo (1996)

    Google Scholar 

  42. Doshi, J., Reneker, D.H.: Electrospinning process and applications of electrospunfibers. J. Electrostat. 35(2–3), 151–160 (1995)

    Article  Google Scholar 

  43. Vilchez, A., Acevedo, F., Cea, M., Seeger, M., Navia, R.: Applications of electrospun nanofibers with antioxidant properties: a review. Nanomater. MDPI 10(175), 1–25 (2020)

    Google Scholar 

  44. Fenn, J.B., Mann, M., Meng, C.K., Wong, S.D.F., Whitehouse, C.M.: Electrospray ionization for mass spectrometry of large biomolecules. Science 246(4926), 64–71 (1989)

    Article  Google Scholar 

  45. Versaci, M., Morabito, F.C.: Fuzzy time series approach for disruption prediction in tokamak reactors. IEEE Trans. Magn. 39(3), 1503–1506 (2003)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. DICEAM Department, Cittadella Universitaria, Via Graziella Feo di Vito, 89122, Reggio Calabria, Italy

    Mario Versaci & Francesco Carlo Morabito

Authors
  1. Mario Versaci
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Francesco Carlo Morabito
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mario Versaci.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

A Proof of Theorem 1

A Proof of Theorem 1

Remark 9

Let us consider three fuzzy images, \(F({\bf{I}}_{norm})_j\), \(j=1,2,3\), whose gray levels are \(\hat{a_{ij}}\), \(\hat{b_{ij}}\), and \(\hat{c_{ij}}\) \(\in X\), respectively. Let us say, for simplicity, that

$$\begin{aligned} m_{F({\bf{I}}_{norm})_1}(\hat{a_{ij}})-m_{F({\bf{I}}_{norm})_2}(\hat{b_{ij}})= & {} \alpha ,\nonumber \\ m_{F({\bf{I}}_{norm})_2}(\hat{b_{ij}})-m_{F({\bf{I}}_{norm})_3}(\hat{c_{ij}})= & {} \beta ,\nonumber \\ m_{F({\bf{I}}_{norm})_3}(\hat{c_{ij}})-m_{F({\bf{I}}_{norm})_1}(\hat{a_{ij}})= & {} \gamma , \end{aligned}$$
(26)

such that \(\alpha +\beta +\gamma =0.\) Further, with (26) taken into account, \(d({F({\bf{I}}_{norm})_1},{F({\bf{I}}_{norm})_2})\), \(d({F({\bf{I}}_{norm})_2},{F({\bf{I}}_{norm})_3})\), and

\(d({F({\bf{I}}_{norm})_3},{F({\bf{I}}_{norm})_1})\), as the generic addends of \(D({F({\bf{I}}_{norm})_1},{F({\bf{I}}_{norm})_2})\), \(D({F({\bf{I}}_{norm})_2},{F({\bf{I}}_{norm})_3})\), and \(D({F({\bf{I}}_{norm})_1},{F({\bf{I}}_{norm})_1})\), can be written as follows:

\(d({F({\bf{I}}_{norm})_1},{F({\bf{I}}_{norm})_2})=2- (1-\alpha ))\rm{{e}}^{\alpha }- (1+\alpha )\rm{{e}}^{-\alpha }\), \(d({F({\bf{I}}_{norm})_2},{F({\bf{I}}_{norm})_3})=2- (1-\beta ))\rm{{e}}^{\beta }- (1+\beta ) )\rm{{e}}^{-\gamma }\), \(d({F({\bf{I}}_{norm})_3},{F({\bf{I}}_{norm})_1})=2- (1-\gamma ))\rm{{e}}^{\gamma }- (1+\gamma ) )\rm{{e}}^{-\gamma }\). If they have the values

$$\begin{aligned} d({F({\bf{I}}_{norm})_1},{F({\bf{I}}_{norm})_2})\ge & {} 0, \end{aligned}$$
(27)
$$\begin{aligned} d({F({\bf{I}}_{norm})_1},{F({\bf{I}}_{norm})_2})= & {} 0\;\Leftrightarrow A=B, \end{aligned}$$
(28)
$$\begin{aligned} d({F({\bf{I}}_{norm})_1},{F({\bf{I}}_{norm})_2})& = d({F({\bf{I}}_{norm})_2},{F({\bf{I}}_{norm})_1}), \end{aligned}$$
(29)
$$\begin{aligned} d({F({\bf{I}}_{norm})_1},{F({\bf{I}}_{norm})_2})& \le d({F({\bf{I}}_{norm})_2},{F({\bf{I}}_{norm})_3}) + d({F({\bf{I}}_{norm})_3},{F({\bf{I}}_{norm})_1}), \end{aligned}$$
(30)

then, with the double summation operator applied to them, (8), (9), (10), and (11) apply.

We prove (27).

If (27) is true, then \(d(F({\bf{I}}_{norm})_1,F({\bf{I}}_{norm})_2)=2-(1-\alpha )\rm{{e}}^{\alpha }-(1+\alpha )\rm{{e}}^{-\alpha }\ge 0\), from which \(2\ge (1-\alpha )\rm{{e}}^{\alpha }+(1+\alpha )\rm{{e}}^{-\alpha }=\rm{{e}}^{\alpha }-\alpha \rm{{e}}^{\alpha }+\rm{{e}}^{-\alpha }\alpha \rm{{e}}^{-\alpha }=(\rm{{e}}^{\alpha }+\rm{{e}}^{-\alpha })-\alpha (\rm{{e}}^{\alpha }-\rm{{e}}^{-\alpha }).\) Then, \(2\ge 2\cosh (\alpha )-2\alpha \sinh (\alpha )\) and, again, \(1\ge \cosh (\alpha )-\alpha \sinh (\alpha ).\) Now, we set \(f(\alpha )=\cosh (\alpha )-(\alpha )\sinh (\alpha )\). However, aiming to search for the minimum value of \(f(\alpha )\), we impose \(f'(\alpha )=\sinh (\alpha )-\sinh (\alpha )-\alpha \cosh (\alpha )=0\) to achieve \(\alpha \cosh (\alpha )=0.\) \(\cosh (\alpha )=0\) is never null, hence the stationary value of \(f(\alpha )\) that one has for \(\alpha =0\). Again, if \(\alpha =0\), \(f(\alpha )=1\), while, if \(\alpha =1\), \(f(\alpha )=\frac{1}{e}<1\). From this, \(\alpha\) is a point of maximum for \(f(\alpha )\). Thus, \(1\ge \cosh (\alpha )-\alpha \sinh (\alpha )\) is always true. Then, given Remark 9, inequality (8) is also verified.

We prove (28).

The sufficient condition is easy to prove. In fact, if \(F({\bf{I}}_{norm})_1=F({\bf{I}}_{norm})_2\), then \(\alpha =0\). Thus, \(d(F({\bf{I}}_{norm})_1,F({\bf{I}}_{norm})_2)=2-(1-\alpha )\rm{{e}}^{\alpha }-(1+\alpha )\rm{{e}}^{-\alpha }=0\). Vice versa, to prove the necessary condition, we impose \(d(F({\bf{I}}_{norm})_1,F({\bf{I}}_{norm})_2)=0\), obtaining \(2=(1-\alpha )\rm{{e}}^{\alpha }-(1+\alpha )\rm{{e}}^{-\alpha }=0.\) Further, it is reasonable to write the following chain of equalities: \(2=(1-\alpha )\rm{{e}}^{\alpha }+(1+\alpha )\rm{{e}}^{-\alpha }=\rm{{e}}^{\alpha }-\alpha \rm{{e}}^{\alpha }+\rm{{e}}^{-\alpha }+\alpha \rm{{e}}^{-\alpha }=\rm{{e}}^{\alpha }+\rm{{e}}^{-\alpha }-\alpha (\rm{{e}}^{\alpha }-\rm{{e}}^{-\alpha })-\alpha (\rm{{e}}^{\alpha }-\rm{{e}}^{-\alpha })=\rm{{e}}^{\alpha }+\rm{{e}}^{-\alpha }-\alpha (\rm{{e}}^{\alpha }-\rm{{e}}^{-\alpha })=2\cosh (\alpha )-2(\alpha )\sinh (\alpha )\) to achieve \(1=\cosh (\alpha )-\alpha \sinh (\alpha )\) verified iff \(\alpha =0\). Thus, (28) is verified. Finally, by Remark 9, (9) is verified.

We prove (29).

\(d(F({\bf{I}}_{norm})_1,F({\bf{I}}_{norm})_2)=2-(1-\alpha )\rm{{e}}^{\alpha }-(1+\alpha )\rm{{e}}^{-\alpha }=2-(1-\alpha )\rm{{e}}^{-\alpha }-(1+\alpha )\rm{{e}}^{-\alpha }=d(F({\bf{I}}_{norm})_2,F({\bf{I}}_{norm})_1)\).

We prove (30).

\(d(F({\bf{I}}_{norm})_1,F({\bf{I}}_{norm})_2)=2-(1-\alpha )\rm{{e}}^{\alpha }-(1+\alpha )\rm{{e}}^{-\alpha }\le 2-(1-\beta )\rm{{e}}^{\beta }-(1+\beta )\rm{{e}}^{-\beta }+2-(1-\gamma )\rm{{e}}^{\gamma }-(1+\gamma )\rm{{e}}^{-\gamma }\), from which

$$\begin{aligned} &d(F({\bf{I}}_{norm})_1,F({\bf{I}}_{norm})_2) \le 2-(1+\alpha +\gamma )\rm{{e}}^{-\alpha -\gamma }\nonumber \\&\quad -\,(1-\alpha -\gamma )\rm{{e}}^{\alpha +\gamma }+\nonumber \\&\quad \times\,\gamma \rm{{e}}^{\alpha +\gamma }+2-(1-\gamma )\rm{{e}}^{\gamma }-(1+\gamma )\rm{{e}}^{-\gamma }. \end{aligned}$$
(31)

By (31), we can write \(d(F({\bf{I}}_{norm})_1,F({\bf{I}}_{norm})_2)\le 2-(1+\alpha )^{-\alpha -\gamma }-\gamma \rm{{e}}^{-\alpha -\gamma }-(1-\alpha )\rm{{e}}^{\alpha +\gamma }+\gamma \rm{{e}}^{\alpha +\gamma }+2-(1-\gamma )\rm{{e}}^{\gamma }-(1+\alpha )\rm{{e}}^{-\alpha }.\) Further, by adding and subtracting \((1+\alpha )\rm{{e}}^{-\alpha }+(1-\alpha )\rm{{e}}^{\alpha }\), we obtain

$$\begin{aligned}&d(F({\bf{I}}_{norm})_1,F({\bf{I}}_{norm})_2)\le 2-(1+\alpha )\rm{{e}}^{-\alpha }-(1-\alpha )\rm{{e}}^{\alpha }+(1+\alpha )\rm{{e}}^{-\alpha }+(1-\alpha )\rm{{e}}^{\alpha }-(1+\alpha )\rm{{e}}^{-\alpha )-\gamma }-\gamma \rm{{e}}^{-\alpha -\gamma }-(1-\gamma )\rm{{e}}^{\alpha +\gamma }+\gamma \rm{{e}}^{\alpha +\gamma }+2-(1-\gamma )\rm{{e}}^{\gamma }-(1+\gamma )\rm{{e}}^{-\gamma }. \end{aligned}$$
(32)

In (32), \(2-(1+\alpha )\rm{{e}}^{-\alpha }-(1-\alpha )\rm{{e}}^{\alpha }=d(F({\bf{I}}_{norm})_1,F({\bf{I}}_{norm})_2)\), while \(2-(1-\gamma )\rm{{e}}^{\gamma }-(1+\gamma )\rm{{e}}^{-\gamma }=d(C,A).\) Then, (32) becomes

$$\begin{aligned} d(F({\bf{I}}_{norm})_1,F({\bf{I}}_{norm})_2)&\le\,d(F({\bf{I}}_{norm})_1,F({\bf{I}}_{norm})_2)+(1+\alpha )\rm{{e}}^{-\alpha }\\&\quad+\,(1-\alpha )\rm{{e}}^{\alpha }-(1+\alpha )\rm{{e}}^{-\alpha -\gamma }- \gamma \rm{{e}}^{-\alpha -\gamma }\\&\quad-\,(1-\alpha )\rm{{e}}^{\alpha +\gamma }+d(F({\bf{I}}_{norm})_3,F({\bf{I}}_{norm})_1), \end{aligned}$$
(33)

thereby reducing the problem to show that \((1+\alpha )\rm{{e}}^{-\alpha }+(1-\alpha )\rm{{e}}^{\alpha }-(1+\alpha )\rm{{e}}^{-\alpha -\gamma }-\gamma \rm{{e}}^{-\alpha -\gamma }-(1-\alpha )\rm{{e}}^{\alpha +\gamma }\) in (33) is not negative. However, \((1+\alpha )\rm{{e}}^{-\alpha }\ge 0\) and \((1-\alpha )\rm{{e}}^{\alpha }\ge 0.\). Thus, it remains to be shown that \(-(1+\alpha )\rm{{e}}^{-\alpha -\gamma }-\gamma )\rm{{e}}^{-\alpha -\gamma }-(1-\alpha )\rm{{e}}^{\alpha +\gamma }+\gamma \rm{{e}}^{\alpha +\gamma }\ge 0\). If, absurdly, it were \(-(1+\alpha )\rm{{e}}^{-\alpha -\gamma }-\gamma \rm{{e}}^{-\alpha -\gamma }-(1-\alpha )\rm{{e}}^{\alpha +\gamma }+\gamma \rm{{e}}^{\alpha +\gamma }< 0,\) we would get

$$\begin{aligned} \gamma \rm{{e}}^{\alpha +\gamma } < (1+\alpha )\rm{{e}}^{-\alpha -\gamma }+\gamma \rm{{e}}^{-\alpha -\gamma }+(1-\alpha )\rm{{e}}^{\alpha +\gamma }=(1+\alpha +\gamma )\rm{{e}}^{-\alpha -\gamma }+(1-\alpha )\rm{{e}}^{\alpha +\gamma }, \end{aligned}$$
(34)

from which \(\gamma <(1+\alpha +\gamma )\frac{\rm{{e}}^{-\alpha -\gamma }}{\rm{{e}}^{\alpha +\gamma }}+(1-\alpha ).\) If (34) is always true, it is necessary that \(\sup \{\gamma \}<\inf \{ (1+\alpha +\gamma )\frac{\rm{{e}}^{-\alpha -\gamma }}{\rm{{e}}^{\alpha +\gamma }}+(1-\alpha )\}\). However, \(\inf \{ 1-m_B(b_{ij})-m_A(a_{ij}) \}=0\) for which \(\inf \{1+\alpha +\gamma \}=0\) and \(\sup \{\gamma \}=1\) which results in a false inequality. Thus, (30) is true and, by Remark 9, (11) holds.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Versaci, M., Morabito, F.C. Image Edge Detection: A New Approach Based on Fuzzy Entropy and Fuzzy Divergence. Int. J. Fuzzy Syst. 23, 918–936 (2021). https://doi.org/10.1007/s40815-020-01030-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40815-020-01030-5

Keywords

Search

Navigation