Global Image Thresholding Adaptive NeuroFuzzy Inference System Trained with Fuzzy Inclusion and Entropy Measures
Abstract
:1. Introduction
2. ANFIS, Global Thresholding Techniques and the Main Parts of Our Previous Research
2.1. Global Thresholding Methods and Neural Networks in Image Segmentation
2.2. Fuzzy Subsethood and Entropy Measures
$S(A,B)=1$if and only if$A\subseteq B$in Zadeh’s sense,  (S1) 
If$P\subseteq A$in Zadeh’s sense, then$S(A,{A}^{c})=0\iff A=X$,  (S2) 
If$B\subseteq {A}_{1}\subseteq {A}_{2}$, then$S({A}_{1},B)\ge $$S({A}_{2},B)$,  (S3) 
and if${B}_{1}\subseteq {B}_{2}$, then$S(A,{B}_{1})\le S(A,{B}_{2}).$ 
If $P\subseteq {A}_{1}\subseteq {A}_{2}$, then $S({A}_{1},{A}_{1}^{c})\ge $ $S({A}_{2},{A}_{2}^{c})$  (S3${}^{\prime}$) 
and if ${B}_{1}\subseteq {B}_{2}$, then $S(A,{B}_{1})\le S(A,{B}_{2})$ 
2.3. Global and Local Thresholding Using Fuzzy Inclusion and Entropy Measures
Algorithm 1: (Global) 

 Group 1 N with ${s}_{1}\le {s}_{2}$ and ${s}_{1}{s}_{2}>0.75$ (very dark),
 Group 2 N with ${s}_{1}\le {s}_{2}$ and $0.5<{s}_{1}{s}_{2}\le 0.75,$
 Group 3 N with ${s}_{1}\le {s}_{2}$ and $0.25<{s}_{1}{s}_{2}\le 0.5,$
 Group 4 N with ${s}_{1}\le {s}_{2}$ and ${s}_{1}{s}_{2}\le 0.25,$
 Group 5 N with ${s}_{1}>{s}_{2}$ and ${s}_{1}{s}_{2}>0.75,$ (very bright)
 Group 6 N with ${s}_{1}>{s}_{2}$ and $0.5<{s}_{1}{s}_{2}\le 0.75,$
 Group 7 N with ${s}_{1}>{s}_{2}$ and $0.25<{s}_{1}{s}_{2}\le 0.5,$
 Group 8 N with ${s}_{1}>{s}_{2}$ and ${s}_{1}{s}_{2}\le 0.25.$
Algorithm 2: (Local) 

2.4. Why ANFIS
3. First Global Thresholding ANFIS
3.1. Data Construction and Initial Evaluation
3.2. G(lobal) ANFIS 1—Adjusting Otsu’s Targets
4. Second Global Thresholding ANFIS—Experimenting on Public Databases
4.1. Public Databases Which We Used
4.2. Data Set Construction of G(lobal) ANFIS 2
4.3. Testing of G ANFIS 2
 211 cases where ${J}_{g}(GT,{B}_{2})>{J}_{g}(GT,{B}_{1})$ (the difference ${J}_{g}(GT,{B}_{2}){J}_{g}(GT,{B}_{1})$ varied from 0.76 to 0.01).
 24 cases where ${J}_{g}(GT,{B}_{2})<{J}_{g}(GT,{B}_{1})$ (the difference ${J}_{g}(GT,{B}_{1}){J}_{g}(GT,{B}_{2})$ varied from 0.17 to 0.01).
 6 cases where the difference of ${J}_{g}(GT,{B}_{1})$ and ${J}_{g}(GT,{B}_{2})$ was zero.
 202 cases where $SSI(GT,{B}_{2})>SSI(GT,{B}_{1})$ (the difference $SSI(GT,{B}_{2})SSI(GT,{B}_{1})$ varied from 0.53 to 0.01),
 20 cases where $SSI(GT,{B}_{2})<SSI(GT,{B}_{1})$ (the difference $SSI(GT,{B}_{1})SSI(GT,{B}_{2})$ varied from 0.11 to 0.01),
 19 cases where the difference of $SSI(GT,{B}_{1})$ and $SSI(GT,{B}_{2})$ was zero.
5. Summary and Some Further Remarks
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Abbreviations
ANN  Artificial Neural Network 
ANFIS  Adaptive NeuroFuzzy Inference System 
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Lines of data matrix 1 used as testing samples  

2  5  7  11  14  18  20  21  25  33 
34  35  36  41  43  50  63  65  87  89 
101  111  115  123  124  128  132  137  142  152 
155  177  178  184  190  193  201  204  209  210 
213  220  222  229  231  238  239  240  247  251 
Sample  ${\mathbf{J}}_{\mathbf{g}}(\mathbf{GT},{\mathbf{B}}_{\mathbf{2}})$  ${\mathbf{J}}_{\mathbf{g}}(\mathbf{GT},{\mathbf{B}}_{\mathbf{1}})$  Difference  Sample  ${\mathbf{J}}_{\mathbf{g}}(\mathbf{GT},{\mathbf{B}}_{\mathbf{2}})$  ${\mathbf{J}}_{\mathbf{g}}(\mathbf{GT},{\mathbf{B}}_{\mathbf{1}})$  Difference 

292066  0.87  0.11  0.76  90076  0.83  0.09  0.74 
21077  0.81  0.14  0.67  135037  0.91  0.29  0.62 
27059  0.72  0.13  0.59  108041  0.86  0.30  0.56 
188063  0.90  0.39  0.51  106020  0.68  0.18  0.50 
101087  0.82  0.33  0.49  109053  0.75  0.26  0.49 
69020  0.88  0.39  0.48  46076  0.84  0.37  0.47 
20008  0.67  0.21  0.46  385028  0.71  0.25  0.46 
216066  0.81  0.36  0.45  100075  0.76  0.30  0.46 
368078  0.68  0.25  0.43  103070  0.70  0.28  0.42 
274007  0.74  0.33  0.41  55075  0.83  0.42  0.41 
155060  0.72  0.31  0.41  216053  0.72  0.31  0.41 
38082  0.86  0.47  0.39  187003  0.64  0.25  0.39 
163085  0.77  0.38  0.39  23080  0.76  0.37  0.39 
229036  0.68  0.30  0.38  138078  0.73  0.35  0.38 
22013  0.62  0.25  0.37  183055  0.74  0.38  0.36 
227046  0.70  0.33  0.37  216081  0.63  0.27  0.36 
65132  0.72  0.36  0.36  41069  0.81  0.46  0.35 
45077  0.72  0.36  0.36  105025  0.86  0.51  0.35 
159008  0.72  0.37  0.35  178054  0.88  0.54  0.34 
376043  0.60  0.26  0.34  69040  0.75  0.42  0.33 
…  …  …  …  …  …  …  … 
227092  0.54  0.69  −0.15  198023  0.12  0.29  −0.17 
Sample  $\mathbf{SSI}(\mathbf{GT},{\mathbf{B}}_{\mathbf{2}})$  $\mathbf{SSI}(\mathbf{GT},{\mathbf{B}}_{\mathbf{1}})$  Difference  Sample  $\mathbf{SSI}(\mathbf{GT},{\mathbf{B}}_{\mathbf{2}})$  $\mathbf{SSI}(\mathbf{GT},{\mathbf{B}}_{\mathbf{1}})$  Difference 

108041  0.59  0.06  0.53  90076  0.58  0.06  0.52 
135037  0.65  0.18  0.47  188063  0.61  0.16  0.45 
61060  0.82  0.42  0.40  21077  0.43  0.06  0.37 
27059  0.39  0.04  0.35  100075  0.44  0.09  0.35 
69020  0.41  0.07  0.34  292066  0.40  0.07  0.33 
101087  0.57  0.25  0.32  65132  0.42  0.10  0.32 
38082  0.42  0.10  0.32  46076  0.56  0.25  0.31 
105025  0.53  0.22  0.31  55075  0.53  0.24  0.29 
94079  0.42  0.15  0.27  138032  0.33  0.06  0.27 
156065  0.40  0.13  0.27  368078  0.35  0.08  0.27 
163085  0.37  0.11  0.26  239096  0.42  0.15  0.27 
106020  0.32  0.06  0.26  385028  0.33  0.08  0.25 
159008  0.42  0.16  0.26  48055  0.57  0.32  0.25 
109053  0.30  0.05  0.25  130026  0.31  0.06  0.25 
41069  0.29  0.05  0.24  103070  0.35  0.10  0.25 
176039  0.40  0.16  0.24  187003  0.30  0.07  0.23 
225017  0.29  0.06  0.23  254054  0.30  0.07  0.23 
143090  0.46  0.23  0.23  216053  0.38  0.16  0.22 
62096  0.44  0.23  0.21  183055  0.40  0.18  0.22 
178054  0.49  0.28  0.21  24063  0.55  0.35  0.20 
…  …  …  …  …  …  …  … 
227092  0.27  0.38  −0.11  151087  0.39  0.50  −0.11 
© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
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Bogiatzis, A.; Papadopoulos, B. Global Image Thresholding Adaptive NeuroFuzzy Inference System Trained with Fuzzy Inclusion and Entropy Measures. Symmetry 2019, 11, 286. https://doi.org/10.3390/sym11020286
Bogiatzis A, Papadopoulos B. Global Image Thresholding Adaptive NeuroFuzzy Inference System Trained with Fuzzy Inclusion and Entropy Measures. Symmetry. 2019; 11(2):286. https://doi.org/10.3390/sym11020286
Chicago/Turabian StyleBogiatzis, Athanasios, and Basil Papadopoulos. 2019. "Global Image Thresholding Adaptive NeuroFuzzy Inference System Trained with Fuzzy Inclusion and Entropy Measures" Symmetry 11, no. 2: 286. https://doi.org/10.3390/sym11020286