# Application of Structural Entropy and Spatial Filling Factor in Colonoscopy Image Classification

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## Abstract

**:**

## 1. Introduction

## 2. Antecedents

#### 2.1. Average, Standard Deviation, Gradients

#### 2.2. Edge Density

#### 2.3. The Rényi Entropy Based Structural Entropy and the Spatial Filling Factor

#### 2.4. Wavelets

#### 2.5. Initial Set of Antecedents

#### 2.6. Colour Spaces HSV and RGB

## 3. Fuzzy Inference and Interpolation

#### 3.1. Fuzzy Sets and Inference

#### 3.2. Fuzzy Rule Interpolation

## 4. Experimental Setup

#### 4.1. Using All the Initial Antecedents

#### 4.2. Using Only the Rényi Entropy-Based Antecedents in the Rulebase

#### 4.3. Antecedents Resulting in Considerably Differing Histograms for Images with and without Polyps

#### 4.4. Antecedents with Histograms Resulting in Considerable Distance between the Histogram Centres for Images with and without Polyps

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Figures of Histograms and Rules

**Table A1.**The rule membership function parameters for the first 33 HSV antecedents. The following notation was used. The first column, i means the antecedent number, ${x}_{i0min}$ and ${x}_{i0max}$ are the infimum and supremum of the support of the rule corresponding to the non-polyp case, similaly, ${x}_{i1min}$ and ${x}_{i1max}$ are the infimum and supremum of the support of the triangular membership function of the polyp containing tiles. The columns ${x}_{i0core}$ and ${x}_{i1core}$ give the center points, i.e., the $\alpha =1$ $\alpha $-cuts.

i | ${\mathit{x}}_{\mathit{i}0\mathit{min}}$ | ${\mathit{x}}_{\mathit{i}0\mathit{core}}$ | ${\mathit{x}}_{\mathit{i}0\mathit{max}}$ | ${\mathit{x}}_{\mathit{i}1\mathit{min}}$ | ${\mathit{x}}_{\mathit{i}1\mathit{core}}$ | ${\mathit{x}}_{\mathit{i}1\mathit{max}}$ |
---|---|---|---|---|---|---|

1 | 0 | 0.0707 | 0.1414 | 0.0303 | 0.0606 | 0.1212 |

2 | 0 | 0 | 0.2323 | 0 | 0 | 0.0404 |

3 | 0 | 0.5556 | 0.9697 | 0.0909 | 0.5556 | 0.8485 |

4 | 0 | 0.0303 | 0.3434 | 0 | 0.0404 | 0.3939 |

5 | 0.0101 | 0.0707 | 0.9293 | 0.0202 | 0.6263 | 1.0000 |

6 | 0 | 0.0202 | 0.2424 | 0 | 0.0505 | 0.3030 |

7 | 0 | 0.1919 | 0.7576 | 0 | 0.5758 | 0.8283 |

8 | 0 | 0.5859 | 0.8283 | 0.1515 | 0.6465 | 0.8889 |

9 | 0 | 0.0606 | 0.7677 | 0.0303 | 0.3636 | 0.7273 |

10 | 0 | 0 | 0.0101 | 0 | 0 | 0.0101 |

11 | 0.9899 | 1.0000 | 1.0000 | 0.9899 | 1.0000 | 1.0000 |

12 | 0 | 0 | 0.0707 | 0 | 0.0101 | 0.0909 |

13 | 0.9293 | 1.0000 | 1.0000 | 0.9091 | 0.9899 | 1.0000 |

14 | 0 | 0 | 0.0909 | 0 | 0.0101 | 0.1111 |

15 | 0.9091 | 1.0000 | 1.0000 | 0.8990 | 0.9899 | 1.0000 |

16 | 0 | 0.0707 | 0.1414 | 0.0303 | 0.0606 | 0.1212 |

17 | 0 | 0 | 0.2424 | 0 | 0 | 0.0808 |

18 | 0 | 0.5556 | 0.9697 | 0.0909 | 0.5556 | 0.8485 |

19 | 0 | 0.0303 | 0.2929 | 0 | 0.0404 | 0.3939 |

20 | 0.0101 | 0.0707 | 0.9293 | 0.0202 | 0.6263 | 1.0000 |

21 | 0 | 0.0202 | 0.2222 | 0 | 0.0303 | 0.2929 |

22 | 0 | 0.2727 | 0.7677 | 0 | 0.2828 | 0.8485 |

23 | 0 | 0.1313 | 0.8586 | 0.1313 | 0.4949 | 0.8182 |

24 | 0 | 0.1313 | 0.5960 | 0.1212 | 0.2828 | 0.7374 |

25 | 0 | 0 | 0.0101 | 0 | 0 | 0 |

26 | 0.9899 | 1.0000 | 1.0000 | 0.9899 | 1.0000 | 1.0000 |

27 | 0 | 0.0101 | 0.0909 | 0 | 0.0101 | 0.1010 |

28 | 0.8889 | 0.9899 | 1.0000 | 0.8990 | 0.9899 | 1.0000 |

29 | 0 | 0 | 0.1414 | 0 | 0.0101 | 0.2020 |

30 | 0.8586 | 0.9899 | 1.0000 | 0.8485 | 0.9899 | 1.0000 |

31 | 0.2626 | 0.2727 | 0.2828 | 0.3939 | 0.4040 | 0.4242 |

32 | 0 | 0 | 0.0202 | 0 | 0 | 0.0101 |

33 | 0.4343 | 0.4848 | 0.5657 | 0.3636 | 0.4242 | 0.5051 |

**Table A2.**The rule membership function parameters for the second 33 HSV antecedents. The following notation was used. The first column, i means the antecedent number, ${x}_{i0min}$ and ${x}_{i0max}$ are the infimum and supremum of the support of the rule corresponding to the non-polyp case, similarly, ${x}_{i1min}$ and ${x}_{i1max}$ are the infimum and supremum of the support of the triangular membership function of the polyp containing tiles. The columns ${x}_{i0core}$ and ${x}_{i1core}$ give the center points, i.e., the $\alpha =1$ $\alpha $-cuts.

i | ${\mathit{x}}_{\mathit{i}0\mathit{min}}$ | ${\mathit{x}}_{\mathit{i}0\mathit{core}}$ | ${\mathit{x}}_{\mathit{i}0\mathit{max}}$ | ${\mathit{x}}_{\mathit{i}1\mathit{min}}$ | ${\mathit{x}}_{\mathit{i}1\mathit{core}}$ | ${\mathit{x}}_{\mathit{i}1\mathit{max}}$ |
---|---|---|---|---|---|---|

34 | 0 | 0.0101 | 0.3838 | 0 | 0.0404 | 0.3333 |

35 | 0.6061 | 0.7071 | 0.7475 | 0.4747 | 0.6263 | 0.7374 |

36 | 0 | 0.0202 | 0.1313 | 0 | 0.0101 | 0.2626 |

37 | 0 | 0.2424 | 0.7677 | 0 | 0.2525 | 0.8687 |

38 | 0 | 0 | 0.9495 | 0.0707 | 0.5657 | 0.9596 |

39 | 0 | 0 | 0.9192 | 0.1212 | 0.6263 | 0.9293 |

40 | 0 | 0 | 0.0202 | 0 | 0 | 0.0101 |

41 | 0.9798 | 1.0000 | 1.0000 | 0.9899 | 1.0000 | 1.0000 |

42 | 0 | 0 | 0.0606 | 0 | 0 | 0.0606 |

43 | 0.9394 | 1.0000 | 1.0000 | 0.9293 | 1.0000 | 1.0000 |

44 | 0 | 0 | 0.0303 | 0 | 0.0101 | 0.1414 |

45 | 0.9697 | 1.0000 | 1.0000 | 0.8889 | 1.0000 | 1.0000 |

46 | 0.5152 | 0.5354 | 0.5455 | 0.4242 | 0.4343 | 0.4545 |

47 | 0 | 0 | 0.0101 | 0 | 0 | 0.0101 |

48 | 0.2222 | 0.2626 | 0.3030 | 0.4141 | 0.4747 | 0.5354 |

49 | 0 | 0.0101 | 0.2424 | 0 | 0.0202 | 0.2626 |

50 | 0.4141 | 0.5354 | 0.5960 | 0.3737 | 0.5051 | 0.6061 |

51 | 0 | 0.0101 | 0.1212 | 0 | 0.0101 | 0.2323 |

52 | 0 | 0.2424 | 0.8081 | 0 | 0.3838 | 0.9091 |

53 | 0 | 0 | 0.9495 | 0.1717 | 0.6465 | 0.9495 |

54 | 0 | 0.2323 | 0.9293 | 0.2222 | 0.5657 | 0.9293 |

55 | 0 | 0 | 0.0202 | 0 | 0 | 0.0101 |

56 | 0.9798 | 1.0000 | 1.0000 | 0.9899 | 1.0000 | 1.0000 |

57 | 0 | 0 | 0.0707 | 0 | 0 | 0.0505 |

58 | 0.9192 | 1.0000 | 1.0000 | 0.9394 | 1.0000 | 1.0000 |

59 | 0 | 0 | 0.0202 | 0 | 0 | 0.0404 |

60 | 0.9798 | 1.0000 | 1.0000 | 0.9596 | 1.0000 | 1.0000 |

61 | 0.4848 | 0.4949 | 0.4949 | 0.3434 | 0.3434 | 0.3535 |

62 | 0 | 0 | 0.0101 | 0 | 0 | 0.0101 |

63 | 0.4444 | 0.4545 | 0.4747 | 0.4343 | 0.4545 | 0.4848 |

64 | 0 | 0.0101 | 0.1414 | 0 | 0.0101 | 0.1313 |

65 | 0.6061 | 0.6263 | 0.6465 | 0.6667 | 0.6970 | 0.7273 |

66 | 0 | 0.0101 | 0.1010 | 0 | 0.0101 | 0.1515 |

**Table A3.**The rule membership function parameters for the thirs 33 HSV antecedents. The following notation was used. The first column, i means the antecedent number, ${x}_{i0min}$ and ${x}_{i0max}$ are the infimum and supremum of the support of the rule corresponding to the non-polyp case, similarly, ${x}_{i1min}$ and ${x}_{i1max}$ are the infimum and supremum of the support of the triangular membership function of the polyp containing tiles. The columns ${x}_{i0core}$ and ${x}_{i1core}$ give the center points, i.e., the $\alpha =1$ $\alpha $-cuts.

i | ${\mathit{x}}_{\mathit{i}0\mathit{min}}$ | ${\mathit{x}}_{\mathit{i}0\mathit{core}}$ | ${\mathit{x}}_{\mathit{i}0\mathit{max}}$ | ${\mathit{x}}_{\mathit{i}1\mathit{min}}$ | ${\mathit{x}}_{\mathit{i}1\mathit{core}}$ | ${\mathit{x}}_{\mathit{i}1\mathit{max}}$ |
---|---|---|---|---|---|---|

67 | 0 | 0 | 0.7475 | 0 | 0 | 0.8283 |

68 | 0 | 0 | 0.9495 | 0.1010 | 0.5152 | 0.9596 |

69 | 0 | 0 | 0.9596 | 0.1111 | 0.6263 | 0.9596 |

70 | 0 | 0 | 0.0202 | 0 | 0 | 0.0202 |

71 | 0.9798 | 1.0000 | 1.0000 | 0.9798 | 1.0000 | 1.0000 |

72 | 0 | 0 | 0.0404 | 0 | 0 | 0.0303 |

73 | 0.9596 | 1.0000 | 1.0000 | 0.9697 | 1.0000 | 1.0000 |

74 | 0 | 0 | 0.0101 | 0 | 0 | 0.0404 |

75 | 0.9899 | 1.0000 | 1.0000 | 0.9596 | 1.0000 | 1.0000 |

76 | 0 | 0 | 0.1313 | 0 | 0 | 0.0707 |

77 | 0 | 0 | 0.0202 | 0 | 0 | 0.0101 |

78 | 0 | 0.1717 | 0.4242 | 0.0505 | 0.2020 | 0.5051 |

79 | 0 | 0.0101 | 0.3535 | 0 | 0.0404 | 0.5556 |

80 | 0 | 0.0606 | 0.2323 | 0.0505 | 0.1212 | 0.5051 |

81 | 0 | 0.0101 | 0.1515 | 0 | 0.0101 | 0.3636 |

82 | 0.6162 | 0.7980 | 0.8182 | 0.4646 | 0.6970 | 0.7980 |

83 | 0 | 0.1717 | 0.5556 | 0.0606 | 0.1111 | 0.4545 |

84 | 0.5657 | 0.6465 | 0.6970 | 0.3535 | 0.5051 | 0.6667 |

85 | 0 | 0.1616 | 0.5859 | 0.0505 | 0.1111 | 0.4141 |

86 | 0.1313 | 0.7071 | 0.8384 | 0.1313 | 0.7273 | 0.8788 |

87 | 0 | 0.1717 | 0.5960 | 0.0505 | 0.1414 | 0.5152 |

88 | 0.4747 | 0.4949 | 0.5253 | 0.5354 | 0.5556 | 0.5758 |

89 | 0 | 0 | 0.1919 | 0 | 0 | 0.0101 |

90 | 0.4141 | 0.4646 | 0.5152 | 0.3535 | 0.4343 | 0.5051 |

91 | 0 | 0.0202 | 0.3535 | 0 | 0.0303 | 0.3333 |

92 | 0.4242 | 0.4949 | 0.6162 | 0.4141 | 0.5253 | 0.6667 |

93 | 0 | 0.0202 | 0.1818 | 0 | 0.0101 | 0.3030 |

94 | 0.5253 | 0.5455 | 0.6061 | 0.7172 | 0.7475 | 0.7778 |

95 | 0 | 0 | 0.0202 | 0 | 0 | 0.0202 |

96 | 0.3434 | 0.4040 | 0.4747 | 0.3535 | 0.4242 | 0.5152 |

97 | 0 | 0.0202 | 0.4343 | 0 | 0.0505 | 0.4545 |

98 | 0.3838 | 0.4444 | 0.5657 | 0.3636 | 0.4444 | 0.6566 |

99 | 0 | 0.0101 | 0.1414 | 0.0101 | 0.0202 | 0.3333 |

**Figure A1.**The measured histograms of the HSV antecedent parameters. The histograms are normalized in both directions. The measured values are represented by black circles in the case of the tiles with polyp (positive case), while blue asterisk means the number of tiles without polyp (negative case). The rules are also plotted for the histogram based triangular membership functions with the same colour (black for positive and blue for negative), just dashed lines were used, while the skirted triangle and half Gaussian histogram fitted rules were shown in magenta colour in the case of the tiles with polyp, and cyan for the polypless tiles.

**Figure A2.**The measured histograms of the HSV antecedent parameters. The histograms are normalized in both directions. The measured values are represented by black circles in the case of the tiles with polyp (positive case), while blue asterisk means the number of tiles without polyp (negative case). The rules are also plotted for the histogram based triangular membership functions with the same colour (black for positive and blue for negative), just dashed lines were used, while the skirted triangle and half Gaussian histogram fitted rules were shown in magenta colour in the case of the tiles with polyp, and cyan for the polypless tiles.

**Figure A3.**The measured histograms of the HSV antecedent parameters. The histograms are normalized in both directions. The measured values are represented by black circles in the case of the tiles with polyp (positive case), while blue asterisk means the number of tiles without polyp (negative case). The rules are also plotted for the histogram based triangular membership functions with the same colour (black for positive and blue for negative), just dashed lines were used, while the skirted triangle and half Gaussian histogram fitted rules were shown in magenta colour in the case of the tiles with polyp, and cyan for the polypless tiles.

**Figure A4.**The measured histograms of the HSV antecedent parameters. The histograms are normalized in both directions. The measured values are represented by black circles in the case of the tiles with polyp (positive case), while blue asterisk means the number of tiles without polyp (negative case). The rules are also plotted for the histogram based triangular membership functions with the same colour (black for positive and blue for negative), just dashed lines were used, while the skirted triangle and half Gaussian histogram fitted rules were shown in magenta colour in the case of the tiles with polyp, and cyan for the polypless tiles.

**Figure A5.**The measured histograms of the HSV antecedent parameters. The histograms are normalized in both directions. The measured values are represented by black circles in the case of the tiles with polyp (positive case), while blue asterisk means the number of tiles without polyp (negative case). The rules are also plotted for the histogram based triangular membership functions with the same colour (black for positive and blue for negative), just dashed lines were used, while the skirted triangle and half Gaussian histogram fitted rules were shown in magenta colour in the case of the tiles with polyp, and cyan for the polypless tiles.

**Figure A6.**The measured histograms of the RGB antecedent parameters. The histograms are normalized in both directions. The measured values are represented by black circles in the case of the tiles with polyp (positive case), while blue asterisk means the number of tiles without polyp (negative case). The rules are also plotted for the histogram based triangular membership functions with the same colour (black for positive and blue for negative), just dashed lines were used, while the skirted triangle and half Gaussian histogram fitted rules were shown in magenta colour in the case of the tiles with polyp, and cyan for the polypless tiles.

**Figure A7.**The measured histograms of the RGB antecedent parameters. The histograms are normalized in both directions. The measured values are represented by black circles in the case of the tiles with polyp (positive case), while blue asterisk means the number of tiles without polyp (negative case). The rules are also plotted for the histogram based triangular membership functions with the same colour (black for positive and blue for negative), just dashed lines were used, while the skirted triangle and half Gaussian histogram fitted rules were shown in magenta colour in the case of the tiles with polyp, and cyan for the polypless tiles.

**Figure A8.**The measured histograms of the RGB antecedent parameters. The histograms are normalized in both directions. The measured values are represented by black circles in the case of the tiles with polyp (positive case), while blue asterisk means the number of tiles without polyp (negative case). The rules are also plotted for the histogram based triangular membership functions with the same colour (black for positive and blue for negative), just dashed lines were used, while the skirted triangle and half Gaussian histogram fitted rules were shown in magenta colour in the case of the tiles with polyp, and cyan for the polypless tiles.

**Figure A9.**The measured histograms of the RGB antecedent parameters. The histograms are normalized in both directions. The measured values are represented by black circles in the case of the tiles with polyp (positive case), while blue asterisk means the number of tiles without polyp (negative case). The rules are also plotted for the histogram based triangular membership functions with the same colour (black for positive and blue for negative), just dashed lines were used, while the skirted triangle and half Gaussian histogram fitted rules were shown in magenta colour in the case of the tiles with polyp, and cyan for the polypless tiles.

**Figure A10.**The measured histograms of the RGB antecedent parameters. The histograms are normalized in both directions. The measured values are represented by black circles in the case of the tiles with polyp (positive case), while blue asterisk means the number of tiles without polyp (negative case). The rules are also plotted for the histogram based triangular membership functions with the same colour (black for positive and blue for negative), just dashed lines were used, while the skirted triangle and half Gaussian histogram fitted rules were shown in magenta colour in the case of the tiles with polyp, and cyan for the polypless tiles.

## References

- Bosman, F.T. Chapter 5.5: Colorectal Cancer. In World Cancer Report the International Agency for Research on Cancer; Stewart, B.W., Wild, C.P., Eds.; World Health Organization: Geneva, Switzerland, 2014; p. 392402. ISBN 978-92-832-0443-5. [Google Scholar]
- Fenlon, H.M.; Nunes, D.P.; Schroy, P.C.; Barish, M.A.; Clarke, P.D.; Ferrucci, J.T. A Comparison of Virtual and Conventional Colonoscopy for the Detection of Colorectal Polyps. N. Engl. J. Med.
**1999**, 341, 1496–1503. [Google Scholar] [CrossRef] - Pickhardt, P.J.; Choi, J.R.; Hwang, I.; Butler, J.A.; Puckett, M.L.; Hildebrandt, H.A.; Wong, R.K.; Nugent, P.A.; Mysliwiec, P.A.; Schindler, W.R. Computed Tomographic Virtual Colonoscopy to Screen for Colorectal Neoplasia in Asymptomatic Adults. N. Engl. J. Med.
**2003**, 349, 2191–2200. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Menardo, G. Sensitivity of diagnostic examinations for colorectal polyps. Tech. Coloproctol.
**2004**, 8 (Suppl. S2), S273–S275. [Google Scholar] [CrossRef] - Tischendorf, J.J.W.; Wasmuth, H.E.; Koch, A.; Hecker, H.; Trautwein, C.; Winograd, R. Value of magnifying chromoendoscopy and narrow band imaging (NBI) in classifying colorectal polyps: A prospective controlled study. Endoscopy
**2007**, 39, 1092–1096. [Google Scholar] [CrossRef] [PubMed] - Song, H.J.; Shim, K.N. Current status and future perspectives of capsule endoskopy. Intest. Res.
**2016**, 14, 21–29. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Enns, R.A.; Hookey, L.; Armstrong, D.; Bernstein, C.N.; Heitman, S.J.; Teshima, C.; Leontiadis, G.I.; Tse, F.; Sadowski, D. Clinical practice guidelines for the use of video capsule endoscopy. Gastroenterology
**2017**, 152, 497–514. [Google Scholar] [CrossRef] [Green Version] - Søreide, K.; Nedrebø, B.S.; Reite, A.; Thorsen, K.; Kørner, H. Endoscopy Morphology, Morphometry and Molecular Markers: Predicting Cancer Risk in Colorectal Adenoma. Expert Rev. Mol. Diagn.
**2009**, 9, 125137. [Google Scholar] [CrossRef] - Jass, J.R. Classication of colorectal cancer based on correlation of clinical, morphological and molecular features. Histopathology
**2006**, 50, 113130. [Google Scholar] - Kudo, S.; Hirota, S.; Nakajima, T.; Hosobe, S.; Kusaka, H.; Kobayashi, T.; Himori, M.; Yagyuu, A. Colorectal tumours and pit pattern. J. Clin. Pathol.
**1994**, 47, 880–885. [Google Scholar] [CrossRef] [Green Version] - Kudo, S.; Tamura, S.; Nakajima, T.; Yamano, H.O.; Kusaka, H.; Watanabe, H. Diagnosis of colorectal tumorous lesions by magnifying endoscopy. Gastrointest. Endosc.
**1996**, 44, 8–14. [Google Scholar] [CrossRef] - Kudo, S.; Rubio, C.A.; Teixeira, C.R.; Kashida, H.; Kogure, E. Pit pattern in colorectal neoplasia: Endoscopic magnifying view. Endoscopy
**2001**, 33, 367–373. [Google Scholar] [CrossRef] [PubMed] - Bernal, J.; Sánchez, F.J.; Vilariño, F. Towards Automatic Polyp Detection with a Polyp Appearance Model. Pattern Recognit
**2012**, 45, 3166–3182. [Google Scholar] [CrossRef] - Bernal, J.; Tajkbaksh, N.; Sánchez, F.J.; Matuszewski, B.; Chen, H.; Yu, L.; Angermann, Q.; Romain, O.; Rustad, B. Comparative Validation of Polyp Detection Methods in Video Colonoscopy: Results from the MICCAI 2015 Endoscopic Vision Challenge. IEEE Trans. Med. Imaging
**2017**, 36, 1231–1249. [Google Scholar] [CrossRef] [PubMed] - Zadeh, L.A. Fuzzy sets. Inf. Control
**1965**, 8, 338–353. [Google Scholar] [CrossRef] [Green Version] - Zadeh, L.A. Fuzzy algorithms. Inf. Control
**1968**, 12, 94–102. [Google Scholar] [CrossRef] [Green Version] - Kóczy, L.T.; Hirota, K. Approximate reasoning by linear rule interpolation and general approximation. Int. J. Approx. Reason.
**1993**, 9, 197–225. [Google Scholar] [CrossRef] [Green Version] - Mamdani, E.H.; Assilian, S. An experiment in linguistic synthesis with a fuzzy logic controller. Int. J. Man-Mach. Stud.
**1975**, 7, 1–13. [Google Scholar] [CrossRef] - Nagy, S.Z.; Sziová, B.; Kóczy, L.T. The effect of image feature qualifiers on fuzzy colorectal polyp detection schemes using KH interpolation—Towards hierarchical fuzzy classification of coloscopic still images. In Proceedings of the FuzzIEEE 2018, Rio de Janeiro, Brazil, 8–13 July 2018; pp. 1–7. [Google Scholar]
- Nagy, S.Z.; Lilik, F.; Kóczy, L.T. Applicability of various wavelet families in fuzzy classification of access networks’ telecommunication lines. In Proceedings of the 2017 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), Naples, Italy, 9–12 July 2017. [Google Scholar]
- Nagy, S.Z.; Sziová, B.; Kóczy, L.T. The effect of wavelet analyis on entropy based fuzzy classification of colonoscopy images. In Proceedings of the 5th International Workshop on Advanced Computational Intelligence and Intelligent Informatics (IWACIII 2017), Beijing, China, 2–5 November 2017. [Google Scholar]
- Sziová, B.; Nagy, S.Z.; Kóczy, L.T. The Effects of Preprocessing on Colorectal Polyp Detecting by Fuzzy Algorithm. In Recent Developments and the New Direction in Soft-Computing Foundations and Applications; Springer: Cham, Switzerland, 2018; pp. 347–357. [Google Scholar]
- Bernal, J.; Sánchez, F.; Esparrach, G.; Gil, D.; Rodríguez, C.; Vilariño, F. WM-DOVA maps for accurate polyp highlighting in colonoscopy: Validation vs. saliency maps from physicians. Comput. Med. Imaging Graph.
**2015**, 43, 99–111. [Google Scholar] [CrossRef] - Silva, J.S.; Histace, A.; Romain, O.; Dray, X.; Grando, B. Towards embedded detection of polyps in WCE images for early diagnosis of colorectal cancer. Int. J. Comput. Assist. Radiol. Surg.
**2014**, 9, 283–293. [Google Scholar] [CrossRef] [PubMed] - Yuji, I.; Akira, H.; Yoshinori, A.; Bhuyan, M.; Robert, J.; Kunio, K. Automatic Detection of Polyp Using Hessian Filter and HOG Features. Procedia Comput. Sci.
**2015**, 60, 730–739. [Google Scholar] - Soumelidis, A.; Fazekas, Z.; Schipp, F. Geometrical description of quasi-hemispherical and calotte-like surfaces using discretised argument-transformed Chebyshev-polynomials. In Proceedings of the 44th IEEE Conference on Decision and Control, Seville, Spain, 15 December 2005. [Google Scholar]
- Wu, F.; KinTak, U. Low-Light image enhancement algorithm based on HSI color space. In Proceedings of the 10th International Congress on Image and Signal Processing, BioMedical Engineering and Informatics (CISP-BMEI), Shanghai, China, 14–16 October 2017. [Google Scholar] [CrossRef]
- Canny, J. A Computational Approach to Edge Detection. IEEE Trans. Pattern Anal. Mach. Intell.
**1986**, PAMI-8, 679–698. [Google Scholar] [CrossRef] - Nagy, S.Z.; Sziová, B.; Pipek, J. On Structural Entropy and Spatial Filling Factor Analysis of Colonoscopy Pictures. Entropy
**2019**, 21, 256. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Von Neumann, J. Thermodynamik quantenmechanischer Gesamtheiten. Nachr. Ges. Wiss. Gött. Math.-Phys. Kl.
**1927**, 102, 273–291. [Google Scholar] - Shannon, C.E. A mathematic theory of communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] [Green Version] - Rényi, A. On measures of information and entropy. In Proceedings of the fourth Berkeley Symposium on Mathematics, Statistics and Probability, Berkeley, CA, USA, 20 June–30 July 1960; pp. 547–561. [Google Scholar]
- Amigó, J.M.; Balogh, S.G.; Hernández, S. A Brief Review of Generalized Entropies. Entropy
**2018**, 20, 813. [Google Scholar] [CrossRef] [Green Version] - Stantchev, I. Structural Entropy: A New Approach for Systems Structure’s Analysis. In Cybernetics and Systems ’86; Trappl, R., Ed.; Springer: Dordrecht, The Netherlands, 1986; pp. 139–186. [Google Scholar] [CrossRef]
- Mojzes, I.; Dominkovics, C.S.; Harsányi, G.; Nagy, S.Z.; Pipek, J.; Dobos, L. Heat treatment parameters effecting the fractal dimensions of AuGe metallization on GaAs. Appl. Phys. Lett.
**2007**, 91, 073107. [Google Scholar] [CrossRef] [Green Version] - Pipek, J.; Varga, I. Universal classification scheme for the spatial-localization properties of one-particle states in finite, d-dimensional systems. Phys. Rev. A
**1992**, 46, 3148–3163. [Google Scholar] [CrossRef] [PubMed] - Varga, I.; Pipek, J. Rényi entropies characterizing the shape and the extension of the phase space representation of quantum wave functions in disordered systems. Phys. Rev. E
**2003**, 68, 026202. [Google Scholar] [CrossRef] [Green Version] - Bonyár, A.; Molnár, L.M.; Harsányi, G. Localization factor: A new parameter for the quantitative characterization of surface structure with atomic force microscopy (AFM). Micron
**2012**, 43, 305–310. [Google Scholar] [CrossRef] - Bonyár, A. AFM characterization of the shape of surface structures with localization factor. Micron
**2016**, 87, 1–9. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Fodor, J.C. A remark on constructing t-norms. Fuzzy Sets Syst.
**1991**, 41, 195–199. [Google Scholar] [CrossRef] - Weber, S. A general concept of fuzzy connectives, negations and implications based on t-norms and t-conorms. Fuzzy Sets Syst.
**1983**, 11, 115–134. [Google Scholar] [CrossRef] - Sziová, B.; Ismail, R.; Lilik, F.; Kóczy, L.T.; Nagy, S.Z. Fuzzy rulebase parameter determination for stabilized KH interpolation based detection of colorectal polyps on colonoscopy images. In Proceedings of the 2020 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), Glasgow, UK, 19–24 July 2020; pp. 1–6. [Google Scholar] [CrossRef]
- Kóczy, L.T.; Hirota, K. Interpolative reasoning with insufficient evidence in sparse fuzzy rule bases. Inf. Sci.
**1993**, 71, 169–201. [Google Scholar] [CrossRef] - Tikk, D.; Joó, I.; Kóczy, L.T.; Várlaki, P.; Moser, B.; Gedeon, T.D. Stability of interpolative fuzzy KH-controllers. Fuzzy Sets Syst.
**2002**, 125, 105–119. [Google Scholar] [CrossRef] - Takacs, O.; Varkonyi-Koczy, A.R. SVD-based complexity reduction of rule-bases with nonlinear antecedent fuzzy sets. IEEE Trans. Instrum. Meas.
**2002**, 51, 217–221. [Google Scholar] [CrossRef] - Brandao, P.; Zisimopoulos, O.; Mazomenos, E.; Ciuti, G.; Bernal, J.; Visentini-Scarzanella, M.; Menciassi, A.; Dario, P.; Koulaouzidis, A.; Arezzo, A.; et al. Towards a computed-aided diagnosis system in colonoscopy: Automatic polyp segmentation using convolution neural networks. J. Med. Robot. Res.
**2021**, 3, 1–10. [Google Scholar] [CrossRef] [Green Version]

**Figure 2.**Structural entropy (${S}_{Str}={S}_{1}-{S}_{2}$) vs. spatial filling factor (−$logq={S}_{0}-{S}_{2}$) for various types of probability distributions. The thick solid line shows the theoretical limit: all the distributions have to be below this line, i.e., ${S}_{Str}\le logq$. The other curves show some examples of trend lines.

**Figure 3.**A picture presented in greyscale from database [24], and its respective wavelet transforms. In this case only the LH wavelet transformed version carries perceptable information.

**Figure 4.**Graphical representation of the Mamadani-Assilian type inference. The antecedents are denoted by ${A}_{ij}$, their actual measured value by ${x}_{i}$, their membership value in the rule corresponding to the consequent ${B}_{j}$ with ${w}_{ij}$. The defuzzified decision is shown as ${B}^{*}$.

**Figure 5.**The histograms of a training set of measured data. Black circles mark the no polyp results, while the blue circles the tiles with polyp. The rulebases resulting from all three of the methods are plotted. The black and blue continious poly-lines–used as above–denote the mean centered rules, and the dashed lines the histogram based rules. Also, for the histogram based rules the Gaussian and half Gaussian membership functions with extended support are plotted with cyan and magenta colours. The half Gaussian extended rule membership function consists of two half Gaussians fitted to the asymmetrical triangle membership functions.

**Figure 6.**Workflow chart. The first two boxes represent the development of the method, the boxes with italic type letters represent those parts of method which can be automatically performed without human interaction, the others need human control or in the case of the detail medical analysis it should be performed and overseen by human medical experts.

**Table 1.**The indices for the initial antecedent parameters. The indices are given for both the RGB and the HSV colour representations, e.g., antecedents 1 and 2 are the mean and standard deviation of the first colour channel, i.e., of the red (R) and of the hue (H) channel, respectively. The wavelet transformed images are denoted by their respective low-pass and high-pass filters, i.e., LL, LH, HL, and HH.

Index | Antecedent Name |
---|---|

1–2 | mean and standard deviation, R/H |

3–4 | mean and standard deviation, G/S |

5–6 | mean and standard deviation, B/V |

7 | edge density, R/H |

8 | edge density, G/S |

9 | edge density, B/V |

10–11 | ${S}_{str}$, $lnq$, R/H |

12–13 | ${S}_{str}$, $lnq$, G/S |

14–15 | ${S}_{str}$, $lnq$, B/V |

16–30 | similar to 1–15, wavelet transform LL |

31–45 | similar to 1–15, wavelet transform LH |

46–60 | similar to 1–15, wavelet transform HL |

61–75 | similar to 1–15, wavelet transform HH |

76–77 | gradient magnitude’s mean and standard deviation, R/H |

78–79 | gradient magnitude’s mean and standard deviation, G/S |

80–81 | gradient magnitude’s mean and standard deviation, B/V |

82–87 | similar to 76–81, gradient direction |

88–93 | similar to 76–81, gradient x component |

94–99 | similar to 76–81, gradient y component |

**Table 2.**Antecedents where this total distance was larger than $0.2$ in decreasing order of the total distance.

51, 4, 61, 19, 85, 29, 73, 27, 35, 34, 32, 82, 97, 43, 72, 64, 13, 12, 94, 28, 36, 99, 78, 49, 84, 91, |

31, 48, 80, 63, 79, 45, 74, 75, 44 |

12, 28, 36, 44, 45, 49, 74, 75, 78, 79, 80, 91, 99 |

**Table 4.**Antecedents, where the centers of the histograms for the two consequents’ training set were fairly distant from each other.

2, 3, 5, 10, 11, 12, 13, 17, 18, 20, 25, 26, 27, 28, 37, 38, 39, 52, |

53, 54, 62, 63, 64, 68, 69, 72, 73, 77, 79, 83, 85, 86, 89, 91, 95, 97 |

**Table 5.**The antecedents of the reduced set. The colour channels were denoted the following way: H: hue, S: saturation, V: value or intensity. The last group containts the gradients. As to different cases were studied: one, which included these gradients, the other which did not, the last row was denoted by *.

Antecedent Group | Antecedent |
---|---|

Original tile | standard deviation H mean of S and V structural entropy of H and S $lnq$ of H and S |

Low-pass–low-pass wavelet transform | edge density of H, S and V |

Low-pass–high-pass wavelet transform | edge density of H, S and V |

High-pass–low-pass wavelet transform | edge density of H, S and V |

High-pass–high-pass wavelet transform | standard deviation of H and S edge densities of S and V structural entropies of S |

(Gradients) * | (standard deviation of H and S) |

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**MDPI and ACS Style**

Sziová, B.; Nagy, S.; Fazekas, Z.
Application of Structural Entropy and Spatial Filling Factor in Colonoscopy Image Classification. *Entropy* **2021**, *23*, 936.
https://doi.org/10.3390/e23080936

**AMA Style**

Sziová B, Nagy S, Fazekas Z.
Application of Structural Entropy and Spatial Filling Factor in Colonoscopy Image Classification. *Entropy*. 2021; 23(8):936.
https://doi.org/10.3390/e23080936

**Chicago/Turabian Style**

Sziová, Brigita, Szilvia Nagy, and Zoltán Fazekas.
2021. "Application of Structural Entropy and Spatial Filling Factor in Colonoscopy Image Classification" *Entropy* 23, no. 8: 936.
https://doi.org/10.3390/e23080936