# Application of Two-Dimensional Entropy Measures to Detect the Radiographic Signs of Tooth Resorption and Hypercementosis in an Equine Model

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Horses

#### 2.2. Radiographs Classification

#### 2.3. Digital Radiograph Processing

#### 2.3.1. Filtering

#### 2.3.2. Extraction of the Entropy–Based Measures

#### 2.4. Statistical Analysis

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Dixon, P.M.; Dacre, I. A review of equine dental disorders. Vet. J.
**2005**, 169, 165–187. [Google Scholar] [CrossRef] [PubMed] - Dixon, P.M. The Gross, Histological, and Ultrastructural Anatomy of Equine Teeth and Their Relationship to Disease. In Proceedings of the 49th Annual Convention of the American Association of Equine Practitioners, New Orleans, LA, USA, 21–25 November 2003; Volume 48, pp. 421–437. [Google Scholar]
- Brigham, E.J.; Duncanson, G.R. An equine postmortem dental study: 50 cases. Equine Vet. Educ.
**2000**, 12, 59–62. [Google Scholar] [CrossRef] - Limone, L. General clinical, oral and dental examination. In Equine Dentistry and Maxillofacial Surgery; Cambridge Scholars Publishing: Newcastle upon Tyne, UK, 2022; p. 302. [Google Scholar]
- Rehrl, S.; Schröder, W.; Müller, C.; Staszyk, C.; Lischer, C. Radiological prevalence of equine odontoclastic tooth resorption and hypercementosis. Equine Vet. J.
**2018**, 50, 481–487. [Google Scholar] [CrossRef] [PubMed] - Henry, T.J.; Puchalski, S.M.; Arzi, B.; Kass, P.H.; Verstraete, F.J.M. Radiographic evaluation in clinical practice of the types and stage of incisor tooth resorption and hypercementosis in horses. Equine Vet. J.
**2016**, 49, 486–492. [Google Scholar] [CrossRef] [PubMed] - Easley, J. A new look at dental radiography. In Proceedings of the 48th Annual Convention of the American Association of Equine Practitioners, Orlando, FL, USA, 4–8 December 2002; Volume 48, pp. 412–420. [Google Scholar]
- Barakzai, S.Z.; Dixon, P.M. A study of open-mouthed oblique radiographic projections for evaluating lesions of the erupted (clinical) crown. Equine Vet. Edu.
**2003**, 15, 143–148. [Google Scholar] [CrossRef] - Greet, T.R.C. Oral and dental trauma. In Equine Dentistry, 1st ed.; Baker, G.J., Easley, J., Eds.; W.B. Saunders: London, UK, 1999; pp. 60–69. [Google Scholar]
- Dixon, P.M.; Tremaine, W.H.; Pickles, K.; Kuhns, L.; Hawe, C.; McCann, J.; McGorum, B.; Railton, D.I.; Brammer, S. Equine dental disease Part 1: A longterm study of 400 cases: Disorders of incisor, canine and first premolar teeth. Equine Vet. J.
**1999**, 31, 369–377. [Google Scholar] [CrossRef] - Staszyk, C.; Bienert, A.; Kreutzer, R.; Wohlsein, P.; Simhofer, H. Equine odontoclastic tooth resorption and hypercementosis. Vet. J.
**2008**, 178, 372–379. [Google Scholar] [CrossRef] [PubMed] - Pearce, C.J. Recent developments in equine dentistry. N. Z. Vet. J.
**2020**, 68, 178–186. [Google Scholar] [CrossRef] - Barrett, M.F.; Easley, J.T. Acquisition and interpretation of radiographs of the equine skull. Equine Vet. Educ.
**2013**, 25, 643–652. [Google Scholar] [CrossRef] - Moore, N.T.; Schroeder, W.; Staszyk, C. Equine odontoclastic tooth resorption and hypercementosis affecting all cheek teeth in two horses: Clinical and histopathological findings. Equine Vet. Educ.
**2016**, 28, 123–130. [Google Scholar] [CrossRef] - Górski, K.; Tremaine, H.; Obrochta, B.; Buczkowska, R.; Turek, B.; Bereznowski, A.; Rakowska, A.; Polkowska, I. EOTRH syndrome in polish half-bred horses-two clinical cases. J. Equine Vet. Sci.
**2021**, 101, 103428. [Google Scholar] [CrossRef] [PubMed] - Saccomanno, S.; Passarelli, P.C.B.; Oliva, B.; Grippaudo, C. Comparison between two radiological methods for assessment of tooth root resorption: An in vitro study. BioMed Res. Int.
**2018**, 2018, 5152172. [Google Scholar] [CrossRef] - Górski, K.; Borowska, M.; Stefanik, E.; Polkowska, I.; Turek, B.; Bereznowski, A.; Domino, M. Selection of Filtering and Image Texture Analysis in the Radiographic Images Processing of Horses’ Incisor Teeth Affected by the EOTRH Syndrome. Sensors
**2022**, 22, 2920. [Google Scholar] [CrossRef] - Manso-Díaz, G.; García-López, J.M.; Maranda, L.; Taeymans, O. The role of head computed tomography in equine practice. Equine Vet. Educ.
**2015**, 27, 136–145. [Google Scholar] [CrossRef] - Baratt, R.M. Dental Radiography and Radiographic Signs of Equine Dental Disease. Vet. Clin. N. Am. Equine Pract.
**2020**, 36, 445–476. [Google Scholar] [CrossRef] [PubMed] - Dakin, S.G.; Lam, R.; Rees, E.; Mumby, C.; West, C.; Weller, R. Technical Set-up and Radiation Exposure for Standing Computed Tomography of the Equine Head: Standing CT of the Equine Head. Equine Vet. Educ.
**2014**, 26, 208–215. [Google Scholar] [CrossRef] - van der Stelt, P.F. Filmless imaging: The uses of digital radiography in dental practice. J. Am. Dent. Assoc.
**2005**, 136, 1379–1387. [Google Scholar] [CrossRef] - Tan, T.; Platel, B.; Mus, R.; Tabar, L.; Mann, R.M.; Karssemeijer, N. Computer-aided detection of cancer in automated 3-D breast ultrasound. IEEE TMI
**2013**, 32, 1698–1706. [Google Scholar] [CrossRef] [PubMed] - Vidal, P.L.; de Moura, J.; Novo, J.; Ortega, M. Multi-stage transfer learning for lung segmentation using portable X-ray devices for patients with COVID-19. Expert Syst. Appl.
**2021**, 173, 114677. [Google Scholar] [CrossRef] - Humeau-Heurtier, A. Texture feature extraction methods: A survey. IEEE Access
**2019**, 7, 8975–9000. [Google Scholar] [CrossRef] - Silva, L.E.; Duque, J.J.; Felipe, J.C.; Murta, L.O., Jr.; Humeau-Heurtier, A. Two-dimensional multiscale entropy analysis: Applications to image texture evaluation. Signal Process.
**2018**, 147, 224–232. [Google Scholar] [CrossRef] - Zarychta, P. Application of fuzzy image concept to medical images matching. In Information Technology in Biomedicine. ITIB 2018. Advances in Intelligent Systems and Computing, 1st ed.; Pietka, E., Badura, P., Kawa, J., Wieclawek, W., Eds.; Springer: Cham, Switzerland, 2019; Volume 762, pp. 27–38. [Google Scholar]
- Borowska, M.; Maśko, M.; Jasiński, T.; Domino, M. The Role of Two-Dimensional Entropies in IRT-Based Pregnancy Determination Evaluated on the Equine Model. In Information Technology in Biomedicine. ITIB 2022. Advances in Intelligent Systems and Computing, 1st ed.; Pietka, E., Badura, P., Kawa, J., Wieclawek, W., Eds.; Springer: Cham, Switzerland, 2022; Volume 1429, pp. 54–65. [Google Scholar]
- Domino, M.; Borowska, M.; Zdrojkowski, Ł.; Jasiński, T.; Sikorska, U.; Skibniewski, M.; Maśko, M. Application of the Two-Dimensional Entropy Measures in the Infrared Thermography-Based Detection of Rider: Horse Bodyweight Ratio in Horseback Riding. Sensors
**2022**, 22, 6052. [Google Scholar] [CrossRef] - Da Silva, L.E.; Senra Filho, A.C.; Fazan, V.P.; Felipe, J.C.; Murta, L.O., Jr. Two-dimensional sample entropy analysis of rat sural nerve aging. In Proceedings of the 2014 36th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, Chicago, IL, USA, 26–30 August 2014; pp. 3345–3348. [Google Scholar]
- Hilal, M.; Berthin, C.; Martin, L.; Azami, H.; Humeau-Heurtier, A. Bidimensional multiscale fuzzy entropy and its application to pseudoxanthoma elasticum. IEEE Trans. Biomed. Eng.
**2019**, 67, 2015–2022. [Google Scholar] [CrossRef] [PubMed] - Ribeiro, H.V.; Zunino, L.; Lenzi, E.K.; Santoro, P.A.; Mendes, R.S. Complexity-entropy causality plane as a complexity measure for two-dimensional patterns. PLoS ONE
**2012**, 7, e40689. [Google Scholar] [CrossRef] - Azami, H.; da Silva, L.E.V.; Omoto, A.C.M.; Humeau-Heurtier, A. Two-dimensional dispersion entropy: An information-theoretic method for irregularity analysis of images. Signal. Process. Image Commun.
**2019**, 75, 178–187. [Google Scholar] [CrossRef] - Azami, H.; Escudero, J.; Humeau-Heurtier, A. Bidimensional Distribution Entropy to Analyze the Irregularity of Small-Sized Textures. IEEE Signal. Proc. Lett.
**2017**, 24, 1338–1342. [Google Scholar] [CrossRef] - Radostits, O.M.; Gay, C.; Hinchcliff, K.W.; Constable, P.D. (Eds.) Veterinary Medicine E-Book: A Textbook of the Diseases of Cattle, Horses, Sheep, Pigs and Goats; Elsevier: Amsterdam, The Netherlands, 2006. [Google Scholar]
- Salem, S.E.; Townsend, N.B.; Refaai, W.; Gomaa, M.; Archer, D.C. Prevalence of oro-dental pathology in a working horse population in Egypt and its relation to equine health. Equine Vet. J.
**2017**, 49, 26–33. [Google Scholar] [CrossRef] - Hüls, I.; Bienert, A.; Staszyk, C. Equine odontoclastic tooth resorption and hyper-cementosis (EOTRH): Röntgenologische und makroskopisch-anatomische Befunde. In Proceedings of the 10 Jahrestagung der Internationalen Gesellschaft zur Funktionsverbesserung der Pferdezähne, Wiesbaden, Germany, 3–4 March 2012. [Google Scholar]
- Floyd, M.R. The modified Triadan system: Nomenclature for veterinary dentistry. J. Vet. Dent.
**1991**, 8, 18–19. [Google Scholar] [CrossRef] [PubMed] - van Griethuysen, J.J.M.; Fedorov, A.; Parmar, C.; Hosny, A.; Aucoin, N.; Narayan, V.; Beets-Tan, R.G.H.; Fillon-Robin, J.C.; Pieper, S.; Aerts, H.J.W.L. Computational radiomics system to decode the radiographic phenotype. Cancer Res.
**2017**, 77, e104–e107. [Google Scholar] [CrossRef] - Lowekamp, B.C.; Chen, D.T.; Ibáñez, L.; Blezek, D. The design of SimpleITK. Front. Neuroinform.
**2013**, 7, 45. [Google Scholar] [CrossRef] - Yaniv, Z.; Lowekamp, B.C.; Johnson, H.J.; Beare, R. SimpleITK image-analysis notebooks: A collaborative environment for education and reproducible research. J. Digit. Imaging
**2018**, 31, 290–303. [Google Scholar] [CrossRef] - Lim, J.S. Two-Dimensional Signal and Image Processing, 1st ed.; Prentice Hall: Englewood Cliffs, NJ, USA, 1990. [Google Scholar]
- Gonzalez, R.C.; Eddins, S.L.; Woods, R.E. Digital Image Publishing Using MATLAB, 1st ed.; Prentice Hall: Upper Saddle River, NJ, USA, 2004. [Google Scholar]
- Flood, M.W.; Grimm, B. EntropyHub: An open-source toolkit for entropic time series analysis. PLoS ONE
**2021**, 16, e0259448. [Google Scholar] - Silva, L.E.V.; Senra Filho, A.C.S.; Fazan, V.P.S.; Felipe, J.C.; Murta Junior, L.O. Two-dimensional sample entropy: Assessing image texture through irregularity. Biomed. Phys. Eng. Express
**2016**, 2, 045002. [Google Scholar] [CrossRef] - Furlong, R.; Hilal, M.; O’brien, V.; Humeau-Heurtier, A. Parameter Analysis of Multiscale Two-Dimensional Fuzzy and Dispersion Entropy Measures Using Machine Learning Classification. Entropy
**2021**, 23, 1303. [Google Scholar] [CrossRef] [PubMed] - Morel, C.; Humeau-Heurtier, A. Multiscale permutation entropy for two-dimensional patterns. Pattern Recognit. Lett.
**2021**, 150, 139–146. [Google Scholar] [CrossRef] - He, J.; Shang, P.; Zhang, Y. PID: A PDF-induced distance based on permutation cross-distribution entropy. Nonlinear Dyn.
**2019**, 97, 1329–1342. [Google Scholar] [CrossRef] - Dohoo, I.; Martin, W.; Stryhn, H. Veterinary Epidemiologic Research, 2nd ed.; VER Inc.: Charlottetown, PE, Canada, 2009. [Google Scholar]
- Sykora, S.; Pieber, K.; Simhofer, H.; Hackl, V.; Brodesser, D.; Brandt, S. Isolation of Treponema and Tannerella spp. from equine odontoclastic tooth resorption and hypercementosis related periodontal disease. Equine Vet. J.
**2014**, 46, 358–363. [Google Scholar] [CrossRef] - Zhang, H.; Hung, C.L.; Min, G.; Guo, J.P.; Liu, M.; Hu, X. GPU-accelerated GLRLM algorithm for feature extraction of MRI. Sci. Rep.
**2019**, 9, 10883. [Google Scholar] [CrossRef] - Smedley, R.C.; Earley, E.T.; Galloway, S.S.; Baratt, R.M.; Rawlinson, J.E. Equine odon-toclastic tooth resorption and hypercementosis: Histopathologic features. Vet. Pathol.
**2015**, 52, 903–909. [Google Scholar] [CrossRef] - Szczypiński, P.; Klepaczko, A.; Pazurek, M.; Daniel, P. Texture and color based image segmentation and pathology detection in capsule endoscopy videos. Comput. Methods Programs Biomed.
**2014**, 113, 396–411. [Google Scholar] [CrossRef] - Szczypinski, P.M.; Klepaczko, A.; Kociołek, M. QMaZda—Software tools for image analysis and pattern recognition. In Proceedings of the 2017 Signal Processing: Algorithms, Architectures, Arrangements, and Applications (SPA), Poznan, Poland, 20–22 October 2017; pp. 217–221. [Google Scholar]
- Depeursinge, A.; Al-Kadi, O.S.; Mitchell, J.R. Biomedical Texture Analysis: Fundamentals, Tools and Challenges; Academic Press: Cambridge, MA, USA, 2017. [Google Scholar]
- Al-Ameen, Z.; Sulong, G.; Gapar, M.D.; Johar, M.D. Reducing the Gaussian blur artifact from CT medical images by employing a combination of sharpening filters and iterative deblurring algorithms. J. Theor. Appl. Inf. Technol.
**2012**, 46, 31–36. [Google Scholar] - Heidari, M.; Mirniaharikandehei, S.; Khuzani, A.Z.; Danala, G.; Qiu, Y.; Zheng, B. Improving the performance of CNN to predict the likelihood of COVID-19 using chest X-ray images with preprocessing algorithms. Int. J. Med. Inform.
**2020**, 144, 104284. [Google Scholar] [CrossRef] [PubMed] - Jusman, Y.; Tamarena, R.I.; Puspita, S.; Saleh, E.; Kanafiah, S.N.A.M. Analysis of features extraction performance to differentiate of dental caries types using gray level co-occurrence matrix algorithm. In Proceedings of the 2020 10th IEEE International Conference on Control System, Computing and Engineering (ICCSCE), Penang, Malaysia, 21–22 August 2020; pp. 148–152. [Google Scholar]
- Nagarajan, M.B.; Coan, P.; Huber, M.B.; Diemoz, P.C.; Glaser, C.; Wismüller, A. Computer-aided diagnosis for phase-contrast X-ray computed tomography: Quantitative characterization of human patellar cartilage with high-dimensional geometric features. J. Digit. Imaging
**2014**, 27, 98–107. [Google Scholar] [CrossRef] [PubMed] - Kociołek, M.; Strzelecki, M.; Obuchowicz, R. Does image normalization and intensity resolution impact texture classification? Comput. Med. Imaging Graph.
**2020**, 81, 101716. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Scheme of radiographic–based detection of the signs of the Equine Odontoclastic Tooth Resorption and Hypercementosis (EOTRH) syndrome. A detailed dental examination (

**A**); a standard radiography (

**B**); segmentation of input radiographs with regions of interest (ROIs) of the first upper right incisor tooth (101) and the first upper left incisor tooth (201) marked with yellow lines (

**C**); filtering of input radiographs by three filters—Normalize, Median, and Laplacian Sharpening (

**D**); a texture analysis of output radiographs after filtering using entropy–based measures (five measures: SampEn2D—two–dimensional sample entropy, FuzzEn2D—two–dimensional fuzzy entropy, PermEn2D—two–dimensional permutation entropy, DispEn2D—two–dimensional dispersion entropy, DistEn2D—two–dimensional distribution entropy) and Gray–Level Co–occurrence Matrix (six selected features: Cluster Prominence, Contrast, Difference Average, Difference Entropy, Difference Variance, Inverse Variance) (

**E**).

**Figure 2.**The comparison of the entropy measures between the EOTRH grades (0–3). The following entropy measures are considered: SampEn2D—two–dimensional sample entropy (

**A**,

**F**,

**K**), FuzzEn2D—two–dimensional fuzzy entropy (

**B**,

**G**,

**L**), PermEn2D—two–dimensional permutation entropy (

**C**,

**H**,

**M**), DispEn2D—two–dimensional dispersion entropy (

**D**,

**I**,

**N**), DistEn2D—two–dimensional distribution entropy (

**E**,

**J**,

**O**). The output radiographs filtered by Normalize (

**A**–

**E**), Median (

**F**–

**J**), and Laplacian Sharpening (

**K**–

**O**) filtering algorithms are separated by dashed horizontal lines. Lower case letters (a–c) indicate differences between groups for p < 0.05 independently for each measure. The significant increase with the EOTRH grades is marked with a red line. Single realizations are marked with dots.

**Figure 3.**The comparison of the entropy measures between the filtering algorithms. The following entropy measures are considered: SampEn2D—two–dimensional sample entropy (

**A**,

**F**,

**K**,

**P**), FuzzEn2D—two–dimensional fuzzy entropy (

**B**,

**G**,

**L**,

**Q**), PermEn2D—two–dimensional permutation entropy (

**C**,

**H**,

**M**,

**R**), DispEn2D—two–dimensional dispersion entropy (

**D**,

**I**,

**N**,

**S**), DistEn2D—two–dimensional distribution entropy (

**E**,

**J**,

**O**,

**T**). The radiographs classified to EOTRH 0 grade (

**A**–

**E**), EOTRH 1 grade (

**F**–

**J**), EOTRH 2 grade (

**K**–

**O**), and EOTRH 3 grade (

**P**–

**T**) are separated by dashed horizontal lines. Lower case letters (a–c) indicate differences between groups for p < 0.05 independently for each measure. Single realizations are marked with dots.

**Figure 4.**The comparison of the selected Gray–Level Co–occurrence Matrix (GLCM) features between the EOTRH grades (0–3). The following GLCM features are considered: Cluster Prominence (

**A**,

**G**,

**M**), Contrast (

**B**,

**H**,

**N**), Difference Average (

**C**,

**I**,

**O**), Difference Entropy (

**D**,

**J**,

**P**), Difference Variance (

**E**,

**K**,

**Q**), Inverse Variance (

**F**,

**L**,

**R**). The output radiographs filtered by Normalize (

**A**–

**F**), Median (

**G**–

**L**), and Laplacian Sharpening (

**M**–

**R**) filtering algorithms are separated by dashed horizontal lines. Lower case letters (a–c) indicate differences between groups for p < 0.05 independently for each feature. The significant increase with the EOTRH grades is marked with a red line. Single realizations are marked with dots.

**Figure 5.**The comparison of the selected Gray–Level Co–occurrence Matrix (GLCM) features between the filtering algorithms. The following GLCM features are considered: Cluster Prominence (

**A**,

**G**,

**M**,

**S**), Contrast (

**B**,

**H**,

**N**,

**T**), Difference Average (

**C**,

**I**,

**O**,

**U**), Difference Entropy (

**D**,

**J**,

**P**,

**V**), Difference Variance (

**E**,

**K**,

**Q**,

**W**), Inverse Variance (

**F**,

**L**,

**R**,

**X**). The radiographs classified to EOTRH 0 grade (

**A**–

**F**), EOTRH 1 grade (

**G**–

**L**), EOTRH 2 grade (

**M**–

**R**), and EOTRH 3 grade (

**S**–

**X**) are separated by dashed horizontal lines. Lower case letters (a–c) indicate differences between groups for p < 0.05 independently for each measure. Single realizations are marked with dots.

**Figure 6.**Comparison of selected entropy measure (DistEn2D–two–dimensional distribution entropy) and selected Gray–Level Co–occurrence Matrix (GLCM) features (Cluster Prominence (

**A**), Contrast (

**B**), Difference Average (

**C**), Difference Entropy (

**D**), Difference Variance (

**E**), Inverse Variance (

**F**)) throughout the EOTRH grades. Measure and features were extracted from the output radiographs after Normalize filtering. Similarity was tested using linear regressions. A p < 0.05 was considered significant. If the difference between slopes was not significant, a single slope measurement was calculated. Plot where the slope value of the entropy measure was higher than the slope value of the GLCM features was marked by dashed frames.

**Table 1.**The comparison of details (linearity of the filter, type of output image, and result of filtering) of three filtering algorithms (Normalize filter, Median filter, and Laplacian Sharpening filter) used in the study.

Filter | Linearity | Output Image | Result |
---|---|---|---|

Normalize filter [41] | Linear filter | A rescaled image in which the pixels have zero mean and unit variance | An increase in the contrast of the image |

Median filter [41] | Non–linear filter | A recalculated image in which the pixels are represented by the medians of the pixels in the neighbourhood of the input pixel | A reduction in the noise |

Laplacian Sharpening filter [42] | Non–linear filter | A produced image in which the pixels are convoluted with a Laplacian operator | A change of the regions of rapid intensity and highlights the edges |

**Table 2.**The comparison of details (definition, relations between the values and the irregularity/complexity of the image, and application of measures) of five entropy measures (SampEn2D, FuzzEn2D, PermEn2D, DispEn2D, DistEn2D) used in the study.

Entropy Measures | Definition | Values | Application |
---|---|---|---|

SampEn2D [29,44] | The negative natural logarithm of the probability of similarity of patterns of length m with patterns of length m + 1
$$\mathrm{SampEn}2\mathrm{D}=-\mathrm{ln}\frac{{\Phi}^{m+1}}{{\Phi}^{m}}$$
| Low: regular patterns or periodic structures, as they have the same number of patterns for both m and m + 1 High: irregular patterns | A measure of the irregularity in the pixel patterns |

FuzzEn2D [30,45] | The negative natural logarithm of the conditional probability
$$\mathrm{FuzzyEn}2\mathrm{D}=-\mathrm{ln}\frac{{\Phi}^{m+1}\left(r\right)}{{\Phi}^{m}\left(r\right)}$$
| Low: regular patterns or periodic structures High: irregular patterns or non-periodic structures | A measure of the irregularity in pixel patterns but using a continuous exponential function to determine the degree of similarity |

PermEn2D [31,46] | The concept of counting permutation patterns $\pi $, where the permutation patterns are obtained after ordering the positions of the initial image patterns
$$\mathrm{PermEn}2\mathrm{D}=-\frac{1}{\left(n-{d}_{n}+1\right)\left(m-{d}_{m}+1\right)}{\displaystyle {\displaystyle \sum}_{\pi =1}^{{d}_{n}!\times {d}_{m}!}}p\left(\pi \right)\mathrm{ln}p\left(\pi \right)$$
| Low: regular patterns with the pixels always appearing in the same order High: irregular patterns with the highly disordered image pixels | An identification of irregular structure of the image |

DispEn2D [32,45] | The conception of using the sigmoid function relies on mapped to $c$ classes and the values of image pixels form ${z}_{i,j}^{c}=round\left(c\times v\left(i,j\right)+0.5\right)$, where $v\left(i,j\right)$
$$\mathrm{DispEn}2\mathrm{D}=-\frac{1}{\left(n-{d}_{n}+1\right)\left(m-{d}_{m}+1\right)}{\displaystyle {\displaystyle \sum}_{\pi =1}^{{d}_{n}!\times {d}_{m}!}}p\left({\pi}_{v}\right)\mathrm{ln}p\left({\pi}_{v}\right)$$
| Low: regular patterns with the low probability of dispersion patterns High: irregular image with the high probability of dispersion patterns | An assessment of the regularity of images with no indeterminacy of small–sized images |

DistEn2D [33,47] | The amount of similarity between two windows by measuring the distance between the corresponding windows based on the distance matrix used to estimate the empirical probability density function $\left(ePDF\right)$
$$\mathrm{DistEn}2\mathrm{D}=-{\displaystyle {{\displaystyle \sum}}_{t=1}^{M}}{p}_{t}lo{g}_{2}\left({p}_{t}\right)$$
| Low: regular patterns of the small size images High: irregular patterns of the small size images | A quantitative description of the irregularities of the images, taking into account the small size of the image |

**Table 3.**The accuracy (Se—sensitivity; Sp—specificity; PPV—positive predictive value; NPV—negative predictive value) of the detection of EOTRH 0 and EOTRH 3 based on the selected entropy measure (DistEn2D—two–dimensional distribution entropy) and the selected Gray–Level Co–occurrence Matrix (GLCM) features (ClusterProminence; Contrast; DifferenceAverage; DifferenceEntropy; DifferenceVariance; Inverse Variance) extracted from the output images filtered by Normalize filter. Three thresholds (mean; mean + SD; mean + 2SD) were used.

Measures | DistEn2D | Cluster Prominence | Contrast | Difference Average | Difference Entropy | Difference Variance | Inverse Variance |
---|---|---|---|---|---|---|---|

Threshold | mean | ||||||

Se | 0.50 | 0.25 | 0.25 | 0.25 | 0.27 | 0.25 | 0.25 |

Sp | 0.95 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 |

PPV | 0.67 | 0.94 | 0.94 | 0.94 | 0.94 | 0.94 | 0.94 |

NPV | 0.90 | 0.70 | 0.70 | 0.70 | 0.70 | 0.70 | 0.70 |

Threshold | mean + SD | ||||||

Se | 0.13 | 0.17 | 0.17 | 0.17 | 0.22 | 0.17 | 0.17 |

Sp | 1.00 | 0.99 | 0.99 | 0.99 | 0.98 | 0.99 | 0.99 |

PPV | 1.00 | 0.91 | 0.91 | 0.91 | 0.93 | 0.91 | 0.91 |

NPV | 0.84 | 0.68 | 0.68 | 0.68 | 0.58 | 0.68 | 0.68 |

Threshold | mean + 2SD | ||||||

Se | 0.00 | 0.03 | 0.07 | 0.07 | 0.07 | 0.07 | 0.07 |

Sp | 1.00 | 0.99 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

PPV | - | 0.67 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

NPV | 0.82 | 0.65 | 0.66 | 0.66 | 0.66 | 0.66 | 0.66 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Górski, K.; Borowska, M.; Stefanik, E.; Polkowska, I.; Turek, B.; Bereznowski, A.; Domino, M.
Application of Two-Dimensional Entropy Measures to Detect the Radiographic Signs of Tooth Resorption and Hypercementosis in an Equine Model. *Biomedicines* **2022**, *10*, 2914.
https://doi.org/10.3390/biomedicines10112914

**AMA Style**

Górski K, Borowska M, Stefanik E, Polkowska I, Turek B, Bereznowski A, Domino M.
Application of Two-Dimensional Entropy Measures to Detect the Radiographic Signs of Tooth Resorption and Hypercementosis in an Equine Model. *Biomedicines*. 2022; 10(11):2914.
https://doi.org/10.3390/biomedicines10112914

**Chicago/Turabian Style**

Górski, Kamil, Marta Borowska, Elżbieta Stefanik, Izabela Polkowska, Bernard Turek, Andrzej Bereznowski, and Małgorzata Domino.
2022. "Application of Two-Dimensional Entropy Measures to Detect the Radiographic Signs of Tooth Resorption and Hypercementosis in an Equine Model" *Biomedicines* 10, no. 11: 2914.
https://doi.org/10.3390/biomedicines10112914