Calculating Hausdorff Dimension in Higher Dimensional Spaces

Fernández–Martínez, M.; Guirao, Juan Luis García; Granero, Miguel Ángel Sánchez

doi:10.3390/sym11040564

Cited by 8 publications

(3 citation statements)

References 17 publications

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“…Hausdorff dimension is the best way to measure fractal dimension of a bounded subset of R n since it considers all the possible coverings (of a given diameter) that the bounded subset may admit, and it possesses better analytical properties than the box dimension. We refer to the works of Fernández-Martínez et al [14][15][16][17] who proposed a way to deal with the calculation of Hausdorff dimension in applications for compact Euclidean subsets including the higher dimensional case in more general settings. Their approach combines both theoretical results along with techniques from machine learning, thus leading to the first-known attempt to calculate Hausdorff dimension in applications.…”

Section: Fractal Dimensionmentioning

confidence: 99%

Fractals: An Eclectic Survey, Part-I

et al. 2022

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Fractals are geometric shapes and patterns that may repeat their geometry at smaller or larger scales. It is well established that fractals can describe shapes and surfaces that cannot be represented by the classical Euclidean geometry. An eclectic survey of fractals is presented in two parts encompassing applications of fractals in a variety of diverse and innovative fields. The goal of the first part is to focus on the glossary of fractals, their mathematical description, aesthetic, artistic, and architectural applications, while the second part is focused on engineering, industry, commercial, and futuristic applications of fractals.

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Section: Fractal Dimensionmentioning

confidence: 99%

Fractals: An Eclectic Survey, Part-I

et al. 2022

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“…For a complete treatment of Hausdorff dimension we refer to the book by Edgar 4 . A computational approach to calculate the Hausdorff dimension of compact Euclidean subsets was given by Fernández-Martínez and Sánchez-Granero 9 and we refer to the recent work by Fernández-Martínez et al 12 for calculation of the Hausdorff dimension in higher dimensional Euclidean spaces.…”

Section: Fractal Dimension: a Brief Reviewmentioning

confidence: 99%

Fractal dimension of coastline of Australia

Husain

Reddy

Bisht

et al. 2021

Sci Rep

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Coastlines are irregular in nature having (random) fractal geometry and are formed by various natural activities. Fractal dimension is a measure of degree of geometric irregularity present in the coastline. A novel multicore parallel processing algorithm is presented to calculate the fractal dimension of coastline of Australia. The reliability of the coastline length of Australia is addressed by recovering the power law from our computational results. For simulations, the algorithm is implemented on a parallel computer for multi-core processing using the QGIS software, R-programming language and Python codes.

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“…The calculations involving the fractal dimensions of the binary CBCTs were performed by means of the following expression (cf. [12,13,20]):…”

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confidence: 99%

Anatomical Considerations and Study of the Fractal Dimension around the Posterior Superior Alveolar Artery

et al. 2020

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The Posterior Superior Alveolar Artery (PSAA) provides vascular support to molars, gingiva, and maxillary sinus. A tear of the PSAA may cause profuse hemorrhages which may lead to complications at a surgical level. As such, it becomes crucial to anatomically analyse several features regarding the PSAA as well as the area surrounding it. In this paper, we are particularly interested in the study of the complexity of the periodontal tissue structure which appears close to the location of the PSAA. A total amount of 400 cone beam computed tomography (CBCT) scans (two per subject) were performed to explore the presence of the PSAA, the thickness of the Schneider’s membrane, and the existence of septa. Several parameters were evaluated including the location of the artery in the maxillary sinus, the distance from the PSAA to the alveolar ridge, the thickness of the membrane, the diameter of the cavities produced by the septa, and the fractal dimension of the trabecular tissue that surrounds the PSAA. They were found strong linear relationships between Distal and Central Measures (a Pearson’s R 2 = 0.9952 ), Mesial and Central Measures ( R 2 = 0.9950 ), and Distal and Mesial Measure ( R 2 = 0.997 ). We hypothesised that the loss of dental pieces would imply a distinct complexity of the trabecular tissue structure surrounding the PSAA. In this way, a p-value equal to 0.001 was provided by the Mann-Whitney test, which supports our hypothesis. Furthermore, the mean of the fractal dimensions of the group of edentulous patients (equal to 1.56 ) was found to be lower than the one of the group of non-edentulous patients (equal to 1.61 ) with small standard deviations in both cases. Our study suggests that accurate calculations of the fractal dimension combined with the use of CBCT do provide valuable information regarding the area that surrounds the PSAA.

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Calculating Hausdorff Dimension in Higher Dimensional Spaces

Cited by 8 publications

References 17 publications

Fractals: An Eclectic Survey, Part-I

Fractals: An Eclectic Survey, Part-I

Fractal dimension of coastline of Australia

Anatomical Considerations and Study of the Fractal Dimension around the Posterior Superior Alveolar Artery

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