Analysing roughness of surface through fractal dimension: A review

Nayak, Soumya Ranjan; Mishra, Jibitesh; Palai, G.

doi:10.1016/j.imavis.2019.06.015

Cited by 94 publications

(35 citation statements)

References 113 publications

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“…The box-counting method was selected for the following reasons. From previous studies, several researchers commented that the box-counting method exhibited inconsistency whereby the box size and different scales also led to different FD [14,38]. The level of confidence of this method reached 99.9%, more than 99% from previous reports [39,40].…”

Section: Fractal Dimension Calculationmentioning

confidence: 98%

“…This theory was first proposed by Mandelbrot in 1967 to describe the exact realistic length of the coastline of Britain [13]. This method has been widely used in many other areas, for example, to evaluate the condition of clouds, mountains, grain boundary, coal, river drainage networks, active faults, plutonic bodies, and other related disciplines [14][15][16][17][18][19][20][21]. Using Mandelbrot's research on the coastline dimension, many scientists have calculated the FD of coastlines all around the world [22][23][24][25][26].…”

Section: Introductionmentioning

confidence: 99%

“…The divider, spectrum, and box-counting methods are the most common methods used to compute FD [14]. In 1967, Mandelbrot introduced the divider method as a way to calculate FD [13].…”

Section: Introductionmentioning

confidence: 99%

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Coastline Fractal Dimension of Mainland, Island, and Estuaries Using Multi-temporal Landsat Remote Sensing Data from 1978 to 2018: A Case Study of the Pearl River Estuary Area

Wang

2020

Remote Sensing

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The Pearl River Estuary Area was selected for this study. For the past 40 years, it has been one of the most complex coasts in China, yet few studies have analyzed the complexity and variations of the area’s different coastlines. In this investigation, the coastlines of the Pearl River Estuary Area were extracted from multi-temporal Landsat remote sensing data from 1978, 1988, 1997, 2008, and 2018. The coastline of this area was classified into mainland, island, and estuarine. To obtain more detailed results of the mainland and island, we regarded this area as the main body, rezoned into different parts. The box-counting dimension was applied to compute the bidimensional (2D) fractal dimension. Coastline length and the fractal dimension of different types of coastline and different parts of the main body were calculated and compared. The fractal dimension of the Pearl River Estuary Area was found to have increased significantly, from 1.228 to 1.263, and coastline length also increased during the study period. The island and mainland showed the most complex coastlines, while estuaries showed the least complexity during the past forty years. A positive correlation was found between length and 2D-fractal dimension in some parts of the study area. Land reclamation had the strongest influence on fractal dimension variations.

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Section: Fractal Dimension Calculationmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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Coastline Fractal Dimension of Mainland, Island, and Estuaries Using Multi-temporal Landsat Remote Sensing Data from 1978 to 2018: A Case Study of the Pearl River Estuary Area

Wang

2020

Remote Sensing

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“…In Euclidean space, if the Hausdorff dimension of a set F is greater than its topological dimension, the set F is a fractal set [ 8 ]. Since the definition is not applicable in practical application, he later proposed that the shape the components of which resemble the whole to some extent are called fractal [ 9 ].…”

Section: Basic Theory Of Fractalmentioning

confidence: 99%

Review about the Application of Fractal Theory in the Research of Packaging Materials

Duan

Mao

et al. 2021

Materials

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The work is intended to summarize the recent progress in the work of fractal theory in packaging material to provide important insights into applied research on fractal in packaging materials. The fractal analysis methods employed for inorganic materials such as metal alloys and ceramics, polymers, and their composites are reviewed from the aspects of fractal feature extraction and fractal dimension calculation methods. Through the fractal dimension of packaging materials and the fractal in their preparation process, the relationship between the fractal characteristic parameters and the properties of packaging materials is discussed. The fractal analysis method can qualitatively and quantitatively characterize the fractal characteristics, microstructure, and properties of a large number of various types of packaging materials. The method of using fractal theory to probe the preparation and properties of packaging materials is universal; the relationship between the properties of packaging materials and fractal dimension will be a critical trend of fractal theory in the research on properties of packaging materials.

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“…FD is, ultimately, a mathematical entity capable of determining the complexity of two-dimensional objects. The computation analysis of complex objects largely benefits from being understood under the lenses of fractal geometry (Nayak et al 2019 ).…”

Section: Introductionmentioning

confidence: 99%

Fractal dimension analysis as an easy computational approach to improve breast cancer histopathological diagnosis

Silva¹,

Monteiro²,

Moreira

et al. 2021

Appl. Microsc.

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Histopathology is a well-established standard diagnosis employed for the majority of malignancies, including breast cancer. Nevertheless, despite training and standardization, it is considered operator-dependent and errors are still a concern. Fractal dimension analysis is a computational image processing technique that allows assessing the degree of complexity in patterns. We aimed here at providing a robust and easily attainable method for introducing computer-assisted techniques to histopathology laboratories. Slides from two databases were used: A) Breast Cancer Histopathological; and B) Grand Challenge on Breast Cancer Histology. Set A contained 2480 images from 24 patients with benign alterations, and 5429 images from 58 patients with breast cancer. Set B comprised 100 images of each type: normal tissue, benign alterations, in situ carcinoma, and invasive carcinoma. All images were analyzed with the FracLac algorithm in the ImageJ computational environment to yield the box count fractal dimension (Db) results. Images on set A on 40x magnification were statistically different (p = 0.0003), whereas images on 400x did not present differences in their means. On set B, the mean Db values presented promissing statistical differences when comparing. Normal and/or benign images to in situ and/or invasive carcinoma (all p < 0.0001). Interestingly, there was no difference when comparing normal tissue to benign alterations. These data corroborate with previous work in which fractal analysis allowed differentiating malignancies. Computer-aided diagnosis algorithms may beneficiate from using Db data; specific Db cut-off values may yield ~ 99% specificity in diagnosing breast cancer. Furthermore, the fact that it allows assessing tissue complexity, this tool may be used to understand the progression of the histological alterations in cancer.

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Analysing roughness of surface through fractal dimension: A review

Cited by 94 publications

References 113 publications

Coastline Fractal Dimension of Mainland, Island, and Estuaries Using Multi-temporal Landsat Remote Sensing Data from 1978 to 2018: A Case Study of the Pearl River Estuary Area

Coastline Fractal Dimension of Mainland, Island, and Estuaries Using Multi-temporal Landsat Remote Sensing Data from 1978 to 2018: A Case Study of the Pearl River Estuary Area

Review about the Application of Fractal Theory in the Research of Packaging Materials

Fractal dimension analysis as an easy computational approach to improve breast cancer histopathological diagnosis

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