box counting
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2022 ◽  
Vol 70 (2) ◽  
pp. 3049-3066
Author(s):  
S. Sathiya Devi ◽  
S. Vidivelli
Keyword(s):  

2022 ◽  
Vol 2159 (1) ◽  
pp. 012011
Author(s):  
J Villamizar ◽  
L Uribe ◽  
A Cerquera ◽  
E Prada ◽  
D Prada ◽  
...  

Abstract Alzheimer’s disease is a neurodegenerative cognitive, affective, and behavioral disorder aligned to the aging process and other coronary diseases. To contribute to the early diagnosis of the disease, a neuroimaging treatment is implemented through a preprocessing to subsequently calculate the fractal dimension associated with these images in order to propose an alternative to the one proposed in medical physics through positron emission tomography. In this work, a comparative analysis is made of a previous work using the Box Counting methodology versus the calculation of the fractal dimension by means of software developed by the researchers based on the same method. The differences between the fractal dimensions of the neuroimages of control patients and patients with the presence of the disease are maintained showing a lower value of fractal dimension in patients with the disease due to the physical deterioration of the brain.


2021 ◽  
Vol 26 (2) ◽  
pp. 127-136
Author(s):  
Ram Krishna Tiwari ◽  
Harihar Paudyal

To establish the relations between b-value and fractal dimension (D0) for the earthquake distribution, we study the regional variations of those parameters in the central Himalaya region. The earthquake catalog of 989 events (Mc = 4.0) from 1994.01.31 to 2020.10.28 was analyzed in the study. The study region is divided into two sub-regions (I) Region A: 27.3°N -30.3°N and 80°E -84.8°E (western Nepal and vicinity) and (II) Region B: 26.4°N -28.6°N and 84.8°E -88.4°E (eastern Nepal and vicinity). The b-value observed is within the range between 0.92 to 1.02 for region A and 0.64 to 0.74 for region B showing the homogeneous nature of the variation. The seismic a-value for those regions ranges respectively between 5.385 to 6.007 and 4.565 to 5.218. The low b-values and low seismicity noted for region B may be related with less heterogeneity and high strength in the crust. The high seismicity with average b-values obtained for region A may be related with high heterogeneity and low strength in the crust. The fractal dimension ≥1.74 for region A and ≥ 1.82 for region B indicate that the earthquakes were distributed over two-dimensional embedding space. The observed correlation between D0 and b is negative for western Nepal and positive for eastern Nepal while the correlation between D0 and a/b value is just opposite for the respective regions. The findings identify both regions as high-stress regions. The results coming from the study agree with the results of the preceding works and reveal information about the local disparity of stress and change in tectonic complexity in the central Himalaya region.


Morphologia ◽  
2021 ◽  
Vol 15 (3) ◽  
pp. 196-206
Author(s):  
N.I. Maryenko ◽  
O.Yu. Stepanenko

Background. Fractal analysis is an informative and objective method of mathematical analysis that can complement existing methods of morphometry and provides a comprehensive quantitative assessment of the spatial configuration of irregular anatomical structures. Objective: a comparative analysis of fractal analysis methods used for morphometry in biomedical research. Methods. A comprehensive analysis of morphological studies, based on fractal analysis. Results. Different types of medical images with different preprocessing algorithms can be used for fractal analysis. The parameter determined by fractal analysis is the fractal dimension, which is a measure of the complexity of the spatial configuration and the degree of filling of space with a certain geometric object. The most known methods of fractal analysis are the following: box counting, caliper, pixel dilation, "mass-radius", cumulative intersection, grid intercept. The box counting method and its modifications is the most commonly used method due to the simplicity and versatility. Different methods of fractal analysis have a similar principle: fractal measures (different geometric figures) of a certain size completely cover the structure in the image, size of fractal measure is iteratively changed, and the minimum number of fractal measures covering the structure is calculated. Methods of fractal analysis differ in the type of fractal measure, which can be a linear segment, a square of a fractal grid, a cube, a circle, a sphere etc. Conclusion. The choice of the method of fractal analysis and image preprocessing method depends on the studied structure, features of its spatial configuration, the type of image used for the analysis, and the aim of the study.


Author(s):  
Ştefan Ţălu ◽  
Pedro Luiz Guzzo ◽  
Bandar Astinchap ◽  
Hamta Ghanbaripour

Abstract Since synthetic quartz is essential to produce 3-D resonators for numerous applications in precision electronics, in this work the surface topography of cylindrical quartz bars is investigated using the multifractal technique. The cylindrical bars were manufactured with ultrasonic machining using with five SiC grits ranging from 6 to 50 µm. The machined surfaces were initially characterized by contact profilometry and scanning electron microscopy (SEM). The multifractality of the machined surfaces was scrutinized using a box-counting method applied to the images obtained with 500X magnification. The multifractal spectrum indicated that the fractal dimension f(α) and the width of the fractal spectrum Δα are dependent on the grit size, but this dependence is not monotonic. The lowest (negative) value for Δf(α) was found for 25 µm grits indicating that for these grits the lower frequency events (grooves with tens µm width occurring along the USM direction) controls the surface topography much more than high frequency events related to brittle microcracking. The abrasive wear due to the continuous slurry recycling in lateral tool-workpiece interfaces contributed to smooth the groove texture as well as the sharpness of microscopic indentations, which remained observed on the surfaces machined with 50 µm grits. The opposite paths observed for the arithmetical mean deviation of the measured profile (Ra) and Δf(α) parameters with the cutting rate measured for each grit size were valuable to differentiate flat-rough and unlevelled-rough topographies in quartz bars.


2021 ◽  
Vol 5 (4) ◽  
pp. 78
Author(s):  
Anis Malekzadeh ◽  
Assef Zare ◽  
Mahdi Yaghoobi ◽  
Roohallah Alizadehsani

This paper proposes a new method for epileptic seizure detection in electroencephalography (EEG) signals using nonlinear features based on fractal dimension (FD) and a deep learning (DL) model. Firstly, Bonn and Freiburg datasets were used to perform experiments. The Bonn dataset consists of binary and multi-class classification problems, and the Freiburg dataset consists of two-class EEG classification problems. In the preprocessing step, all datasets were prepossessed using a Butterworth band pass filter with 0.5–60 Hz cut-off frequency. Then, the EEG signals of the datasets were segmented into different time windows. In this section, dual-tree complex wavelet transform (DT-CWT) was used to decompose the EEG signals into the different sub-bands. In the following section, in order to feature extraction, various FD techniques were used, including Higuchi (HFD), Katz (KFD), Petrosian (PFD), Hurst exponent (HE), detrended fluctuation analysis (DFA), Sevcik, box counting (BC), multiresolution box-counting (MBC), Margaos-Sun (MSFD), multifractal DFA (MF-DFA), and recurrence quantification analysis (RQA). In the next step, the minimum redundancy maximum relevance (mRMR) technique was used for feature selection. Finally, the k-nearest neighbors (KNN), support vector machine (SVM), and convolutional autoencoder (CNN-AE) were used for the classification step. In the classification step, the K-fold cross-validation with k = 10 was employed to demonstrate the effectiveness of the classifier methods. The experiment results show that the proposed CNN-AE method achieved an accuracy of 99.736% and 99.176% for the Bonn and Freiburg datasets, respectively.


Energies ◽  
2021 ◽  
Vol 14 (24) ◽  
pp. 8253
Author(s):  
Xiaolong Li ◽  
Chen Cao ◽  
Xin Lin

Successive flashover would result in carbonized tracking on insulator surface and cause deterioration to the insulation. Thus, investigation of the tracking can be beneficial in understanding flashover characteristics during long-term operation. In this paper, DC flashover was operated on the insulator, and the image of tracking after successive discharge were captured. Improved differential box-counting method (IDBM) was applied to analyze these images based on fractal theory. Weighted item was suggested during the counting procedure for rectangle image with margin covered by cut-size box. Fractal dimension of the tracking was calculated according to the suggested method. It is claimed that the suggested method could estimate the discharge propagation property and deterioration characteristics on the insulator surface. Moreover, IDBM showed advantages in image pre-processing and deterioration property revealed compared to traditional box-counting method attributing to the consideration of color depth. This image analysis method shows universality in dealing with tracking image and could provide additional information to flashover voltage. This paper suggested a potential approach for the investigation of discharge mechanism and corresponding deterioration in future research.


2021 ◽  
Vol 5 (4) ◽  
pp. 244
Author(s):  
Wolfgang E. Lorenz ◽  
Matthias Kulcke

This article contributes to clarifying the questions of whether and how fractal geometry, i.e., some of its main properties, are suitable to characterize architectural designs. This is done in reference to complexity-related aesthetic qualities in architecture, taking advantage of the measurability of one of them; the fractal dimension. Research in this area so far, has focused on 2-dimensional elevation plans. The authors present several methods to be used on a variety of source formats, among them a recent method to analyze pictures taken from buildings, i.e., 2.5-dimensional representations, to discuss the potential that lies within their combination. Color analysis methods will provide further information on the significance of a multilayered production and observation of results in this realm. In this publication results from the box-counting method are combined with a coordinate-based method for analyzing redundancy of proportions and their interrelations as well as the potential to include further layers of comparison are discussed. It presents a new area of box-counting implementation, a methodologically redesigned gradient analysis and its new algorithm as well as the combination of both. This research shows that in future systems it will be crucial to integrate several strategies to measure balanced aesthetic complexity in architecture.


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