CONSECUTIVE DIFFERENCES AS A METHOD OF SIGNAL FRACTAL ANALYSIS

We propose a new method for calculating fractal dimension (DF) of a signal y(t), based on coefficients [Formula: see text], mean absolute values of its nth order derivatives (consecutive finite differences for sampled signals). We found that logarithms of [Formula: see text], n = 2,3,…,n max , exhibited linear dependence on n: [Formula: see text] with stable slopes and Y-intercepts proportional to signal DF values. Using a family of Weierstrass functions, we established a link between Y-intercepts and signal fractal dimension: [Formula: see text] and calculated parameters A(n max ) and B(n max ) for n max = 3,…,7. Compared to Higuchi's algorithm, advantages of this method include greater speed and eliminating the need to choose value for k max , since the smallest error was obtained with n max = 3.

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Bouguer density determination by fractal analysis

Geophysics ◽

10.1190/1.1442909 ◽

1990 ◽

Vol 55 (7) ◽

pp. 932-935 ◽

Cited By ~ 12

Author(s):

Freyr Thorarinsson ◽

Stefan G. Magnusson

Keyword(s):

Fractal Dimension ◽

Fractal Analysis ◽

Gravity Data ◽

Bouguer Anomaly ◽

New Method ◽

Data Sets ◽

Isostatic Compensation ◽

Topographic Features ◽

Density Determination ◽

Density Values

Density values for the Bouguer reduction of two gravity data sets from Iceland are determined using a new method based on minimization of the roughness of the Bouguer anomaly surface. The fractal dimension of the surface is used as a gauge of the roughness. The analysis shows the size of topographic features supported by crust without isostatic compensation to be 25 to 30 km in southwest Iceland and 9 to 10 km inside the active rifting zone. The densities selected for these areas are 2490 and [Formula: see text], respectively.

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FRACTAL ANALYSIS OF AMPHIBOLE AGGREGATION GROWTH FROM A BASALTIC MELT AND RESIDUAL MELT AT HIGH PRESSURE AND HIGH TEMPERATURE

Fractals ◽

10.1142/s0218348x18500329 ◽

2018 ◽

Vol 26 (03) ◽

pp. 1850032

Author(s):

XIANXU HU ◽

BO ZHANG ◽

QIZHE TANG ◽

JINGUI XU ◽

DAWEI FAN ◽

...

Keyword(s):

High Pressure ◽

Fractal Dimension ◽

High Temperature ◽

Fractal Analysis ◽

Crystallization Temperature ◽

Texture Evolution ◽

Dominant Factor ◽

Dimension Formula ◽

Basaltic Melt ◽

Residual Melt

The aim of this work is to quantitatively explore the texture evolution of amphibole aggregation and residual melt with pressure and temperature. The amphibole aggregation growth from a basaltic melt and the residual melt at high pressure (0.6–2.6[Formula: see text]GPa) and high temperature (860–970[Formula: see text]C) exhibit statistical self-similarity which made us consider studying such characteristic by fractal analysis. The bi-phase box counting method was applied for fractal analysis of each product to identify the fractal phase and the fractal dimension was estimated. In the experimental products, the residual melt is identified as the fractal and amphibole as the Euclidean except for one experiment. The results show that the residual melt can be quantified by the fractal dimension [Formula: see text] within the range of 1.782–1.848. The temperature has a significant effect on the morphology of amphibole and the fractal dimension of the residual melt. The higher the crystallization temperature is, the more regular the amphibole grains are. At lower temperature (from 860[Formula: see text]C to 915[Formula: see text]C), the fractal dimension of the residual melt decreased with the increasing crystallization temperature, but at higher temperature (970[Formula: see text]C), the fractal phase changed to amphibole and the fractal dimension of amphibole is 1.816. The pressure may be the dominant factor that controls the morphology of the mineral aggregation and the residual melt. The fractal dimension of melt decreased linearly with the increasing pressure and if the linear relationship between the fractal dimension and pressure can be further verified in the future, it can be used as a potential geological barometer.

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Heterogeneity of Cerebral Blood Flow: a Fractal Approach

Nuklearmedizin ◽

10.1055/s-0038-1632242 ◽

2000 ◽

Vol 39 (02) ◽

pp. 37-42 ◽

Cited By ~ 14

Author(s):

P. Hartikainen ◽

J. T. Kuikka

Keyword(s):

Blood Flow ◽

Cerebral Blood Flow ◽

Fractal Dimension ◽

Frontal Lobe ◽

Fractal Analysis ◽

Relative Dispersion ◽

Emission Computed Tomography ◽

Brain Blood Flow ◽

Healthy Controls ◽

Fractal Approach

Summary Aim: We demonstrate the heterogeneity of regional cerebral blood flow using a fractal approach and singlephoton emission computed tomography (SPECT). Method: Tc-99m-labelled ethylcysteine dimer was injected intravenously in 10 healthy controls and in 10 patients with dementia of frontal lobe type. The head was imaged with a gamma camera and transaxial, sagittal and coronal slices were reconstructed. Two hundred fifty-six symmetrical regions of interest (ROIs) were drawn onto each hemisphere of functioning brain matter. Fractal analysis was used to examine the spatial heterogeneity of blood flow as a function of the number of ROIs. Results: Relative dispersion (= coefficient of variation of the regional flows) was fractal-like in healthy subjects and could be characterized by a fractal dimension of 1.17 ± 0.05 (mean ± SD) for the left hemisphere and 1.15 ± 0.04 for the right hemisphere, respectively. The fractal dimension of 1.0 reflects completely homogeneous blood flow and 1.5 indicates a random blood flow distribution. Patients with dementia of frontal lobe type had a significantly lower fractal dimension of 1.04 ± 0.03 than in healthy controls. Conclusion: Within the limits of spatial resolution of SPECT, the heterogeneity of brain blood flow is well characterized by a fractal dimension. Fractal analysis may help brain scientists to assess age-, sex- and laterality-related anatomic and physiological changes of brain blood flow and possibly to improve precision of diagnostic information available for patient care.

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Optimal Density Determination of Bouguer Anomaly Using Fractal Analysis (Case of study: Charak area,IRAN)

JOURNAL OF ADVANCES IN PHYSICS ◽

10.24297/jap.v1i1.2140 ◽

2005 ◽

Vol 1 (1) ◽

pp. 21-24

Author(s):

Hamid Reza Samadi

Keyword(s):

Surface Roughness ◽

Fractal Dimension ◽

Fractal Analysis ◽

Fractal Geometry ◽

Host Rock ◽

Bouguer Anomaly ◽

Research Area ◽

Density Difference ◽

Optimal Density

In exploration geophysics the main and initial aim is to determine density of under-research goals which have certain density difference with the host rock. Therefore, we state a method in this paper to determine the density of bouguer plate, the so-called variogram method based on fractal geometry. This method is based on minimizing surface roughness of bouguer anomaly. The fractal dimension of surface has been used as surface roughness of bouguer anomaly. Using this method, the optimal density of Charak area insouth of Hormozgan province can be determined which is 2/7 g/cfor the under-research area. This determined density has been used to correct and investigate its results about the isostasy of the studied area and results well-coincided with the geology of the area and dug exploratory holes in the text area

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Fractal Dimension and Retinal Pathology: A Meta-Analysis

Applied Sciences ◽

10.3390/app11052376 ◽

2021 ◽

Vol 11 (5) ◽

pp. 2376

Author(s):

Sam Yu ◽

Vasudevan Lakshminarayanan

Keyword(s):

Fractal Dimension ◽

Fractal Analysis ◽

Meta Analysis ◽

Scientific Evidence ◽

Retinal Disease ◽

Clinical Value ◽

Disease Category ◽

Normal Individuals ◽

Retinal Pathology ◽

Meta Analyses

Due to the fractal nature of retinal blood vessels, the retinal fractal dimension is a natural parameter for researchers to explore and has garnered interest as a potential diagnostic tool. This review aims to summarize the current scientific evidence regarding the relationship between fractal dimension and retinal pathology and thus assess the clinical value of retinal fractal dimension. Following the PRISMA guidelines, a literature search for research articles was conducted in several internet databases (EMBASE, MEDLINE, Web of Science, Scopus). This led to a result of 28 studies included in the final review, which were analyzed via meta-analysis to determine whether the fractal dimension changes significantly in retinal disease versus normal individuals. From the meta-analysis, summary effect sizes and 95% confidence intervals were derived for each disease category. The results for diabetic retinopathy and myopia suggest decreased retinal fractal dimension for those pathologies with the association for other diseases such as diabetes mellitus, hypertension, and glaucoma remaining uncertain. Due to heterogeneity in imaging/fractal analysis setups used between studies, it is recommended that standardized retinal fractal analysis procedures be implemented in order to facilitate future meta-analyses.

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LATTICE DYNAMICS OF THE BINARY APERIODIC CHAINS OF ATOMS I: FRACTAL DIMENSION OF PHONON SPECTRA

International Journal of Modern Physics B ◽

10.1142/s0217979295000628 ◽

1995 ◽

Vol 09 (12) ◽

pp. 1429-1451 ◽

Cited By ~ 4

Author(s):

WŁODZIMIERZ SALEJDA

Keyword(s):

Fractal Dimension ◽

Lattice Dynamics ◽

Nearest Neighbor ◽

Average Distance ◽

Local Maximum ◽

Fractal Dimensions ◽

Dimension Formula ◽

Phonon Spectra ◽

Wide Range ◽

Silver Mean

The microscopic harmonic model of lattice dynamics of the binary chains of atoms is formulated and studied numerically. The dependence of spring constants of the nearest-neighbor (NN) interactions on the average distance between atoms are taken into account. The covering fractal dimensions [Formula: see text] of the Cantor-set-like phonon spec-tra (PS) of generalized Fibonacci and non-Fibonaccian aperiodic chains containing of 16384≤N≤33461 atoms are determined numerically. The dependence of [Formula: see text] on the strength Q of NN interactions and on R=mH/mL, where mH and mL denotes the mass of heavy and light atoms, respectively, are calculated for a wide range of Q and R. In particular we found: (1) The fractal dimension [Formula: see text] of the PS for the so-called goldenmean, silver-mean, bronze-mean, dodecagonal and Severin chain shows a local maximum at increasing magnitude of Q and R>1; (2) At sufficiently large Q we observe power-like diminishing of [Formula: see text] i.e. [Formula: see text], where α=−0.14±0.02 and α=−0.10±0.02 for the above specified chains and so-called octagonal, copper-mean, nickel-mean, Thue-Morse, Rudin-Shapiro chain, respectively.

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Outer Cutoff Value for the Box-Counting Method for Fractal Analysis of the Nucleus Using Kirsch Edge Detection

Acta Cytologica ◽

10.1159/000512096 ◽

2020 ◽

pp. 1-8

Author(s):

Haruhiko Yoshioka ◽

Kouki Minami ◽

Hirokazu Odashima ◽

Keita Miyakawa ◽

Kayo Horie ◽

...

Keyword(s):

Fractal Dimension ◽

Edge Detection ◽

Fractal Analysis ◽

Nuclear Size ◽

Original Position ◽

Cytological Diagnosis ◽

Cutoff Value ◽

Position Bias ◽

Box Counting ◽

Nuclear Rotation

Objective: The complexity of chromatin (i.e., irregular geometry and distribution) is one of the important factors considered in the cytological diagnosis of cancer. Fractal analysis with Kirsch edge detection is a known technique to detect irregular geometry and distribution in an image. We examined the outer cutoff value for the box-counting (BC) method for fractal analysis of the complexity of chromatin using Kirsch edge detection. Materials: The following images were used for the analysis: (1) image of the nucleus for Kirsch edge detection measuring 97 × 122 pix (10.7 × 13.4 μm) with a Feret diameter of chromatin mesh (n = 50) measuring 17.3 ± 1.8 pix (1.9 ± 0.5 μm) and chromatin network distance (n = 50) measuring 4.4 ± 1.6 pix (0.49 ± 0.18 μm), and (2) sample images for Kirsch edge detection with varying diameters (10.4, 15.9, and 18.1 μm) and network width of 0.4 μm. Methods: Three types of bias that can affect the outcomes of fractal analysis in cytological diagnosis were defined. (1) Nuclear position bias: images of 9 different positions generated by shifting the original position of the nucleus in the middle of a 256 × 256 pix (28.1 μm) square frame in 8 compass directions. (2) Nuclear rotation bias: images of 8 different rotations obtained by rotating the original position of the nucleus in 45° increments (0°, 45°, 90°, 135°, 180°, 225°, 270°, and 315°). (3) Nuclear size bias: images of varying size (diameter: 190 pix [10.4 μm], 290 pix [15.9 μm], and 330 pix [18.1 μm]) with the same mesh pattern (network width: 8 pix [0.4 μm]) within a 512 × 512 pix square. Different outer cutoff values for the BC method (256, 128, 64, 32, 16, and 8 pix) were applied for each bias to assess the fractal dimension and to compare the coefficient of variation (CV). Results: The BC method with the outer cutoff value of 32 pix resulted in the least variation of fractal dimension. Specifically, with the cutoff value of 32 pix, the CV of nuclear position bias, nuclear rotation bias, and nuclear size bias were <1% (0.1, 0.4, and 0.3%, respectively), with no significant difference between the position and rotation bias (p = 0.19). Our study suggests that the BC method with the outer cutoff value of 32 pix is suitable for the analysis of the complexity of chromatin with chromatin mesh.

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THE METHOD OF ESTIMATION FRACTAL DIMENSION OF THE TELECOMMUNICATION NETWORK

Электросвязь ◽

10.34832/elsv.2021.22.9.003 ◽

2021 ◽

Author(s):

Ф.Х. НАХЛИ ◽

А.И. ПАРАМОНОВ

Keyword(s):

Fractal Dimension ◽

Comparative Analysis ◽

Telecommunication Network ◽

Shortest Paths ◽

Network Connectivity ◽

Telecommunication Networks ◽

New Method ◽

Topological Properties

Анализируется фрактальная размерность (ФР) сети связи и ее использование для исследования и планирования сетей связи. Рассматривается применение метода «выращивания кластера» для оценки ФР и предлагается новый метод определения ФР сети, основанный на оценивании связности сети путем поиска кратчайших путей. Показано, что оценка ФР сети является дополнительной характеристикой, отражающей топологические свойства сети. Дается сравнительный анализ предложенного метода и «выращивания кластера». Полученные результаты позволяют выбрать метод и получить оценки ФР сети в зависимости от ее особенностей. The paper analyzes the fractal dimension of the network and its use for telecommunication networks research and planning. The analysis of the "cluster growing" method for assessing the fractal dimension is given and a new method for assessing the fractal dimensionof anetwork is proposed, based onassessing the network connectivity by finding the shortest paths. The article shows that the assessment of the fractal dimension of the network is an additional characteristic that reflects the topological properties of the network. Comparative analysis of the proposed method and "cluster growing" is given. The results obtained make it possible to select a method and obtain estimates of the fractal dimension of the network, depending on its features.

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Analisis Dimensi Fraktal Kejadian Gempa Dl Laut Banda Indonesia

Jurnal Kelautan Tropis ◽

10.14710/jkt.v19i2.837 ◽

2016 ◽

Vol 19 (2) ◽

pp. 108

Author(s):

Sugeng Widada

Keyword(s):

Fractal Dimension ◽

Fractal Analysis ◽

Fractal Approach ◽

Study Support ◽

Tectonic System

The Banda Sea region is an active earthquakes area which indicated by mean monthly incident of quakes more than 220. The condition is caused the area being located in the triple jucntion. Earthquakes system in this region which occur during September 2015 up to October 2016 is analyzed by fractal approach to investigate the subduction system.Earthquakes system is chaotic, so can be quantified using fractal concept. Quantify result of Banda Sea earthquakes system using Aki method is fractal dimension 2.08. It indicates that the slab was fractured by some fault in form an angle or upright possition with the subduction strike. Such a thing also be proven by the fact that the length zone of slab moved during each earthquake is not same, the variation is about 6 – 1,056 m. Based on the fractal analysis, also be identified that about 6.25 magnitute six earthquakes are expected each year. The result of study support the previous studies which propose that the tectonic system in Banda Sea region is very complex. Keywards: Earthuakes system, fractal, Banda Sea Kawasan Laut Banda merupakan daerah aktif gempa yang ditunjukan dengan kejadian gempa rata-rata bulanan Iebih dan 220. Keadaan ini dapat dimengerti mengingat kawasan tersebut merupakan pertemuan tiga buah lempeng yang bergerak. Pola kegempaan di daerah tesebut yang tejadi pada September 2015 hingga Oktober 2016 dicoba dianalisa menggunakan pendekatan fraktal untuk mengetahui pola subduksi di daerah tersebut. Pola kegempaan merupakan suatu kejadian yang chaos, sehingga dapat dilakukan kuantisasi berdasarkan konsep fraktal. Hasil kuantisasi pola gempa Laut Banda meggunakan metode Aki diperoleh dimensi fraktal 2,08. Hal ini menunjukan bahwa slab yang menunjam dan bergerak sehingga menimbulkan gempa terbagi dalarn beberapa bagian melalui suatu sesar yang menyududut / tegak lurus jurus subduksi. Keadaan ini dikuatkan oleh hasil perhitungan panjang daerah yang bergerak untuk setiap kejadian gempa tidak sama, yaitu bervariasi dari 6 – 1.056 m. Berdasarkan analisa fraktal tersebut juga diketahui bahwa gempa dengan magnitudo 6,25 akan terjadi 6 kali dalam satu tahun. Hasil penelitian ini mendukung hasil penelitian terdahulu yang menyatakan bahwa tatanan tektonik di daerah Laut Banda sangat kompleks. Kata Kunci: Pole gempa, fraktal, Laut Banda

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Application of Fractal Theory in Aggregate Gradation Research

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.204-208.1923 ◽

2012 ◽

Vol 204-208 ◽

pp. 1923-1928

Author(s):

Bo Tan ◽

Rui Hua Yang ◽

Yan Ting Lai

Keyword(s):

Fractal Dimension ◽

Fractal Theory ◽

Asphalt Mixture ◽

Fractal Dimensions ◽

Aggregate Gradation ◽

Structure Characteristics ◽

Aggregate Distribution ◽

Dimension Formula ◽

Mass Fractal ◽

Fractal Characteristics

The paper presents the fractal dimension formula of distribution of asphalt mixture aggregate diameter by the deducing mass fractal characteristics function. Taking AC-20 and SMA-20 as examples, selected 6 groups of representative grading curves within the grading envelope proposed by the present specification, and calculated their fractal dimensions. The asphalt mixture gradation has fractal dimension D (D∈(1,3)), and the fractal of continuous gradation is single while the fractal of gap-gradation shows multi-fractal with 4.75 as the dividing point. Fractal dimension of aggregate gradation of asphalt mixture reflect the structure characteristics of aggregate distribution, that is, finer is aggregate, bigger is the fractal dimension.

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