Fractal Dimension Estimation Research Articles

Fractal Time Series: Background, Estimation Methods, and Performances.

Over the past 40years, from its classical application in the characterization of geometrical objects, fractal analysis has been progressively applied to study time series in several different disciplines. In neuroscience, starting from identifying the fractal properties of neuronal and brain architecture, attention has shifted to evaluating brain signals in the time domain. Classical linear methods applied to analyzing neurophysiological signals can lead to classifying irregular components as noise, with a potential loss of information. Thus, characterizing fractal properties, namely, self-similarity, scale invariance, and fractal dimension (FD), can provide relevant information on these signals in physiological and pathological conditions. Several methods have been proposed to estimate the fractal properties of these neurophysiological signals. However, the effects of signal characteristics (e.g., its stationarity) and other signal parameters, such as sampling frequency, amplitude, and noise level, have partially been tested. In this chapter, we first outline the main properties of fractals in the domain of space (fractal geometry) and time (fractal time series). Then, after providing an overview of the available methods to estimate the FD, we test them on synthetic time series (STS) with different sampling frequencies, signal amplitudes, and noise levels. Finally, we describe and discuss the performances of each method and the effect of signal parameters on the accuracy of FD estimation.

Fractal Dimension Calculation Methods and Their Use in the Textile Industry

From the methods of calculating the fractal dimension, the prism method, the cube method, the method of counting cells, instructions on the methods of calculating the fractal dimension using set theory and the block diagrams of algorithms were developed. A software tool was created based on the Hausdorff method of calculating the fractal dimension. Fractal measurement determined by this software can be used in medicine for changes in the vascular system, in the accurate diagnosis of tumors in the brain, in the textile industry, in the textural analysis of the permeability and roughness of fabrics.

Effect of polypropylene fiber and nano-silica on the compressive strength and frost resistance of recycled brick aggregate concrete

Abstracts Wasted clay bricks as coarse aggregate of recycled concrete is an effective solution to save energy and reduce CO2 emissions in the construction industry. However, the mechanical properties and frost resistance of recycled brick aggregate (RBA) concrete are inferior to those of ordinary concrete, which limits its widespread application. In this research, the effects of RBA, polypropylene fiber (PPF) and nano-silica (NS) on the mechanical properties and frost resistance of concrete were investigated. The effect of RBA, PPF, and NS on the compressive strength was quantitatively analyzed, and microstructural analysis and fractal dimension calculation of the concrete were performed. The results show that the concrete compressive strength decreased with the increase in RBA replacement rate, and it was effectively improved by adding PPF and NS (PPF-NS). The compressive strength first increased and then decreased with the increase in PPF and NS. The improvement effect of 0.12% PPF and 2% NS on the compressive strength of 50% replacement rate of RBA concrete was most effective. The gray relational degrees between the compressive strength and RBA, PPF, and NS were 0.6578, 0.8297, and 0.5941, respectively. The frost resistance of PPF-NS modified concrete was better than that of ordinary concrete, mainly manifested in its superior apparent phenomena, mass loss, and strength loss. Compared with normal concrete, the microstructure was denser and the fractal dimension of the cross-section was higher for RBA concrete modified with PPF-NS before and after freeze–thaw cycles.

No evidence for fractal scaling in canopy surfaces across a diverse range of forest types

Abstract Canopy structural complexity is a key emergent property of forest ecosystems that directly relates to their ability to store carbon, cycle water and nutrients, and provide habitat for biodiversity. However, we lack a general framework for quantifying the structural complexity of forest canopies, and it remains to be seen if there are simple rules that explain the huge variation in canopy structure that we observe across the world's forests. A leading candidate for characterizing differences in structural complexity among forest ecosystems are fractal scaling laws, or measures of statistical self‐similarity. If forest canopies were fractal in nature, their structural attributes could be distilled into a single coefficient—their fractal dimension—which could then be used to directly compare structural differences within and across biomes. To test this idea, we used airborne laser scanning (ALS) data acquired across nine landscapes in Australia that span a huge environmental gradient and include everything from dry shrublands, tropical savannas, dense rainforests, to 90 m tall Mountain Ash forests. Using the ALS data, we built high‐resolution 3D canopy height models of each landscape so that we could quantify how closely they followed fractal scaling laws, based on a comprehensive set of fractal dimension estimators. Across all ecosystem types, fractal scaling assumptions were consistently violated, with clear and systematic differences between real‐world forest canopies and simulated fractal surfaces. However, deviations from fractality did vary predictably among sites, with forests in less arid environments, dominated by tall trees with large crowns, exhibiting a higher degree of self‐similarity across scales. Synthesis: Our study conclusively shows that canopy surfaces are not fractal beyond the scale of individual tree crowns. Nevertheless, we still observed ecologically meaningful and generalisable patterns in forest structure across scales, which closely reflect biophysical and physiological constraints on tree size and architecture. These patterns point the way towards a more general framework for characterizing ecological complexity.

Graphene oxide coated fly ash for reinforcing dynamic tensile behaviours of cementitious composites

By virtue of its low cost, high workability, wide availability and suitability for large-volume fabrication, cement-based grouting is the most applied material in underground projects. However, due to the weak tensile strength and high-frequency impact dynamic load disturbance in underground engineering, cement-based grouting is vulnerable to tensile damage, seriously influencing the stability and safety of underground projects. Hence, in this work, the graphene oxide (GO) coated fly ash was adopted to reinforce the dynamic tensile behaviour of the cement-based matrixes to fabricate high-performance, cost-effective and environmentally friendly cementitious composites. The test results show that the GO-coated fly ash could significantly enhance the dynamic tensile performance of the cementitious composites. With only 0.08 wt% GO mixing, the dynamic tensile properties can be strengthened by 19.5–88.5%. After GO nanosheets accelerate the cement hydration process and improve the chemical composition of hydration products, GO nanosheets exhibit macroscopic nucleation and "crack-bridging" effects on the cement matrixes, and the cracks propagation of the hardened specimens would be suppressed under dynamic loading, thus enhancing the tensile resistance of the specimen. The fractal dimension calculation results prove that the GO-coated fly ash-modified cementitious composites have higher integrity after tensile impact, about 2.8–11.9% higher than plain cement-based slurry. Finally, the theoretical analysis reveals that GO nanosheets can enhance the tensile behaviours of the cement matrix by optimizing the dynamic load coefficient. Nevertheless, with the crack propagation, the reinforcing efficiency of GO would be gradually weakened. The findings of this study can promote an understanding of the enhancing mechanisms of GO on the dynamic tensile behaviour of cementitious composites and inspire the potential application of GO-coated fly ash to reduce cement usage in grouting engineering.

A new result on the fractal dimension estimates of random attractor for non-autonomous random 2D stochastic dynamical type systems

We prove some conditions for bounding the fractal dimension of random invariant sets of non-autonomous random dynamical systems on separable Banach spaces. Then we apply these conditions to prove the finiteness of fractal dimension of random attractor for stochastic 2D hydrodynamical type equations with linear additive white noise in bounded domains or unbounded domains satisfying the Poincaré inequality.

An Algorithm for Crack Detection, Segmentation, and Fractal Dimension Estimation in Low-Light Environments by Fusing FFT and Convolutional Neural Network

The segmentation of crack detection and severity assessment in low-light environments presents a formidable challenge. To address this, we propose a novel dual encoder structure, denoted as DSD-Net, which integrates fast Fourier transform with a convolutional neural network. In this framework, we incorporate an information extraction module and an attention feature fusion module to effectively capture contextual global information and extract pertinent local features. Furthermore, we introduce a fractal dimension estimation method into the network, seamlessly integrated as an end-to-end task, augmenting the proficiency of professionals in detecting crack pathology within low-light settings. Subsequently, we curate a specialized dataset comprising instances of crack pathology in low-light conditions to facilitate the training and evaluation of the DSD-Net algorithm. Comparative experimentation attests to the commendable performance of DSD-Net in low-light environments, exhibiting superlative precision (88.5%), recall (85.3%), and F1 score (86.9%) in the detection task. Notably, DSD-Net exhibits a diminutive Model Size (35.3 MB) and elevated Frame Per Second (80.4 f/s), thereby endowing it with the potential to be seamlessly integrated into edge detection devices, thus amplifying its practical utility.

Structural and Optical Properties of N+ Implanted V2O5 Thin Film on Glass Substrate

In this study, we investigate the effect of nitrogen (16 keV N+) ion implantation on V2O5 thin films with varying thicknesses, focusing on their structural, compositional, and optical characteristics. Through X-ray diffraction, we reveal the coexistence of orthorhombic and tetragonal phases, while UV–vis spectroscopy determines the band gap. Atomic force microscopy unveils the intriguing surface morphology, with the thickest film displaying high smoothness. Notably, grain formation and growth dynamics are intricately linked to roughness (α) and fractal dimension (Df) calculations. Additionally, wetting properties are connected to fractal growth in ion-implanted films. This work underscores the stable optical band gap of V2O5 thin films for optical applications and offers insights into managing wetting behaviors in nanofluidics through fractal patterns, potentially impacting nanotechnology and material engineering.

Approach to characterize rock fracture surface: Insight from roughness and fractal dimension

The characterization of rock fracture is essential for revealing and understanding the fracture mechanism. In this research, the fracture of sandstone is characterized from the perspective of the 3D morphology of fracture surface. The 3D morphology is quantified by two fractal dimension (D) calculation methods: cubic covering method (CCM) and surface area method (SAM). The applicability of both methods is verified by two special cases of which one is a 2D plane and another one is a novel surface model with a dimension of 3. However, in actual application, one shortcoming of the CCM rooted in the algorithm is exposed, which introduces a distorted D for the surface with a small degree of undulation. On the contrary, relying on a new algorithm proposed to calculate the fracture surface area, the SAM can completely avoid the distortion problem. Based on the SAM, the 3D morphologies of a series of fracture surfaces of sandstone in the conventional triaxial test are quantified. It is observed that D has a strong correlation with the fracture mechanism. With the transition of failure mechanism from tension fracture to shear fracture, D presents a decreasing trend. The reason underlying this correlation is analyzed. In Discussion, several improved CCMs are introduced, and whether the improvements have solved the distortion problem is assessed. The accuracy of the SAM is also evaluated using the Takagi fractal surfaces.

Study on the Characteristics of Composite Recycled Aggregate Concrete Based on Box–Behnken Design Response Surface Model

This study investigates the influence of recycled fine aggregates (RFA) and waste concrete powder (WCP) on the compressive strength of concrete. The response surface methodology is employed, considering three factors: the content of WCP, the water–cement ratio, and the replacement ratio of recycled fine aggregates. Compressive tests are conducted at different ages (3 days, 28 days, and 90 days). A statistical approach is used to establish a response surface model for compressive strength and to verify its fitting with experimental results. By combining qualitative and quantitative analyses, including morphological analysis of SEM images, stratified binarization statistics, and fractal dimension calculations, the mechanisms of the effects of singly and doubly mixed RFA and WCP on the compressive strength of concrete are analyzed. The results show that an increase in the content of WCP from 5% to 15% gradually improves the compressive strength of the concrete. The water–cement ratio significantly affects the compressive strength, with an optimal ratio of 0.43. Replacement with less than 33% fine aggregates does not significantly reduce the compressive strength of the concrete. The concrete specimens with singly mixed fine aggregates exhibit the largest pore area after stratified grayscale binarization, while those with doubly mixed aggregates have the largest area of hydrated calcium silicate. Fractal analysis of the binarized images confirms a positive correlation between the fractal dimension of hydrated calcium silicate and compressive strength and a negative correlation between the fractal dimension of pores and compressive strength.

Estimation of fractal dimension of trees using LiDAR point data with sequential data decimation

Fractal dimensions of trees provide key ecological information regarding tree and forest stand structure. Although LiDAR scanning has the potential to reduce the intensive labor required to estimate fractal dimensions with manual measurements, neither an accuracy assessment comparing with true dimensions nor a guide or method for accurately measuring the dimensions using LiDAR scanners has been presented. This study examined the accuracy of fractal dimensions estimated using terrestrial LiDAR scanning (TLS) and unmanned aerial LiDAR scanning (ULS) through computer simulations and developed an approach to reduce the estimation bias. The true fractal dimensions of digitized trees of red pine (Pinus densiflora), ubame oak (Quercus phillyreoides), and giant timber bamboo (Phyllostachys bambusoides) in 18 site scenarios were calculated by applying the box-counting method to the tree object data. The true fractal dimensions were compared with the estimated dimensions in four TLS and ULS scenarios. Furthermore, a method to reduce the estimation bias was developed by introducing sequential point decimation and modeling the acquisition rate decay, thereby extrapolating the number of voxels before decay. Simple box-counting on highly dense and precise TLS data collected from four positions could retrieve the true fractal dimensions for all 18 site scenarios. However, sparse and less accurate TLS underestimated fractal dimensions in 8 out of 18 cases, and ULS overestimated the dimensions in 29 of 36 cases. The change in the relationship between the number of voxels and mesh size on a log-log plot with sequential decimation could be used to judge whether a set of TLS data can retrieve accurate fractal dimensions by applying simple box-counting to the TLS data. The developed method improved the estimations of fractal dimensions for 30 of the 37 cases where the simple box-counting on the TLS and ULS data provided biased estimations. In contrast, the estimation bias increased in 27 of the 35 cases where the simple box-counting on the TLS and ULS data successfully retrieved the true fractal dimensions.

Statistical performance of local attractor dimension estimators in non-Axiom A dynamical systems.

We investigate various estimators based on extreme value theory (EVT) for determining the local fractal dimension of chaotic dynamical systems. In the limit of an infinitely long time series of an ergodic system, the average of the local fractal dimension is the system's global attractor dimension. The latter is an important quantity that relates to the number of effective degrees of freedom of the underlying dynamical system, and its estimation has been a central topic in the dynamical systems literature since the 1980s. In this work, we propose a framework that combines phase space recurrence analysis with EVT to estimate the local fractal dimension around a particular state of interest. While the EVT framework allows for the analysis of high-dimensional complex systems, such as the Earth's climate, its effectiveness depends on robust statistical parameter estimation for the assumed extreme value distribution. In this study, we conduct a critical review of several EVT-based local fractal dimension estimators, analyzing and comparing their performance across a range of systems. Our results offer valuable insights for researchers employing the EVT-based estimates of the local fractal dimension, aiding in the selection of an appropriate estimator for their specific applications.

The existence conditions for global exponential attractor of non-autonomous evolution equations and applications

Global exponential attractor theory is an effective tool for studying the dynamic of dissipative evolution equations. However, the existence of global exponential attractor for non-autonomous dynamic system is not only a difficult problem, but also an unsolved problem. In this paper, according to the structure of evolution equations, we propose two equivalent conditions for the existence of global exponential attractor of non-autonomous evolution equations. Using these two equivalent conditions, we prove the existence of bounded absorbing set, the existence of global exponentially attracting set for non-autonomous dissipative evolution equations, and study the finite fractal dimension estimation for the global exponentially attracting set of non-autonomous dissipative evolution equations. Finally, as applications, the existence of global exponential attractor is proved for non-autonomous semilinear parabolic equation and the epidemic model. Our results also provide a convenient and effective tool to study the global exponential attractor.

Study on the Damage Mechanism of Sandstone under Different Water Content States

Understanding the failure mechanisms of rocks that are exposed to different water contents is important for rock stability in rock engineering applications, and the quantitative analysis of rock behavior is necessary for predicting and preventing the occurrence of rock failure due to water effects. Mechanical tests using real-time acoustic emission (AE) technology were carried out to reveal the damage evolution in sandstone rocks in a dried state, natural state, and saturated state, which includes a quantitative analysis of AE characteristics and cracking properties. The testing results indicate that with the growth of water content, sandstone rocks show a decreasing trend in strength and tend to experience gentle damage with relatively fewer fractures. The crack morphology of the main fracture surfaces is quantitatively described, including a fractal dimension calculation and cracking length measurements. As the water content rises, when rock failure occurs, a higher AE b-value can be obtained, revealing an increasing proportion of large-scale cracks. The fractal dimension of the acoustic emission hit rate shows that the evolution of rock damage and deformation has self-similarity, that is, the transformation from order to disorder to order, and it is affected by different water contents. The AE waveforms of the sandstone have two dominant frequency bands (0~75 kHz and 75~150 kHz) no matter which water-bearing state they are in. The increase in rock water content has resulted in the decline of AE waveforms located in the range of 200–300 kHz, whereas the rise of AE waveforms is located in the range of 0–50 kHz. The findings of this study deepen our understanding of the mechanism behind rock failure and provide a meaningful reference for disaster assessment and control.

SURFACE ROUGHNESS CHANGE IN THE HEAT TREATMENT OF TI-6AL-4V EXTRA LOW INTERSTITIAL ALLOY

The heat treatment improves the casting results structure and changes the alloys microstructure. The fractal dimension of the Ti-6Al-4V Extra Low Interstitial (ELI) alloy changes with the alteration of surface roughness under various heat treatment conditions. This study used three primary raw materials: first-grade Titanium, pure aluminum, and pure vanadium. The SEM images of the sample were translated into topographic data. The Fractal Dimension (FD) calculation results that the surfaces topography appearing throughout the heat treatment temperatures and has fractal features. The Heat Treatment (HT) process shows a close relationship between heat treatment temperature and fractal dimension. Surface roughness after heat treatment at 1050oC is smoother than at 850oC and 950oC.

Fractal recognition and contact characteristics of ionic electroactive polymer interface based on microstructure analysis

Interface research is one of the basic topics in the study of mass transfer dynamics and actuation deformation processes. The main objective of this paper is to conduct in-depth research on the interface characteristics of ionic electroactive polymers, and this problem has not been studied to a large extent until now. The formation of the interface is closely related to the factors such as substrate pretreatment, the electroless plating process and the adsorption of ions. The interface has an irregular and self-similar microstructure, fractal theory can well analyze its intrinsic fractal characteristics. Based on the three-dimensional rough interface, the fractal characteristics of six two-dimensional cross-sections were identified. The fractal dimension, relative error, correlation coefficient and scale-free interval obtained by these six methods were quantitatively compared. The interfacial morphological characteristics of the ion-activated polymer were analyzed experimentally and the optimal fractal dimension calculation method was determined. A fractal contact model of the ionic electroactive polymer was developed and the calculation of the peel strength of the interface was proposed. The results show that RMSM is a feasible and effective method for the fractal characterization of the profiles of ionically electroactive polymers. The interfacial contact model based on fractal features can effectively characterize the peeling strength of the interface, which can provide a reference for later microscopic fractal studies on the theories of mass transfer process, actuation deformation and ion migration.

Alpha magnitude

Magnitude is an isometric invariant for metric spaces that was introduced by Leinster around 2010, and is currently the object of intense research, since it has been shown to encode many known invariants of metric spaces. In recent work, Govc and Hepworth introduced persistent magnitude, a numerical invariant of a filtered simplicial complex associated to a metric space. Inspired by Govc and Hepworth's definition, we introduce alpha magnitude and investigate some of its key properties. Heuristic observations lead us to conjecture a relationship with the Minkowski dimension of compact subspaces of Euclidean space. Finally, alpha magnitude presents computational advantages over both magnitude as well as Rips magnitude, and we thus propose it as a new measure for the estimation of fractal dimensions of real-world data sets that is easily computable.

A Multi-Spectral Fractal Image Model and Its Associated Fractal Dimension Estimator

We propose both a probabilistic fractal model and fractal dimension estimator for multi-spectral images. The model is based on the widely known fractional Brownian motion fractal model, which is extended to the case of images with multiple spectral bands. The model is validated mathematically under the assumption of statistical independence of the spectral components. Using this model, we generate several synthetic multi-spectral fractal images of varying complexity, with seven statistically independent spectral bands at specific wavelengths in the visible domain. The fractal dimension estimator is based on the widely used probabilistic box-counting classical approach extended to the multivariate domain of multi-spectral images. We validate the estimator on the previously generated synthetic multi-spectral images having fractal properties. Furthermore, we deploy the proposed multi-spectral fractal image estimator for the complexity assessment of real remotely sensed data sets and show the usefulness of the proposed approach.

Information–Thermodynamic Method for the Study of Proliferation of Organized Cellular Structure

The aim of the article was to develop an innovative method for the study of cell proliferation based on the information-thermodynamic approach, including the mathematical ratio-the entropy of cell proliferation and an algorithm for the calculation of fractal dimension of the cellular structure. Approbation of this method with pulsed electromagnetic impact on culture in vitro was implemented. It is shown on the basis of experimental data that the organized cellular structure of juvenile human fibroblasts is a fractal. The method makes it possible to determine the stability of the effect on cell proliferation. The prospects for the application of the developed method are discussed.

Thermo-responsive behavior of glycerol-plasticized oleogels stabilized by zein

A series of zein-stabilized oleogels were prepared and plasticized by the addition of glycerol. The rheological and microstructural properties of oleogels were studied as a function of glycerol and zein concentration. The plasticized zein oleogel was best characterized as gels with a weak-link regime. The fractal dimension calculation as an indication of structure density indicated that oleogels at low glycerol content had a loose protein network that became denser with further addition of glycerol. Increasing zein concentration formed a denser oleogel network. The oleogels showed a thermo-responsive rheological property in the range of 25–100 °C that was reversible based on glycerol/zein ratios. The transition from solid-like to liquid-like behavior of oleogels by heating started from 75 °C, where G′ could drop up to two orders of magnitude. Besides, the structural analysis did not show any indication of protein secondary structural denaturation during heating.