Showing papers on "Fractal dimension published in 2019"

Modeling attractors of chaotic dynamical systems with fractal–fractional operators

Abstract: In this paper, newly proposed differential and integral operators called the fractal–fractional derivatives and integrals have been used to predict chaotic behavior of some attractors from applied mathematics. These new differential operators are convolution of fractal derivative with power law, exponential decay law with Delta-Dirac property and the generalized Mittag-Leffler function with Delta-Dirac property, respectively. These new operators will be able to capture self-similarities in the chaotic attractors under investigation. We presented, in general, three numerical schemes that can be used to solve such systems of nonlinear differential equations. The general conditions for the existence and the uniqueness of the exact solutions are obtained. These new differential and integral operators applied in five selected chaotic attractors with numerical simulations for varying values of fractional order α and the fractal dimension τ have yielded very interesting results. It is to be believed that this research study will open new doors for in-depth investigation in dynamical systems of varying nature.

On two-scale dimension and its applications

Abstract: Dimension or scale is everything. When a thing is observed by different scales, different results can be obtained. Two scales are enough for most of practical problems, and a new definition of a two-scale dimension instead of the fractal dimension is given to deal with discontinuous problems. Fractal theory considers a self-similarity pattern, which cannot be found in any a real problem, while the two-scale theory observes each problem with two scales, the large scale is for an approximate continuous problem, where the classic calculus can be fully applied, and on the smaller scale, the effect of the porous structure on the properties can be easily elucidated. This paper sheds a new light on applications of fractal theory to real problems.

Analysis of pore size distribution and fractal dimension in tight sandstone with mercury intrusion porosimetry

Abstract: The pore size distribution (PSD) and fractal dimension of tight sandstone are studied with mercury intrusion porosimetry (MIP). Four different fractal models are used to calculate fractal dimensions from mercury intrusion capillary pressure (MICP) and mercury extrusion capillary pressure (MECP) respectively, and the physical meanings of the fractal dimensions calculated from different fractal models are discussed. The fractal characteristics of tight sandstone pore spaces with unimodal and bimodal PSD are compared. The correlations between fractal dimensions and tight sandstone petrophysical properties are researched, and the optimal fractal model for formation evaluation is recommended. Finally, the key parameters influencing tight sandstone permeability are investigated.

Diffusion- and reaction-limited cluster aggregation revisited

Abstract: We simulated irreversible aggregation of non-interacting particles and of particles interacting via repulsive and attractive potentials explicitly implementing the rotational diffusion of aggregating clusters. Our study confirms that the attraction between particles influences neither the aggregation mechanism nor the structure of the aggregates, which are identical to those of non-interacting particles. In contrast, repulsive particles form more compact aggregates and their fractal dimension and aggregation times increase with the decrease of the temperature. A comparison of the fractal dimensions obtained for non-rotating clusters of non-interacting particles and for rotating clusters of repulsive particles provides an explanation for the conformity of the respective values obtained earlier in the well established model of diffusion-limited cluster aggregation neglecting rotational diffusion and in experiments on colloidal particles.

Differential box counting methods for estimating fractal dimension of gray-scale images: A survey

Abstract: Fractal dimension is extensively in use as features in computer vision applications to characterize roughness and self-similarity of objects in an image for many years. These features have been adopted successfully mainly in texture segmentation and classification. Differential box counting method is one of the widely accepted approaches, those exist in literature to estimate fractal dimension of an image. In this work, we comprehensively reviewed the available differential box counting methods. First, the differential box counting method is discussed in detail along with its computer vision applications and drawbacks. Second, various variants of differential box counting method are thoroughly studied and grouped using different parameters of differential box counting method. Third, the synthetic and real-world databases, considered for demonstrating experimental results by the state-of-the-art methods have been presented. Fourth, some of the state-of-the-art methods have been implemented and corresponding results obtained in this study are reported. Fifth, three evaluation metrics have also been reviewed. However, these metrics work only for synthetic fractal Brownian motion images because the theoretical fractal dimension values for these images are known and have been used as a set of ground truths. Finally, we concluded the status of differential box counting methods and explored the possible future directions.

A physical interpretation of fractional viscoelasticity based on the fractal structure of media: Theory and experimental validation

Abstract: In this work, a physical connection between the fractional time derivative and fractal geometry of fractal media is developed and applied to viscoelasticity and thermal diffusion in elastomers. Integral to this formulation is the application of both the fractal dimension and the spectral dimension which characterizes diffusion in fractal media. The methodology extends the generalized molecular theory of Rouse and Zimm where generalized Gaussian structures (GGSs) replace the Rouse matrix with the generalized Gaussian Rouse matrix (GRM). Importantly, the Zimm model is extended to fractal media where the new relaxation formulation contains internal state variables that naturally depend on the fractional time derivative of deformation. Through the use of thermodynamic laws in fractal media, we derive the linear fractional model of viscoelasticity based on both spectral and fractal dimensions. This derivation shows how the order of the fractional derivative in the linear fractional model of viscoelasticity is a rate dependent material property that is strongly correlated with fractal and spectral dimensions in fractal media. To validate the correlation between fractional rates and fractal material structure, we measure the viscoelasticity and thermal diffusion of two different dielectric elastomers: Very High Bond (VHB) 4910 and VHB 4949. Using Bayesian uncertainty quantification (UQ) based on uniaxial stress–strain measurements, the fractional order of the derivative in the linear fractional model of viscoelasticity is quantified. Two dimensional fractal dimensions are also independently quantified using the box counting method. Lastly, the diffusion equation in fractal media is inferred from experiments using Bayesian UQ to quantify the spectral dimension by heating the polymer locally with a laser beam and quantifying thermal diffusion. Comparing theory to experiments, a strong correlation is found between the viscoelastic fractional order obtained from stress–strain measurements in comparisons with independent predictions of fractional viscoelasticity based on the fractal structure and fractional thermal diffusion rates.

Differential and integral operators with constant fractional order and variable fractional dimension

Abstract: The complexities of nature have pushed humankind to construct complex mathematical formula that can be used to capture such natural occurrence. Very recently the concept differential and integral operators with fractional order and fractal dimension were introduced. The concept has opened new doors for investigations. In this paper, we present a step forward, where the constant fractal dimension is replaced by variable dimension. We present in detail some properties of this new operators, we suggested a new numerical approach that can be used to solve differential and integral equations associate to this operators. We presented some examples and simulations are presented to underpin the strength of the new operators. We strongly believe that this paper will open many new doors of investigation toward modeling real world problems.

Pore structure and fractal characteristics of Ordovician Majiagou carbonate reservoirs in Ordos Basin, China

Abstract: Fractal analysis was performed on carbonate core plug samples from the Ordovician Majiagou carbonate reservoirs in the Ordos Basin using mercury intrusion capillary pressure (MICP), nuclear magnetic resonance (NMR), and the x-ray computed tomography (CT) measurements to improve our understanding of the pore structure characteristics. The relationships between pore structure parameters and the fractal dimensions were investigated. The pore systems are dominated by secondary intercrystalline pores and enlarged dissolution pores as well as microfractures. The fractal curves from MICP analysis break into two segments at the Swanson’s parameter. The small pore-throat systems can be described by the fractal theory, whereas pores connected by relatively large throats (greater than the pore-throat radius at the Pittman’s hyperbola’s apex) are not cylindrical in shape, cannot be described by a capillary tube model, and tend to have apparent fractal dimensions larger than 3.0. The fact that the entirety of the capillary curve cannot be fit by a single fractal dimension implies that there are multiple pore systems present with different fractal dimensions. The CT analysis shows that the pores are dispersed in the three-dimensional spaces mainly with elliptical shapes. The NMR measurements are sensitive to pore-body size and MICP probes pore-throat dimensions, the latter being complementary to the pore-body–size distribution. None of the CT, MICP, and NMR techniques provide “right” or “wrong” answers to the pore-throat systems, but they probe different aspects of the pore systems. This study assumes the pore shapes to be spherical in general, and then the fractal dimension is calculated from the NMR transverse relaxation time (T2) spectrum. The fractal dimensions of all the samples are calculated, and the accuracy of the fractal model is verified by the high regression coefficients. Almost all the pore systems can be described by fractal theory, and the fractal dimensions are strongly correlated with the T2 values separating the immovable fluid and the free fluid. Microfractures may bias T2 toward larger values, making it hard to derive fractal dimensions from NMR measurements. The coexistence of small pores (pore radius 50 μm) results in a heterogeneous pore distribution and a high fractal dimension. Reservoir quality increases with the complexity degree of the microscopic pore structure. Conversely, samples that are dominated by small pore systems tend to have a lower fractal dimension and a less complex pore structure.

What is the dimension of your binary data

Abstract: Many 0/1 datasets have a very large number of variables; on the other hand, they are sparse and the dependency structure of the variables is simpler than the number of variables would suggest. Defining the effective dimensionality of such a dataset is a nontrivial problem. We consider the problem of defining a robust measure of dimension for 0/1 datasets, and show that the basic idea of fractal dimension can be adapted for binary data. However, as such the fractal dimension is difficult to interpret. Hence we introduce the concept of normalized fractal dimension. For a dataset $D$, its normalized fractal dimension is the number of columns in a dataset $D'$ with independent columns and having the same (unnormalized) fractal dimension as $D$. The normalized fractal dimension measures the degree of dependency structure of the data. We study the properties of the normalized fractal dimension and discuss its computation. We give empirical results on the normalized fractal dimension, comparing it against baseline measures such as PCA. We also study the relationship of the dimension of the whole dataset and the dimensions of subgroups formed by clustering. The results indicate interesting differences between and within datasets.

GIS-based landslide susceptibility mapping using hybrid integration approaches of fractal dimension with index of entropy and support vector machine

Abstract: The loess area in the northern part of Baoji City, Shaanxi Province, China is a region with frequently landslide occurrences. The main aim of this study is to quantitatively predict the extent of landslides using the index of entropy model (IOE), the support vector machine model (SVM) and two hybrid models namely the F-IOE model and the F-SVM model constructed by fractal dimension. First, a total of 179 landslides were identified and landslide inventory map was produced, with 70% (125) of the landslides which was optimized by 10-fold cross-validation being used for training purpose and the remaining 30% (54) of landslides being used for validation purpose. Subsequently, slope angle, slope aspect, altitude, rainfall, plan curvature, distance to rivers, land use, distance to roads, distance to faults, normalized difference vegetation index (NDVI), lithology, and profile curvature were considered as landslide conditioning factors and all factor layers were resampled to a uniform resolution. Then the information gain ratio of each conditioning factors was evaluated. Next, the fractal dimension for each conditioning factors was calculated and the training dataset was used to build four landslide susceptibility models. In the end, the receiver operating characteristic (ROC) curves and three statistical indexes involving positive predictive rate (PPR), negative predictive rate (NPR) and accuracy (ACC) were applied to validate and compare the performance of these four models. The results showed that the F-SVM model had the highest PPR, NPR, ACC and AUC values for training and validation datasets, respectively, followed by the F-IOE model. Finally, it is concluded that the F-SVM model performed best in all models, the hybrid model built by fractal dimension has advantages than original model, and can provide reference for local landslide prevention and decision making.

Integrating Three-Dimensional Benthic Habitat Characterization Techniques into Ecological Monitoring of Coral Reefs

Abstract: Long-term ecological monitoring of reef fish populations often requires the simultaneous collection of data on benthic habitats in order to account for the effects of these variables on fish assemblage structure. Here, we described an approach to benthic surveys that uses photogrammetric techniques to facilitate the extraction of quantitative metrics for characterization of benthic habitats from the resulting three-dimensional (3D) reconstruction of coral reefs. Out of 92 sites surveyed in the Northwestern Hawaiian Islands, photographs from 85 sites achieved complete alignment and successfully produced 3D reconstructions and digital elevation models (DEMs). Habitat metrics extracted from the DEMs were generally correlated with one another, with the exception of curvature measures, indicating that complexity and curvature measures should be treated separately when quantifying the habitat structure. Fractal dimension D64, calculated by changing resolutions of the DEMs from 1 cm to 64 cm, had the best correlations with other habitat metrics. Fractal dimension was also less affected by changes in orientations of the models compared to surface complexity or slope. These results showed that fractal dimension can be used as a single measure of complexity for the characterization of coral reef habitats. Further investigations into metrics for 3D characterization of habitats should consider relevant spatial scales and focus on obtaining variables that can complement fractal dimension in the characterization of reef habitats.

Atypical temporal-scale-specific fractal changes in Alzheimer's disease EEG and their relevance to cognitive decline.

Abstract: Recent advances in nonlinear analytic methods for electroencephalography have clarified the reduced complexity of spatiotemporal dynamics in brain activity observed in Alzheimer's disease (AD). However, there are far fewer studies exploring temporal scale dependent fractal properties in AD, despite the importance of studying the dynamics of brain activity within physiologically relevant frequency ranges. Higuchi's fractal dimension is a widely used index for evaluating fractality in brain activity, but temporal-scale-specific characteristics are lost due to its requirement of averaging over the entire range of temporal scales. In this study, we adapted Higuchi's fractal algorithm into a method for investigating temporal-scale-specific fractal properties. We then compared the values of the temporal-scale-specific fractal dimension between healthy control (HC) and AD patient groups. Our data indicate that relative to the HC group, the AD group demonstrated reduced fractality at both slow and fast temporal scales. Moreover, we confirmed that the fractality at fast temporal scales correlates with cognitive decline. These properties might serve as a basis for a useful approach to characterizing temporal neural dynamics in AD or other neurodegenerative disorders.