Fractal Dimension Estimation Methods Research Articles

The Influence of the Structural Parameters of Nanoporous Alumina Matrices on Optical Properties

In this work, two types of nanoporous alumina membranes were prepared and tested. Structural features of the samples obtained by using different acids were investigated by scanning electron microscopy (SEM). And further SEM-images were analyzed by different types of fractal dimension estimation methods. The transmission and scattering of accelerated He+ ions were studied in experiments on the ion irradiation of dielectric channels based on porous alumina. An ion accelerator was used as a source of the He+ beam with an energy of 1.7 MeV. Ion scattering was studied by Rutherford backscattering spectrometry. Helium transition through nanoporous alumina at various angles between the normal to the sample and the beam direction were observed. It is shown that the porous structure of anodic aluminum oxide is excellent as a dielectric matrix of nanocapillaries. Owing to the small angle scattering, it allows for the transportation of the accelerated charged particles through the dielectric capillaries, and, as a result, the localization of high energy ion irradiation effects. Additionally, according to the transmission of UV–V is spectra, the energy gaps of samples obtained were calculated.

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.

Novel Path Counting-Based Method for Fractal Dimension Estimation of the Ultra-Dense Networks

Next-generation networks, including the Internet of Things (IoT), fifth-generation cellular systems (5G), and sixth-generation cellular systems (6G), suffer from the dramatic increase of the number of deployed devices. This puts high constraints and challenges on the design of such networks. Structural changing of the network is one of such challenges that affect the network performance, including the required quality of service (QoS). The fractal dimension (FD) is considered one of the main indicators used to represent the structure of the communication network. To this end, this work analyzes the FD of the network and its use for telecommunication networks investigation and planning. The cluster growing method for assessing the FD is introduced and analyzed. The article proposes a novel method for estimating the FD of a communication network, based on assessing the network’s connectivity, by searching for the shortest routes. Unlike the cluster growing method, the proposed method does not require multiple iterations, which reduces the number of calculations, and increases the stability of the results obtained. Thus, the proposed method requires less computational cost than the cluster growing method and achieves higher stability. The method is quite simple to implement and can be used in the tasks of research and planning of modern and promising communication networks. The developed method is evaluated for two different network structures and compared with the cluster growing method. Results validate the developed method.

A Fractional Integral and Fractal Dimension-Based Deep Learning Approach for Pavement Crack Detection in Transportation Service Management

With artificial intelligence prevailing in intelligent transportation system, pavement crack detection with deep learning has aroused wide attentions in both academia and transportation sector. Nevertheless, it still remains a challenge to accomplish crack detection due to the complexity in pavement background. Motivated by latest advents in computer vision research, a fractional integral-based filtering method is advocated to remove pavement noise, and a fractal dimension estimation method has also emerged to present shape feature at pixel level, with the multi-scale feature architecture. Therefore, we try to propose a deep learning method, integrating fractional integral with fractal dimension, for crack detection in transportation service management. Firstly, the crack image is taken as input in the bottom-up architecture to extract fractal dimension on multi-scale levels, and a per-level feature unit is built to incorporate maps to make context information flow. Secondly, after fed into a convolutional filter for dimension resizing, all the resized feature maps are next fused at each level to comprise a group network. Finally, extensive experiments are executed on different crack datasets, exhibiting that the proposed method surpasses existing cutting-edge ones in terms of both generalizability and accuracy, with the benefits from not only fractional integral filtering, but also multi-scale fractal dimension features.

Rolling bearing fault diagnosis utilizing variational mode decomposition based fractal dimension estimation method

A novel fractal dimension estimation method based on VMD is proposed in this paper. VMD is utilized to decompose the multi-component signal into several components. Multi-dimensional super-body volume is defined and calculated based on the decomposed components. Fractal dimension is then estimated by the least square method. Simulation results verify that fractal dimension estimation accuracy of the proposed method outperform box counting method and detrended fluctuation analysis. Furthermore, with this novel method, fractal characteristics of vibration signals form rolling bearing are studied. Achievements indicate that vibration signals are characterized by double-scale fractal features. Thus, two fractal dimensions corresponding to the small and large time scales respectively are extracted as feature parameters of vibration signals. Finally, double-scale fractal dimensions are employed for rolling bearing fault diagnosis. Classification results indicate that double-scale fractal dimensions extracted by VMD are capable of expressing fractal characteristics of vibration signals and diagnosing the rolling bearing faults.

Techniques used to calculate shale fractal dimensions involve uncertainties and imprecisions that require more careful consideration

Surface roughness of shales has a key influence on the petroleum resources they are able to store and the fraction of them that can be recovered. The fractal dimension quantifies the degree of roughness and is influenced primarily by the pore surfaces within the shale that typically include micro-, meso-and macro-pores. Isotherms generated by gas adsorption experiments are the common data source used to derive estimates of fractal dimension. The Frenkel-Halsey-Hill fractal technique is the most widely applied fractal dimension estimation method. Other methods can derive fractal dimension from isotherm data but typically the values they generate are different from the Frenkel-Halsey-Hill derived fractal dimension values. Moreover, those differences can vary significantly depending on the type of shales involved. Those shales displaying more complex pore-scale distributions including extensive micro-porosity components tend to be associated with the greatest discrepancies. A comparison of three fractal dimension calculation methods applied to shales reveals aspects of their calculation and interpretation methods that explain the differences in the fractal dimension values they generate. This study identifies the uncertainties that should be taken into account when applying the methods and the appropriate curve fitting optimization configurations that should be evaluated. Taking these factors into account leads to more realistic selections of appropriate fractal dimension values from gas adsorption isotherms of organic-rich shales. Cited as : Wood, D.A. Techniques used to calculate shale fractal dimensions involve uncertainties and imprecisions that require more careful consideration. Advances in Geo-Energy Research, 2021, 5(2): 153-165, doi: 10.46690/ager.2021.02.05

An effective method to compute the box-counting dimension based on the mathematical definition and intervals

Fractal dimension is an appropriate indicator to describe the complexity of a certain geometry, and box-counting analysis is proved to be an effective and appropriate method for fractal dimension estimation which is widely used. However, traditional box-counting methods based on images may not always accurate especially in small box scale due to the restriction of pixels. This paper aims to introduce a method based on generating fractal and determining boxes by rigid mathematical definition to eliminate the deviation of each box scale and to carry out more dependable results. Several samples of the simple and recursive fractal are analyzed to verify the accuracy of the box-counting method based on mathematical definition and interval in this paper.

Fractal dimension of synthesized and natural color images in Lab space

Fractal dimension (FD) is a useful metric for the analysis of natural images that exhibit a high degree of complexity, randomness and irregularity in color and texture. Several approaches exist in the literature to measure FD of gray-scale images. The aim of this study is to introduce a FD estimation method for color images with color proximity in Lab space. The proposed method uses a xy-plane partitioning–shifting mechanism, where the divisors of image size are used as grid sizes. The proposed method simulates on synthesized color fractal Brownian motion (FBM) images, publicly available Brodatz database, Google color fractal images and noisy Brodatz database. The random midpoint displacement algorithm for the formation of gray-scale images is extended in this work to synthesize color FBM images. Noisy Brodatz database is obtained by adding salt-and-pepper noise with different noise densities to understand the behavior of FD. The experimental results illustrate that the proposed method is effective and efficient and outperforms the three state-of-the-art methods by observing the values of two proposed metrics, namely average error and average computed FD. A new mathematical expression for estimating FD of a color image is demonstrated, which relies on the number of edge pixels of individual color channel using multiple linear regression.

Understanding the fracture behavior of brittle and ductile multi-flawed rocks by uniaxial loading by digital image correlation

Fracturing behavior of ductile multi-flawed rock mass is very different from those of brittle multi-flawed rock mass. However, there is a very vague distinction of the fracture mechanism of ductile multi-flawed rock mass from that of brittle multi-flawed rock mass due to the absence of quality experimental data. To evaluate the fracturing behavior and further reveal the failure mechanism of brittle and ductile multi-flawed rock masses, a comparison study on brittle and ductile laboratory rock-like specimens that contain nine pre-existing flaws subjected to uniaxial compression is conducted. The advanced optical technique, Digital Image Correlation (DIC), is adopted to capture the real-time displacement and strain fields on the surface of the rock-like specimens. The fractal dimension estimation method is introduced to quantitatively track the fracture process. The fracture process, the crack coalescence mode and the fracturing mechanism are obtained as well as the stress-strain response. For ductile multi-flawed rock-like specimens, four crack initiation modes and nine types of crack coalescence are observed. For brittle multi-flawed rock like specimens, four crack initiation modes and eight types of crack coalescence are observed. Comparison of fracturing behavior indicates that crack initiation mode transforms from shear crack to tensile crack as the brittleness index increases; the coalescence mode transforms from shear crack coalescence to the coalescence of tensile crack and shear crack; then to the tensile crack coalescence as the brittleness index increases. Observations on the ultimate failure modes indicate that the failure mode transforms from shear failure to the mixed tension-shear failure, then to splitting failure as the brittleness index increases. Most macrocracks in brittle multi-flawed rock-like specimens develop from crack coalescence occurring in a “columnar” pattern, while most macrocracks in ductile multi-flawed rock-like specimens develop from shear cracks. Moreover, the effects of the brittleness index on the complete stress-strain curve, the crack initiation stress and the peak strength are also investigated in detail.

Surface Quality Evaluation in Orthogonal Turn-Milling Based on Box-Counting Method for Image Fractal Dimension Estimation

Turn-milling is a relatively new cutting process combining the both advantages of turning and milling, during which the tool and the workpiece rotate simultaneously. Due to the small axial cutting force, turn-milling is especially suitable for the machining of miniature parts. Usually, the surface quality is characterized by roughness, but the size of miniature parts is difficult to meet the requirements of the sampling length, which will have a huge influence on the characterization results. This paper introduces the box-counting method for image fractal dimension estimation to evaluate the surface quality of miniature parts during orthogonal turn-milling. The fractal dimension is an independent parameter for scale, which is not influenced by the sampling length. The effect of cutting parameters on surface quality of miniature parts during turn-milling is also studied.

SYNTHETIC MINORITY OVERSAMPLING TECHNIQUE AND FRACTAL DIMENSION FOR IDENTIFYING MULTIPLE SCLEROSIS

Multiple sclerosis (MS) is a severe brain disease. Early detection can provide timely treatment. Fractal dimension can provide statistical index of pattern changes with scale at a given brain image. In this study, our team used susceptibility weighted imaging technique to obtain 676 MS slices and 880 healthy slices. We used synthetic minority oversampling technique to process the unbalanced dataset. Then, we used Canny edge detector to extract distinguishing edges. The Minkowski–Bouligand dimension was a fractal dimension estimation method and used to extract features from edges. Single hidden layer neural network was used as the classifier. Finally, we proposed a three-segment representation biogeography-based optimization to train the classifier. Our method achieved a sensitivity of 97.78±1.29%, a specificity of 97.82±1.60% and an accuracy of 97.80±1.40%. The proposed method is superior to seven state-of-the-art methods in terms of sensitivity and accuracy.

Fractal Dimension of Irregular Region of Interest Application to Corn Phenology Characterization

Analysis of multitemporal remote sensing imagery offers a reliable and cost-effective means for monitoring crops on a broad-scale and provides consistent temporal measurements. Fractal geometry has been used as a quantitative description of spatial complexity of remote sensing images. Yet, corn field of spatial irregularity alters the fractal dimension of the landscape, which shall be suppressed in the estimation. In this paper, we propose a method for computing fractal dimension from irregular region of interests that minimizes the contribution from 2-D spatial irregularity. Our method was evaluated with normalized difference vegetation index products derived from moderate resolution imaging spectroradiometer and satellite pour l’observation de la terre VEGETATION sensors from three states in the U.S. The experimental results using the time series demonstrated that our proposed fractal dimension estimation method exhibited great consistency and invariance to the change of image spectral characteristics, spatial resolution, and the degree of pixel mixing. In contrast to entropy and variance, the spectral characteristics of different imaging devices exhibited lower impact to the fractal dimension, which also implies its scale invariance. With respect to the detection rate of the first peak, fractal dimension achieved the best consistency. The proposed method for computing fractal dimension provides a critical and reliable measure for studying phenological patterns.

FRACTAL DIMENSION ESTIMATION OF RGB COLOR IMAGES USING MAXIMUM COLOR DISTANCE

Natural images exhibit a high degree of complexity, randomness and irregularity in color and texture, however fractal can be an effective tool to describe various irregular phenomena in nature. Fractal dimensions are important because they can be defined in connection with real-world data, and they can be measured approximately by means of experiments. In this paper, we proposed a fractal dimension estimation method for RGB color images. In the proposed method, we present a hyper-surface partition method which considers the hyper-surface as continuous and divide the image into nonoverlapped blocks. We also defined a counting method in color domain. To validate the proposed method, experiments were carried on two types of color images: synthesized fractal images and natural RGB color images. The experimental results demonstrate that the proposed method is effective and efficient. The behaviors of the proposed method on the rescale images are also shown in the paper. And it can be performed as a reliable FD estimation approach for the RGB color images.

Classification of Mammograms into Normal, Benign and Malignant based on Fractal Features

Modern life style of women has made them more vulnerable to breast cancer and it is considered as the largest cause of mortality among women. This paper presents a novel method to classify mammograms into normal ones, with benign and malignant microcalcifications, and with malignant and benign tumors using fractal features derived from fractal dimension. Here, three fractal dimension estimation methods such as Differential Box Counting (DBC), Triangular Prism Surface Area (TPSA) and Blanket methods are used for computing the six fractal features utilized for the classification. The new fractal feature f6 obtained using TPSA method is found to be the best with 100% classification accuracy. The average value of f6 is found to be 0.1110, 0.2875, 0.4743, 0.5271 and 0.8558, for normal, benign masses, benign and malignant microcalcifications and malignant masses respectively. The classification performance of the different features was analyzed using the Receiver Operating Characteristics (ROC).

Classification of electromyography signals using relevance vector machines and fractal dimension

Surface electromyography (EMG) signals have been studied extensively in the last years aiming at the automatic classification of hand gestures and movements as well as the early identification of latent neuromuscular disorders. In this paper, we investigate the potentials of the conjoint use of relevance vector machines (RVM) and fractal dimension (FD) for automatically identifying EMG signals related to different classes of limb motion. The adoption of FD as the mechanism for feature extraction is justified by the fact that EMG signals usually show traces of self-similarity. In particular, four well-known FD estimation methods, namely box-counting, Higuchi's, Katz's and Sevcik's methods, have been considered in this study. With respect to RVM, besides the standard formulation for binary classification, we also investigate the performance of two recently proposed variants, namely constructive mRVM and top-down mRVM, that deal specifically with multiclass problems. These classifiers operate solely over the features extracted by the FD estimation methods, and since the number of such features is relatively small, the efficiency of the classifier induction process is ensured. Results of experiments conducted on a publicly available dataset involving seven distinct types of limb motions are reported whereby we assess the performance of different configurations of the proposed RVM+FD approach. Overall, the results evidence that kernel machines equipped with the FD feature values can be useful for achieving good levels of classification performance. In particular, we have empirically observed that the features extracted by the Katz's method is of better quality than the features generated by other methods.

Identification of Crack Location in Beam Structures Using Wavelet Transform and Fractal Dimension

Identification of structural crack location has become an intensely investigated subject due to its practical importance. In this paper, a hybrid method is presented to detect crack locations using wavelet transform and fractal dimension (FD) for beam structures. Wavelet transform is employed to decompose the mode shape of the cracked beam. In many cases, small crack location cannot be identified from approximation signal and detailed signals. And FD estimation method is applied to calculate FD parameters of detailed signals. The crack locations will be detected accurately by FD singularity of the detailed signals. The effectiveness of the proposed method is validated by numerical simulations and experimental investigations for a cantilever beam. The results indicate that the proposed method is feasible and can been extended to more complex structures.

Assessing fractal dimension methods as feature extractors for EMG signal classification

The study of electromyographic (EMG) signals has gained increased attention in the last decades since the proper analysis and processing of these signals can be instrumental for the diagnosis of neuromuscular diseases and the adaptive control of prosthetic devices. As a consequence, various pattern recognition approaches, consisting of different modules for feature extraction and classification of EMG signals, have been proposed. In this paper, we conduct a systematic empirical study on the use of Fractal Dimension (FD) estimation methods as feature extractors from EMG signals. The usage of FD as feature extraction mechanism is justified by the fact that EMG signals usually show traces of self-similarity and by the ability of FD to characterize and measure the complexity inherent to different types of muscle contraction. In total, eight different methods for calculating the FD of an EMG waveform are considered here, and their performance as feature extractors is comparatively assessed taking into account nine well-known classifiers of different types and complexities. Results of experiments conducted on a dataset involving seven distinct types of limb motions are reported whereby we could observe that the normalized version of the Katz׳s estimation method and the Hurst exponent significantly outperform the others according to a class separability measure and five well-known accuracy measures calculated over the induced classifiers.

Intrinsic Dimensionality Estimation for High-dimensional Data Sets: New Approaches for the Computation of Correlation Dimension

The analysis of high–dimensional data is usually challenging since many standard modelling approaches tend to break down due to the so–called “curse of dimensionality”. Dimension reduction techniques, which reduce the data set (explicitly or implicitly) to a smaller number of variables, make the data analysis more efficient and are furthermore useful for visualization purposes. However, most dimension reduction techniques require fixing the intrinsic dimension of the low-dimensional subspace in advance. The intrinsic dimension can be estimated by fractal dimension estimation methods, which exploit the intrinsic geometry of a data set. The most popular concept from this family of methods is the correlation dimension, which requires estimation of the correlation integral for a ball of radius tending to 0. In this paper we propose approaches to approximate the correlation integral in this limit. Experimental results on real world and simulated data are used to demonstrate the algorithms and compare to other methodology. A simulation study which verifies the effectiveness of the proposed methods is also provided.

Closed contour fractal dimension estimation by the Fourier transform

This work proposes a novel technique for the numerical calculus of the fractal dimension of fractal objects which can be represented as a closed contour. The proposed method maps the fractal contour onto a complex signal and calculates its fractal dimension using the Fourier transform. The Fourier power spectrum is obtained and an exponential relation is verified between the power and the frequency. From the parameter (exponent) of the relation, is obtained the fractal dimension. The method is compared to other classical fractal dimension estimation methods in the literature, e.g., Bouligand–Minkowski, box-counting and classical Fourier. The comparison is achieved by the calculus of the fractal dimension of fractal contours whose dimensions are well-known analytically. The results showed the high precision and robustness of the proposed technique.

Evaluation of fractal dimension estimation methods for feature extraction in motor imagery based brain computer interface

Abstract A brain computer interface (BCI) enables direct communication between a brain and a computer translating brain activity into computer commands using preprocessing, feature extraction and classification operations. Feature extraction is crucial as it has a substantial effect on the classification accuracy and speed. While fractal dimension has been successfully used in various domains to characterize data exhibiting fractal properties, its usage in motor imagery based BCI has been more recent. There are several fractal dimension estimation methods, some of which are not applicable to all types of data exhibiting fractal properties. In this study, commonly used fractal dimension estimation methods to characterize time series (Katz’s method, Higuchi’s method and the rescaled range method) were evaluated for feature extraction in motor imagery based BCI by conducting offline analyses of a two class motor imagery dataset. Different classifiers (fuzzy k nearest neighbors (FKNN), support vector machine and linear discriminant analysis) were tested in combination with these methods to determine the methodology with the best performance. This methodology was then modified by implementing the time dependent fractal dimension (TDFD), differential fractal dimension and differential signals methods to determine if the results could be further improved. Katz’s method with FKNN resulted in the highest classification accuracy (of 85%), and further improvements (by 3%) were achieved by implementing the TDFD method. The results point to Katz’s method with FKNN as a favorable methodology for motor imagery based BCI and warrant further research to implement this methodology in online analysis of motor imagery data and analysis of other signals.