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.

Mathematical Incorrectness of So-Called Higuchi‘s Fractal Dimension

The so-called Higuchi’s method of fractal dimension estimation is widely used and the term Higuchi’s fractal dimension even occurs in many publications. This paper deals with this method from a mathematical point of view. Terms distance and dimension and its basic properties are explained and Higuchi’s dimension according the original source is defined. The definition of Higuchi’s dimension was compared with the mathematical definition of distance and dimension. It is shown, that the definition of Higuchi’s dimension does not satisfy axioms of distance and dimension. The so-called Higuchi’s method and Higuchi’s dimension are mathematically incorrect. Therefore, all results achieved by this method are scientifically unreliable.

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.

Anderson relaxation test for intrinsic dimension selection in model-based clustering

Parsimonious finite mixture models often require the a priori selection of desired model dimensionality. For example, projection-based parsimonious models demand the dimension of the subspace for projection. Other models ask for their own structural restrictions on parameters. The subspace clustering framework is a projection-based parsimonious model for various finite mixtures, including the Gaussian variant. The existing dimension selection methods for subspace clustering are ad-hoc or potentially computationally prohibitive, creating a need for a principled, yet computationally lightweight, approach. In light of this problem, a hypothesis test-based intrinsic dimension estimation method called the Anderson Relaxation Test (ART) is introduced, and its performance is examined in both simulated and real data settings.

Intrinsic dimension estimation method based on correlation dimension and kNN method

In practical problems, high-dimensional data usually has a low-dimensional structure, or the data is located on a low-dimensional manifold. The dimension of this manifold is called the intrinsic dimension of the data. There are many intrinsic dimension estimation methods, among which methods based on the correlation dimension have received extensive attention. However, correlation dimension-based estimation methods often provide a dimension lower than the true intrinsic dimension of the dataset. To explore the reasons behind underestimation, the probabilities of underestimation, overestimation and proper estimation are analyzed using order statistics. The analysis results show that the probability of underestimation is much higher than that of the other two cases, and is verified by simulation experiments. Based on the above analysis, a new method for the estimation of the intrinsic dimension is proposed based on the correlation dimension and k-nearest neighbor method (kNN), which effectively reduces the underestimation. This method is implemented using two algorithms, namely a search algorithm and a matching algorithm. Comprehensive experimental studies on simulation datasets and real datasets show that the proposed algorithms are more effective than the comparison methods.

Generalized least‐squares in dimension expansion method for nonstationary processes

AbstractIn this article, we consider the problem of modeling nonstationary spatial random processes. Bornn et al.(2012) proposed a dimension expansion method, a novel technique for modeling nonstationary processes, aiming to find a dimensionally sparse projection in which the originally nonstationary field exhibits stationarity. However, their dimension expansion approach is a lasso‐penalized least‐squares method that does not account for the covariance structure of the empirical semivariogram. We thus propose a general latent dimension estimation method by replacing the least‐squares method with generalized least‐squares (GLS). Furthermore, we improve the GLS method by weighted least‐squares, which is more computationally efficient and accurate. The performance of the proposed methods is demonstrated through simulations and real data examples.

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.

Design of Thrust-Optimized Scramjet Nozzle and Its Concise Estimation Method

This Paper is a computational investigation of a scramjet nozzle design based on maximum thrust theory and its feasibility analysis. A new design method for a two-dimensional complete thrust-optimized single expansion ramp nozzle under aerodynamic constraints by applying the method of characteristics is proposed. The carefully chosen verification and validation benchmarks confirm the accuracy, effectiveness, and superiority of the current method. Results indicate that the introduced design method has comprehensive advantages over the traditional approach under design/off-design conditions. Specifically, the maximum increase in axial thrust reaches 6.38%, and the maximum increase in lift exceeds 180%. Necessary conditions of the thrust-optimized nozzle design are deduced combining Prandtl–Meyer relation and corner conditions, which provides a direct judgment criterion for the rationality of initial design conditions, and leads to a concise dimension estimation method. Relative errors of no more than 5% demonstrate the credibility of this approach through a comparison with the theoretical solutions. The refined structure topology of the thrust-optimized single expansion ramp nozzle further reveals the equivalent relations between the aerodynamic and geometric design conditions based on the estimation results.

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

Dimension of reservoir computers.

A reservoir computer is a complex dynamical system, often created by coupling nonlinear nodes in a network. The nodes are all driven by a common driving signal. In this work, three dimension estimation methods, false nearest neighbor, covariance dimension, and Kaplan-Yorke dimension, are used to estimate the dimension of the reservoir dynamical system. It is shown that the signals in the reservoir system exist on a relatively low dimensional surface. Changing the spectral radius of the reservoir network can increase the fractal dimension of the reservoir signals, leading to an increase in a testing error.

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.

An improved LLE-based cluster security approach for nonlinear system fault diagnosis

This paper proposes a novel improved LLE-based (local linear embedding) approach (TLLE) for solving the fault diagnosis problem of the nonlinear system. Firstly, tangent space distance is introduced to LLE approach, which can satisfy the local linearity of LLE and better preserve the local manifold features of the original data simultaneously. Then, to solve the problem of inner dimension in LLE approach is hard to estimate, the method of intrinsic dimension estimation based on fractal dimension is employed in the approach by means of linear fitting. Furthermore, a fault diagnosis scheme is presented based on the TLLE approach. We combine fault state with special distribution to complete the fault diagnosis, which can simplify the computation obviously and improve the real-time capability of the approach. Finally, numerical simulations of the TE process data are performed to illustrate the effectiveness of the proposed fault diagnosis scheme.

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.

Local Intrinsic Dimension Estimation by Generalized Linear Modeling.

We propose a method for intrinsic dimension estimation. By fitting the power of distance from an inspection point and the number of samples included inside a ball with a radius equal to the distance, to a regression model, we estimate the goodness of fit. Then, by using the maximum likelihood method, we estimate the local intrinsic dimension around the inspection point. The proposed method is shown to be comparable to conventional methods in global intrinsic dimension estimation experiments. Furthermore, we experimentally show that the proposed method outperforms a conventional local dimension estimation method.

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.

Quad-polarized synthetic aperture radar and multispectral data classification using classification and regression tree and support vector machine–based data fusion system

Interpretation of synthetic aperture radar (SAR) data processing is difficult because the geometry and spectral range of SAR are different from optical imagery. Consequently, SAR imaging can be a complementary data to multispectral (MS) optical remote sensing techniques because it does not depend on solar illumination and weather conditions. This study presents a multisensor fusion of SAR and MS data based on the use of classification and regression tree (CART) and support vector machine (SVM) through a decision fusion system. First, different feature extraction strategies were applied on SAR and MS data to produce more spectral and textural information. To overcome the redundancy and correlation between features, an intrinsic dimension estimation method based on noise-whitened Harsanyi, Farrand, and Chang determines the proper dimension of the features. Then, principal component analysis and independent component analysis were utilized on stacked feature space of two data. Afterward, SVM and CART classified each reduced feature space. Finally, a fusion strategy was utilized to fuse the classification results. To show the effectiveness of the proposed methodology, single classification on each data was compared to the obtained results. A coregistered Radarsat-2 and WorldView-2 data set from San Francisco, USA, was available to examine the effectiveness of the proposed method. The results show that combinations of SAR data with optical sensor based on the proposed methodology improve the classification results for most of the classes. The proposed fusion method provided approximately 93.24% and 95.44% for two different areas of the data.

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.

Development of a mixed pixel filter for improved dimension estimation using AMCW laser scanner

Accurate dimension estimation is desired in many fields, but the traditional dimension estimation methods are time-consuming and labor-intensive. In the recent decades, 3D laser scanners have become popular for dimension estimation due to their high measurement speed and accuracy. Nonetheless, scan data obtained by amplitude-modulated continuous-wave (AMCW) laser scanners suffer from erroneous data called mixed pixels, which can influence the accuracy of dimension estimation. This study develops a mixed pixel filter for improved dimension estimation using AMCW laser scanners. The distance measurement of mixed pixels is firstly formulated based on the working principle of laser scanners. Then, a mixed pixel filter that can minimize the classification errors between valid points and mixed pixels is developed. Validation experiments were conducted to verify the formulation of the distance measurement of mixed pixels and to examine the performance of the proposed mixed pixel filter. Experimental results show that, for a specimen with dimensions of 840mm×300mm, the overall errors of the dimensions estimated after applying the proposed filter are 1.9mm and 1.0mm for two different scanning resolutions, respectively. These errors are much smaller than the errors (4.8mm and 3.5mm) obtained by the scanner’s built-in filter.

Computational performance optimization of support vector machine based on support vectors

The computational performance of support vector machine (SVM) mainly depends on the size and dimension of training sample set. Because of the importance of support vectors in the determination of SVM classification hyperplane, a kind of method for computational performance optimization of SVM based on support vectors is proposed. On one hand, at the same time of the selection of super-parameters of SVM, according to Karush-Kuhn-Tucker condition and on the precondition of no loss of potential support vectors, we eliminate non-support vectors from training sample set to reduce sample size and thereby to reduce the computation complexity of SVM. On the other hand, we propose a simple intrinsic dimension estimation method for SVM training sample set by analyzing the correlation between number of support vectors and intrinsic dimension. Comparative experimental results indicate the proposed method can effectively improve computational performance.