Local Fractal Dimension Research Articles

Change Detection in Point Clouds Using 3D Fractal Dimension

The management of large point clouds obtained by LiDAR sensors is an important topic in recent years due to the widespread use of this technology in a wide variety of applications and the increasing volume of data captured. One of the main applications of LIDAR systems is the study of the temporal evolution of the real environment. In open environments, it is important to know the evolution of erosive processes or landscape transformation. In the context of civil engineering and urban environments, it is useful for monitoring urban dynamics and growth, and changes during the construction of buildings or infrastructure facilities. The main problem with change detection (CD) methods is erroneous detection due to precision errors or the use of different capture devices at different times. This work presents a method to compare large point clouds, based on the study of the local fractal dimension of point clouds at multiple scales. Our method is robust in the presence of environmental and sensor factors that produce abnormal results with other methods. Furthermore, it is more stable than others in cases where there is no significant displacement of points but there is a local alteration of the structure of the point cloud. Furthermore, the precision can be adapted to the complexity and density of the point cloud. Finally, our solution is faster than other CD methods such as distance-based methods and can run at O(1) under some conditions, which is important when working with large datasets. All these improvements make the proposed method more suitable than the others to solve complex problems with LiDAR data, such as storage, time series data management, visualization, etc.

Determination of local fractal dimension of the catalyst support MgAl2O4 surface

In this paper, a procedure for estimating local fractal dimensionality values for greyscale images was developed. This procedure was tested on synthesised fractal and multifractal Brownian surfaces. The use of the method of detrended fluctuation analysis to determine the Hurst index for samples with a size not exceeding 2000 points was analysed. The optimal values of the sample size and the scale parameter have been obtained, allowing the Hurst index to be calculated with an error of less than 20%. The procedure was also tested on greyscale images, which are 8-bit representations of a synthesised fractal surface. The peculiarities of the application of the procedure for calculating the local fractal dimension for images obtained by scanning electron microscopy have been analysed. The values of the local fractal dimension of the oxide support MgAl2O4 are calculated using SEM images at different magnifications. It has been shown that, taking into account the fractal dimensionality calculation, different morphologies of the particle surface are observed at different scales of image magnification. At the highest magnification, the particle surface is found to consist of embedded smooth micron particles. The surface morphology of the MgAl2O4 particle depends on the micro- and mesopores between the embedded particles, this surface can be characterised by a fractal surface with FD2 = 2.3–2.4.

YOLO-ECG: multi-stage ECG arrhythmia classification using deep learning based YOLO network for portable monitoring

Cardiovascular disease is the leading cause of death and more than half million people were died around the world. However, cardiovascular health monitoring is crucial for effective heart disease diagnosis and management. In this paper, a novel deep learning-based YOLO-ECG model is proposed to ECG arrhythmia classification method for portable monitoring. Initially, the ECG signals are gathered using 12-lead electrodes in the real time and these signals are denoised using two-dimensional stationary wavelet transform (2D-SWT). In SWT, zeros are inserted between filter taps rather than decimal points to eliminate repetitions and increase robustness. The denoised ECG signals are fed into the deep learning-based YOLO network with Gaussian error linear unit (GELU) activation function for detecting the ECG abnormalities of arrythmia. ECG waveforms are analyzed for the local fractal dimension at each sample point before heartbeat waveforms are extracted within a set length window. A squeeze and excitation attention (SEAN) module is introduced in the YOLO network for selecting size of 1D convolution kernel, and the dimension is preserved during local cross-channel interactions, decrease network complexity and enhance model efficiency. The classification findings demonstrate that the proposed YOLO-ECG model performs better by ECG recordings from the MIT-BIH arrhythmia dataset. From the experimental analysis, the proposed YOLO-ECG model yields the overall accuracy of 99.16% for efficient classification of arrythmia ECG signals.

Brain Volumetric changes in patients with Spinocerebellar Ataxia Type 2

Aim: Spinocerebellar ataxia type 2 (SCA2) is a trinucleotide repeat disorder characterized by mutations in the ataxin-2 gene. Although the primary symptoms of this disease are related to cerebellar involvement, such as ataxia and dysarthria, cognitive disorders associated with supratentorial involvement, depression, and extrapyramidal symptoms are also common. The aim of this study was to investigate volumetric changes in patients with SCA2. Materials and Methods: Nine SCA2 patients and sixteen age and gender-matched healthy controls were included in the study. Voxel-based morphometry (VBM) was used to explore changes in whole brain gray and white matter, while surface-based morphometry (SBM) was employed to investigate changes in cortical thickness (CT), local gyrification index (LGI), sulcal depth (SD), and fractal dimension (FD). Deep learning methods were utilized for the automatic segmentation of subcortical gray matter structures and extrapyramidal system structures. Comparisons between groups were made using GLM, T-Test, and Mann-Whitney U test. Results: VBM results showed widespread gray matter loss in the cerebellum and widespread white matter loss in the brain stem (Cluster size: 25918 voxels for gray matter, 24557 voxels for white matter). SBM findings indicated a decrease in cortical thickness (Cluster size:19276 vertices for left hemisphere; 17276 vertices for right hemisphere), more pronounced in the supratentorial frontoparietal region, while FD, LGI, and SD did not differ between groups. Volume loss was observed in the subthalamic nucleus, globus pallidus internus and externus, and red nucleus in SCA2 patients. Conclusion: SCA2 patients exhibit white and gray matter volume loss in the infratentorial region. Additionally, they show a decrease in frontoparietal cortical thickness, while FD, LGI, and SD do not show any changes. Moreover, atrophy is also observed in the extrapyramidal system structures.

Liver cancer: A geometric approach to fractals and scaling analysis

In the study of liver cancer, invasive techniques such as computed tomography (CT) and magnetic resonance imaging (MRI) are used. In the present work, an in vivo 3D study will be carried out using MRI and CT images, using scaling dynamics, that is, calculating critical exponents of local roughness and fractal dimension. The results obtained for stages IA, IIA, and IIIA in liver cancer correspond to what was proposed by F. Family and T. Vicsek (1991).

Local fractal dimension of collagen detects increased spatial complexity in fibrosis

Increase of collagen content and reorganization characterizes fibrosis but quantifying the latter remains challenging. Spatially complex structures are often analyzed via the fractal dimension; however, established methods for calculating this quantity either provide a single dimension for an entire object or a spatially distributed dimension that only considers binary images. These neglect valuable information related to collagen density in images of fibrotic tissue. We sought to develop a fractal analysis that can be applied to 3-dimensional (3D) images of fibrotic tissue. A fractal dimension map for each image was calculated by determining a single fractal dimension for a small area surrounding each image pixel, using fiber thickness as the third dimension. We found that this local fractal dimension increased with age and with progression of fibrosis regardless of collagen content. Our new method of distributed 3D fractal analysis can thus distinguish between changes in collagen content and organization induced by fibrosis.

Adaptive super-resolution image reconstruction based on fractal theory

Image super-resolution (SR) reconstruction has been a significant research field in image processing, although the current approaches to dealing with complicated image reconstruction still have some issues. In order to overcome these problems, traditional approaches apply fractal geometry to image super-resolution reconstruction, either in textured or non-textured regions, treating the image as a single fractal set. However, it won't be possible to classify the image specifically, such as edge, common texture, and complex texture, unless identifying the texture complexity of all regions in the image. For the texture region, this paper considers the fractal as a perturbation function of rational interpolation and constructs a new interpolation model with scale factor parameter. The parameter is related to the local fractal dimension (LFD). This paper defines a new function between scale factor and local fractal dimension so as to select the optimal value adaptively. Considering the image edge quality, this paper combines the non-texture regions and edge regions, applies an improved edge interpolation algorithm and sets a new pixel mapping method to meet the needs of different zoom factors. Based on the above, this paper proposes an adaptive super-resolution reconstruction algorithm. Particularly, region segmentation with different texture complexity is a foundational step of our algorithm, this paper proposes an image segmentation algorithm based on the research of local fractal dimension which combines the multi-scale analysis capability of wavelets with the multi-scale self-similarity feature of fractals. The experimental results show that our algorithm can obtain accurate texture details, smooth edges and low noise subjectively, and achieve the best evaluation objectively. This paper lays the theoretical foundation for fractal applications in super-resolution image reconstruction.

Bifractality of fractal scale-free networks.

The presence of large-scale real-world networks with various architectures has motivated active research towards a unified understanding of diverse topologies of networks. Such studies have revealed that many networks with scale-free and fractal properties exhibit the structural multifractality, some of which are actually bifractal. Bifractality is a particular case of the multifractal property, where only two local fractal dimensions d_{f}^{min} and d_{f}^{max}(>d_{f}^{min}) suffice to explain the structural inhomogeneity of a network. In this work we investigate analytically and numerically the multifractal property of a wide range of fractal scale-free networks (FSFNs) including deterministic hierarchical, stochastic hierarchical, nonhierarchical, and real-world FSFNs. Then we demonstrate how commonly FSFNs exhibit the bifractal property. The results show that all these networks possess the bifractal nature. We conjecture from our findings that any FSFN is bifractal. Furthermore, we find that in the thermodynamic limit the lower local fractal dimension d_{f}^{min} describes substructures around infinitely high-degree hub nodes and finite-degree nodes at finite distances from these hub nodes, whereas d_{f}^{max} characterizes local fractality around finite-degree nodes infinitely far from the infinite-degree hub nodes. Since the bifractal nature of FSFNs may strongly influence time-dependent phenomena on FSFNs, our results will be useful for understanding dynamics such as information diffusion and synchronization on FSFNs from a unified perspective.

Breast cancer: A fractal approach, scaling analysis and quantum cluster algorithm

Tumor growth in the breast is a complex process, understanding this type of phenomenon is important in order to establish adequate strategies at diagnostic and therapeutic levels, beginning with the evaluation morphology, through adequate descriptors. Scaling analysis is used to extract parameters such as exponents of local roughness and fractal dimension, which characterize tumor growth. The results show that invasive lobular and ductal tumors in situ behave as proposed by Family-Vicsek (Family and Vicsek, 1991), while benign tumors behave differently.

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.

Fractal analysis of microstructures in Portland cement pastes – Effect of curing conditions

Cement is used for the construction and renovation of structures such as buildings, bridges, and viaducts owing to its cohesive properties, low cost, versatility, and moldability. The conditions and curing time affect the microstructural characteristics and properties of cementitious materials in their various constitutions. Morphological changes occur in the microstructures of cementitious matrices with the progression of the hydration process and according to the type of cure, which affects the mechanical properties and durability of the material. Therefore, cylindrical specimens were fabricated to investigate the effect of curing conditions (ambient or submerged) on the morphology and compressive strength of hardened pastes cured for 7 and 28 days. The samples were subjected to the compressive strength test, and the surface of the fractured samples was analyzed using microscopic techniques. The quantifications of the morphological characteristics of the hydrated cement were correlated with the mechanical tests, and the local and global fractal dimensions were found to increase with time. This increase was accompanied by an increase in the compressive strength, which inhibited the propagation of the cracks and the development of broken fractal figures due to greater resistance to fracture.

The roughness of martian topography: A metre-scale fractal analysis of six selected areas

Studies of the roughness of natural surfaces (landscapes) provide useful information for planetary geology. This paper covers the mapping and analysis of the spatial variability of the surface roughness of Martian topography at a high spatial resolution (metre-scale). The methodology provides new images of the Martian surface texture at the metre-scale that can assist in the interpretation of geological events, processes and formations. It can also assist in geological mapping and in the evaluation of sites that merit further exploration. Digital elevation models, generated by stereo-pair HiRISE images, of six different terrains (aeolian, volcanic, hydrated, cratered, reticulate and sublimated) were used to characterize the metre-scale terrain roughness of representative test sites on Mars. Surface roughness was evaluated by using the local fractal dimension and the results show that the mean of the local fractal dimension ranges from 2.17 in reticulate terrain to 2.71 in sublimated terrain in the southern polar cap. The roughness of the sublimated terrain is significantly higher than the roughness of typical terrains on Earth. Basically, the roughness of the Martian terrain at the metre-scale depends on the rugosity of the landscape, which can be quantified as the number of metric-scale closed depressions and mounds present on the terrain. The information provided by the spatial variability patterns of metre-scale roughness maps provides a significant resource for local planetary geology research at high resolution scale.

Segmentation of Concrete Cracks by Using Fractal Dimension and UHK-Net

Concrete wall surfaces are prone to cracking for a long time, which affects the stability of concrete structures and may even lead to collapse accidents. In view of this, it is necessary to recognize and distinguish the concrete cracks. Then, the stability of concrete will be known. In this paper, we propose a novel approach by fusing fractal dimension and UHK-Net deep learning network to conduct the semantic recognition of concrete cracks. We first use the local fractal dimensions to study the concrete cracking and roughly determine the location of concrete crack. Then, we use the U-Net Haar-like (UHK-Net) network to construct the crack segmentation network. Ultimately, the different types of concrete crack images are used to verify the advantage of the proposed method by comparing with FCN, U-Net, YOLO v5 network. Results show that the proposed method can not only characterize the dark crack images, but also distinguish small and fine crack images. The pixel accuracy (PA), mean pixel accuracy (MPA), and mean intersection over union (MIoU) of crack segmentation determined by the proposed method are all greater than 90%.

Visualizing the spatiotemporal pattern of yolk sac membrane vascular network by enhanced local fractal analysis.

To establish methods for providing a comprehensive and detailed description of the spatial distribution of the vascular networks, and to reveal the spatiotemporal pattern of the yolk sac membrane vascular network during the angiogenic procedure. Addressing the limitations in the conventional local fractal analysis, an improved approach, named scanning average local fractal dimension, was proposed. This method was conducted on 6high-resolution vascular images of the yolk sac membrane for 3 eggs at two stages (E3 and E4) to characterize the spatial distribution of the complexity of the vascular network. With the proposed method, the spatial distribution of the complexity of the yolk sac membrane vascular network was visualized. From E3 to E4, the local fractal dimension increased in 3 eggs, 1.80±0.02 vs. 1.85±0.02, 1.72±0.03 vs. 1.83±0.02, and 1.77±0.03 vs. 1.82±0.02, respectively. The mean local fractal dimension in the most distal area from the embryo proper was the lowest at E3 while the highest at E4. At E3, the most peaks of the local fractal dimension were located in the vein territories and shifted to artery territories at E4. The spatial distribution of the complexity of the yolk sac membrane vascular network exhibited diverse patterns at different stages. In addition from E3 to E4, the increment of complexity at the intersection areas between arteries and sinus terminalis was with the most advance. This is consistent with the physiologic evidence. The present work provides a potential approach for investigating the spatiotemporal pattern of the angiogenic process.

Migration of hyper-fractal analysis from visual basic 6 to C# .Net

In Grossu et al. (Comput. Phys. Commun. 184 (2013) 1812–1813) we presented Hyper-Fractal Analysis, a visual tool for estimating the fuzzy fractal dimension of images and 4D objects. As Visual Basic 6 could be considered an outdated language, with limited Object-Oriented Programming capabilities, migrating the application to C# .Net was treated in high priority. Following the goal of creating a highly reusable fractal analysis library, the code was also refactored to SOLID. Together with various improvements, the.Net version is also providing new tools for iso-fractal areas identification. The project success was confirmed by a comparative old/new version study. On the other hand, the most relevant functionalities were covered by unit tests. New version program summaryProgram Title: Hyper-Fractal Analysis v05CPC Library link to program files:https://doi.org/10.17632/z9knmny56p.1Licensing provisions: GPLv2Programming language: C# 7.3 /.Net Framework 4.7.1Journal reference of previous version: Computer Physics Communications 184, Issue 7 (2013) 1812–1813Does the new version supersede the previous version?: YesNature of problem: Estimating the fractal dimension of images and multi-dimensional objects.Solution method: Optimized version of the fuzzy box-counting algorithm.Reasons for the new version: Migration from VB6 to C# .Net, adding new tools for iso-fractal areas identification.Summary of revisions:•The main goal of this work was creating a highly reusable fractal analysis library. Following this purpose Hyper-Fractal Analysis v04 [1] was migrated from VB6 (an outdated language, with limited OOP capabilities) to C# .Net [2]. This opportunity was also used for refactoring the code to SOLID [3].•The box-counting algorithm [4] implemented in the previous version is limited to 4D objects. In current version the number of dimensions is limited only by the available resources.•The previous fuzzy [5,6] box-counting implementation is accepting only 0-255 shades of gray windows. The new version allows defining parametrizable fuzzy band-pass filters.•An important attention was also paid on developing new tools for identification of iso-fractal areas (the sets of points characterized by the same local fractal dimension [7]). This functionality is expected to be of particular interest in medical image analysis [7-13].•The Windows Forms GUI (Fig. 1) was completely reconsidered on top of the Model View Presenter pattern [14].•For an increased reliability and for minimizing the regression risks during future developments the most relevant functionalities were covered by unit tests [15].

Fractal analysis of the martian landscape: A study of kilometre-scale topographic roughness

Natural and engineered surfaces can be modelled as fractal objects described by a non-integer dimension (i.e., their fractal dimension). This approach has been used extensively to characterize surface complexity in terms of rugosity or roughness. In this paper, the local fractal dimension is proposed as a means of mapping the spatial variability of the surface roughness of the Martian landscape. The local fractal dimension complements measures of topographic roughness proposed by other authors and provides new images of the Martian surface texture at the kilometric scale that can be used to interpret geological events, geological processes and geological formations. It can also assist in geological mapping and in the detection and prioritisation of sites that merit further exploration. Mars Orbiter Laser Altimeter data were used to characterize the kilometre-scale terrain roughness of Mars. The procedure is general and can also be applied to digital elevation models with higher resolution (for example, digital elevation models generated by stereo-pair HiRISE images) to assess roughness at the metric scale. Sixteen test sites were selected and used to illustrate the patterns of the spatial variability of landscape roughness (local fractal dimension maps) at the kilometric scale. The results show that the mean of the local fractal dimension ranges from 2.036 in Argyre Planitia to 2.403 at the North Pole. The results also show that the mean roughness values of the Martian landscapes are less than those of the Earth's continents. Basically, the roughness of the Martian terrain at the kilometric scale depends, inter alia, on the number of kilometric-scale landforms present on the terrain. It is thus logical that the local fractal dimension is higher on Earth, a geologically active planet, than on Mars. Only the North Pole test site (an area larger than the North polar ice cap) shows a roughness histogram similar to those of terrestrial sites. The abundance of information embedded in the spatial variability patterns in the roughness maps provides a significant resource for planetary geology research.

An Improved Tool Wear Monitoring Method Using Local Image and Fractal Dimension of Workpiece

Tool wear is a key factor that dominates the surface quality and distinctly influences the generated workpiece surface texture. In order to realize accurate evaluation of the tool wear from the generated workpiece surface after machining process, a new tool wear monitoring method is developed by fractal dimension of the acquired workpiece surface digital image. A self-made simple apparatus is employed to capture the local digital images around the region of interest. In addition, a skew correction method based on local fast Fourier transformation energy is also proposed for the surface texture direction adjustment. Furthermore, the tool wear quantitative evaluation was derived based on fractal dimension utilizing its high reliability for inherent irregularity description. The proposed tool wear monitoring method has verified its feasibility as well as its effectiveness in actual milling experiments using the material of AISI 1045 in a vertical machining center. Testing results demonstrate that the proposed method was capable of tool wear condition evaluation.

Multifractal scaling analyses of urban street network structure: The cases of twelve megacities in China

Traffic networks have been proved to be fractal systems. However, previous studies mainly focused on monofractal networks, while complex systems are of multifractal structure. This paper is devoted to exploring the general regularities of multifractal scaling processes in the street network of 12 Chinese cities. The city clustering algorithm is employed to identify urban boundaries for defining comparable study areas; box-counting method and the direct determination method are utilized to extract spatial data; the least squares calculation is employed to estimate the global and local multifractal parameters. The results showed multifractal structure of urban street networks. The global multifractal dimension spectrums are inverse S-shaped curves, while the local singularity spectrums are asymmetric unimodal curves. If the moment order q approaches negative infinity, the generalized correlation dimension will seriously exceed the embedding space dimension 2, and the local fractal dimension curve displays an abnormal decrease for most cities. The scaling relation of local fractal dimension gradually breaks if the q value is too high, but the different levels of the network always keep the scaling reflecting singularity exponent. The main conclusions are as follows. First, urban street networks follow multifractal scaling law, and scaling precedes local fractal structure. Second, the patterns of traffic networks take on characteristics of spatial concentration, but they also show the implied trend of spatial deconcentration. Third, the development space of central area and network intensive areas is limited, while the fringe zone and network sparse areas show the phenomenon of disordered evolution. This work may be revealing for understanding and further research on complex spatial networks by using multifractal theory.

No-Reference Image Quality Assessment with Global Statistical Features.

The perceptual quality of digital images is often deteriorated during storage, compression, and transmission. The most reliable way of assessing image quality is to ask people to provide their opinions on a number of test images. However, this is an expensive and time-consuming process which cannot be applied in real-time systems. In this study, a novel no-reference image quality assessment method is proposed. The introduced method uses a set of novel quality-aware features which globally characterizes the statistics of a given test image, such as extended local fractal dimension distribution feature, extended first digit distribution features using different domains, Bilaplacian features, image moments, and a wide variety of perceptual features. Experimental results are demonstrated on five publicly available benchmark image quality assessment databases: CSIQ, MDID, KADID-10k, LIVE In the Wild, and KonIQ-10k.

Noisy Single Image Super-Resolution Based on Local Fractal Feature Analysis

Generally, most existing super-resolution (SR) methods do not consider noise, which treats SR reconstruction and denoising as two separate problems and performs separately. However, noise is inevitably introduced in the imaging process. Based on analysis of the degraded model, in this paper, the problems of interpolation and denoising are modeled to estimate the noiseless and missing images under the same framework. By applying local fractal dimension (LFD) into image local feature analysis, a noisy single-image SR method is proposed. For each noisy image, we first construct a rational fractal interpolation model containing scaling factors, which can effectively maintain the inherent properties of the data. Furthermore, the original image structure can be well preserved by applying the interpolation model. Considering the local characteristics of the image, scaling factors are calculated on the basis of the LFDs. Then, through further local feature analysis of the interpolated image, a denoising method based on LFD is proposed for recovering a noiseless image. Finally, a high-quality high-resolution image is obtained. Experimental results demonstrate that our method outperforms the state-of-the-art methods both quantitatively and qualitatively.