Multiscale Fractal Dimension Research Articles

Exploring the recurrent states of football teams' tactical organization on the pitch during Brazilian official matches.

Football teams' tactical organization on the pitch is usually represented by the surface area. Considering the different shapes adopted by the teams during the match, the role of the tactical variability for success is lacking. The aim of this study was to explore and to evaluate the association between recurrent states of tactical organization and technical performance during football matches. A total of 28 teams of Brazilian First Division Championships were analysed. Teams' surface area shapes were represented by the maximum value of the Multiscale Fractal Dimension in each timestamp, producing a time series. Recurrences of states of tactical organization were determined via recurrence plots and recurrence quantitative analysis during attacking and defending phases, and considering the whole match. The outcomes were correlated with nine traditional technical performance indicators. The main results showed that structural recurrence or variability on tactical organization is associated with performance success during the defending and attacking actions. Recurrence plot and measures based on the recurrence density proved to be valuable tools to represent teams' dynamics.

Medical Image Fusion Based on Local Saliency Energy and Multi-scale Fractal Dimension.

At present, there are some problems in multimodal medical image fusion, such as texture detail loss, leading to edge contour blurring and image energy loss, leading to contrast reduction. To solve these problems and obtain higher-quality fusion images, this study proposes an image fusion method based on local saliency energy and multi-scale fractal dimension. First, by using a non-subsampled contourlet transform, the medical image was divided into 4 layers of high-pass subbands and 1 layer of low-pass subband. Second, in order to fuse the high-pass subbands of layers 2 to 4, the fusion rules based on a multi-scale morphological gradient and an activity measure were used as external stimuli in pulse coupled neural network. Third, a fusion rule based on the improved multi-scale fractal dimension and new local saliency energy was proposed, respectively, for the low-pass subband and the 1st closest to the low-pass subband. Layerhigh pass sub-bands were fused. Lastly, the fused image was created by performing the inverse non-subsampled contourlet transform on the fused sub-bands. On three multimodal medical image datasets, the proposed method was compared with 7 other fusion methods using 5 common objective evaluation metrics. Experiments showed that this method can protect the contrast and edge of fusion image well and has strong competitiveness in both subjective and objective evaluation.

Enhancing Image Quality Through a Novel Multiscale Fractal Dimension Formulated by the Characteristic Function

Enhancing Image Quality Through a Novel Multiscale Fractal Dimension Formulated by the Characteristic Function

Multivariate Multiscale Higuchi Fractal Dimension and Its Application to Mechanical Signals

Fractal dimension, as a common nonlinear dynamics metric, is extensively applied in biomedicine, fault diagnosis, underwater acoustics, etc. However, traditional fractal dimension can only analyze the complexity of the time series given a single channel at a particular scale. To characterize the complexity of multichannel time series, multichannel information processing was introduced, and multivariate Higuchi fractal dimension (MvHFD) was proposed. To further analyze the complexity at multiple scales, multivariate multiscale Higuchi fractal dimension (MvmHFD) was proposed by introducing multiscale processing algorithms as a technology that not only improved the use of fractal dimension in the analysis of multichannel information, but also characterized the complexity of the time series at multiple scales in the studied time series data. The effectiveness and feasibility of MvHFD and MvmHFD were verified by simulated signal experiments and real signal experiments, in which the simulation experiments tested the stability, computational efficiency, and signal separation performance of MvHFD and MvmHFD, and the real signal experiments tested the effect of MvmHFD on the recognition of multi-channel mechanical signals. The experimental results show that compared to other indicators, A achieves a recognition rate of 100% for signals in three features, which is at least 17.2% higher than for other metrics.

Variable-step multi-scale fractal dimension and its application to ship radiated noise

Fractal dimension (FD) is a common tool for detecting dynamic changes in signals, however, it can only reflect the signal information at a single scale and has certain limitations. Currently, several scholars have proposed a few indexes based on multi-scale coarse-grained processing (MCP) and refined composite MCP (RCMCP). Nevertheless, the coarse-grained characteristics of MCP and RCMCP lead to their inadequate extraction of information from time series in the calculation process. To address this issue, variable-step multi-scale FD (VSMFD) was proposed, which not only reflects signal information from multiple scales, but also is more accurate and comprehensive when extracting signal information. The results of three sets of simulation experiments indicate that VSMFD is the most sensitive to the dynamic transform of chirped signals, and has the best separability for simulated noise signals and chaotic signals. In addition, both sets of ship radiated noise experiments indicate that VSMFD performs better in distinguishing ship radiated noise than indexes based on MCP and RCMCP, and has broad application prospects in the field of underwater acoustic signal processing.

Hierarchical Refined Composite Multi-Scale Fractal Dimension and Its Application in Feature Extraction of Ship-Radiated Noise

The fractal dimension (FD) is a classical nonlinear dynamic index that can effectively reflect the dynamic transformation of a signal. However, FD can only reflect signal information of a single scale in the whole frequency band. To solve this problem, we combine refined composite multi-scale processing with FD and propose the refined composite multi-scale FD (RCMFD), which can reflect the information of signals at a multi-scale. Furthermore, hierarchical RCMFD (HRCMFD) is proposed by introducing hierarchical analysis, which successfully represents the multi-scale information of signals in each sub-frequency band. Moreover, two ship-radiated noise (SRN) multi-feature extraction methods based on RCMFD and HRCMFD are proposed. The simulation results indicate that RCMFD and HRCMFD can effectively discriminate different simulated signals. The experimental results show that the proposed two-feature extraction methods are more effective for distinguishing six types of SRN than other feature-extraction methods. The HRCMFD-based multi-feature extraction method has the best performance, and the recognition rate reaches 99.7% under the combination of five features.

An approach for damage detection in noisy environments using DOG multi-scale space and fractal dimension

ABSTRACT The most prominent features of fractal dimension analysis are high sensitivity to damage and instant determination of damage location. However, an intrinsic deficiency is that it is difficult to identify damage in a structure while suppressing noise simultaneously. In addition, the vibration measurement with low-spatial-resolution is also the main factor restricting the accurate damage location. In this paper, a novel damage detection method based on fractal dimension is proposed, in order to provide the high-spatial-resolution vibration information, the phase-based optical flow is introduced, which is a vision-based measurement method that can calculate the structural vibration with high-spatial-resolution. Then, to detect damage in a structure while suppressing noise, an advanced signal analysis method is proposed, with this method, the fractal dimension (FRAC) is mapped into Difference of Gaussian (DOG) multi-scale space, producing the FRAC-DOG to reveal damage-induced singularities in mode shapes. Finally, a data fusion technique is applied to the proposed method to provide a reasonable damage detection result that maximises the elimination of uncertainty. The numerical and experimental results demonstrate that the proposed method can detect damage with high-precision in noisy environments by comparing it with the existing damage detection methods.

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.

A Multi-Scale Fractal Approach for Coal Permeability Estimation via MIP and NMR Methods

Permeability in porous media has an important role in many engineering applications, which depends mainly on the pore size, distribution, and connectivity of porous media. As the pore structure distribution of coal has a multi-scale fractal dimension characteristic, this study aimed to propose a multi-scale fractal dimension characteristics units model (MFU) to describe the pore structure distribution by analyzing the multi-scale fractal dimension characteristics of coal pore media. Then, a multi-scale fractal permeability model was established based on MFU. The pore structure distribution was obtained by mercury injection porosimetry (MIP) and nuclear magnetic resonance (NMR) experiments. Based on MIP and NMR experimental data, the permeability contribution of different pore diameters were calculated. The results show that the permeability contribution of the micropore was minimal and can be ignored. The permeability contribution of mesopores was about 1–5%, and the permeability contribution of macropores was about 95–99%, which plays a decisive role in the seepage process. The calculated results, based on multi-scale fractal permeability model and the experimental permeability data, are in the same order of magnitude. The permeability prediction based on proposed model is better than classical single fractal permeability model.

Diversity-Robust Acoustic Feature Signatures Based on Multiscale Fractal Dimension for Similarity Search of Environmental Sounds

This paper proposes new acoustic feature signatures based on the multiscale fractal dimension (MFD), which are robust against the diversity of environmental sounds, for the content-based similarity search. The diversity of sound sources and acoustic compositions is a typical feature of environmental sounds. Several acoustic features have been proposed for environmental sounds. Among them is the widely-used Mel-Frequency Cepstral Coefficients (MFCCs), which describes frequency-domain features. However, in addition to these features in the frequency domain, environmental sounds have other important features in the time domain with various time scales. In our previous paper, we proposed enhanced multiscale fractal dimension signature (EMFD) for environmental sounds. This paper extends EMFD by using the kernel density estimation method, which results in better performance of the similarity search tasks. Furthermore, it newly proposes another acoustic feature signature based on MFD, namely very-long-range multiscale fractal dimension signature (MFD-VL). The MFD-VL signature describes several features of the time-varying envelope for long periods of time. The MFD-VL signature has stability and robustness against background noise and small fluctuations in the parameters of sound sources, which are produced in field recordings. We discuss the effectiveness of these signatures in the similarity sound search by comparing with acoustic features proposed in the DCASE 2018 challenges. Due to the unique descriptiveness of our proposed signatures, we confirmed the signatures are effective when they are used with other acoustic features.

Multiscale fractal dimension applied to tactical analysis in football: A novel approach to evaluate the shapes of team organization on the pitch.

The aim of this study was to evaluate different shape descriptors applied to images of polygons that represent the organization of football teams on the pitch. The effectiveness of different shape descriptors (area/perimeter, fractal area, circularity, maximum fractal, rectangularity, multiscale fractal curve—MFC), and the concatenation of all shape descriptors (except MFC), denominated Alldescriptors (AllD)) was evaluated and applied to polygons corresponding to the shapes represented by the convex hull obtained from players’ 2D coordinates. A content-based image retrieval system (CBIR) was applied for 25 users (mean age of 31.9 ± 8.4 years) to evaluate the relevant images. Measures of effectiveness were used to evaluate the shape descriptors (P@n and R@n). The MFD (P@5, 0.46±0.37 and P@10, 0.40±0.31, p < 0.001; R@5, 0.14±0.13 and R@10, 0.24±0.19, p < 0.001) and AllD (P@5 = 0.43±0.36 and P@10 = 0.39±0.32, p < 0.001; R@5 = 0.13±0.11 and R@10 = 0.24±0.20, p < 0.001) descriptors presented higher values of effectiveness. As a practical demonstration, the best evaluated shape descriptor (MFC) was applied for tactical analysis of an official match. K-means clustering technique was applied, and different shapes of organization could be identified throughout the match. The MFC was the most effective shape descriptor in relation to all others, making it possible to apply this descriptor in the analysis of professional football matches.

Multiscale fractal dimension analysis of a reduced order model of coupled ocean–atmosphere dynamics

Abstract. Atmosphere and ocean dynamics display many complex features and are characterized by a wide variety of processes and couplings across different timescales. Here we demonstrate the application of multivariate empirical mode decomposition (MEMD) to investigate the multivariate and multiscale properties of a reduced order model of the ocean–atmosphere coupled dynamics. MEMD provides a decomposition of the original multivariate time series into a series of oscillating patterns with time-dependent amplitude and phase by exploiting the local features of the data and without any a priori assumptions on the decomposition basis. Moreover, each oscillating pattern, usually named multivariate intrinsic mode function (MIMF), represents a local source of information that can be used to explore the behavior of fractal features at different scales by defining a sort of multiscale and multivariate generalized fractal dimensions. With these two complementary approaches, we show that the ocean–atmosphere dynamics presents a rich variety of features, with different multifractal properties for the ocean and the atmosphere at different timescales. For weak ocean–atmosphere coupling, the resulting dimensions of the two model components are very different, while for strong coupling for which coupled modes develop, the scaling properties are more similar especially at longer timescales. The latter result reflects the presence of a coherent coupled dynamics. Finally, we also compare our model results with those obtained from reanalysis data demonstrating that the latter exhibit a similar qualitative behavior in terms of multiscale dimensions and the existence of a scale dependency of the statistics of the phase-space density of points for different regions, which is related to the different drivers and processes occurring at different timescales in the coupled atmosphere–ocean system. Our approach can therefore be used to diagnose the strength of coupling in real applications.

Test-retest reliability of multiscale fractal dimension measurements of plantar pressure maps during dynamic tasks

The tools used to evaluate foot types are divergent since they adopt classic linear analyzes, based on anthropometric or image measurements, which do not dynamically contemplate the variability of foot shape. The use of newer techniques such as multiscale fractal dimension (MFD) may be a key to this type of problem. However, for these measures to be used safely and consistently, it is essential to evaluate their reliability. The aim of this study was to evaluate the test-retest reliability of MFD measurements of adult plantar pressure maps during gait, as well as the standard error of measurement (SEM), and minimal detectable change (MDC90). Seventy-two subjects were included in the test-retest, with a one week interval. The plantar pressure maps were constructed using a pressure platform. The data were processed in a routine for extracting the MFD curve measurements (maximum and integral values). The Intraclass Correlation Coefficient results (ICC3,k) were excellent for both measurements (maximum value 0.96, 95% confidence interval [0.93–0.97], and integral 0.95 [0.92–0.97]) with low SEM and MDC90 values below 10% of the mean. The application of MFD to the plantar pressure data generated by the pressure platform is reliable and could allow exploration of the complexity of foot shapes, enabling their classification.

A Chaotic Feature Extraction Based on SMMF and CMMFD for Early Fault Diagnosis of Rolling Bearing

Various failure mechanisms of rolling bearing under different working conditions involve important nonlinear dynamic characteristics. And the incipient fault detection of the complex and non-stationary rolling bearing signal is difficult, especially with multiple interference source components. To address this issue, a novel fault feature extraction method which integrates chaos theory and mathematical morphology is proposed. Firstly, the smooth multi-scale morphological filtering (SMMF) is adopted to reduce the noise in chaotic rolling bearing signals without destroying its original nonlinear dynamic structure. In addition, a method of generalized estimation of morphological fractal dimension called composite multi-scale morphological fractal dimension (CMMFD) is proposed, and applied to quantify the complexity of those orbits reconstructed from rolling bearing signals. The CMMFD improves the identification ability of the original morphological fractal dimension (MFD). Then, CMMFD features are extracted from SMMF denoised signal. Different conditions of rolling bearing signals are distinguished in this feature space. Finally, the CMMFD features are input into a Hidden Markov Model (HMM) training and diagnosis to distinguish different faults of rolling bearing signals. The new diagnosis approach based on SMMF and CMMFD is compared with some existing methods in analysis of simulated signals and experimental rolling bearing signals. Accuracy and efficiency of different approaches are contrasted by analyzing the results of simulation and experiments. Comparative results demonstrate that the proposed diagnosis approach based on SMMF and CMMFD can reliably recognize different fault types at the early stage and have more accurate results.

Pedestrian network generation based on crowdsourced tracking data

ABSTRACT Pedestrian networks play an important role in various applications, such as pedestrian navigation services and mobility modeling. This paper presents a novel method to extract pedestrian networks from crowdsourced tracking data based on a two-layer framework. This framework includes a walking pattern classification layer and a pedestrian network generation layer. In the first layer, we propose a multi-scale fractal dimension (MFD) algorithm in order to recognize the two different types of walking patterns: walking with a clear destination (WCD) or walking without a clear destination (WOCD). In the second layer, we generate the pedestrian network by combining the pedestrian regions and pedestrian paths. The pedestrian regions are extracted based on a modified connected component analysis (CCA) algorithm from the WOCD traces. We generate the pedestrian paths using a kernel density estimation (KDE)-based point clustering algorithm from the WCD traces. The pedestrian network generation results using two actual crowdsourced datasets show that the proposed method has good performance in both geometrical correctness and topological correctness.

Multi-Scale Fractal Dimension to Quantify Heart Rate Variability and Systolic Blood Pressure Variability: A Postural Stress Analysis

This paper presents multi-scale fractal dimension (MFD) for the complexity assessment of heart rate variability (HRV) and systolic blood pressure variability (SBPV) under postural stress. A comparison of MFD with multi-scale approximate entropy (MAE) at multiple time scales ([Formula: see text]–10) is presented. The simulations were conducted on 50 random noise (RN), fractional Brownian motion (fBm), and MIX(p) [ 1 ] series along with standard EuroBaVar dataset. Also, 50 healthy volunteers, 20 post-acute myocardial infarction (AMI) patients participated in the study. It is observed that MFD and MAE decrease with [Formula: see text] for healthy volunteers and remained constant for post AMI and BRS failure patients. Average MFD of HRV in supine position is always significantly different than standing position scale except [Formula: see text], whereas the average MAE of HRV shows no significant difference ([Formula: see text]) between supine and standing postures at each [Formula: see text] except [Formula: see text] for healthy volunteers. Average MFD of HRV in supine position is significantly different than the standing position for [Formula: see text]–7, whereas the average MAE of HRV shows significant difference ([Formula: see text]) between supine and standing postures at [Formula: see text] for post AMI, and patients with impaired baroreflex sensitivity (BRS). Average MFD of SBPV in position is significantly different ([Formula: see text]) than standing position at each [Formula: see text], whereas the average MAE of SBPV shows no significant difference ([Formula: see text]) between supine and standing postures at each [Formula: see text] except [Formula: see text]. The complexity of HRV in standing position is less than that of supine position as shown by lower MFD and MAE values in standing than that of supine position, however, the complexity of SBPV decreases from standing to supine position indicating prevalence of sympathetic control.

Experiment and interpretation of geomorphological detection by multi-beam sonar in Huguang-yan Maar Lake

The Huguangyan Maar Lake is caused by a volcanic eruption. The original sediments at the bottom of the lake are natural yearbooks for the evolution of the earth’s climate and environment. A multi-beam sonar technique is used to scan the full coverage on the lake surface to get fine samples of echoes form the lake bottom. The information of the bathymetric data of the lake bottom is accurately described and the results of three-dimensional geomorphology imaging are given. The results show that the lake is divided into two parts, east and west, by a north-south underwater volcanic wall, and the maximum depth of water is over 22 meters. We apply a texture feature extraction method based on multi-scale fractal dimension to describe the characteristics of the sediments and roughness. Multi-scale fractal dimensions algorithm is used to extract waveform characteristics of depth samples in different directions. General distribution of sediment at the bottom of the lake is distinguished by the width of multi-scale fractal spectrum. The results obtained can be helpful for the estimation of the physical properties of sediments.

Research on Degradation State Recognition of Planetary Gear Based on Multiscale Information Dimension of SSD and CNN

Planetary gear is the key part of the transmission system for large complex electromechanical equipment, and in general, a series of degradation states are undergone and evolved into a local fatal fault in its full life cycle. So it is of great significance to recognize the degradation state of planetary gear for the purpose of maintenance repair, predicting development trend, and avoiding sudden fault. This paper proposed a degradation state recognition method of planetary gear based on multiscale information dimension of singular spectrum decomposition (SSD) and convolutional neural network (CNN). SSD can automatically realize the embedding dimension selection and component grouping segmentation, and the original vibration signal being nonlinear and nonstationary can be decomposed into a series of singular spectrum decomposition components (SSDCs), adaptively. Then, the multiscale information dimension which combines multiscale analysis and fractal information dimension is proposed for quantifying and extracting the feature information contained in each SSDC. Finally, CNN is used to achieve the effective recognition of the degradation state of planetary gear. The experimental results show that the proposed method can accurately recognize the degradation state of planetary gear, and the overall recognition rate is up to 97.2%, of which the recognition rate of normal planetary gear reaches 100%.

Angular descriptors of complex networks: A novel approach for boundary shape analysis

Abstract We introduce a method for shape recognition based on the angular analysis of Complex Networks. Our method models shapes as Complex Networks defining a more descriptive representation of the inner angularity of the shape’s perimeter. The result is a set of measures that better describe shapes if compared to previous approaches that use only the vertices’ degree. We extract the angle between the Complex Network edges, and then we analyze their distribution along with a network dynamic evolution. The proposed approach, named Angular Descriptors of Complex Networks (ADCN), presents a high discriminatory power, as evidenced by experiments conducted in five datasets. It is rotation invariant, presents high robustness against scale changes and degradation levels, overcoming traditional methods such as Zernike moments, Multiscale Fractal dimension, Fourier, Curvature and the degree-based descriptors of Complex Networks.

The power of 3D fractal dimensions for comparative shape and structural complexity analyses of irregularly shaped organisms

Abstract The increasing interest in 3D morphological approaches for addressing evolutionary and ecological questions has driven the need for reliable, efficient, and easily accessible methods for shape quantification and complexity characterization. This applies in particular to irregularly shaped organisms, such as stony corals, which are challenging to study using traditional morphometric analyses. A potential approach to assess shape and spatial complexity of 3D documented organisms are 3D fractal dimension analyses that combine information from various spatial scales, thus enabling a holistic shape quantification. Therefore, the latest 3D scanning technology and fractal dimension analyses were used to test the performance of this novel combination of methods for quantifying inter‐ and intraspecific differences in shape and spatial complexity of highly irregularly shaped organisms. As model taxa, six widespread species of stony corals were used, belonging to the genera Acropora, Pocillopora and Porites. First, high‐resolution shape models of coral colonies were generated using 3D structured light scanning. Then the univariate fractal dimension D and the multivariate multiscale fractal dimension as well as two traditional univariate morphological measures (surface–volume ratio and rugosity) were calculated. Finally all parameters were tested for significant interspecific and intraspecific differences, and the discriminative power of the two fractal dimension and the two traditional measures were assessed. The results show that 3D shape analyses based on fractal dimensions can be used to quantify both inter‐ and intraspecific variations in irregularly shaped organisms such as corals. Compared to traditional methods, fractal dimensions performed at least as good at the interspecific level and considerably better at the intraspecific level. This study demonstrates that 3D fractal dimension analyses are an efficient and easily applicable method for shape quantification and complexity characterization of irregularly shaped organisms, such as stony corals. Given the specific properties of fractal dimensions, they may either serve as an alternative complexity measure or as an additional characteristic for studying irregularly shaped organisms. Ultimately, fractal dimensions may provide a more integrative understanding of morphological variation within an ecological and evolutionary context, and could open up new opportunities for semi‐automatic or automatic species determination based on 3D morphological images.