Evaluating the fractal dimension of profiles

Fractal basins as a mechanism for the nimble brain

An interesting feature of the brain is its ability to respond to disparate sensory signals from the environment in unique ways depending on the environmental context or current brain state. In dynamical systems, this is an example of multi-stability, the ability to switch between multiple stable states corresponding to specific patterns of brain activity/connectivity. In this article, we describe chimera states, which are patterns consisting of mixed synchrony and incoherence, in a brain-inspired dynamical systems model composed of a network with weak individual interactions and chaotic/periodic local dynamics. We illustrate the mechanism using synthetic time series interacting on a realistic anatomical brain network derived from human diffusion tensor imaging. We introduce the so-called vector pattern state (VPS) as an efficient way of identifying chimera states and mapping basin structures. Clustering similar VPSs for different initial conditions, we show that coexisting attractors of such states reveal intricately "mingled" fractal basin boundaries that are immediately reachable. This could explain the nimble brain's ability to rapidly switch patterns between coexisting attractors.

Box dimension of the border of Kingdom of Saudi Arabia

Fractal dimension unlike topological dimension is (usually) a non-integer number which measures complexity, roughness, or irregularity of an object with respect to the space in which the set lies. It is used to characterize highly irregular objects in nature containing statistical self-similarity such as mountains, snowflakes, clouds, coastlines, borders etc. In this article, box dimension (a version of fractal dimension) of the border of Kingdom of Saudi Arabia (KSA) is computed using a multicore parallel processing algorithm based on the classical box-counting method. A power law relation is obtained from numerical simulations which relates the length of the border with the scale size and provides a very close estimate of the actual length of the KSA border within the scaling regions and scaling effects on the length of KSA border are considered. The algorithm presented in the article is shown to be highly scalable and efficient and the speedup of the algorithm is computed using Amdahl's and Gustafson's laws. For simulations, a high performance parallel computer is employed using Python codes and QGIS software.

Fractal dimension based geographical clustering of COVID-19 time series data

Understanding the local dynamics of COVID-19 transmission calls for an approach that characterizes the incidence curve in a small geographical unit. Given that incidence curves exhibit considerable day-to-day variation, the fractal structure of the time series dynamics is investigated for the Flanders and Brussels Regions of Belgium. For each statistical sector, the smallest administrative geographical entity in Belgium, fractal dimensions of COVID-19 incidence rates, based on rolling time spans of 7, 14, and 21 days were estimated using four different estimators: box-count, Hall-Wood, variogram, and madogram. We found varying patterns of fractal dimensions across time and location. The fractal dimension is further summarized by its mean, variance, and autocorrelation over time. These summary statistics are then used to cluster regions with different incidence rate patterns using k-means clustering. Fractal dimension analysis of COVID-19 incidence thus offers important insight into the past, current, and arguably future evolution of an infectious disease outbreak.

AFM advanced modes for dental and biomedical applications

Several analytical methods have been employed to elucidate bonding mechanisms between dental hard tissues, luting agents and restorative materials. Atomic Force Microscopy (AFM) imaging that has been extensively used in materials science, but its full capabilities are poorly explored by dental research community. In fact, commonly used to obtain topographic images of different surfaces, it turns out that AFM is an underestimated technique considering that there are dozens of basic and advanced modes that are scarcely used to explain properties of biomaterials. Thus, this paper addresses the use of phase-contrast imaging, force-distance curves, nanomechanical and Kelvin probe force techniques during AFM analysis to explore topological, nanomechanical and electrical properties of Y-TZP samples modified by different surface treatments, which has been widely used to promote adhesive enhancements to such substrate. The AFM methods are capable of access erstwhile inaccessible properties of Y-TZP which allowed us to describe its adhesive properties correctly. Thus, AFM technique emerges as a key tool to investigate the complex nature of biomaterials and highlighting its inherent interdisciplinarity that can be successfully used for bridging fragmented disciplines such as solid-state physics, microbiology and dental sciences.

Fractal analysis highlights analogies in arenaceous tubes of Sabellaria alveolata (Metazoa, Polychaeta) and agglutinated tests of foraminifera (Protista)

Bioconstructions of Sabellaria alveolata (Polychaeta Sabellariidae) from southern Sicily (Central Mediterranean) were sampled and analysed through a multidisciplinary approach in order to unravel the construction pattern of arenaceous tubes and explore possible analogies existing between the worm tubes and the agglutinated tests of benthic foraminifera (Protista). Scanning Electron Microscopy and Energy Dispersive Spectroscopy analyses were carried out on entire tubes as well as sectioned ones. Results show that arenaceous tubes are built following a rigorous architectural framework, based on selection and methodical arrangement of the agglutinated grains, and show surprising analogies with the test microstructure previously observed in agglutinated foraminifera. The grain distribution detected in both model species bioconstructions was analysed using a fractal numerical model (Hausdorff fractal dimension). Collected data show that in both organisms the grains were distributed according to a fractal model, indicating that the evolutionary process may have led to finding the same optimal constructive strategy across organisms with an independent evolutionary history, notwithstanding different geometrical scales. Furthermore, in sectioned tubes we observed microplastic fragments agglutinated within the arenaceous wall and in the inter-tube area. This unexpected finding shows that marine animals can be affected by microplastic pollution not only in soft tissues, but also engineered hard structures, and suggests the problem is more pervasive than estimated so far.

Lessons for Data-Driven Modelling from Harmonics in the Norwegian Grid

With the advancing integration of fluctuating renewables, a more dynamic demand-side, and a grid running closer to its operational limits, future power system operators require new tools to anticipate unwanted events. Advances in machine learning and availability of data suggest great potential in using data-driven approaches, but these will only ever be as good as the data they are based on. To lay the ground-work for future data-driven modelling, we establish a baseline state by analysing the statistical distribution of voltage measurements from three sites in the Norwegian power grid (22, 66, and 300 kV). Measurements span four years, are line and phase voltages, are cycle-by-cycle, and include all (even and odd) harmonics up to the 96 order. They are based on four years of historical data from three Elspec Power Quality Analyzers (corresponding to one trillion samples), which we have extracted, processed, and analyzed. We find that: (i) the distribution of harmonics depends on phase and voltage level; (ii) there is little power beyond the 13 harmonic; (iii) there is temporal clumping of extreme values; and (iv) there is seasonality on different time-scales. For machine learning based modelling these findings suggest that: (i) models should be trained in two steps (first with data from all sites, then adapted to site-level); (ii) including harmonics beyond the 13 is unlikely to increase model performance, and that modelling should include features that (iii) encode the state of the grid, as well as (iv) seasonality.

EPSPatNet86: eight-pointed star pattern learning network for detection ADHD disorder using EEG signals

Objective.The main objective of this work is to present a hand-modelled one-dimensional signal classification system to detect Attention-Deficit Hyperactivity Disorder (ADHD) disorder using electroencephalography (EEG) signals.Approach.A novel handcrafted feature extraction method is presented in this research. Our proposed method uses a directed graph and an eight-pointed star pattern (EPSPat). Also, tunable q wavelet transforms (TQWT), wavelet packet decomposition (WPD), statistical extractor, iterative Chi2 (IChi2) selector, and the k-nearest neighbors (kNN) classifier have been utilized to develop the EPSPat based learning model. This network uses two wavelet decomposition methods (TQWT and WPD), and 85 wavelet coefficient bands are extracted. The proposed EPSPat and statistical feature creator generate features from the 85 wavelet coefficient bands and the original EEG signal. The learning network is termed EPSPatNet86. The main purpose of the presented EPSPatNet86 is to detect abnormalities of the EEG signals. Therefore, 85 wavelet subbands have been generated to extract features. The created 86 feature vectors have been evaluated using the Chi2 selector and the kNN classifier in the loss value calculation phase. The final features vector is created by employing a minimum loss-valued eight feature vectors. The IChi2 selector selects the best feature vector, which is fed to the kNN classifier. An EEG signal dataset has been used to demonstrate the presented model's EEG signal classification ability. We have used an ADHD EEG dataset since ADHD is a commonly seen brain-related ailment.Main results.Our developed EPSPatNet86 model can detect the ADHD EEG signals with 97.19% and 87.60% accuracy using 10-fold cross and subject-wise validations, respectively.Significance.The calculated results demonstrate that the presented EPSPatNet86 attained satisfactory EEG classification ability. Results show that we can apply our developed EPSPatNet86 model to other EEG signal datasets to detect abnormalities.

Investigation of Fractal Carbon Nanotube Networks for Biophilic Neural Sensing Applications

We propose a carbon-nanotube-based neural sensor designed to exploit the electrical sensitivity of an inhomogeneous fractal network of conducting channels. This network forms the active layer of a multi-electrode field effect transistor that in future applications will be gated by the electrical potential associated with neuronal signals. Using a combination of simulated and fabricated networks, we show that thin films of randomly-arranged carbon nanotubes (CNTs) self-assemble into a network featuring statistical fractal characteristics. The extent to which the network's non-linear responses will generate a superior detection of the neuron's signal is expected to depend on both the CNT electrical properties and the geometric properties of the assembled network. We therefore perform exploratory experiments that use metallic gates to mimic the potentials generated by neurons. We demonstrate that the fractal scaling properties of the network, along with their intrinsic asymmetry, generate electrical signatures that depend on the potential's location. We discuss how these properties can be exploited for future neural sensors.

Automatic segmentation, feature extraction and comparison of healthy and stroke cerebral vasculature

Accurate segmentation of cerebral vasculature and a quantitative assessment of its morphology is critical to various diagnostic and therapeutic purposes and is pertinent to studying brain health and disease. However, this is still a challenging task due to the complexity of the vascular imaging data. We propose an automated method for cerebral vascular segmentation without the need of any manual intervention as well as a method to skeletonize the binary segmented map to extract vascular geometric features and characterize vessel structure. We combine a Hessian-based probabilistic vessel-enhancing filtering with an active-contour-based technique to segment magnetic resonance and computed tomography angiograms (MRA and CTA) and subsequently extract the vessel centerlines and diameters to calculate the geometrical properties of the vasculature. Our method was validated using a 3D phantom of the Circle-of-Willis region, demonstrating 84% mean Dice similarity coefficient (DSC) and 85% mean Pearson's correlation coefficient (PCC) with minimal modified Hausdorff distance (MHD) error (3 surface pixels at most), and showed superior performance compared to existing segmentation algorithms upon quantitative comparison using DSC, PCC and MHD. We subsequently applied our algorithm to a dataset of 40 subjects, including 1) MRA scans of healthy subjects (n = 10, age = 30 ± 9), 2) MRA scans of stroke patients (n = 10, age = 51 ± 15), 3) CTA scans of healthy subjects (n = 10, age = 62 ± 12), and 4) CTA scans of stroke patients (n = 10, age = 68 ± 11), and obtained a quantitative comparison between the stroke and normal vasculature for both imaging modalities. The vascular network in stroke patients compared to age-adjusted healthy subjects was found to have a significantly (p < 0.05) higher tortuosity (3.24 ± 0.88 rad/cm vs. 7.17 ± 1.61 rad/cm for MRA, and 4.36 ± 1.32 rad/cm vs. 7.80 ± 0.92 rad/cm for CTA), higher fractal dimension (1.36 ± 0.28 vs. 1.71 ± 0.14 for MRA, and 1.56 ± 0.05 vs. 1.69 ± 0.20 for CTA), lower total length (3.46 ± 0.99 m vs. 2.20 ± 0.67 m for CTA), lower total volume (61.80 ± 18.79 ml vs. 34.43 ± 22.9 ml for CTA), lower average diameter (2.4 ± 0.21 mm vs. 2.18 ± 0.07 mm for CTA), and lower average branch length (4.81 ± 1.97 mm vs. 8.68 ± 2.03 mm for MRA), respectively. We additionally studied the change in vascular features with respect to aging and imaging modality. While we observed differences between features as a result of aging, statistical analysis did not show any significant differences, whereas we found that the number of branches were significantly different (p < 0.05) between the two imaging modalities (201 ± 73 for MRA vs. 189 ± 69 for CTA). Our segmentation and feature extraction algorithm can be applied on any imaging modality and can be used in the future to automatically obtain the 3D segmented vasculature for diagnosis and treatment planning as well as to study morphological changes due to stroke and other cerebrovascular diseases (CVD) in the clinic.

Capillary Water Absorption and Micro Pore Connectivity of Concrete with Fractal Analysis

This study focuses on the relationship between the complexity of pore structure and capillary water absorption of concrete, as well as the connection behavior of concrete in specific directions. In this paper, the water absorption of concrete with different binders was tested during the curing process, and the pore structure of concrete was investigated by mercury intrusion porosimetry (MIP). The results show that the water absorption of concrete with mineral admixtures is lower, mainly due to the existence of reasonable pore structure. The effect of slag on concrete modification is more remarkable comparing with fly ash. In addition, the analysis shows that the pore with different diameters has different fractal characteristics. The connectivity probability and water absorption of unidirectional chaotic pore are linearly correlated with the pore diameter of 50–550 nm, and the correlation coefficient reaches a very significant level, and detailed analysis was undertaken to interpret these results based on fractal theory.

Geospatial immune variability illuminates differential evolution of lung adenocarcinoma

Remarkable progress in molecular analyses has improved our understanding of the evolution of cancer cells toward immune escape1-5. However, the spatial configurations of immune and stromal cells, which may shed light on the evolution of immune escape across tumor geographical locations, remain unaddressed. We integrated multiregion exome and RNA-sequencing (RNA-seq) data with spatial histology mapped by deep learning in 100 patients with non-small cell lung cancer from the TRACERx cohort6. Cancer subclones derived from immune cold regions were more closely related in mutation space, diversifying more recently than subclones from immune hot regions. In TRACERx and in an independent multisample cohort of 970 patients with lung adenocarcinoma, tumors with more than one immune cold region had a higher risk of relapse, independently of tumor size, stage and number of samples per patient. In lung adenocarcinoma, but not lung squamous cell carcinoma, geometrical irregularity and complexity of the cancer-stromal cell interface significantly increased in tumor regions without disruption of antigen presentation. Decreased lymphocyte accumulation in adjacent stroma was observed in tumors with low clonal neoantigen burden. Collectively, immune geospatial variability elucidates tumor ecological constraints that may shape the emergence of immune-evading subclones and aggressive clinical phenotypes.

Airway fractal dimension predicts respiratory morbidity and mortality in COPD

BACKGROUND

Chronic obstructive pulmonary disease (COPD) is characterized by airway remodeling. Characterization of airway changes on computed tomography has been challenging due to the complexity of the recurring branching patterns, and this can be better measured using fractal dimensions.

METHODS

We analyzed segmented airway trees of 8,135 participants enrolled in the COPDGene cohort. The fractal complexity of the segmented airway tree was measured by the Airway Fractal Dimension (AFD) using the Minkowski-Bougliand box-counting dimension. We examined associations between AFD and lung function and respiratory morbidity using multivariable regression analyses. We further estimated the extent of peribronchial emphysema (%) within 5 mm of the airway tree, as this is likely to affect AFD. We classified participants into 4 groups based on median AFD, percentage of peribronchial emphysema, and estimated survival.

RESULTS

AFD was significantly associated with forced expiratory volume in one second (FEV1; P < 0.001) and FEV1/forced vital capacity (FEV1/FVC; P < 0.001) after adjusting for age, race, sex, smoking status, pack-years of smoking, BMI, CT emphysema, air trapping, airway thickness, and CT scanner type. On multivariable analysis, AFD was also associated with respiratory quality of life and 6-minute walk distance, as well as exacerbations, lung function decline, and mortality on longitudinal follow-up. We identified a subset of participants with AFD below the median and peribronchial emphysema above the median who had worse survival compared with participants with high AFD and low peribronchial emphysema (adjusted hazards ratio [HR]: 2.72; 95% CI: 2.20-3.35; P < 0.001), a substantial number of whom were not identified by traditional spirometry severity grades.

CONCLUSION

Airway fractal dimension as a measure of airway branching complexity and remodeling in smokers is associated with respiratory morbidity and lung function change, offers prognostic information additional to traditional CT measures of airway wall thickness, and can be used to estimate mortality risk.

TRIAL REGISTRATION

ClinicalTrials.gov identifier: NCT00608764.

FUNDING

This study was supported by NIH K23 HL133438 (SPB) and the COPDGene study (NIH Grant Numbers R01 HL089897 and R01 HL089856). The COPDGene project is also supported by the COPD Foundation through contributions made to an Industry Advisory Board comprised of AstraZeneca, Boehringer Ingelheim, Novartis, Pfizer, Siemens, Sunovion and GlaxoSmithKline.

Definition of fractal topography to essential understanding of scale-invariance

Fractal behavior is scale-invariant and widely characterized by fractal dimension. However, the cor-respondence between them is that fractal behavior uniquely determines a fractal dimension while a fractal dimension can be related to many possible fractal behaviors. Therefore, fractal behavior is independent of the fractal generator and its geometries, spatial pattern, and statistical properties in addition to scale. To mathematically describe fractal behavior, we propose a novel concept of fractal topography defined by two scale-invariant parameters, scaling lacunarity (P) and scaling coverage (F). The scaling lacunarity is defined as the scale ratio between two successive fractal generators, whereas the scaling coverage is defined as the number ratio between them. Consequently, a strictly scale-invariant definition for self-similar fractals can be derived as D = log F /log P. To reflect the direction-dependence of fractal behaviors, we introduce another parameter H_xy, a general Hurst exponent, which is analytically expressed by H_xy = log P_x/log P_y where P_x and P_y are the scaling lacunarities in the x and y directions, respectively. Thus, a unified definition of fractal dimension is proposed for arbitrary self-similar and self-affine fractals by averaging the fractal dimensions of all directions in a d-dimensional space, which . Our definitions provide a theoretical, mechanistic basis for understanding the essentials of the scale-invariant property that reduces the complexity of modeling fractals.

The remarkable coherence between two Italian far away recording stations points to a role of acoustic emissions from crustal rocks for earthquake analysis

We observed a remarkable near-to-unity correlation between the time series of Acoustic Emissions (AEs) collected at two stations approximately 300 km apart from each other and located along the Apennine belt (Italy). This finding prompted us to verify the hypothesis that AE signals can carry with them an indication of anomalies in a crustal stress trend, possibly related to earthquake occurrences. Thus, we checked the ability of Recurrence Quantification Analysis and Fractal Analysis as applied to AE to identify signal phase transitions before the crisis occurs. The sharp drop of the Percent of Determinism after its maximum value, and simultaneously with minimum values of the Fractal Dimension (D), few days before some seismic events take place, seems to point to the relevance of the proposed approach as precursor detection.

Patterns of temporal scaling of groundwater level fluctuation

We studied the fractal scaling behavior of groundwater level fluctuation for various types of aquifers in Puerto Rico using the methods of (1) detrended fluctuation analysis (DFA) to examine the monofractality and (2) wavelet transform maximum modulus (WTMM) to analyze the multifractality. The DFA results show that fractals exist in groundwater fluctuations of all the aquifers with scaling patterns that are anti-persistent (1 < β < 1.5; 1.32 ± 0.12, 18 wells) or persistent (β > 1.5; 1.62 ± 0.07, 4 wells). The multi-fractal analysis confirmed the need to characterize these highly complex processes with multifractality, which originated from the stochastic distribution of the irregularly-shaped fluctuations. The singularity spectra of the fluctuation processes in each well were site specific. We found a general elevational effect with smaller fractal scaling coefficients in the shallower wells, except for the Northern Karst Aquifer Upper System. High spatial variability of fractal scaling of groundwater level fluctuations in the karst aquifer is due to the coupled effects of anthropogenic perturbations, precipitation, elevation and particularly the high heterogeneous hydrogeological conditions.

Quantitative Assessment of Mouse Mammary Gland Morphology Using Automated Digital Image Processing and TEB Detection

The assessment of rodent mammary gland morphology is largely used to study the molecular mechanisms driving breast development and to analyze the impact of various endocrine disruptors with putative pathological implications. In this work, we propose a methodology relying on fully automated digital image analysis methods including image processing and quantification of the whole ductal tree and of the terminal end buds as well. It allows to accurately and objectively measure both growth parameters and fine morphological glandular structures. Mammary gland elongation was characterized by 2 parameters: the length and the epithelial area of the ductal tree. Ductal tree fine structures were characterized by: 1) branch end-point density, 2) branching density, and 3) branch length distribution. The proposed methodology was compared with quantification methods classically used in the literature. This procedure can be transposed to several software and thus largely used by scientists studying rodent mammary gland morphology.

Temporal Structure of Human Gaze Dynamics Is Invariant During Free Viewing

We investigate the dynamic structure of human gaze and present an experimental study of the frequency components of the change in gaze position over time during free viewing of computer-generated fractal images. We show that changes in gaze position are scale-invariant in time with statistical properties that are characteristic of a random walk process. We quantify and track changes in the temporal structure using a well-defined scaling parameter called the Hurst exponent, H. We find H is robust regardless of the spatial complexity generated by the fractal images. In addition, we find the Hurst exponent is invariant across all participants, including those with distinct changes to higher order visual processes due to neural degeneration. The value we find for H of 0.57 shows that the gaze dynamics during free viewing of fractal images are consistent with a random walk process with persistent movements. Our research suggests the human visual system may have a common strategy that drives the dynamics of human gaze during exploration.

Lung cancer-a fractal viewpoint

Fractals are mathematical constructs that show self-similarity over a range of scales and non-integer (fractal) dimensions. Owing to these properties, fractal geometry can be used to efficiently estimate the geometrical complexity, and the irregularity of shapes and patterns observed in lung tumour growth (over space or time), whereas the use of traditional Euclidean geometry in such calculations is more challenging. The application of fractal analysis in biomedical imaging and time series has shown considerable promise for measuring processes as varied as heart and respiratory rates, neuronal cell characterization, and vascular development. Despite the advantages of fractal mathematics and numerous studies demonstrating its applicability to lung cancer research, many researchers and clinicians remain unaware of its potential. Therefore, this Review aims to introduce the fundamental basis of fractals and to illustrate how analysis of fractal dimension (FD) and associated measurements, such as lacunarity (texture) can be performed. We describe the fractal nature of the lung and explain why this organ is particularly suited to fractal analysis. Studies that have used fractal analyses to quantify changes in nuclear and chromatin FD in primary and metastatic tumour cells, and clinical imaging studies that correlated changes in the FD of tumours on CT and/or PET images with tumour growth and treatment responses are reviewed. Moreover, the potential use of these techniques in the diagnosis and therapeutic management of lung cancer are discussed.

Is a two-dimensional generalization of the Higuchi algorithm really necessary?

Fractal dimensions of data series, particularly time series can be estimated very well by using Higuchi's algorithm. Without phase space constructions, the fractal dimension of a one-dimensional data stream is calculated. Higuchi's method is well accepted and widely applied, because it is very reliable and easy to implement. A generalization of the genuine 1D algorithm to two dimensions would be desirable in order to investigate digital images. In this study, we propose several 2D generalization algorithms and evaluate differences between them. Additionally, a comparison to previously published pseudo 2D generalizations, and to the Fourier and the Blanket method are presented. The algorithms were tested on artificially generated grey value and red-green-blue colour images. It turned out that the proposed 2D generalized Higuchi algorithms are very robust, but differences in between the generalizations as well as differences to the pseudo 2D algorithms are astonishingly small.

Analysis of Fibonacci gratings and their diffraction patterns

Aperiodic and fractal optical elements are proving to be promising candidates in image-forming devices. In this paper, we analyze the diffraction patterns of Fibonacci gratings (FbGs), which are prototypical examples of aperiodicity. They exhibit novel characteristics such as redundancy and robustness that keep their imaging characteristics intact even when there is significant loss of information. FbGs also contain fractal signatures and are characterized by a fractal dimension. Our study suggests that aperiodic gratings may be better than their fractal counterparts in technologies based on such architectures. We also identify the demarcating features of aperiodic and fractal diffraction, which have been rather fuzzy in the literature so far.

Fractal signatures in the aperiodic Fibonacci grating

The Fibonacci grating (FbG) is an archetypal example of aperiodicity and self-similarity. While aperiodicity distinguishes it from a fractal, self-similarity identifies it with a fractal. Our paper investigates the outcome of these complementary features on the FbG diffraction profile (FbGDP). We find that the FbGDP has unique characteristics (e.g., no reduction in intensity with increasing generations), in addition to fractal signatures (e.g., a non-integer fractal dimension). These make the Fibonacci architecture potentially useful in image forming devices and other emerging technologies.

Accuracy and precision of fractal dimension measured on model surfaces

OBJECTIVES

To develop a method, which is precise, accurate, and insensitive to the angle of inclination for determining the fractal dimensional increment (D*) of a surface.

METHODS

Brownian interpolation was used to generate simulated ceramic fracture surfaces having known D* values of 0.1, 0.2, 0.3, and 0.4 with 10 surfaces at each D* value. Each surface was inclined at four angles (0°, 3°, 5°, and 7°) from horizontal. The 160 (40×4) surfaces were analyzed by a variety of methods including Minkowski Cover (MC), Root Mean Square Roughness vs. Area (RMS), Kolmogorov Box (KB), Hurst Exponent (HE), Slit Island Box (SIB), and Slit Island Richardson (SIR). The coefficient of variation (CV) and mean error were used to identify the methods with best precision (lowest CV) and accuracy (lowest mean error), respectively, and three-way ANOVA followed by Turkey's HSD (α=0.05) was used to identify significant effects.

RESULTS

CV was significantly affected by fractal dimension (p=0.002) and method (p<0.001) but not by angle of inclination (p=0.765). The CV value for MC was lower than those for other methods (p≤0.05). Mean error was significantly affected by three-way interaction between fractal dimension, method, and angle of inclination (p<0.001). The mean error for KB was higher than those for other methods (p≤0.05) for inclined surfaces.

SIGNIFICANCE

MC was determined to have the best combination of precision, accuracy, and lack of sensitivity to angle of inclination for Brownian interpolation surfaces having D* values in the range commonly reported for ceramic fracture surfaces.

Fractal dynamics in chaotic quantum transport

Despite several experiments on chaotic quantum transport in two-dimensional systems such as semiconductor quantum dots, corresponding quantum simulations within a real-space model have been out of reach so far. Here we carry out quantum transport calculations in real space and real time for a two-dimensional stadium cavity that shows chaotic dynamics. By applying a large set of magnetic fields we obtain a complete picture of magnetoconductance that indicates fractal scaling. In the calculations of the fractality we use detrended fluctuation analysis-a widely used method in time-series analysis-and show its usefulness in the interpretation of the conductance curves. Comparison with a standard method to extract the fractal dimension leads to consistent results that in turn qualitatively agree with the previous experimental data.

Robustness of Cantor diffractals

Diffractals are electromagnetic waves diffracted by a fractal aperture. In an earlier paper, we reported an important property of Cantor diffractals, that of redundancy [R. Verma et. al., Opt. Express 20, 8250 (2012)]. In this paper, we report another important property, that of robustness. The question we address is: How much disorder in the Cantor grating can be accommodated by diffractals to continue to yield faithfully its fractal dimension and generator? This answer is of consequence in a number of physical problems involving fractal architecture.

Fractal dimension and vessel complexity in patients with cerebral arteriovenous malformations

The fractal dimension (FD) can be used as a measure for morphological complexity in biological systems. The aim of this study was to test the usefulness of this quantitative parameter in the context of cerebral vascular complexity. Fractal analysis was applied on ten patients with cerebral arteriovenous malformations (AVM) and ten healthy controls. Maximum intensity projections from Time-of-Flight MRI scans were analyzed using different measurements of FD, the Box-counting dimension, the Minkowski dimension and generalized dimensions evaluated by means of multifractal analysis. The physiological significance of this parameter was investigated by comparing values of FD first, with the maximum slope of contrast media transit obtained from dynamic contrast-enhanced MRI data and second, with the nidus size obtained from X-ray angiography data. We found that for all methods, the Box-counting dimension, the Minkowski dimension and the generalized dimensions FD was significantly higher in the hemisphere with AVM compared to the hemisphere without AVM indicating that FD is a sensitive parameter to capture vascular complexity. Furthermore we found a high correlation between FD and the maximum slope of contrast media transit and between FD and the size of the central nidus pointing out the physiological relevance of FD. The proposed method may therefore serve as an additional objective parameter, which can be assessed automatically and might assist in the complex workup of AVMs.

Evolution of fractal structures in dislocation ensembles during plastic deformation

Based on the irreversible thermodynamics approach to dislocation plasticity of metals, a simple description of the dislocation density evolution and strain hardening was suggested. An analytical expression for the fractal dimension (FD) of a cellular (or tangled) dislocation structure evolving in the course of plastic deformation was obtained on the basis of the dislocation model proposed. This makes it possible to trace the variation of FD of the dislocation cell structure with strain by just measuring the macroscopic stress-strain curve. The FD behavior predicted in this way showed good agreement with the experimentally measured FD evolution at different stages of deformation of a Ni single crystal and a Cu polycrystal. One new result following from the present model is that the FD of the bulk dislocation structure in a deforming metal peaks at a certain strain close to the onset of necking. The significance of fractal analysis as an informative index to follow the spatial evolution of dislocation structures approaching the critical state is highlighted.

Redundancy in Cantor diffractals

Cantor diffractals are waves that have encountered a Cantor grating. In this paper, we report an important property of Cantor diffractals, namely that of redundancy. We observe that the Fraunhofer diffraction pattern comprises of several bands, each containing complete information about the fractal aperture. This redundancy allows for a faithful reconstruction of the Cantor grating by an inverse Fourier transformation of an arbitrary band.

Geometrical exponents of contour loops on synthetic multifractal rough surfaces: multiplicative hierarchical cascade p model

In this paper, we study many geometrical properties of contour loops to characterize the morphology of synthetic multifractal rough surfaces, which are generated by multiplicative hierarchical cascading processes. To this end, two different classes of multifractal rough surfaces are numerically simulated. As the first group, singular measure multifractal rough surfaces are generated by using the p model. The smoothened multifractal rough surface then is simulated by convolving the first group with a so-called Hurst exponent, H*. The generalized multifractal dimension of isoheight lines (contours), D(q), correlation exponent of contours, x(l), cumulative distributions of areas, ξ, and perimeters, η, are calculated for both synthetic multifractal rough surfaces. Our results show that for both mentioned classes, hyperscaling relations for contour loops are the same as that of monofractal systems. In contrast to singular measure multifractal rough surfaces, H* plays a leading role in smoothened multifractal rough surfaces. All computed geometrical exponents for the first class depend not only on its Hurst exponent but also on the set of p values. But in spite of multifractal nature of smoothened surfaces (second class), the corresponding geometrical exponents are controlled by H*, the same as what happens for monofractal rough surfaces.

Fractal characteristics of May-Grünwald-Giemsa stained chromatin are independent prognostic factors for survival in multiple myeloma

BACKGROUND

The use of computerized image analysis for the study of nuclear texture features has provided important prognostic information for several neoplasias. Recently fractal characteristics of the chromatin structure in routinely stained smears have shown to be independent prognostic factors in acute leukemia. In the present study we investigated the influence of the fractal dimension (FD) of chromatin on survival of patients with multiple myeloma.

METHODOLOGY

We analyzed 67 newly diagnosed patients from our Institution treated in the Brazilian Multiple Myeloma Study Group. Diagnostic work-up consisted of peripheral blood counts, bone marrow cytology, bone radiograms, serum biochemistry and cytogenetics. The International Staging System (ISS) was used. In every patient, at least 40 digital nuclear images from diagnostic May-Grünwald-Giemsa stained bone marrow smears were acquired and transformed into pseudo-3D images. FD was determined by the Minkowski-Bouligand method extended to three dimensions. Goodness-of-fit of FD was estimated by the R(2) values in the log-log plots. The influence of diagnostic features on overall survival was analyzed in Cox regressions. Patients that underwent autologous bone marrow transplantation were censored at the day of transplantation.

PRINCIPAL FINDINGS

Median age was 56 years. According to ISS, 14% of the patients were stage I, 39% were stage II and 47% were stage III. Additional features of a bad prognosis were observed in 46% of the cases. When stratifying for ISS, both FD and its goodness-of-fit were significant prognostic factors in univariate analyses. Patients with higher FD values or lower goodness-of-fit showed a worse outcome. In the multivariate Cox-regression, FD, R(2), and ISS stage entered the final model, which showed to be stable in a bootstrap resampling study.

CONCLUSIONS

Fractal characteristics of the chromatin texture in routine cytological preparations revealed relevant prognostic information in patients with multiple myeloma.

A Modified Box-Counting Method to Estimate the Fractal Dimensions

A fractal is a property of self-similarity, each small part of the fractal object is similar to the whole body. The traditional box-counting method (TBCM) to estimate fractal dimension can not reflect the self-similar property of the fractal and leads to two major problems, the border effect and noninteger values of box size. The modified box-counting method (MBCM), proposed in this study, not only eliminate the shortcomings of the TBCM, but also reflects the physical meaning about the self-similar of the fractal. The applications of MBCM shows a good estimation compared with the theoretical ones, which the biggest difference is smaller than 5%.

Front instabilities and invasiveness of simulated avascular tumors

We study the interface morphology of a 2D simulation of an avascular tumor composed of identical cells growing in an homogeneous healthy tissue matrix (TM), in order to understand the origin of the morphological changes often observed during real tumor growth. We use the Glazier-Graner-Hogeweg model, which treats tumor cells as extended, deformable objects, to study the effects of two parameters: a dimensionless diffusion-limitation parameter defined as the ratio of the tumor consumption rate to the substrate transport rate, and the tumor-TM surface tension. We model TM as a nondiffusing field, neglecting the TM pressure and haptotactic repulsion acting on a real growing tumor; thus, our model is appropriate for studying tumors with highly motile cells, e.g., gliomas. We show that the diffusion-limitation parameter determines whether the growing tumor develops a smooth (noninvasive) or fingered (invasive) interface, and that the sensitivity of tumor morphology to tumor-TM surface tension increases with the size of the dimensionless diffusion-limitation parameter. For large diffusion-limitation parameters, we find a transition (missed in previous work) between dendritic structures, produced when tumor-TM surface tension is high, and seaweed-like structures, produced when tumor-TM surface tension is low. This observation leads to a direct analogy between the mathematics and dynamics of tumors and those observed in nonbiological directional solidification. Our results are also consistent with the biological observation that hypoxia promotes invasive growth of tumor cells by inducing higher levels of receptors for scatter factors that weaken cell-cell adhesion and increase cell motility. These findings suggest that tumor morphology may have value in predicting the efficiency of antiangiogenic therapy in individual patients.

Investigations of human EEG response to viewing fractal patterns

Owing to the prevalence of fractal patterns in natural scenery and their growing impact on cultures around the world, fractals constitute a common feature of our daily visual experiences, raising an important question: what responses do fractals induce in the observer? We monitored subjects' EEG while they were viewing fractals with different fractal dimensions, and the results show that significant effects could be found in the EEG even by employing relatively simple silhouette images. Patterns with a fractal dimension of 1.3 elicited the most interesting EEG, with the highest alpha in the frontal lobes but also the highest beta in the parietal area, pointing to a complicated interplay between different parts of the brain when experiencing this pattern.

Analysis of simulated scanning of atomic-scale silicon surface by atomic force microscopy

This study constructs a contact-mode atomic force microscopy (AFM) simulation measurement model with constant force mode to simulate and analyze the outline scanning measurement by AFM. The simulation method is that when the probe passes the surface of sample, the action force of the atom of sample received by the atom of the probe can be calculated by using Morse potential. Through calculation, the equivalent force on the cantilever of probe can be acquired. By using the deflection angle equation for the cantilever of probe developed and inferred by this study, the deflection angle of receiving action force can be calculated. On the measurement point, as the deflection angle reaches a fixed deflection angle, the scan height of this simulation model can be acquired. By scanning in the right order, the scan curve of the simulation model can be obtained. By using this simulation measurement model, this study simulates and analyzes the scanning of atomic-scale surface outline. Meanwhile, focusing on the tip radii of different probes, the concept of sensitivity analysis is employed to investigate the effects of the tip radius of probe on the atomic-scale surface outline. As a result, it is found from the simulation on the atomic-scale surface that within the simulation scope of this study, when the tip radius of probe is greater than 12 nm, the effects of single atom on the scan curve of AFM can be better decreased or eliminated.

A simple digital image processing system to aid in melanoma diagnosis in an everyday melanocytic skin lesion unit: a preliminary report

BACKGROUND

For early melanoma diagnosis, experienced dermatologists have an accuracy of 64-80% using clinical diagnostic criteria, usually the ABCD rule, while automated melanoma diagnosis systems are still considered to be experimental and serve as adjuncts to the naked-eye expert prediction. In an attempt to aid in early melanoma diagnosis, we developed an image processing program with the aim to discriminate melanoma from melanocytic nevi, establishing a mathematical model to come up with a melanoma probability.

METHODS

Digital images of 132 melanocytic skin lesions (23 melanomas and 109 melanocytic nevi) were studied in features of geometry, color, and color texture. A total of 43 variables were studied for all lesions, e.g., geometry, color texture, sharpness of border, and color variables. Univariate logistic regression analysis followed by "-2 log likelihood" test and Spearman's rank correlation coefficient were used to eliminate inappropriate variables, as the presence of multi-collinearity among variables could cause severe problems in any stepwise variable selection method. Initially, "-2 log likelihood" and nonparametric Spearman's rho picked five variables to be included in a multivariate model of prediction. The five-variable model was then reduced to three variables and the performance of each model was tested. The "jackknife" method was performed in order to validate the model with the three variables and its accuracy was weighed vs. the five-variable model by receiver-operating characteristics (ROC) curve plotting. It was concluded that the reduced model did not compromise discriminatory power.

RESULTS

Not all variables contributed much to the model, therefore they were progressively eliminated and the model was finally reduced to three covariates of significance. A predictive equation was calculated, incorporating parameters of geometry, color, and color texture as independent covariates for the prediction of melanoma. The proposed model provides melanoma probability with a 60.9% sensitivity and 95.4% specificity of prediction, an overall accuracy of 89.4% (probability level 0.5), and 8% false-negative results.

CONCLUSIONS

Through a digital image processing system and the development of a mathematical model of prediction, discrimination between melanomas and melanocytic nevi seems feasible with a high rate of accuracy using multivariate logistic regression analysis. The proposed model is an alternative method to aid in early melanoma diagnosis. Expensive and sophisticated equipment is not required and it can be easily implemented in a reasonably priced portable programmable computer, in order to predict previously undiagnosed skin melanoma before histopathology results confirm diagnosis.

Long-term signal detection, segmentation and summarization using wavelets and fractal dimension: a bioacoustics application in gastrointestinal-motility monitoring

The current paper describes a wavelet-based method for long-term processing and analysis of gastrointestinal sounds (GIS). Windowing techniques are used to select sequential blocks of the prolonged multi-channel recordings and proceed to various wavelet-domain processing stages. De-noising, significant-activity detection, automated segmentation and extraction of summary curves are applied in an integrated mode, allowing for enhanced content manipulation and analysis. The proposed analysis scheme combines flexible long-term graphical representation tools, while maintaining the ability of quick browsing via visualization and auralization of the detected short-term events. This work is part of a project aiming to implement non-invasive diagnosis over gastrointestinal-motility (GIM) physiology. However, the proposed techniques might be applied to any study of long-term bioacoustics time series.

Anatomy and flow in normal and ischemic microvasculature based on a novel temporal fractal dimension analysis algorithm using contrast enhanced ultrasound

Strategies for improvement of blood flow by promoting new vessel growth in ischemic tissue are being developed. Recently, contrast-enhanced ultrasound (CEU) imaging has been used to assess tissue perfusion in models of ischemia-related angiogenesis, growth-factor mediated angiogenesis, and tumor angiogenesis. In these studies, microvascular flow is measured in order to assess the total impact of adaptations at different vascular levels. High-resolution methods for imaging larger vessels have been developed in order to derive "angiograms" of arteries, veins, and medium to large microvessels. We describe a novel method of vascular bed (microvessel and arterial) characterization of vessel anatomy and flow simultaneously, using serial measurement of the fractal dimension (FD) of a temporal sequence of CEU images. This method is proposed as an experimental methodology to distinguish ischemic from nonischemic tissue. Moreover, an improved approach for extracting the FD unique to this application is introduced.

Use of color texture in determining the nature of melanocytic skin lesions—a qualitative and quantitative approach

Melanocytic nevi are recognized as precursors of melanoma. Aiding in early recognition of melanoma, we estimated color texture parameters, fractal dimension and lacunarity of melanoma and other melanocytic nevi. Digital images of the lesions were processed. Graphic three-dimensional pseudoelevation images of the lesions and surrounding skin were produced to identify irregularities in color texture within the lesions. Estimation of lacunarity and fractal dimension followed in order to produce a numerical estimate of the coarseness of color texture. Clinicians readily perceive the resulting "geographical" images. Irregularity in the anaglyph, which might veil malignancy, is effortlessly identified through these images, and therefore an early excision of a suspect lesion is indicated.

A practical method to experimentally evaluate the Hausdorff dimension: an alternative phase-transition-based methodology

We introduce a methodology to estimate numerically the Hausdorff dimension of a geometric set. This practical method has been conceived as a subsequent tool of another context study, associated to our concern to distinguish between various fractal sets. Its conception is natural since it can be related to the original idea involved in the definitions of Hausdorff measure and Hausdorff dimension. It is based on the critical behavior of the measure spectrum functions of the set around its Hausdorff dimension value. We illustrate on several well-known examples, the ability of this method to accurately estimate the Hausdorff dimension. Also, we show how the transition property, exhibited by the quantities used as substitutes of the Hausdorff measure in the corresponding fractal dimension relationships, can be used to accurately estimate the fractal dimension. To show the potential of our method, we also report the results of Hausdorff dimension measurements on some typical examples, compared to a direct application of the scaling relation involved in the box-counting dimension definition.

The influence of edge detection algorithms on the estimation of the fractal dimension of binary digital images

The boundary of a fractal object, represented in a two-dimensional space, is theoretically a line with an infinitely small width. In digital images this boundary or contour is limited to the pixel resolution of the image and the width of the line commonly depends on the edge detection algorithm used. The Minkowski dimension was evaluated by using three different edge detection algorithms (Sobel, Roberts, and Laplace operator). These three operators were investigated because they are very widely used and because their edge detection result is very distinct concerning the line width. Very common fractals (Sierpinski carpet and Koch islands) were investigated as well as the binary images from a cancer invasion assay taken with a confocal laser scanning microscope. The fractal dimension is directly proportional to the width of the contour line and the fact, that in practice very often the investigated objects are fractals only within a limited resolution range is considered too.