Fractal Box Counting Research Articles

IMPLEMENTASI DIMENSI FRAKTAL BOX COUNTING DAN K-MEANS DALAM KLASIFIKASI JENIS KUPU-KUPU (LEPIDOPTERA) BERDASARKAN BENTUK SAYAP

Butterflies are insects that have aesthetic beauty and are also important assets in the field of nature protection. In nature, butterflies have a crucial role in maintaining ecosystems, especially in the process of pollination of flowering plants which contributes to the natural propagation of plants. Butterfly wings have an important role in identifying butterfly species based on size, shape and color. In mathematics, there is the concept of fractals which is a very suitable method for modeling natural phenomena and used as a tool to represent natural objects. In this study, a classification of butterfly species was carried out based on the shape of the butterfly's wings. A total of 100 butterfly images were used in this study, the first step was to convert the RGB images to grayscale and then segment the butterfly wings. The area is then used to identify butterfly wing type patterns using edge detection using the Canny method. Furthermore, by using the fractal box counting method, the resulting dimensional values are used in the clustering process. The results of the dimension values will be classified using the K-Means grouping method with five groups, namely Adonis Butterfly, Julia Butterfly, Brown Siproeta Butterfly, Metal Marks Butterfly, and African Giant Swallowtail Butterfly, showing an accuracy of 89%.

Stochastic-Fractal Analysis Modeling of Salts Precipitation from Aqueous Solution

Electrolytes are of interest because thin plate coatings are normally obtained from aqueous solutions. The properties of the surface are important because various properties such as resistance or durability depend on it. To understand the phenomenological processes, it is better to analyze simpler processes such as sodium chloride. In this paper, a model is proposed to predict the temporal behavior of the fractal dimension of the patterns formed in salts precipitation by solvent evaporation in a scattering surface; for fractal-box counting, ImageJ software was used. The model was obtained by applying stochastic methods and fractal geometry, describing the internal fluctuations caused by precipitation and dissolution on the mesoscopic scale of solid crystalline particles. From adjusting the proposed model to the experimental data, it is possible to estimate the velocity constants related to the microscopic precipitation processes of the particles that form the pattern. The model was validated and used to study the precipitation of carbonate salts and sodium chloride, respectively, obtaining predictions corresponding to the physicochemical properties of these salts. From the adjustment of the proposed models to the observed experimental data, the value of the velocity constants of the precipitation and dissolution processes was also estimated.

Application of Fractal Analysis in Detecting Trabecular Bone Changes in Periapical Radiograph of Patients with Periodontitis.

Introduction Evaluation of detailed features of the supporting bone is an important step in diagnosis and treatment planning for teeth with clinical attachment loss. Fractal analysis can be used as a method for evaluating the complexity of trabecular bone structures. The aim of this study was to evaluate the trabecular bone changes in periapical radiographs of patients with different stages of periodontitis using fractal analysis. Methods This comparative cross-sectional study was performed on patients with and without clinical attachment loss in mandibular first molars. Teeth with clinical attachment loss were divided into mild, moderate, and severe periodontitis groups. Digital periapical radiographs were obtained from the mandibular first molars using the same exposure parameters. DICOM file of the radiographs was exported to ImageJ software for fractal analysis. Three regions of interest (ROIs) were considered in each radiograph: two proximal ROIs mesial and distal to the mandibular first molar and one apical ROI. Fractal dimension (FD) values were calculated using the fractal box counting approach. Statistical analysis was performed using the chi-square test, Mann–Whitney test, intraclass correlation coefficient, and ANOVA (α = 0.05). Results FD values were significantly different between moderate and severe periodontitis and healthy periodontal bone (P < 0.05), except for the distal ROI for moderate periodontitis cases (P=0.280). However, FD values of the supporting bone in periodontally healthy teeth and teeth with mild periodontitis did not show a statistically significant difference (P > 0.05). Conclusion Fractal analysis is a useful tool for evaluation of bone alterations in moderate and severe periodontitis, but was not able to detect the most initial radiographic bone signs of mild periodontitis.

Fractal characteristics of the microstructure of loaded coal based on NMR and μ-CT

ABSTRACT The change in the fracture microstructure of loaded coal affects the gas seepage characteristics of the coal. Using micron CT scanning (μ-CT) and nuclear magnetic resonance (NMR) experiments, the fractal characteristics of coal fracture structure under triaxial compression are studied from two aspects of theoretical and experimental fractal dimension for the first time. VG Studio MAX software was used to reconstruct the three-dimensional (3D) fracture structure of the coal samples, and the 3D fracture structure was modified by the porosity measured by NMR. Through the analysis of the reconstruction model and the T2 relaxation spectrum, the spatial distribution of internal cracks in coal was obtained. The fractal geometry theory, NMR experimental fractal dimension algorithm and 3D box counting method were used to calculate the theoretical and experimental fractal dimensions and then comprehensively analyze the fractal characteristics of cracks in loaded coal. The results show that an external load promotes the development of pores/cracks in coal. With increasing theoretical fractal dimension, CT porosity increases from 0.0630% to 0.101%, and NMR porosity increases from 0.0655% to 0.105%. The relationship between porosity and theoretical fractal dimension can be expressed by y = ax2+ bx+c (fractal dimension is an independent variable). With increasing porosity, the experimental fractal dimension of NMR decreases linearly from 2.975 to 2.951, and the experimental fractal dimension of μ-CT increases linearly from 2.191 to 2.324. The different changing trends of theoretical and experimental fractal dimensions verify the differences in their representational meanings.

Bio Metrological Assessment of Shelf-life of Azadirachta Indica

The concentration and activity of water in Azadirachta indica describes the shelf life of Neem. The physical and chemical properties of many biological materials depends on the level of water content, which affects the maturity of the fruits and vegetables. Biospeckle technique is a tool used for contrast analysis and hence distinguishes the water activities at different stages in shelf-life of the specimen. Decomposing condition of Azadirachta indica is evaluated by Time History of Speckle Pattern (THSP) during different stages with histogram analysis and fractal box counting methods.

Fractal networks induced by movements of random walkers on a tree graph

We show that a fractal, scale-free, and small-world network can grow on a non-fractal Cayley tree through the simple random process of shortcut creation by movements of random walkers. The Cayley tree provides a stage for the walkers to diffuse into a locally one-dimensional structure with the small-world property. Self-organization of fractal graph by adding shortcut edges to the locally one-dimensional structure generated by walkers’ flow from the root vertex to the outermost shell of the Cayley tree is a novel mechanism for generating fractal graphs. Also, we show numerically that the fractal box counting dimension and a power-law exponent describing the degree distribution of the resulting graph are mainly determined by the bifurcation number of the original Cayley tree. In our model, the fractal cluster dimension must change with the size of the evolving graph. This is a general result that resolves an apparent contradiction between the fractality in the context of the cluster-growing method and the small-world property of the graph.

Fractal box-counting technique for shelf life study of grapes by laser speckle image processing

Background: In agriculture, it is essential to follow the ripening of fruits and vegetables which is a subject of interest economically. The use of optical non-destructive method is the most suitable technique for evaluating the ripening stage of fruits when they ripen off-tree and eventually to predict the optimum storage life. Many methods are recently developed in optical image processing. The texture correlation function and the fractal box counting (FBC) of biospeckle pattern have been described. Materials and Methods: The dynamic speckle patterns known as boiling speckle are produced by a coherent light illumination incident on the active material of biological tissue. A high coherent laser source He-Ne with 632 nm with neutral density filter and charge-coupled device camera are the major components of the experimental setup. The same experiment may be done by a green laser of wavelength 541 nm (green color). The same procedure continues to take the speckle images in periodically. Results and Discussion: With the help of FBC method in image processing technique, we can assess optically rough surface fruits such as grapes in this work. According to the FBC theory, the speckle images (R1) give a binary counting value D = 1.7640 at the initial stage. Gray image converted into binary image which gives the fractal counting value D = 1.7554 at later on phase. It means that the specimen (grapes) becomes dry periodically and its value is affected by the surrounding environmental condition. Conclusion: Using the dynamic speckle phenomenon or biospeckle method, we demonstrated the assessment of shelf life activity of grapes. The visual phase of the autocorrelation function gives reliable results showing the variation in activity because of the internal changes caused by decreasing water content and then becomes dry grape stage. The correlation function of speckle patternand the Fractal Box Counting (FBC) technique has been proposed as a tool to estimate the postharvest stage of grapes. The bioactivities of fruits and vegetable materials are based on Brownian motion, due to subcellular organs. This indicates the senescence or biological aging characteristics. The quantification of the biospeckle phenomena is associated with the moment of inertia, time history of speckle pattern and grey level co-occurrence matrix. We proposed a new FBC technique to assess the shelf life of grapes.

Spark testing to measure carbon content in carbon steels based on fractal box counting

Spark testing is an effective and inexpensive way to classify the steel materials in industrial practice. Especially, it could determine carbon content in a moment. However, this method has reliance upon the skill and experience of operator. Therefore, computer recognition of spark has been developed to determine carbon content for increasing reliability and efficiency. The numerable high-resolution images of sparks emitting at high speed are taken pictures by camera or mobile phone, then input the images into computer, and calculated the fractal dimension of spark based on fractal box counting method. Result shows that the fractal dimension has a good correlation with carbon content, and the similar natural logarithm function is obtained by a least squares fit through plots of carbon content against fractal dimension. Carbon content measurement error is about 0.06%. It means a convenient and accurate way to identify an unknown carbon steel may be achieved.

Role of chemical reaction and drag force during drop impact gelation process

An instantaneous chemical reaction and solidification have been experimentally explored during sodium alginate drop impact on calcium chloride liquid pool through a high-speed imaging system. Here, different concentrations of sodium alginate are used to study the effect on gelation process. As soon as the alginate drop touches the calcium chloride pool, a crosslinking polymerization reaction starts under the liquid pool surface. The consequence of crosslinking is rapid formation of calcium alginate gel which denotes instantaneous freezing of liquid drop. The drop impact solidification immediately results in the formation of a strong gel layer in front of it which is having significant role in spherical shape recovery. This layer formation is favoured by increasing sodium alginate concentration in the drop and thus crater growth is hindered also. The initial reaction time which is determined as soon as the drop touches the surface, is found to be dependent on the drop kinetic energy. However, the drop gelation time reduces with drop concentration and it controls the final shape and morphology of gel. Such reduction in gelation time is inferred from the enhanced rate of crosslinking reaction. The motion of liquid drop through liquid pool while interacting with the surroundings influences the drag around the spherical drop. The shape evolution in the solidifying gel is governed by the dynamics between the fluid drag and chemical reaction. The morphology of the gel surface has also been studied and its roughness variation has been measured from fractal box counting method.

CHARACTERIZATION OF STROKE LESION USING FRACTAL ANALYSIS

Objective: The characterization of stroke lesions is a challenging research issue due to the wide variability in the structure of lesion patterns. The objective of this research work is to characterize the stroke lesion structures using fractal analysis.Methods: To characterize the complex nature of the lesion structures, fractal box counting analysis is presented in this work. Three parameters from fractal dimension (FD) are considered to characterize the nature of the normal and abnormal brain tissues.Results: The experimental results are presented for 15 different datasets. Three different parameters namely FD average, FD deviation, and FDlacunarity are extracted to quantify the properties of the stroke lesion. The observations indicate that there is a significant proportion of separationof feature values between the normal and abnormal brain tissues.Conclusion: This work presents an efficient scheme for characterizing the stroke lesions using fractal parameters. It could be further enhanced by incorporating features extracted from other non-linear techniques.

Random fractal surface analysis of disordered organic thin films

Study of surface morphologies reveals useful information about interfacial contacts and materials properties such as charge carrier mobility, optical absorption and adhesion that directly affect system performance. The total effective interface area and root mean square (rms) of surface height fluctuations are among the common parameters often used to corroborate device performance upon surface modification. In this paper, random fractal surface analysis is used to characterize geometrical features of microstructure morphologies in disordered organic nickel tetrasulfonated phthalocyanine (NiTsPc) films under different solvent treatment. The different but complementary approaches are used, namely the power spectral density analysis, variogram method based on generalized Cauchy process (GCP) and grayscale fractal box counting approach. It is shown that each of these approaches offers certain perspective about the complex morphologies often found in disordered thin films and thus joint interpretation of these approaches offer better characterization of the surface properties, filling the gap left by others. The morphologies are also interpreted in the context of percolation network supported by the photocurrent density measurements.

Fractal analysis of extra-embryonic vessels of chick embryos under the effect of glucosamine and chondroitin sulfates

Like heparan sulfate proteoglycans, some monosaccharides and glycosaminoglycans, such as sulfated glucosamine (GS) and chondroitin (CS), integrate the vascular extracellular matrix and may influence vascular endothelial cell growth. To assess the effects of these substances on blood vessel formation, we used the chick yolk sac membrane (YSM) model and fractal geometry quantification, which provided an objective in vivo method for testing potential agents that promote vasculogenesis and angiogenesis. An image processing method was developed to evaluate YSM capillary vessels after they were implanted in a methylcellulose disk of GS or CS at a concentration between 0.001–0.1mg/disk (performed on 2-day old embryos). This method resulted in a binary image of the microvascular network (white vessels on a black background). Fractal box-counting (DBC) and information (DINF) dimensions were used to quantify the activity of GS and CS in vasculogenesis and angiogenesis. YSM treated with GS (0.001–0.1mg) and CS (0.03–0.1mg) showed an increase in fractal dimensions that corresponded to vitelline vessel growth compared to the control group (vehicle), with GS displaying higher fractal dimension values.

Morphological study of Insoluble Organic Matter from carbonaceous chondrites: Correlation with petrologic grade

The major form of organic material delivered to Earth from an extraterrestrial origin is Insoluble Organic Matter (IOM). A morphological study of IOM in the CR (Renazzo-type) and CM (Mighei-type) carbonaceous chondrites was performed in order to constrain its origins and processing history. IOM residues from the following CR chondrites: GRO 95577 (CR1), Al Rais (CR1/2), EET 92042 (CR2), QUE 99177 (CR3) and the CM chondrites: MET 01070 (CM2.2), Cold Bokkeveld (CM2.3), Murchison (CM2.4) and QUE 97990 (CM2.5) were studied using Annular Dark Field STEM imaging. Characteristic features of the IOM, organic nanoglobules, were manually identified and measured for their abundances and size distributions. The IOM residues were also compared holistically for their degree of average ‘roughness’ or ‘coarsening’ using fractal image analysis. Manually identified nanoglobules have abundances making up less than 10% of the total IOM, which is consistent with previous studies. Their measured abundances do not correlate with petrologic grade. Thus parent body processing did not systematically deplete their abundances. The IOM is however on average ‘smoother’ or ‘coarser’ in the more altered chondrites, demonstrated by a lower fractal dimension using fractal box counting (DB). The DB values for the IOM in the CR chondrites are distinctive: QUE 99177 has the largest DB value (average=1.54±0.004) and GRO 99577 has the lowest (average=1.45±0.011). Al Rais and EET 92042 have IOM with average DB values within this range (average, 1.46±0.009 and 1.50±0.006). The CMs record a similar but less distinctive trend in DB, with QUE 97990 having the largest value (1.52±0.004), MET 01070 the lowest (1.45±0.019), and Cold Bokkeveld (1.50±0.011) and Murchison (1.49±0.017) equivalent to one another within error. The identified nanoglobules in the IOM of the CM chondrites are on average larger than those in the CR chondrites. The ‘coarsening’ or ‘smoother’ texture of the IOM (lower DB) in the more altered chondrites coupled with a tentative increase in the size of large features (identified nanoglobules) demonstrates that the aqueous processes leading to the lower petrologic types also formed the overall IOM morphology. In addition, observations of fluid-like textures more frequently found in the more altered carbonaceous chondrite residues suggests that organic and aqueous fluids determined at least some of these morphologies. The polymerization of organic solutions is consistent with these morphologies. Their formation conditions are more favorable under the containment of carbonaceous chondrite parent bodies.

Feature Analysis and Classification of Particle Data from Two-Dimensional Video Disdrometer

We developed a ground observation system for solid precipitation using two-dimensional video disdrometer (2DVD). Among 16,010 particles observed by the system, around 10% of them were randomly sampled and manually classified into five classes which are snowflake, snowflake-like, intermediate, graupel-like, and graupel. At first, each particle was represented as a vector of 72 features containing fractal dimension and box-count to represent the complexity of particle shape. Feature analysis on the dataset clarified the importance of fractal dimension and box-count features for characterizing particles varying from snowflakes to graupels. On the other hand, performance evaluation of two-class classification by Support Vector Machine (SVM) was conducted. The experimental results revealed that, by selecting only 10 features out of 72, the average accuracy of classifying particles into snowflakes and graupels could reach around 95.4%, which had not been achieved by previous studies.

Simulación de estructuras eritrocitarias con base en la geometria fractal y euclidiana

Antecedentes: La aplicación simultánea de la geometría fractal y euclidiana fundamentó el desarrollo de una metodología diagnóstica de la morfofisiología eritocitaria, con la cual se obtiene un método diagnóstico objetivo y reproducible para la detección de anormalidades patológicas en extendidos de muestras de sangre, útil para determinar la viabilidad de bolsas para trasfusión.Objetivo: determinar posibles trayectorias de evolución entre estructuras eritrocitarias desde normalidad a equinocitos aplicando el diagnóstico previamente desarrollado a partir de la geometría fractal y la euclidiana.Materiales y métodos: con base en la metodología diagnóstica previamente desarrollada, se tomaron 8 eritrocitos normales, con los cuales se desarrollaron simulaciones de tres posibles trayectorias de evolución. Para ello inicialmente se calculó la dimensión fractal, el número de pixeles ocupados por la superficie eritrocitaria y el número de espacios tocados por el borde al superponer una rejilla de 5x5 píxeles en el espacio generalizado fractal de Box counting. Posteriormente se realizaron variaciones geométricas del borde de cada eritrocito, de tal forma que al evaluar simultáneamente el borde y la superficie se mantuvieran proporciones superficie/borde características del estado requerido para cada simulación, de acuerdo con los resultados previos. Resultados: Se obtuvieron simulaciones de posibles trayectorias de normalidad a equinocitos, con medidas cuantitativas, objetivas y reproducibles. Conclusiones: la organización fractal/euclidiana en la arquitectura de los glóbulos rojos, permite el desarrollo de trayectorias acausales de alteración celular hacia equinocitos, de utilidad en el diagnóstico del eritrocito en la práctica clínica diaria.

On Relationships Between Fixation Identification Algorithms and Fractal Box Counting Methods.

Fixation identification algorithms facilitate data comprehension and provide analytical convenience in eye-tracking analysis. However, current fixation algorithms for eye-tracking analysis are heavily dependent on parameter choices, leading to instabilities in results and incompleteness in reporting. This work examines the nature of human scanning patterns during complex scene viewing. We show that standard implementations of the commonly used distance-dispersion algorithm for fixation identification are functionally equivalent to greedy spatiotemporal tiling. We show that modeling the number of fixations as a function of tiling size leads to a measure of fractal dimensionality through box counting. We apply this technique to examine scale-free gaze behaviors in toddlers and adults looking at images of faces and blocks, as well as large number of adults looking at movies or static images. The distributional aspects of the number of fixations may suggest a fractal structure to gaze patterns in free scanning and imply that the incompleteness of standard algorithms may be due to the scale-free behaviors of the underlying scanning distributions. We discuss the nature of this hypothesis, its limitations, and offer directions for future work.

Fractal Analysis on Morphology of Laser Irradiated Vanadium Surfaces Under Different Ambient

Abstract Pulsed laser irradiated vanadium surface morphology under different ambient has been prepared and characterized using fractal dimension analysis method on scanning electron microscopy (SEM) images. In presence of different ambient, self-periodic and self-similar surface patterns (e.g. dots, islands, and pins) were grown and appeared in different shapes. The fractal dimension (FD) of this developed vanadium nanostructure was calculated by fractal box count method (FBM). The calculated fractal dimension (FD, Df) shows dependence on the different type on ambient and the number of laser shots.

Seismic hazards assessment of Kumaun Himalaya and adjacent region

The ongoing continent–continent collision between Indian and Eurasian plates houses a seismic gap in the geologically complex and tectonically active central Himalaya. The seismic gap is characterized by unevenly distributed seismicity. The highly complex geology with equally intricate structural elements of Himalaya offers an almost insurmountable challenge to estimating seismogenic hazard using conventional methods of Physics. Here, we apply integrated unconventional hazard mapping approach of the fractal analysis for the past earthquakes and the box counting fractal dimension of structural elements in order to understand the seismogenesis of the region properly. The study area extends from latitude 28°N–33°N and longitude 76°E–81°E has been divided into twenty-five blocks, and the capacity fractal dimension (D 0) of each block has been calculated using the fractal box counting technique. The study of entire blocks reveal that four blocks are having very low value of D 0 (0.536, 0.550, 0.619 and 0.678). Among these four blocks two are characterized by intense clustering of earthquakes indicated by low value of correlation fractal dimension (D c ) (0.245, 0.836 and 0.946). Further, these two blocks are categorized as highly stressed zones and the remaining two are characterized by intense clustering of structural elements in the study area. Based on the above observations, integrated analysis of the D c of earthquakes and D 0 of structural elements has led to the identification of diagnostic seismic hazard pattern for the four blocks.

Image processing and fractal box counting: user-assisted method for multi-scale porous scaffold characterization

Image analysis has gained new effort in the scientific community due to the chance of investigating morphological properties of three dimensional structures starting from their bi-dimensional gray-scale representation. Such ability makes it particularly interesting for tissue engineering (TE) purposes. Indeed, the capability of obtaining and interpreting images of tissue scaffolds, extracting morphological and structural information, is essential to the characterization and design of engineered porous systems. In this work, the traditional image analysis approach has been coupled with a probabilistic based percolation method to outline a general procedure for analysing tissue scaffold SEM micrographs. To this aim a case study constituted by PCL multi-scaled porous scaffolds was adopted. Moreover, the resulting data were compared with the outputs of conventionally used techniques, such as mercury intrusion porosimetry. Results indicate that image processing methods well fit the porosity features of PCL scaffolds, overcoming the limits of the more invasive porosimetry techniques. Also the cut off resolution of such IP methods was discussed. Moreover, the fractal dimension of percolating clusters, within the pore populations, was addressed as a good indication of the interconnection degree of PCL bi-modal scaffolds. Such findings represent (i) the bases for a novel approach complementary to the conventional experimental procedure used for the morphological analysis of TE scaffolds, in particular offering a valid method for the analysis of soft materials (i.e., gels); also (ii) providing a new perspective for further studies integrating to the structural and morphological data, fluid-dynamics and transport properties modelling.

Comparison between radarsat-1 SAR different data modes for oil spill detection by a fractal box counting algorithm

This work presents a modified formula for the fractal box counting dimension. The method is based on the utilisation of the probability distribution formula in the fractal box count. The purpose of this method is to use it for the discrimination of oil spill areas from the surrounding features, e.g. sea surface and look-alikes, using RADARSAT-1 SAR Wide beam mode (W1), Standard beam mode (S2) and Standard beam mode (S1) data acquisition under different wind speeds. The results show that the new formula is able to discriminate between oil spills and look-alike areas. The results also illustrate that the new fractal formula identifies well the deficiency of oil spills in pairs of S2 data. Further, there are no significant differences between fractal values of look-alikes, low wind zone, and current shear features in different beam modes for acquisition of RADARSAT-1 SAR data. The W1 mode data, however, show an error standard deviation of 0.002, thus performing a better discrimination of oil spills than the S1 and S2 mode data.