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摘要: Because many Chemical Engineering phenomena and processes involve distinctive shapes and structures, fractal dimension (FD) analysis is of great interest to the Chemical Engineering community due to its utility in providing a statistical index of shape complexity, which not only quantifies image structure but also helps explain functional properties. Past box counting (BC) methods for estimation of FDs are inconsistent due to quantization error (QE) introduced from image rotation and translation关键词: Fractal Dimension;Box Counting;Integer Linear Program;Regression Analysis;Optical Images
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摘要: The morphology and structure of soot particles in laminar diffusion flames of anisole/n-heptane have been investigated in this work. Soot morphology has been analyzed by thermophoretic sampling and transmission electron microscopy (TS-TEM). Further, a detailed TEM image process was performed and a set of fractal-based distribution analysis algorithms were proposed and applied in this study. The addition of anisole in n-heptane flames greatly influences the soot formation process, which leads to 关键词: Anisole;n-Heptane;Laminar diffusion flame;Soot particle morphology;Thermophoretic sampling;Box counting
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摘要: Natural porous media have extremely complex pore structures, which greatly affect the material properties. However, it remains a challenge to characterize the pore structure complexity accurately. Fractal geometry is one of the effective tools for this task, which allows for quantitative analysis. In this paper, a new method is proposed for evaluating the pore structure complexity of digital rock models by using the relative value of box-counting fractal dimension. The proposed method transforms关键词: Fractal dimension;Digital rock;Pore structure;Box counting;Complexity
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Book Chapter
摘要: In the study of natural things with a high degree of complexity, randomness, and irregularity in the colour domain, FD is a helpful attribute. This paper dealt with RGB color texture analysis in terms of box-counting (BC) and differential BC approaches. However, the analysis of RGB color images was evaluated in 5D space and images are normally treated as non-uniform metric space; hence it was very difficult to estimate surface roughness. In this study, we’ve reviewed the various color-based box-关键词: Fractal;RGB color images;Box counting;DBC;Texture;Regression
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Journal
Hobbs Bruce E.;Ord Alison;Blenkinsop Thomas;
Journal of Structural GeologyVolume 156, Issue , 2022, PP
摘要: The concept of fractal spatial distributions of mineralisation has been widely proposed since Mandelbrot (1965) who emphasised the stable Pareto-Lévy distribution as the relevant distribution. The concept of a fractal is used as a basis for estimating endowment and for erecting exploration models based on self-organised criticality. This paper explores the proposition that the growth kinetics for a mineralising system are reflected in the probability distributions that describe the spatial patte关键词: Fractals;Generalised Extreme Value (GEV) distributions;Fréchet-;Weibull-;Gumbel-distributions;Box counting;Nearest neighbour-distributions;Mineralisation growth models
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Journal
摘要: Neovascular glaucoma (NVG) is a human eye disease due to diabetes that leads to permanent vision loss. Early detection and treatment of it prevent further vision loss. Hence the development of an automated system is more essential to help the ophthalmologist in detecting NVG at an earlier stage. In this paper, a novel approach is used for detection of Neovascular glaucoma using fractal geometry concepts. Fractal geometry is a branch of mathematics. It is useful in computing fractal features of i关键词: Glaucoma;Fractal Dimension (FD);Box Counting;Segmentation;Texture Features;Retina.
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Book Chapter
摘要: In this part of the book noninvasive studies and methods for mineral exploration are proposed. The proposed methodologies consist in combining known techniques for recording data [deep seismic sounding, magneto acoustic emission (MAE), airborne gamma ray, drone, fault map, etc.] with mathematical or statistical techniques [empirical mode decomposition (EMD), fractal and multifractal analysis, statistical parameters] in order to extract features that make it possible to identify ore deposits. Var关键词: Precambrian lithospheric formation;magneto acoustic emission;bi-dimensional empirical mode decomposition;box counting;fractal and multifractal spectra
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Journal
Research Article
摘要: In the present work, fractal geometry is applied to histological sections of the hepatopancreas tissue of white shrimp (L. vannamei) experimentally infected with Hepatobacter penaei bacteria (NHPB), in order to perform an analysis of the morphology adopted by the different cellular components damaged by this intracellular-type pathogen bacteria. The fractal dimension of the digital images of the histological sections was obtained by using a code written in Matlab, which uses the box counting met关键词: Fractal dimension;Box counting;NHPB;Bioassay
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Journal
摘要: Abstract(#br)Dorsal hand vein pattern is a physiological feature that can distinguish and define one person from another. Feature extraction from images is considered as the most important step in biometric systems. In this study, a fractal technique, which is both an advanced and a complex method, is proposed for feature extraction from the images of hand vessel pattern. In recent years, this approach has been widely used as an active research area in image processing. Therefore, fractal size b关键词: Fractal dimension; Box counting; Hand vein recognition
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Journal
Research Article
摘要: AbstractPublic concerns and official pressures around environmental protection as well as exhausting petroleum resources have brought about preferences in studying and applying environmental-friendly polymers instead of synthetic petroleum-based polymers. For this aim, biopolymers can have outstanding advantages in biodegradability by saving time, energy and effort spent on increasingly costs of polymeric wastes. Natural starch (being studied in this work), as one of the most abundant natural re关键词: Image analysis; Natural starch; Fractal dimension FD; Box counting; Pattern recognition; Computer program
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Journal
摘要: Abstract(#br)This work proposes a method for supervised classification of grayscale texture images using the numerical computation of the box counting fractal dimension. Each pixel is mapped onto a point in a three-dimensional cloud, where the normalized gray level of each pixel is the third coordinate, and we analyze the distribution of points inside a mesh of boxes. Information at different resolutions are captured by varying the size of each box in the mesh. The texture descriptors are provid关键词: Fractal dimension; Box counting; Texture image classification
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Journal
Research Article
Nadia. M. G. Alsaidi;Arkan J. Mohammed;Wael J. Abdulaal;
Mustansiriyah Journal of ScienceVolume 29, Issue 3, 2019, PP 106-112
摘要: Characterizing of visual objects is an important role in pattern recognition that can be performed through shape analysis. Several approaches have been introduced to extract relevant information of a shape. The complexity of the shape is the most widely used approach for this purpose where fractal dimension and generalized fractal dimension are methodologies used to estimate the complexity of the shapes. The box counting dimension is one of the methods that used to estimate fractal dimension. It关键词: Fractal Dimension;Box Counting;Fingerprint;Authentication
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Journal
摘要: Abstract(#br)Agglomerates are often composed of amorphous and irregular primary particles, especially in the food industry. The spatial morphology of this kind of soft agglomerate, here maltodextrin, can be quantified by fractal dimension. Previous research in this regard was focused on simulated agglomerates or studied 2D projected images of real agglomerates. In this work 3D volume images of agglomerates are generated with the help of X-ray computed tomography. Primary particles are distinguis关键词: Maltodextrin agglomerates; 3D X-ray scanning; Fractal dimension; Power law relationship; Box counting; Primary particle shape and size
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Journal
摘要: Abstract(#br)The Fractal Dimension (FD) of digital image represents the roughness in terms of real number with self similarity property in order to correlate with human perception of surface roughness. FD has broadly adopted in various kinds of applications in different field of computer graphics and image processing such as texture analysis, classification, segmentation and many more that establish in various literatures. For estimating FD there are several techniques were presented in gray sca关键词: Fractal dimension; Color images; Box counting; DBC; BC; Euclidian distance
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Journal
摘要: Abstract(#br)The box counting dimension d B of a complex network G , and the generalized dimensions { D q , q ∈ R } of G , have been well studied. However, the Hausdorff dimension d H of a geometric object, which generalizes d B by not assuming equal-diameter boxes, has not previously been extended to G . Similarly, the generalized Hausdorff dimensions { D q H , q ∈ R } of a geometri关键词: Complex networks; Fractal dimensions; Generalized dimensions; Hausdorff dimension; Multifractals; Box counting
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Journal
摘要: Abstract(#br)Detection of the characteristics of the sleep stages, such as sleep spindles and K-complexes in EEG signals, is a challenging task in sleep research as visually detecting them requires high skills and efforts from sleep experts. In this paper, we propose a robust method based on time frequency image (TFI) and fractal dimension (FD) to detect sleep spindles in EEG signals. The EEG signals are divided into segments using a sliding window technique. The window size is set to 0.5 s with关键词: Sleep spindles; Time frequency image; Fractal dimension; Box counting; EEG signals
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Journal
摘要: Abstract(#br)While the widespread emergence of nanoscience and nanotechnology can be dated back to the early eighties, the last decade has witnessed a true coming of age of this research field, with novel nanomaterials constantly finding their way into marketed products. The performance of nanomaterials being dominated by their nanoscale morphology, their quantitative characterization with respect to a number of properties is often crucial. In this context, those imaging techniques able to resol关键词: Atomic Force Microscope; AFM; Nanometric length scale; Fractal dimension; Morphology; Scaling analysis; DNA; PDI8-CN2; Organic films; Submonolayer; Entangle; Height-Height Correlation Function; Persistence length; Box counting; Segmentation; Vectorization; Molecular beam epitaxy; Surface roughness; MBE; Perimeter–area
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Journal
摘要: Abstract(#br)Fractal geometry has been widely used in literature to characterize the mechanical behaviour of quasi-brittle materials. In this work, innovative cementitious composites with carbon-based pyrolyzed micro-aggregates were tested until complete fracture and their fracture behaviour was studied in the light of fractal geometry. Images of the crack paths across the tested specimens were acquired by Scanning Electron Microscopy (SEM) and their fractal dimension was calculated via the box 关键词: Cementitious composites; Crack path; Fractal dimension; Box counting; Fracture energy; Fracture toughness
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Journal
摘要: Multiple sclerosis (MS) is a severe brain disease. Early detection can provide timely treatment. Fractal dimension can provide statistical index of pattern changes with scale at a given brain image. In this study, our team used susceptibility weighted imaging technique to obtain 676 MS slices and 880 healthy slices. We used synthetic minority oversampling technique to process the unbalanced dataset. Then, we used Canny edge detector to extract distinguishing edges. The Minkowski–Bouligand dimens关键词: Synthetic Minority Oversampling Technique; Canny Edge Detector; Fractal Dimension; Minkowski–Bouligand Dimension; Box Counting; Three-Segment Representation; Biogeography-Based Optimization
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Journal
摘要: Abstract(#br)Computing the generalized dimensions D q of a complex network requires covering the network by a minimal number of “boxes” of size s , for a range of s . We show that, unlike the case for a geometric multifractal, for a complex network the shape of the D q vs. q curve can be monotone increasing, or monotone decreasing, or even have both a local maximum and a local minimum, depending on the range of box sizes used to compute D q . We provide insight into this behavior by deriving a s关键词: Complex networks; Generalized dimensions; Multifractals; Box counting
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Journal
摘要: Abstract(#br)Box counting is a simple method to calculate the fractal dimension of an image, but it is poorly understood for estimating phase fraction from the microstructure image. Here, we investigate six recipes to estimate the phase fraction by box counting method (BCM). The optimal recipe is determined by applications in the microstructures of Ni-B alloys, and is further verified by theoretical analysis. As compared to pre-existing methods, BCM is more convenient and can provide accurate ph关键词: Box counting; phase fraction; recipe; image
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Journal
摘要: Abstract(#br)Computing the generalized dimensions D q of a complex network requires covering the network by a minimal number of “boxes” of size s . We show that the current definition of D q is ambiguous, since there are in general multiple minimal coverings of size s . We resolve the ambiguity by first computing, for each s , the minimal covering that is summarized by the lexicographically minimal vector x ( s ) . We show that x ( s ) is unique and easily obtained from any box counting method. 关键词: Complex networks; Generalized dimensions; Multifractals; Box counting; Partition function
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Journal
摘要: AbstractDuring the last years, several methods from the statistical physics of complex systems have been applied to the study of natural language written texts. They have mostly been focused on the detection of long-range correlations, multifractal analysis and the statistics of the content word positions. In the present paper, we show that these statistical aspects of language series are not independent but may exhibit strong interrelations. This is done by means of a two-step investigation. Fi关键词: Multifractality;Box counting;Long-range correlations;Language;Language families;Written texts
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Journal
摘要: Geochemical investigation of koton karfe Oolitic iron ore was carried out in order to provide comprehensive data on the geochemical and mineralogical composition. The investigation involved mineral phase analysis major elements and grade determination. Result the XRF analysis shows high Iron Fe2O3 content ranging from 63.46 to 74.38. Aluminum Al2O3 and silica SiO2 contents are also fairly high ranging from 7.60 - 16.85 and 7.81 -18.55 respectively. Phosphorous content ranges from 0.89 - 1.20 and关键词: Keywords galvanized;fractal;box counting;weight loss.
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Journal
摘要: BACKGROUND/AIM(#br)Fractal dimension (FD) is widely used in medicine and biology as a tool for defining features of structure. This study aimed to compare pathological endometrium (simple-complex hyperplasia and endometrial carcinoma), as well as the endometrial changes, during the phases of the menstrual cycle. The main goal was the objective measurement of fractal dimension and to refrain from subjective evaluation.(#br)MATERIALS AND METHODS(#br)Two thousand cases of endometrial tissue from pa关键词: Fractal dimension; Box counting; Endometrium; Hyperplasia
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Journal
E. Rosenberg;
Community EcologyVolume 17, Issue 1, 2016, PP 24-27
摘要: We extend Kenkel’s model for determining the minimal allowable box size s* to be used in computing the box counting dimension of a self-similar geometric fractal. This minimal size s* is defined in terms of a specified parameter ε which is the deviation of a computed slope from the box counting dimension. We derive an exact implicit equation for s* for any ε. We solve the equation using binary search, compare our results to Kenkel’s, and illustrate how s* varies with ε. A listing of the Python c关键词: Box counting;Fractal dimension;Resolution;Self-similar;Scale
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Journal
Research Article
摘要: Abstract(#br)From a mechanical point of view, the phenomenon of damage in concrete members is represented by surface discontinuities in the form of cracks and crack networks, in progressive form, or volume discontinuities in the form of cavities. Crack width is commonly used as a damage indicator for concrete members, but more detailed information of the corresponding crack network such as its distribution and pattern is not considered in this indicator. To study the influence of aggregate’ grad关键词: Multi-fractal analysis; Crack propogation; Aggregates grading; Box counting; Crack network
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Journal
摘要: Palmprint is a human physiological feature which can distinguish and identify one person from another. In the palmprint recognition biometric systems, the feature extraction is considered as the most important step. In this paper, we use the fractal approach which is both a very advanced and sophisticated method in order to extract the palmprint texture information features. This approach has been widely used in recent years being considered as an active research area in the image processing fie关键词: biometrics; palmprints; texture analysis; fractal dimension; box counting; identification; palmprint recognition; feature extraction; texture features.
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Journal
Research Article
摘要: Monitoring friction stir welding process based on weld images is attempted in this research. Well-known fractal theory is applied to images of the welds and extracted features in terms of fractal dimension are correlated to ultimate tensile strength (UTS) of the joints. The correlation shows a decreasing trend that can be an indicator towards monitoring of weld quality. Apart from fractal theory, wavelet transform is also applied to the acquired images and an indicator is proposed relating the i关键词: Fractal dimension; Image analysis; Box counting; Katz's; Monitoring; Wavelet transform
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Journal
摘要: Abstract Fractal analysis is a powerful tool for the classification of materials. However, until now, there were no efficient tools that could process large or color images due to the processing time required. In this article we present a fast, easy to implement and very easily expandable to any number of dimensions variation, the box merging method. It is applied here to RGB test images which are considered as sets in 5-D space.关键词: box counting; fractal dimension; fractals; image classification; image processing
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