Showing papers on "Fractal analysis published in 2016"

Quantification of Vessel Density in Retinal Optical Coherence Tomography Angiography Images Using Local Fractal Dimension.

Abstract: PURPOSE To evaluate a fully automated local fractal dimension method to quantify vessel density and foveal avascular zone (FAZ) area in optical coherence tomography angiography (OCTA) images. METHODS Fifty-two healthy Asian Indian eyes underwent imaging prospectively with OCTA system. Superficial and deep retinal vascular plexus was imaged. Local fractal analysis was applied to the OCTA images. A scan area of 3 × 3 mm was selected in the superficial and deep retinal layers. Foveal avascular zone area and vessel density were quantified in circular and sectoral zones around the fovea. A unique contour map of vessel density and dropout zones was developed to perform regional comparisons. RESULTS Foveal avascular zone of superficial (0.35 ± 0.013 mm2) and deep (0.49 ± 0.012 mm2) retinal vascular plexus was segmented. The agreement between the manually segmented and local fractal dimension segmented FAZ area was 0.97 (95% confidence interval [CI]: 0.94-0.98) and did not change significantly with age (P = 0.94 and 0.21, respectively). The vessel density was greater in the deep than the superficial retinal vascular plexus (P < 0.0001). When the image was subdivided into sectors around the FAZ, inferior sector had greater vessel density than the others (temporal, superior, and nasal) in both superficial and deep retinal vascular plexus (P < 0.05). These observations were similar to recent studies on animal retinal vasculature map. CONCLUSIONS A novel implementation of local fractal dimension to calculate vessel density and FAZ area was demonstrated. Age did not impact vessel density but sectoral analyses showed greater vessel density in the inferior zone.

Entropy and Fractal Antennas

Abstract: The entropies of Shannon, Renyi and Kolmogorov are analyzed and compared together with their main properties. The entropy of some particular antennas with a pre-fractal shape, also called fractal antennas, is studied. In particular, their entropy is linked with the fractal geometrical shape and the physical performance.

Fractal Analysis of Shale Pore Structure of Continental Gas Shale Reservoir in the Ordos Basin, NW China

Abstract: Describing the characteristics of shale pore structure is vital for the assessment of shale reservoir, which has significant influence on the storage and seepage mechanisms of gas shale. To profoundly understand the shale pore structure characteristics of continental shale reservoir, fractal analysis was performed on 45 continental shale samples from the Ordos Basin, NW China, via low-pressure N2 adsorption experiments. The characteristics of N2 adsorption isotherms revealed that slit-shaped shale pores are dominant among the geometric shapes of shale pores. During N2 molecules adsorption process, different characteristics were displayed at two regions where relative pressures (P/P0) were 0–0.45 and 0.45–1. The Frenkel–Halsey–Hill (FHH) method was used to calculate fractal dimensions (D) at these two regions. In addition, the fractal exponents “(D – 3)/3” and “(D – 3)” were compared adequately. The results show creditable fractal characteristics for continental shale. Fractal exponent D – 3 is more suitab...

Fractal cometary dust – a window into the early Solar system

Abstract: The properties of dust in the protoplanetary disk are key to understanding the formation of planets in our Solar System. Many models of dust growth predict the development of fractal structures that evolve into non-fractal, porous dust pebbles representing the main component for planetesimal accretion. In order to understand comets and their origins, the Rosetta orbiter followed comet 67P/Churyumov-Gerasimenko for over two years and carried a dedicated instrument suite for dust analysis. One of these instruments, the MIDAS atomic force microscope, recorded the 3D topography of micro- to nanometre sized dust. All particles analysed to date have been found to be hierarchical agglomerates. Most show compact packing, however, one is extremely porous. This paper contains a structural description of a compact aggregate and the outstanding porous one. Both particles are tens of micrometres in size and show rather narrow subunit size distributions with noticeably similar mean values of 1.48+0.13−0.59 μm for the porous particle and 1.36+0.15−0.59 μm for the compact. ompact. The porous particle allows a fractal analysis, where a density-density correlation function yields a fractal dimension of Df = 1.70 ± 0.1. GIADA, another dust analysis instrument on-board Rosetta, confirms the existence of a dust population with a similar fractal dimension. The fractal particles are interpreted as pristine agglomerates built in the protoplanetary disk and preserved in the comet. The similar subunits of both fractal and compact dust indicate a common origin which is, given the properties of the fractal, dominated by slow agglomeration of equally sized aggregates known as cluster-cluster agglomeration.

Fractal Analysis on Heterogeneity of Pore–Fractures in Middle–High Rank Coals with NMR

Abstract: Because of the complex nature of coal reservoirs, no determined method can clearly assess their pore–fracture structure. This paper combines a fractal method with the transverse relaxation time (T2) of nuclear magnetic resonance (NMR) to study pore–fractures in a coal reservoir. Based on 10 coal samples with vitrinite reflectance (Ro,m) in the range 1.32–2.43%, the pore size/volume distribution from mercury intrusion porosity (MIP) and NMR was compared, revealing that a strong correspondence exists, and that the correspondence for pore diameter of >100 nm is better than that for pore diameter of 100 ms of T2 values, respectively...

Fractal features of carbon–nickel composite thin films

Abstract: This work analyses the three-dimensional (3-D) surface texture of carbon-nickel (C-Ni) films grown by radio frequency (RF) magnetron co-sputtering on glass substrates. The C-Ni thin films were deposited under different deposition times, from 50 to 600 s, at room temperature. Atomic force microscopy was employed to characterize the 3-D surface texture data in connection with the statistical, and fractal analyses. It has been found that up to 180 s the sputtering occurs in more metal content mode and in greater than 180 s it occurs in more non-metal content mode. This behavior demonstrated a strong link between the structural and morphological properties of C-Ni composite films and facilitates a deeper understanding of structure/property relationships and surface defects in prepared samples. Furthermore, these findings can be applied to research on the mechanisms to prepare and control high-quality C-Ni films.

Understanding Fractal Dimension of Urban Form through Spatial Entropy

Abstract: Spatial patterns and processes of cities can be described with various entropy functions. However, spatial entropy always depends on the scale of measurement, and it is difficult to find a characteristic value for it. In contrast, fractal parameters can be employed to characterize scale-free phenomena. This paper is devoted to exploring the similarities and differences between spatial entropy and fractal dimension in urban description. Drawing an analogy between cities and growing fractals, we illustrate the definitions of fractal dimension based on different entropy concepts. Three representative fractal dimensions in the multifractal dimension set are utilized to make empirical analyses of urban form of two cities. The results show that the entropy values are not determinate, but the fractal dimension value is certain; if the linear size of boxes is small enough (e.g., <1/25), the linear correlation between entropy and fractal dimension is clear. Further empirical analysis indicates that fractal dimension is close to the characteristic values of spatial entropy. This suggests that the physical meaning of fractal dimension can be interpreted by the ideas from entropy and scales and the conclusion is revealing for future spatial analysis of cities. Key words: fractal dimension; entropy; mutlifractals; scaling; urban form; Chinese cities

Quantitative evaluation of midpalatal suture maturation via fractal analysis

Abstract: Objective: The purpose of this study was to determine whether the results of fractal analysis can be used as criteria for midpalatal suture maturation evaluation. Methods: The study included 131 subjects aged over 18 years of age (range 18.1–53.4 years) who underwent cone-beam computed tomography. Skeletonized images of the midpalatal suture were obtained via image processing software and used to calculate fractal dimensions. Correlations between maturation stage and fractal dimensions were calculated using Spearman’s correlation coefficient. Optimal fractal dimension cut-off values were determined using a receiver operating characteristic curve. Results: The distribution of maturation stages of the midpalatal suture according to the cervical vertebrae maturation index was highly variable, and there was a strong negative correlation between maturation stage and fractal dimension (−0.623, p < 0.001). Fractal dimension was a statistically significant indicator of dichotomous results with regard to maturation stage (area under curve = 0.794, p < 0.001). A test in which fractal dimension was used to predict the resulting variable that splits maturation stages into ABC and D or E yielded an optimal fractal dimension cut-off value of 1.0235. Conclusions: There was a strong negative correlation between fractal dimension and midpalatal suture maturation. Fractal analysis is an objective quantitative method, and therefore we suggest that it may be useful for the evaluation of midpalatal suture maturation.

On the relationship between fractal dimension and fractal index

Abstract: For Gaussian processes there is a simple and well-known relationship between the fractal dimension of sample paths and the fractal index of the covariance function. This property is of considerable practical interest, since it forms the basis of several estimators of fractal dimension. Motivated by statistical applications involving non-Gaussian processes, we discuss the relationship in a wider context. We show that the relationship fails in some circumstances, but nevertheless does hold in a variety of cases.

Estimating permeability of porous media based on modified Hagen–Poiseuille flow in tortuous capillaries with variable lengths

Abstract: Fractal analysis of permeability of porous media has received much attention over several decades. Many traditional and recently presented fractal models are derived based on Hagen–Poiseuille (H–P) flow in circular-shaped straight/tortuous capillaries that cut through the model. However, the real capillaries in the nature have different lengths and variably-shaped cross sections. In this study, a fractal scaling law for length distribution of capillaries in porous media was derived and its validity was verified by comparisons of fractal dimensions for length distribution between calculated and theoretical values, and then we proposed a modified H–P equation that considers the fractal properties of capillary length and capillary shape. Based on a discrete element method (DEM) code, a series of capillary network models were established and their equivalent permeability was calculated by solving the modified H–P equation. Some empirical expressions were given, which could be utilized to predict the magnitudes of capillary number, equivalent permeability of porous media, the smallest side length of capillary networks at the representative elementary volume size, ratio of capillary network permeability to fracture network permeability, dimensionless permeability, where dimensionless permeability is defined as the ratio of equivalent permeability of models consist of variably-shaped capillaries to that of circular-shaped capillaries. The results show that the proposed fractal length distribution of capillaries, the modified H–P equation, and the developed DEM code can be used to calculate the equivalent permeability of porous media and that the capillaries cannot always be assumed to cut through the model with circular shapes.

Non-local Integrals and Derivatives on Fractal Sets with Applications

Abstract: In this paper, we discussed the non-local derivative on the fractal Cantor set. The scaling properties are given for both local and non-local fractal derivatives. The local and non-local fractal differential equations are solved and compared and related physical models are suggested.

Fractal analysis for studying the evolution of forests

Abstract: Deforestation is an important phenomenon that may create major imbalances in ecosystems. In this study we propose a new mathematical analysis of the forest area dynamic, enabling qualitative as well as quantitative statements and results. Fractal dimensions of the area and the perimeter of a forest were determined using digital images. The difference between fractal dimensions of the area and the perimeter images turned out to be a crucial quantitative parameter. Accordingly, we propose a new fractal fragmentation index, FFI , which is based on this difference and which highlights the degree of compaction or non-compaction of the forest area in order to interpret geographic features. Particularly, this method was applied to forests, where large areas have been legally or illegally deforested. However, these methods can easily be used for other ecological or geographical investigations based on digital images, including deforestation of rainforests.

An improved method to estimate the fractal dimension of colour images

Abstract: Summary Fractal dimension (FD) is an important feature of fractal geometry has many applications in various fields including image processing, image analysis, texture segmentation, shape classification and identifying the image features such as roughness and smoothness of an image. There are many techniques to estimate the dimension of fractal surface. The famous technique to calculate fractal dimension is the grid dimension method, which is popularly known as box counting method and some of the other improved methods like differential box counting and improved differential box counting method are used to estimate fractal dimension of grayscale images. The usual way of estimating the roughness or FD of color image involves two steps: (i) converting color image to grayscale and (ii) finding the roughness of generated grayscale image. But due to this conversion, significant color information is lost and leads to inaccurate roughness estimation. To avoid such inaccuracy this paper proposes the development and study of novel technique for estimating fractal dimension of color images. In this study, the improved differential box counting method is applied to the 24 bit representation of RGB color images to extract the roughness of color images. The validation of the proposal is performed by generating twelve different synthesized color images in terms of small variation of intensity value in RGB space and compared with previously three well defined existing methods that are weighted sum, average and desaturation method. The results showed that our proposed method is able to capture the accurate sharp variation of roughness as compared to the existing methods.

Correlation of Fractal Dimension Values with Implant Insertion Torque and Resonance Frequency Values at Implant Recipient Sites.

Abstract: Purpose: Fractal analysis is a mathematical method used to describe the internal architecture of complex structures such as trabecular bone. Fractal analysis of panoramic radiographs of implant recipient sites could help to predict the quality of the bone prior to implant placement. This study investigated the correlations between the fractal dimension values obtained from panoramic radiographs and the insertion torque and resonance frequency values of mandibular implants. Materials and Methods: Thirty patients who received a total of 55 implants of the same brand, diameter, and length in the mandibular premolar and molar regions were included in the study. The same surgical procedures were applied to each patient, and the insertion torque and resonance frequency values were recorded for each implant at the time of placement. The radiographic fractal dimensions of the alveolar bone in the implant recipient area were calculated from preoperative panoramic radiographs using a box-counting algorithm. The insertion torque and resonance frequency values were compared with the fractal dimension values using the Spearman test. Results: All implants were successful, and none were lost during the follow-up period. Linear correlations were observed between the fractal dimension and resonance frequency, between the fractal dimension and insertion torque, and between resonance frequency and insertion torque. Conclusion: These results suggest that the noninvasive measurement of the fractal dimension from panoramic radiographs might help to predict the bone quality, and thus the primary stability of dental implants, before implant surgery.

Investigating the Fractal Characteristics of Pore-Fractures in Bituminous Coals and Anthracites through Fluid Flow Behavior

Abstract: The characteristics of pore-fractures are the key petrophysical properties used to assess and evaluate coalbed methane (CBM) reservoirs and include pore types, structure types, porosity/percentage, and pore-fracture space properties. To study the storage and seepage capability of a CBM reservoir, based on 18 coal samples with the maximum vitrinite reflectance (Ro,max) in the range of 1.06–3.04%, we used the nuclear magnetic resonance (NMR) method and fractal analysis to investigate the effect of coalification on the characteristics of pore-fractures. First, the pore-fracture space in a coal reservoir includes irreducible fluid space and moveable fluid space. We built up the moveable fluid NMR fractal based on the saturated/irreducible fluid NMR fractals. Saturated fluid fractal (Dw), irreducible fluid fractal (Dir), and moveable fluid fractal (DM) have the following relationship: DM > Dir > DW. Additionally, Dir has a huge fluctuating value area with the maximum vitrinite reflectance (Ro,max) varying betw...

Nanoindentation study on apple tissue and isolated cells by atomic force microscopy, image and fractal analysis

Abstract: Atomic force microscopy (AFM) was used to evaluate the mechanical properties of apple tissue ( Malus domestica var. Golden Delicious) and isolated cells. Young's modulus (E) obtained for the tissue was 0.86 ± 0.81 MPa, while for isolated cells an average value of 0.63 ± 0.42 MPa was found. The fractal dimension of the distribution of E (FD E ) was estimated to be 1.81 for tissue and 2.47 for isolated cells, which indicates more anisotropic structures in single cell topography in comparison with the tissue. Finally, an image texture analysis was performed and different values were obtained; for roughness Ra = 49.70 ± 23 and 18.00 ± 3, entropy Ent = 5.26 ± 0.66 and 5.70 ± 0.23 and for the fractal dimension of the texture image FD T = 2.55 ± 0.02 and 2.62 ± 0.03, for tissue and isolated cells respectively. Industrial relevance AFM is rarely used to evaluate the structural and mechanical properties of food materials. AFM studies could provide insight on food properties. For example, the nanoindentation of cells by AFM could be used to follow-up changes to the mechanical properties that occur during the processing and storage of fruits with industrial relevance such as apple. Study of the nanostructure and nanomechanical properties of vegetable cells could lead to a more in-depth understanding of the relationship structure–functionality in foodstuffs. This could be applied to address several issues, such as extending the storage life of climacteric fruits, verifying the structural modifications of the tissues when they are chemically treated and having a better control of mechanical and structural properties. Currently, fractal and image analysis is gaining attention in the food science industry. Their application allows one to obtain quantitative information from nanoindentation data and AFM images which can be related to structural modifications which occur during ripening and processing of climacteric fruits. This methodology could be extended to other food materials.

A survey on fractal dimension for fractal structures

Abstract: Along the years, the foundations of Fractal Geometry have received contributions starting from mathematicians like Cantor, Peano, Hilbert, Hausdorff, Caratheodory, Sierpinski, and Besicovitch, to quote some of them. They were some of the pioneers exploring objects having self-similar patterns or showing anomalous properties with respect to standard analytic attributes. Among the new tools developed to deal with this kind of objects, fractal dimension has become one of the most applied since it constitutes a single quantity which throws useful information concerning fractal patterns on sets. Several years later, fractal structures were introduced from Asymmetric Topology to characterize self-similar symbolic spaces. Our aim in this survey is to collect several results involving distinct definitions of fractal dimension we proved jointly with Prof. M.A. Sanchez-Granero in the context of fractal structures.