Showing papers on "Fractal dimension published in 1999"

Fractal antenna engineering: the theory and design of fractal antenna arrays

Abstract: A fractal is a recursively generated object having a fractional dimension. Many objects, including antennas, can be designed using the recursive nature of a fractal. In this article, we provide a comprehensive overview of recent developments in the field of fractal antenna engineering, with particular emphasis placed on the theory and design of fractal arrays. We introduce some important properties of fractal arrays, including the frequency-independent multi-band characteristics, schemes for realizing low-sidelobe designs, systematic approaches to thinning, and the ability to develop rapid beam-forming algorithms by exploiting the recursive nature of fractals. These arrays have fractional dimensions that are found from the generating subarray used to recursively create the fractal array. Our research is in its infancy, but the results so far are intriguing, and may have future practical applications.

Fractal nature of fat crystal networks

Abstract: The quantification of microstructure in fat crystal networks is studied using the relationship of the shear elastic modulus ${(G}^{\ensuremath{'}})$ to the volume fraction of solid fat (\ensuremath{\Phi}) via the mass fractal dimension (D) of the network. Results from application of a scaling theory (weak-link regime theory), developed for colloidal gels, to the microstructure of fat crystal networks are presented and discussed. A method to measure mass fractal dimensions and chemical length exponents or backbone fractal dimensions (x) from in situ polarized light microscope (PLM) images of the microstructural network of fat crystals is developed and applied to the fat systems studied. Fractal dimensions measured from in situ PLM images of the various fat systems are in good agreement with fractal dimensions measured using rheological measurements and the weak-link regime theory (percent deviations range from 0.40% to 2.50%). The crystallization behavior of the various fat systems is studied using differential scanning calorimetry, and the potential for altering ${G}^{\ensuremath{'}}$ by changing crystallization conditions using the fractal dimension of the network as an indicator is discussed.

Rheological Study on the Fractal Nature of the Protein Gel Structure

Abstract: The effects of ionic strength on fractal structures in heat-induced gels prepared from globular proteins were investigated in the framework of a fractal aggregation of colloidal particles. All gels formed at 90 °C exhibited power law relationships between the storage shear modulus (G‘) and protein concentration. At 25 mM NaCl, the fractal dimension, d (∼2.2), calculated based on the value of the power law exponent agreed with those for reaction-limited cluster−cluster aggregation. Further addition of NaCl (50, 80, 500, 1000 mM) decreased the values of d (∼1.8), which agreed with d for diffusion-limited cluster−cluster aggregation. These results suggest that the predominant effect of an increase in ionic strength on globular protein gelation is ascribed to shielding charges on the surface of the proteins, thereby increasing the reaction probability of protein aggregation. The effective structure-determining rheological properties of heat-induced protein gels are characterized by fractal dimensions deduced ...

When and where is a city fractal

Abstract: We present here an analysis of the development of the Tel Aviv metropolis by using the concept of fractals. The fractal dimension of the entire metropolis, and of its parts, was estimated as a function of time, from 1935 onwards. The central part and the northern tier are fractal at all times. Their fractal dimension increased with time. However, the metropolis as a whole can be said to be fractal only after 1985. There is a general tendency towards fractality, in the sense that the fractal dimension of the different parts converge towards the same value. (This abstract was borrowed from another version of this item.)

Fractal Geometry of Rocks

Abstract: The analysis of small- and ultra-small-angle neutron scattering data for sedimentary rocks shows that the pore-rock fabric interface is a surface fractal (D{sub s}=2.82) over 3 orders of magnitude of the length scale and 10 orders of magnitude in intensity. The fractal dimension and scatterer size obtained from scanning electron microscopy image processing are consistent with neutron scattering data. {copyright} {ital 1999} {ital The American Physical Society}

Mechanical and structural model of fractal networks of fat crystals at low deformations.

Abstract: Fat-crystal networks demonstrate viscoelastic behavior at very small deformations. A structural model of these networks is described and supported by polarized light and atomic-force microscopy. A mechanical model is described which allows the shear elastic modulus (G') of the system to be correlated with forces acting within the network. The fractal arrangement of the network at certain length scales is taken into consideration. It is assumed that the forces acting are due to van der Waals forces. The final expression for G' is related to the volume fraction of solid fat (Phi) via the mass fractal dimension (D) of the network, which agrees with the experimental verification of the scaling behavior of fat-crystal networks [S. S. Narine and A. G. Marangoni, Phys. Rev. E 59, 1908 (1999)]. G' was also found to be inversely proportional to the diameter of the primary particles (sigma approximately equal to 6 microm) within the network (microstructural elements) as well as to the diameter of the microstructures (xi approximately equal to 100 microm) and inversely proportional to the cube of the intermicrostructural element distance (d(0)). This formulation of the elastic modulus agrees well with experimental observations.

Fractal analysis of surface roughness by using spatial data

Abstract: Summary. We develop fractal methodology for data taking the form of surfaces. An advantage of fractal analysis is that it partitions roughness characteristics of a surface into a scale-free component (fractal dimension) and properties that depend purely on scale. Particular emphasis is given to anisotropy where we show that, for many surfaces, the fractal dimension of line transects across a surface must either be constant in every direction or be constant in each direction except one. This virtual direction invariance of fractal dimension provides another canonical feature of fractal analysis, complementing its scale invariance properties and enhancing its attractiveness as a method for summarizing properties of roughness. The dependence of roughness on direction may be explained in terms of scale rather than dimension and can vary with orientation. Scale may be described by a smooth periodic function and may be estimated nonparametrically. Our results and techniques are applied to analyse data on the surfaces of soil and plastic food wrapping. For the soil data, interest centres on the effect of surface roughness on retention of rain-water, and data are recorded as a series of digital images over time. Our analysis captures the way in which both the fractal dimension and the scale change with rainfall, or equivalently with time. The food wrapping data are on a much finer scale than the soil data and are particularly anisotropic. The analysis allows us to determine the manufacturing process which produces the smoothest wrapping, with least tendency for micro-organisms to adhere.

Estimating crowd density with Minkowski fractal dimension

Abstract: The estimation of the number of people in an area under surveillance is very important for the problem of crowd monitoring. When an area reaches an occupation level greater than the projected one, people's safety can be in danger. This paper describes a new technique for crowd density estimation based on Minkowski fractal dimension. The fractal dimension has been widely used to characterize data texture in a large number of physical and biological sciences. The results of our experiments show that fractal dimension can also be used to characterize levels of people congestion in images of crowds. The proposed technique is compared with a statistical and a spectral technique, in a test study of nearly 300 images of a specific area of the Liverpool Street Railway Station, London, UK. Results obtained in this test study are presented.

Multifractals and 1/f noise : wild self-affinity in physics (1963-1976) : selecta volume N

Abstract: Preface.- Panorama of grid-bound self affine variablity.- Sketches of prehistory and history.- Scaling, invariants and fixed points.- Filtering and specifications of self-affinity.- Short pieces.- New model for error clustering on telephone circuits.- Additional tests on clustering.- Self-similarity and conditional stationarity.- 1/f noises and the infrared catastrophe.- Co-indicator functions and related 1/f noises.- Sporadic random functions and conditional spectra: Self-similar examples and limits.- Random sets of multiplicity for trigonometric series.- Sporadic turbulence.- Intermittent free turbulence.- Lognormal hypothesis and distribution of energy dissipation in intermittent turbulence.- Intermittent turbulence in self-similar cascades: Divergence of high moments and dimension of the carrier.- Iterated random multiplications and invariance under randomly weighted averaging.- 'On certain martingales of Benoit Mandelbrot'.- Intermittent turbulence and fractal dimension: the kurtosis and the spectral exponent 5 3 + B.- Fractal dimension, dispersion, and singularities of fluid motion.- Elementary fractals and multifractals.- Bibliography.- Index.

Comparison of slope estimates from low resolution DEMs: scaling issues and a fractal method for their solution

Abstract: Five different algorithms for calculating slope from digital elevation models (DEMs) have been compared from regional to global scales. Though different methods produce different results, the most significant outcome is that slope varies inversely with the DEM grid size. Thus, slopes estimated from coarse resolution data can be considered to produce significant underestimates of the true slope. A fractal theory is adapted to solve this problem. The variogram technique for the definition of fractal parameters is demonstrated to provide a relationship between slope and the spatial resolution of measurement. The variation of fractal parameters is discussed at various scales, and a model is developed to estimate the high resolution slope based on the coarse resolution DEM by using fractal parameters. The fractal parameters are estimated from the standard deviation of elevation in a 3 × 3 window of the DEM to account for local variability in the surface. Standard deviation of elevation is found to be the most invariant property of different scale DEMs of the same area. The model is validated using different resolution DEMs in southern Spain and it is used to estimate the high resolution slope values at global scales based on a coarse resolution DEM. The slopes estimated using the technique outlined are a significant improvement on those estimated directly from the coarse resolution data. Slopes estimated in this way allow the more effective use of available coarse resolution data in regional and global scale modelling studies. Copyright © 1999 John Wiley & Sons, Ltd.

Multi-Scale Fractal Analysis of Image Texture and Pattern

Abstract: Fractals embody important ideas of self-similarity, in which the spatial behavior or appearance of a system is largely independent of scale. Self-similarity is defined as a property of curves or surfaces where each part is indistinguishable from the whole, or where the form of the curve or surface is invariant with respect to scale. An ideal fractal (or monofractal) curve or surface has a constant dimension over all scales, although it may not be an integer value. This is in contrast to Euclidean or topological dimensions, where discrete one, two, and three dimensions describe curves, planes, and volumes. Theoretically, if the digital numbers of a remotely sensed image resemble an ideal fractal surface, then due to the self-similarity property, the fractal dimension of the image will not vary with scale and resolution. However, most geographical phenomena are not strictly self-similar at all scales, but they can often be modeled by a stochastic fractal in which the scaling and self-similarity properties of the fractal have inexact patterns that can be described by statistics. Stochastic fractal sets relax the monofractal self-similarity assumption and measure many scales and resolutions in order to represent the varying form of a phenomenon as a function of local variables across space. In image interpretation, pattern is defined as the overall spatial form of related features, and the repetition of certain forms is a characteristic pattern found in many cultural objects and some natural features. Texture is the visual impression of coarseness or smoothness caused by the variability or uniformity of image tone or color. A potential use of fractals concerns the analysis of image texture. In these situations it is commonly observed that the degree of roughness or inexactness in an image or surface is a function of scale and not of experimental technique. The fractal dimension of remote sensing data could yield quantitative insight on the spatial complexity and information content contained within these data. A software package known as the Image Characterization and Modeling System (ICAMS) was used to explore how fractal dimension is related to surface texture and pattern. The ICAMS software was verified using simulated images of ideal fractal surfaces with specified dimensions. The fractal dimension for areas of homogeneous land cover in the vicinity of Huntsville, Alabama was measured to investigate the relationship between texture and resolution for different land covers.

Clustering and size distributions of fault patterns: Theory and measurements

Abstract: The fractal geometry of fault systems has been mainly characterized by two scaling-laws describing either their spatial distribution (clustering) or their size distribution. However, the relationships between the exponents of both scaling-laws has been poorly investigated. We show theoretically and numerically that the fractal dimension D and the exponent a of the frequency length distribution of fault networks, are related through the relation x=(a−1)/D, where x is the exponent of a new scaling law involving the average distance from a fault to its nearest neighbor of larger length. Measurements of the relevant exponents on the San Andreas fault pattern are in agreement with the theoretical analysis and allows us to test the fragmentation models proposed in the literature. We also found a correlation between the position of a fault and its length so that large faults have their nearest neighbor located at greater distances than small faults.

Fractional calculus and the evolution of fractal phenomena

Abstract: It is argued that the evolution of complex phenomena ought to be described by fractional, differential, stochastic equations whose solutions have scaling properties and are therefore random, fractal functions. To support this argument we demonstrate that the fractional derivative (integral) of a generalized Weierstrass function (GWF) is another fractal function with a greater (lesser) fractal dimension. We also determine that the GWF is a solution to such a fractional differential stochastic equation of motion.

Fractal analysis of radiographs: assessment of trabecular bone structure and prediction of elastic modulus and strength.

Abstract: The purpose of this study was to determine whether fractal dimension of radiographs provide measures of trabecular bone structure which correlate with bone mineral density (BMD) and bone biomechanics, and whether these relationships depend on the technique used to calculate the fractal dimension. Eighty seven cubic specimen of human trabecular bone were obtained from the vertebrae and femur. The cubes were radiographed along all three orientations--superior-inferior (SI), medial-lateral (ML), and anterior-posterior (AP), digitized, corrected for background variations, and fractal based techniques were applied to quantify trabecular structure. Three different techniques namely, semivariance, surface area, and power spectral methods were used. The specimens were tested in compression along three orientations and the Young's modulus (YM) was determined. Compressive strength was measured along the SI direction. Quantitative computed tomography was used to measure trabecular BMD. High-resolution magnetic-resonance images were used to obtain three-dimensional measures of trabecular architecture such as the apparent bone volume fraction, trabecular thickness, spacing, and number. The measures of trabecular structure computed in the different directions showed significant differences (p<0.05). The correlation between BMD, YM, strength, and the fractal dimension were direction and technique dependent. The trends of variation of the fractal dimension with BMD and biomechanical properties also depended on the technique and the range of resolutions over which the data was analyzed. The fractal dimension showed varying trends with bone mineral density changes, and these trends also depended on the range of frequencies over which the fractal dimension was measured. For example, using the power spectral method the fractal dimension increased with BMD when computed over a lower range of spatial frequencies and decreased for higher ranges. However, for the surface area technique the fractal dimension increased with increasing BMD. Fractal measures showed better correlation with trabecular spacing and number, compared to trabecular thickness. In a multivariate regression model inclusion of some of the fractal measures in addition to BMD improved the prediction of strength and elastic modulus. Thus, fractal based texture analysis of radiographs are technique dependent, but may be used to quantify trabecular structure and have a potentially valuable impact in the study of osteoporosis.

Fractal dimensions of speech sounds: computation and application to automatic speech recognition.

Abstract: The dynamics of airflow during speech production may often result in some small or large degree of turbulence. In this paper, the geometry of speech turbulence as reflected in the fragmentation of the time signal is quantified by using fractal models. An efficient algorithm for estimating the short-time fractal dimension of speech signals based on multiscale morphological filtering is described, and its potential for speech segmentation and phonetic classification discussed. Also reported are experimental results on using the short-time fractal dimension of speech signals at multiple scales as additional features in an automatic speech-recognition system using hidden Markov models, which provide a modest improvement in speech-recognition performance.

Compressibility and Fractal Dimension of Fine Coal Particles in Relation to Pore Structure Characterisation Using Mercury Porosimetry

Abstract: Mercury porosimetry has been applied to characterize the pore structure of fine coals particles. Interparticle voids and compressibility effects on the mercury intrusion data were examined. It is found that coal compressibility has a significant effect on mercury porosimetry data when pressure P > 20 MPa. The compressibility of the two coals used was determined to be 3.13 x 10(-10) m(2) N-1 and 2.50 x 10(-10) m(2) N-1 for CA and GO coals, respectively. Fractal dimension analysis provides a fingerprint to distinguish the effect of coal compression from the pore filling process during mercury intrusion. It is shown that fractal dimension can be evaluated from the compressibility corrected pore volume data. Results from the present study suggest that statistic self-similarity of the fractal dimension perspective is limited by certain artificial effects, such as crushing and grinding. Different surface irregularities exist over different pore size ranges, and a single fractal dimension value can only be used to describe the surface irregularity within a limited pore size range. The average fractal dimensions in the pore size range of 6-60 nm were found to be 2.71 and 2.43 for CA and GO coals, respectively.

Order From Chaos

Abstract: B ENOIT is a fractal analysis software product for Windows 95, Windows 98, or Windows NT used to find order and patterns in seemingly chaotic data, particularly where traditional statistical approaches to data analysis fail. It is widely used in disciplines as diverse as biology, chemistry, physics, economics, medicine, and geology. The U.S. Geological Survey, for example, employs fractal analysis to accurately predict the volume of undiscovered deposits of oil and natural gas, on the basis of data from known deposits. BENOIT measures user-supplied data by standard fractal methods. For a fractal, measures change in value as the scale decreases in size because ever-smaller pieces become included in the analysis. Measures are plotted as a function of ruler size on a log-log plot, and a fractal dimension is calculated from the slope of the resulting line. Users select one of 10 analytical measures with the software. Five of the available measures in the program act upon bitmap images in Windows BMP format. These are described as the “self-similar” or two-dimensional (2D) methods, while the remaining group of routines act upon time-series or 1D data. The latter group requires data to be in a simple but specific data format, such as is available in Excel. The program also features a data generator that produces files with a given fractal dimension. Users may find this useful for testing and control purposes. The self-similar or image methods available in BENOIT measure different characteristics of bitmap objects in ways that should be scale-invariant. A real dataset normally has some fractal limit, and outside the limit, the fractal dimension will return a trivial value (1 for time-series or 2 for image data). Upper and lower fractal limits are controlled by the size of the dataset. Self-similar methods available in BENOIT are well known in fractal analysis: box dimension, perimeter-area dimension, information dimension, and ruler dimension. All methods are explained in standard Help files that contain several pages of information for each topic. The 1D analysis routines use “self-affine” methods of analysis. Self-affine fractals differ from self-similiar fractals in that their parts need to be rescaled by different factors in different coordinates to resemble the original. In the roughness-length method, the root-mean-square variation or roughness of the data is calculated for a variety of horizontal scales. The operation provides an estimate of the Hurst exponent, H , in a log-log plot, which is related to the fractal dimension. Standard self-affine methods available include R/S (Rescaled Range) analysis, power spectrum, roughness-length, variogram, and wavelets. Printing of log-log figures is provided, but without many features that would be found in a spreadsheet. Documentation for the program is available online. BENOIT has a highly visual interface, complete with an animated grid or ruler for self-similar fractal methods, and it gives users control of all calculations that the program performs, unlike other fractal software. Benoit is not without flaws. Some operations, such as name registration with the Windows NT 4.0 taskbar and the Open File requester, do not conform to standard Windows conventions. It would be of help to have an outline or flowchart of the operation of the different routines available in BENOIT for newcomers to fractal analysis. In summary, the variety of fractal analysis methods available in BENOIT, together with generally detailed help files and significant user control of operations, make BENOIT a good resource for learning about and using fractal analysis methods.

Shear elasticity and yield stress of silica–silicone physical gels: Fractal approach

Abstract: The elastoplastic behavior of silica–silicone compounds has been characterized by a yield stress σs and an elastic modulus G0. Scaling laws have been established for the changes in rheometrical parameters with the volume fraction Φv of silica: σs∝(Φv)3.3 and G0∝(Φv)4.2 and single master curves have been obtained whatever the silica type used. The mesoscopic structure of the compounds has been studied using x-ray and light scattering and a semidilute fractal structure has been observed, with a fractal dimension D≈1.8 and a characteristic length scale 4 μm. In this paper, a formulation of the nonfluctuating semidilute fractal concept allows us to relate the fractal dimension to the rheological scaling laws: σs∝(Φv)4/(3−D) and G0∝(Φv)5/(3−D).