Showing papers on "Fractal dimension published in 1997"

Techniques in fractal geometry

Abstract: Mathematical Background. Review of Fractal Geometry. Some Techniques for Studying Dimension. Cookie-cutters and Bounded Distortion. The Thermodynamic Formalism. The Ergodic Theorem and Fractals. The Renewal Theorem and Fractals. Martingales and Fractals. Tangent Measures. Dimensions of Measures. Some Multifractal Analysis. Fractals and Differential Equations. References. Index.

Fractals and fractional calculus in continuum mechanics

Abstract: A. Carpinteri: Self-Similarity and Fractality in Microcrack Coalescence and Solid Rupture.- B. Chiaia: Experimental Determination of the Fractal Dimension of Microcrack Patterns and Fracture Surfaces.- P.D. Panagiotopoulos, O.K. Panagouli: Fractal Geometry in Contact Mechanics and Numerical Applications.- R. Lenormand: Fractals and Porous Media: from Pore to Geological Scales.- R. Gorenflo, F. Mainardi: Fractional Calculus: Integral and Differential Equations of Fractional Order.- R. Gorenflo: Fractional Calculus: some Numerical Methods.- F. Mainardi: Fractional Calculus: some Basic Problems in Continuum and Statistical Mechanics.

Use of the fractal dimension for the analysis of electroencephalographic time series

Abstract: Electroencephalogram (EEG) traces corresponding to different physiopathological conditions can be characterized by their fractal dimension, which is a measure of the signal complexity. Generally this dimension is evaluated in the phase space by means of the attractor dimension or other correlated parameters. Nevertheless, to obtain reliable values, long duration intervals are needed and consequently only long-term events can be analysed; also much calculation time is required. To analyse events of brief duration in real-time mode and to apply the results obtained directly in the time domain, thus providing an easier interpretation of fractal dimension behaviour, in this work we optimize and propose a new method for evaluating the fractal dimension. Moreover, we study the robustness of this evaluation in the presence of white or line noises and compare the results with those obtained with conventional spectral methods. The non-linear analysis carried out allows us to investigate relevant EEG events shorter than those detectable by means of other linear and non-linear techniques, thus achieving a better temporal resolution. An interesting link between the spectral distribution and the fractal dimension value is also pointed out.

Fractals in pathology

Abstract: Many natural objects, including most objects studied in pathology, have complex structural characteristics and the complexity of their structures, for example the degree of branching of vessels or the irregularity of a tumour boundary, remains at a constant level over a wide range of magnifications. These structures also have patterns that repeat themselves at different magnifications, a property known as scaling self-similarity. This has important implications for measurement of parameters such as length and area, since Euclidean measurements of these may be invalid. The fractal system of geometry overcomes the limitations of the Euclidean geometry for such objects and measurement of the fractal dimension gives an index of their space-filling properties. The fractal dimension may be measured using image analysis systems and the box-counting, divider (perimeter-stepping) and pixel dilation methods have all been described in the published literature. Fractal analysis has found applications in the detection of coding of coding regions in DNA and measurement of the space-filling properties of tumours, blood vessels and neurones. Fractal concepts have also been usefully incorporated into models of biological processes, including epithelial cell growth, blood vessel growth, periodontal disease and viral infections.

Yield stress thixotropic clay suspension : investigations of structure by light, neutron, and x-ray scattering

Abstract: The characteristic length scales of the structure and fractal behavior of a thixotropic colloidal suspension of synthetic clay were studied by using a combination of small-angle neutron and x-ray scattering and static light scattering. At the same time, macroscopic mechanical behavior at rest was characterized by means of rheometric measurements. Two characteristic length scales were detected in these yield stress suspensions of discotic texture. The first, measuring several tens of nanometres, is linked to a fractal dimension of 3. The second, of the order of 1 mm, is linked to a fractal behavior of dimension D that increases with the particle volume fraction. Consequently, it is suggested that the structure of the dispersions at rest is composed of subunits measuring a few tens of nanometers that combine to form dense aggregates measuring about 1 mm. At larger length scales, these micrometer-sized aggregates are rearranged to form a continuous three-dimensional isotropic structure that has a fractal behavior of dimension D, which gives the gels their texture. The increase of this fractal dimension with the particle volume fraction, the ionic strength, and the gelation time is correlated to a hardening of the mechanical properties of the gels at rest. The gel state is reached above a volume fraction f v for a given ionic strength and gelation time. In the gel phase, a critical volume fraction f vc separates two domains. Gels belonging to the domain f v,f v,f vc have a fractal behavior of dimension D5160.05, suggesting an alignment of the micrometer-sized aggregates that leads to the formation of a mechanically weak fibrous structure. Gels belonging f v.f vc have a fractal dimension D51.860.01, corresponding to a mechanically stronger structure consisting of zones of high and lower particle density. A scaling law enabled these fractal dimensions to be correlated with the effect of the volume fraction on the yield stress. In contrast to what is commonly assumed in relation to clay suspensions, it is suggested here that it is the large length scales, of the order of 1 mm, associated with a fractal arrangement that governs the macroscopic mechanical behavior.

Estimating the Fractal Dimension of a Locally Self-similar Gaussian Process by using Increments

Abstract: SUMMARY Consider the problem of estimating the parameter ca of a stationary Gaussian process with

Fractals in Engineering

Abstract: Pattern Formation in Metal on Metal Epitaxy, P 0 Hwang and P J Behm Scaling in the Frequency-Dependent Admittance of Electrodeposited Fractal Electrodes, A E Larsen et al. Resonant Optics of Fractals, V Shalaev et al. Fractal Surfaces in Engineering: Applications of Dynamic Scaling, F Family Fractals in Fluid Mechanics, K R Sreenivasan Diffusional Analysis of Intermittent Two-Phase Flow Transitions, M Giona et al. Anomalous Relaxation in Noise-Driven Bistable Systems, S J Fraser & R Kaptal Complexity Maps Reveal Clusters in Neuronal Arborizations, B Dubuc et al. Infrared Scene Modeling and Interpolation Using Fractional Levy Stable Motion, S M Kogon & D G Manolakis Solving the Inverse Problem for Function/Image Approximation Using Iterated Function Systems: 1. Theoretical Basis, B Forte & E R Vrscay Solving the Inverse Problem for Function/Image Approximation Using Iterated Function Systems: 11. Algorithm and Computations, B Forte & E R Vrscay Fractal Image Compression, Y Fisher Utilization of Fractal Image Models in Medical Image Processing, W S Kuklinski Universal Multifractals in Seismicity, C Hooge et al. Determination of Fractal Scales on Trabecular Bone X-Ray Images, R Harba et al. Developing a New Method for the Identification of Microorganisms for the Food Industry Using the Fractal Dimension, O Castillo & P Melin. (Part contents).

Fractal effects of surface roughness on the mechanical behavior of rock joints

Abstract: Extensive studies show that the naturally developed rock joint surfaces have the properties of fractals. The surface roughness of rock joints can be well described within the framework of fractal geometry. To give a better understanding of the roughness of rock fracture surfaces in relation to the mechanical properties and behavior of rock joints in loading, a systematic investigation has been carried out. By means of a laser scanning instrument, the fracture surfaces induced in rocks are measured. The fractal characters of rough fracture surfaces of rocks are analysed according to the variogram method, which elaborates explicitly the importance of the geometric parameters — fractal dimension D and the intercept A . Investigation extends to the anisotropy and heterogeneity of rock fracture surfaces, and the scale effect on the fractal estimation. The shear tests on rock joints show a combined effect of the fractal parameters on the mechanical properties and behavior of rock joints. To control the roughness and show the effects in a direct manner, a series of fractal joints with different fractal dimensions are generated on the basis of the Weierstrass-Mandelbrot function and then manufactured in the polycarbonate plates. By using the photoelastic method, the manners of normal and shear deformation, the stress field in the vicinity of the joint surface and the contact behaviour of the fractal joints have been studied in detail under uniaxial compression and direct shear. Based on the experimental results, the dependence of deformation stiffness on the fractal dimensions of rock joints has been recognized. The empirical criteria of shear strength and the evolution law of surface damage during the shear process are developed.

A light scattering study of the fractal aggregation behavior of a model colloidal system

Abstract: The mass fractal dimension of aggregates of colloidal polystyrene latex particles was measured using small-angle static light scattering over a range of electrolyte and particle concentrations. The measured fractal dimensions ranged from 1.78 to 2.20, which are in good agreement with the predicted values of the diffusion-limited cluster−cluster aggregation and reaction-limited cluster−cluster aggregation regimes, respectively. It was found that increasing the salt concentration had the effect of reducing the fractal dimension, indicating a more open aggregate structure. Two regimes of behavior were observed for the fractal dimension as a function of particle concentration. At high salt levels (>1 M KNO3) the particle concentration was seen to have little or no effect, while at low salt levels (<1 M KNO3) an increase in concentration led to a decrease in the fractal dimension. This was attributed to a reduction in the time available for reconfiguring of particles in the aggregates due to an increased parti...

Collision Frequencies between Fractal Aggregates and Small Particles in a Turbulently Sheared Fluid

Abstract: Three groups of aggregates with fractal dimensions of 1.89 ± 0.06, 2.21 ± 0.06, and 2.47 ± 0.10 were generated by coagulation of latex microspheres (2.85 μm) in a Jar-test (paddle-mixing) device. The collision rates between these fractal aggregates (200−1000 μm) and small (1.48 μm) particles were measured in the turbulent shear environ ment of the paddle mixer at mean shear rates of 2.1, 7.3, and 14.7 s-1. Collision frequencies were 5 orders of magnitude higher than predicted by a curvilinear model but 2 orders of magnitude lower than predicted by a rectilinear model. Collision frequencies much higher than predicted by the curvilinear collision kernel were attributed to significant flow through the interior of the fractal aggregates. The fluid shear rate (G) and the aggregate fractal dimension (D) affected the collision frequency function (β) between fractal aggregates and small particles, resulting in β ∼ G1-0.33D. According to this relationship, as D → 0, the ag gregates become infinitely porous and β b...

Multifractal Modeling and Lacunarity Analysis

Abstract: The so-called “gliding box method” of lacunarity analysis has been investigated for implementing multifractal modeling in comparison with the ordinary box-counting method. Newly derived results show that the lacunarity index is associated with the dimension (codimension) of fractal, multifractal and some types of nonfractals in power-law relations involving box size; the exponent of the lacunarity function corresponds to the fractal codimension (E – D) for fractals and nonfractals, and to the correlation codimension (E – τlpar;2)) for multifractals. These results are illustrated with two case studies: De Wijs's zinc concentration values from the Pulacayo sphalerite-quartz vein in Bolivia and Cochran's tree seedlings example. Both yield low lacunarities and slightly depart from translational invariance.

Dimension estimation of discrete-time fractional Brownian motion with applications to image texture classification

Abstract: Fractional Brownian motion (FBM) is a suitable description model for a large number of natural shapes and phenomena. In applications, it is imperative to estimate the fractal dimension from sampled data, namely, discrete-time FBM (DFBM). To this aim, the increment of DFBM, referred to as discrete-time fractional Gaussian noise (DFGN), is invoked as an auxiliary tool. The regular part of DFGN is first filtered out via Levinson's algorithm. The power spectral density of the regular process is found to satisfy a power law that its exponent can be well fitted by a quadratic function of fractal dimension. A new method is then proposed to estimate the fractal dimension of DFBM from the given data set. The computational complexity and statistical properties are investigated. Moreover, the proposed algorithm is robust with respect to amplitude scaling and shifting, as well as time shifting on the data. Finally, the effectiveness of this estimator is demonstrated via a classification problem of natural texture images.

The Study on Bivariate Fractal Interpolation Functions and Creation of Fractal Interpolated Surfaces

Abstract: In this paper, the methods of construction of a fractal surface are introduced, the principle of bivariate fractal interpolation functions is discussed. The theorem of the uniqueness of an iterated function system of bivariate fractal interpolation functions is proved. Moreover, the theorem of fractal dimension of fractal interpolated surface is derived. Based on these theorems, the fractal interpolated surfaces are created by using practical data.

Fractal geometry of bean root systems: correlations between spatial and fractal dimension.

Abstract: An obstacle to the study of root architecture is the difficulty of measuring and quantifying the three-dimensional configuration of roots in soil. The objective of this work was to determine if fractal geometry might be useful in estimating the three-dimensional complexity of root architecture from more accessible measurements. A set of results called projection theorems predict that the fractal dimension (FD) of a projection of a root system should be identical to the FD of roots in three-dimensional space (three-dimensional FD). To test this prediction we employed SimRoot, an explicit geometric simulation model of root growth derived from empirical measurements of common bean (Phaseolus vulgaris L.). We computed the three-dimensional FD, FD of horizontal plane intercepts (planar FD), FD of vertical line intercepts (linear FD), and FD of orthogonal projections onto planes (projected FD). Three-dimensional FD was found to differ from corresponding projected FD, suggesting that the analysis of roots grown in a narrow space or excavated and flattened prior to analysis is problematic. A log-linear relationship was found between FD of roots and spatial dimension. This log-linear relationship suggests that the three-dimensional FD of root systems may be accurately estimated from excavations and tracing of root intersections on exposed planes.

A new interpretation of scattered light measurements at Titan's limb

Abstract: Images of Titan, taken by Voyager 2 at phase angles Φ=140° and Φ=155° have provided radial intensity profiles at the bright and dark limbs, which provide information on the vertical and latitudinal distribution of organic hazes. In previous work, the deduced extinction coefficient, using ad hoc particle sizes, was obtained without help of microphysics, and it appeared difficult to compare it with coefficients computed from theoretical models. We use here our fractal approach of microphysical modeling and optics of agregates to compute intensity profiles of the main haze at the bright limb, and compare to the Voyager observations. Fractal aerosol distributions are obtained using different production altitudes and rates. Scattering and absorption of light are described by an improved model, based on the use of fractal aggregates made of spherical (Mie) particles. We show that the fractal dimension of aggregates has to be Df≈2, as predicted by microphysical arguments. Only a production altitude z0≈385±60 km, corresponding to a monomer radius rm≈0.066 μm, is fully consistent with both phase angle data. We also point out that the production rate of the aerosols decreases by a factor ≈2 between 30°S and the midnorthern latitude and further, increases up to 80°N. The average value of the production rate is Q≈1.4×10−13 kg/m2/s; we give arguments in favor of dynamical processes rather than of a purely microphysical mechanisms to explain such latitudinal variations.

Mean-field approximation of Mie scattering by fractal aggregates of identical spheres

Abstract: We apply the recent exact theory of multiple electromagnetic scattering by sphere aggregates to statistically isotropic finite fractal clusters of identical spheres. In the mean-field approximation the usual Mie expansion of the scattered wave is shown to be still valid, with renormalized Mie coefficients as the multipolar terms. We give an efficient method of computing these coefficients, and we compare this mean-field approach with exact results for silica aggregates of fractal dimension 2.

Fractal and multifractal measures of natural and synthetic fracture networks

Abstract: We have analyzed the fractal and multifractal nature of a series of 17 natural fracture trace maps, representing a wide variety of scales, geological settings, and lithologies, as well as a number of typical synthetic fracture networks in which fracture locations, orientations, and lengths are drawn from various probability distribution functions. Recent studies have shown that multifractal methods can be used to investigate fracture networks at greater depth, since the fractal dimension represents only part of the scaling spectrum characterizing each network. We find that the real and synthetic fracture maps display fractal and multifractal properties. Moreover, the properties of the synthetic networks are very similar to those of the real networks, with nontrivial fractal dimensions and multifractal spectra. We suggest that different fracturing mechanisms can lead to two or more distinct subranges over which a fractal dimension can be defined, while the heterogeneity of the rock, and the nature of the fracturing mechanisms, lead to multifractal properties. We also find that the fractal dimension of a synthetic fracture network is relatively insensitive to parameters such as fracture length and orientation but can be controlled by appropriate choice of the relative fracture density.

Applications of Fractals to Soil Studies

Abstract: Publisher Summary This chapter discusses the applications of fractals for soil analysis. The different types of fractal dimensions are discussed that are used in soil science. The mathematical basis of fractal geometry makes it a potentially useful tool to describe the heterogeneity of soil structure quantitatively. A quantitative description of soil structure would ideally be able to be directly related to the processes occurring within the soil. This chapter discusses the relationship between fractal dimensions of the soil structure and soil physical processes. Soil structure has been characterized by fractal dimensions, estimated either directly from images of soil structure or indirectly from bulk density or mercury porosirnetry data, for example. The fragmentation mechanism is also discussed. The main factor that limits the number of applications that the fragmentation fractal dimension (D f ) has is that D f , is estimated from a distribution of aggregates or particles that bear no resemblance to the original soil matrix. In some of the studies reported soil chemical properties and soil mineralogy have been discussed in relation to the fractal dimension estimated for a particular soil, or they have been used to explain apparent changes in fractal scaling. Further research into relating chemical properties and mineralogy of different soil types to their fractal dimensions may help in the understanding the origin of this type of fractal scaling.