Showing papers on "Fractal dimension published in 2006"

Models for effective density and settling velocity of flocs

Abstract: New models to predict settling velocity and effective density of flocs are proposed. The models are based on the concept of fractal geometry, but with the assumption of variable fractal dimension with the floc size. The best results are obtained when the fractal dimension is estimated by a power law function of the floc diameter. The models are compared with observations from 26 published data sets relating floc size to settling velocity measured under various conditions and at different locations. The floc size covered by the data varies between 1.4 and about 25,500µm. Five commonly used models are also compared to these data and found to reproduce inadequately the full range of the observations. Sensitivity analysis shows that, with the proposed models, the spread in the data may be reproduced by varying the size of primary particles from about 0.05 to 20µm. The new models are proposed for integration into numerical models to simulate sediment transport of cohesive sediments, contaminants, and biologica...

Fractal Distribution of Snow Depth from Lidar Data

Abstract: Snowpack properties vary dramatically over a wide range of spatial scales, from crystal microstructure to regional snow climates. The driving forces of wind, energy balance, and precipitation interact with topography and vegetation to dominate snow depth variability at horizontal scales from 1 to 1000 m. This study uses land surface elevation, vegetation surface elevation, and snow depth data measured using airborne lidar at three sites in north-central Colorado. Fractal dimensions are estimated from the slope of a log-transformed variogram and demonstrate scale-invariant, fractal behavior in the elevation, vegetation, and snow depth datasets. Snow depth and vegetation topography each show two distinct fractal distributions over different scale ranges (multifractal behavior), with short-range fractal dimensions near 2.5 and long-range fractal dimensions around 2.9 at all locations. These fractal ranges are separated by a scale break at 15–40 m, depending on the site, which indicates a process cha...

Fractal analysis of remotely sensed images: A review of methods and applications

Abstract: Mandelbrot's fractal geometry has sparked considerable interest in the remote sensing community since the publication of his highly influential book in 1977. Fractal models have been used in several image processing and pattern recognition applications such as texture analysis and classification. Applications of fractal geometry in remote sensing rely heavily on estimation of the fractal dimension. The fractal dimension (D) is a central construct developed in fractal geometry to describe the geometric complexity of natural phenomena as well as other complex forms. This paper provides a survey of several commonly used methods for estimating the fractal dimension and their applications to remote sensing problems. Methodological issues related to the use of these methods are summarized. Results from empirical studies applying fractal techniques are collected and discussed. Factors affecting the estimation of fractal dimension are outlined. Important issues for future research are also identified and discussed.

Emissions from the laboratory combustion of wildland fuels: Particle morphology and size

Abstract: [1] The morphology of particles emitted by wildland fires contributes to their physical and chemical properties but is rarely determined. As part of a study at the USFS Fire Sciences Laboratory (FSL) investigating properties of particulate matter emitted by fires, we studied the size, morphology, and microstructure of particles emitted from the combustion of eight different wildland fuels (i.e., sagebrush, poplar wood, ponderosa pine wood, ponderosa pine needles, white pine needles, tundra cores, and two grasses) by scanning electron microscopy. Six of these fuels were dry, while two fuels, namely the tundra cores and one of the grasses, had high fuel moisture content. The particle images were analyzed for their density and textural fractal dimensions, their monomer and agglomerate number size distributions, and three different shape descriptors, namely aspect ratio, root form factor, and roundness. The particles were also probed with energy dispersive X-ray spectroscopy confirming their carbonaceous nature. The density fractal dimension of the agglomerates was determined using two different techniques, one taking into account the three-dimensional nature of the particles, yielding values between 1.67 and 1.83, the other taking into account only the two-dimensional orientation, yielding values between 1.68 and 1.74. The textural fractal dimension that describes the roughness of the boundary of the two-dimensional projection of the particle was between 1.10 and 1.19. The maximum length of agglomerates was proportional to a power a of their diameter and the proportionality constant and the three shape descriptors were parameterized as function of the exponent a.

Multifractal analysis of human retinal vessels

Abstract: In this paper, it is shown that vascular structures of the human retina represent geometrical multifractals, characterized by a hierarchy of exponents rather then a single fractal dimension. A number of retinal images from the STARE database are analyzed, corresponding to both normal and pathological states of the retina. In all studied cases, a clearly multifractal behavior is observed, where capacity dimension is always found to be larger then the information dimension, which is in turn always larger then the correlation dimension, all the three being significantly lower then the diffusion limited aggregation (DLA) fractal dimension. We also observe a tendency of images corresponding to the pathological states of the retina to have lower generalized dimensions and a shifted spectrum range, in comparison with the normal cases

Fractal properties and small-scale structure of cosmic string networks

Abstract: We present results from a detailed numerical study of the small-scale and loop production properties of cosmic string networks, based on the largest and highest resolution string simulations to date. We investigate the nontrivial fractal properties of cosmic strings, in particular, the fractal dimension and renormalized string mass per unit length, and we also study velocity correlations. We demonstrate important differences between string networks in flat (Minkowski) spacetime and the two very similar expanding cases. For high resolution matter era network simulations, we provide strong evidence that small-scale structure has converged to 'scaling' on all dynamical length scales, without the need for other radiative damping mechanisms. We also discuss preliminary evidence that the dominant loop production size is also approaching scaling.

Natural rock joint roughness quantification through fractal techniques

Abstract: Accurate quantification of roughness is important in modeling hydro-mechanical behavior of rock joints. A highly refined variogram technique was used to investigate possible existence of anisotropy in natural rock joint roughness. Investigated natural rock joints showed randomly varying roughness anisotropy with the direction. A scale dependant fractal parameter, K v, seems to play a prominent role than the fractal dimension, D r1d, with respect to quantification of roughness of natural rock joints. Because the roughness varies randomly, it is impossible to predict the roughness variation of rock joint surfaces from measurements made in only two perpendicular directions on a particular sample. The parameter D r1d × K v seems to capture the overall roughness characteristics of natural rock joints well. The one-dimensional modified divider technique was extended to two dimensions to quantify the two-dimensional roughness of rock joints. The developed technique was validated by applying to a generated fractional Brownian surface with fractal dimension equal to 2.5. It was found that the calculated fractal parameters quantify the rock joint roughness well. A new technique is introduced to study the effect of scale on two-dimensional roughness variability and anisotropy. The roughness anisotropy and variability reduced with increasing scale.

Reduction of Physiological Stress Using Fractal Art and Architecture

Abstract: The author reviews visual perception studies showing that fractal patterns possess an aesthetic quality based on their visual complexity. Specifically, people display an aesthetic preference for patterns with mid-range fractal dimensions, irrespective of the method used to generate them. The author builds upon these studies by presenting preliminary research indicating that mid-range fractals also affect the observer's physiological condition. The potential for incorporating these fractals into art and architecture as a novel approach to reducing stress is also discussed.

Microstructure and fractal analysis of fat crystal networks

Abstract: This paper reviews the study of the morphology and physical properties of fat crystal networks. Various microscopical and rheological methods can be used to quantify the microstructure of fats, with the ultimate aim of relating structure to mechanical response. Even though a variety of physical models have been proposed to explain the relationship between the mechanical properties of fats and their microstructure, the fractal scaling model most closely describes the experimentally observed behavior. Mass fractal dimensions determined by microscopy and rheology can be used successfully to quantify the microstructure of fats since fractal dimension values are sensitive to the combined effects of crystal size, morphology, and the spatial distribution of mass within the fat crystal network. Methods used to determine the fractal dimension of a fat crystal network such as box counting, particle counting. Fourier transform, light scattering and oil migration are explained in detail here. The relationship between fractal dimensions determined by microscopy and rheology are discussed in light of the fact that different measures of the fractal dimension describe different microstructural features in a fat crystal network.

Electromagnetic fields on fractals

Abstract: Fractals are measurable metric sets with non-integer Hausdorff dimensions. If electric and magnetic fields are defined on fractal and do not exist outside of fractal in Euclidean space, then we can use the fractional generalization of the integral Maxwell equations. The fractional integrals are considered as approximations of integrals on fractals. We prove that fractal can be described as a specific medium.

LES modelling of an unconfined large-scale hydrogen-air deflagration ∗

Abstract: This paper describes the large eddy simulation modelling of unconfined large-scale explosions. The simulations are compared with the largest hydrogen–air deflagration experiment in a 20 m diameter hemispherical polyethylene shell in the open. Two combustion sub-models, one developed on the basis of the renormalization group (RNG) theory and another derived from the fractal theory, were applied. Both sub-models include a sub-grid scale model of the turbulence generated by flame front itself based on Karlovitz’s theory and the observation by Gostintsev et al on a critical distance for transition from laminar to self-similar flame propagation regime. The RNG sub-model employs Yakhot’s formula for turbulent premixed flame propagation velocity. The best fit flame propagation dynamics is obtained for the fractal sub-model with a fractal dimension D = 2.22. The fractal sub-model reproduces the experimentally observed flame acceleration during the whole duration of explosion, accurately simulating the negative phase of the pressure wave but overestimating by 50% the positive phase amplitude. The RNG sub-model is closer to the experiment in predicting the positive phase but under-predicts by 30% the negative phase amplitude. Both sub-models simulate experimental flame propagation up to 20 m and pressure dynamics up to 80 m with reasonable accuracy. (Some figures in this article are in colour only in the electronic version)

Technical factors in fractal analysis of periapical radiographs.

Abstract: Objectives: Fractal analysis quantifies complex geometric structures by generating a fractal dimension, which can measure trabecular bone density. The use of non-standardized radiographic techniques potentially limits the reliability of fractal analysis. The objective of this study was to determine how variations in radiographic technique affect fractal dimension. Methods: Periapical radiographs of maxillary incisors taken on eight dry human skulls at varying angulation, tube potential and impulse settings were subjected to Fourier transform fractal analysis. Results: A significant (p 0.05) on fractal dimension, with the standard deviation of the fractal dimension ranging from ±0.005 to ±0.062 at various radiographic machine settings. Conclusions:...

Improving accuracy and precision in estimating fractal dimension of animal movement paths.

Abstract: It is difficult to watch wild animals while they move, so often biologists analyse characteristics of animal movement paths. One common path characteristic used is tortuousity, measured using the fractal dimension (D). The typical method for estimating fractal D, the divider method, is biased and imprecise. The bias occurs because the path length is truncated. I present a method for minimising the truncation error. The imprecision occurs because sometimes the divider steps land inside the bends of curves, and sometimes they miss the curves. I present three methods for minimising this variation and test the methods with simulated correlated random walks. The traditional divider method significantly overestimates fractal D when paths are short and the range of spatial scales is narrow. The best method to overcome these problems consists of walking the dividers forwards and backwards along the path, and then estimating the path length remaining at the end of the last divider step.

Dirac operators and spectral triples for some fractal sets built on curves

Abstract: We construct spectral triples and, in particular, Dirac operators, for the algebra of continuous functions on certain compact metric spaces. The triples are countable sums of triples where each summand is based on a curve in the space. Several fractals, like a finitely summable infinite tree and the Sierpinski gasket, fit naturally within our framework. In these cases, we show that our spectral triples do describe the geodesic distance and the Minkowski dimension as well as, more generally, the complex fractal dimensions of the space. Furthermore, in the case of the Sierpinski gasket, the associated Dixmier-type trace coincides with the normalized Hausdorff measure of dimension $\log 3/ \log 2$.