Showing papers on "Fractal dimension published in 2011"

Curves and Fractal Dimension

Abstract: Written for mathematicians, engineers, researchers in experimental science, and anyone interested in fractals, this book presents the fundamentals of curve analysis with a new and clear introduction to fractal dimension. It explains the geometrical and analytical properties of trajectories, aggregate contours, geographical coastlines, profiles of rough surfaces, and other curves of finite and fractal length. The approach is through precise definitions from which properties are deduced and applications and computational methods are derived. Written without the traditional heavy symbolism of mathematics texts, this book requires two years of calculus as a prerequisite to understanding. This text also contains material appropriate for graduate coursework in curve analysis and/or fractal dimension.

The Mobility of Fractal Aggregates: A Review

Abstract: A review of the experimental and theoretical literature describing the mobility of fractal aggregates over the previous three decades is presented. Aggregates are those formed via both diffusion and reaction limited cluster-cluster aggregation processes, DLCA and RLCA, which form aggregates with fractal dimensions and prefactors of ca. 1.78 and 1.3 and 2.1 and 0.94, respectively. Emphasis is placed on DLCA aggregates. The entire Knudsen number range from continuum to free molecular is reviewed. The review finds a simple and general consensus description of mobility for the entire range.

Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions

Abstract: Number theory and fractal geometry are combined in this study of the vibrations of fractal strings. The book centres around a notion of complex dimension, originally developed for the proof of the Prime Number Theorem, and extended here to apply to the zeta functions associated with fractals.

Computation of fractal dimension of rock pores based on gray CT images

Abstract: The characterization of pore structure in rocks is relevant in determining their various mechanical behaviors. Digital image processing methods integrated with fractal theory were applied to analyze images of rock slices obtained from industry CT, elucidating the characteristics of rock pore structure and the relationship between porosity and fractal dimensions. The gray values of pixels in CT images of rocks provide comprehensive results with respect to the attenuation coefficients of various materials in corresponding rock elements, and these values also reflect the effect of rock porosity at various scales. A segmentation threshold can be determined by inverse analysis based on the pore ratios that are measured experimentally, and subsequently binary images of rock pores can be obtained to study their topological structures. The fractal dimension of rock pore structure increases with an increase in rock pore ratio, and fractal dimensions might differ even if pore ratios are the same. The more complex the structure of a rock, the larger the fractal dimension becomes. The experimental studies have validated that fractal dimension calculated directly from gray CT images of rocks can give an effective complementary parameter to use alongside pore ratios and they can suitably represent the fractal characteristics of rock pores.

Multiparticle sintering dynamics: from fractal-like aggregates to compact structures.

Abstract: Multiparticle sintering is encountered in almost all high temperature processes for material synthesis (titania, silica, and nickel) and energy generation (e.g., fly ash formation) resulting in aggregates of primary particles (hard- or sinter-bonded agglomerates). This mechanism of particle growth is investigated quantitatively by mass and energy balances during viscous sintering of amorphous aerosol materials (e.g., SiO(2) and polymers) that typically have a distribution of sizes and complex morphology. This model is validated at limited cases of sintering between two (equally or unequally sized) particles, and chains of particles. The evolution of morphology, surface area and radii of gyration of multiparticle aggregates are elucidated for various sizes and initial fractal dimension. For each of these structures that had been generated by diffusion limited (DLA), cluster-cluster (DLCA), and ballistic particle-cluster agglomeration (BPCA) the surface area evolution is monitored and found to scale differently than that of the radius of gyration (moment of inertia). Expressions are proposed for the evolution of fractal dimension and the surface area of aggregates undergoing viscous sintering. These expressions are important in design of aerosol processes with population balance equations (PBE) and/or fluid dynamic simulations for material synthesis or minimization and even suppression of particle formation.

Fractal Dimension of Color Fractal Images

Abstract: Fractal dimension is a very useful metric for the analysis of the images with self-similar content, such as textures. For its computation there exist several approaches, the probabilistic algorithm being accepted as the most elegant approach. However, all the existing methods are defined for 1-D signals or binary images, with extension to grayscale images. Our purpose is to propose a color version of the probabilistic algorithm for the computation of the fractal dimension. To validate this new approach, we also propose an extension of the existing algorithm for the generation of probabilistic fractals, in order to obtain color fractal images. Then we show the results of our experiments and conclude this paper.

Application of percolation theory to microtomography of structured media: percolation threshold, critical exponents, and upscaling.

Abstract: Percolation theory provides a tool for linking microstructure and macroscopic material properties. In this paper, percolation theory is applied to the analysis of microtomographic images for the purpose of deriving scaling laws for upscaling of properties. We have tested the acquisition of quantities such as percolation threshold, crossover length, fractal dimension, and critical exponent of correlation length from microtomography. By inflating or deflating the target phase and percolation analysis, we can get a critical model and an estimation of the percolation threshold. The crossover length is determined from the critical model by numerical simulation. The fractal dimension can be obtained either from the critical model or from the relative size distribution of clusters. Local probabilities of percolation are used to extract the critical exponent of the correlation length. For near-isotropic samples such as sandstone and bread, the approach works very well. For strongly anisotropic samples, such as highly deformed rock (mylonite) and a tree branch, the percolation threshold and fractal dimension can be assessed with accuracy. However, the uncertainty of the correlation length makes it difficult to accurately extract its critical exponents. Therefore, this aspect of percolation theory cannot be reliably used for upscaling properties of strongly anisotropic media. Other methods of upscaling have to be used for such media.

Robust Methodology for Fractal Analysis of the Retinal Vasculature

Abstract: We have developed a robust method to perform retinal vascular fractal analysis from digital retina images. The technique preprocesses the green channel retina images with Gabor wavelet transforms to enhance the retinal images. Fourier Fractal dimension is computed on these preprocessed images and does not require any segmentation of the vessels. This novel technique requires human input only at a single step; the allocation of the optic disk center. We have tested this technique on 380 retina images from healthy individuals aged 50+ years, randomly selected from the Blue Mountains Eye Study population. To assess its reliability in assessing retinal vascular fractals from different allocation of optic center, we performed pair-wise Pearson correlation between the fractal dimension estimates with 100 simulated region of interest for each of the 380 images. There was Gaussian distribution variation in the optic center allocation in each simulation. The resulting mean correlation coefficient (standard deviation) was 0.93 (0.005). The repeatability of this method was found to be better than the earlier box-counting method. Using this method to assess retinal vascular fractals, we have also confirmed a reduction in the retinal vasculature complexity with aging, consistent with observations from other human organ systems.

The relationship between the fractal dimension and shape properties of particles

Abstract: Due to their irregularity, the shape of particles is not accurately described by Euclidian geometry. However, fractal geometry uses the concept of fractal dimension, DR, as a way to describe the shape of particles. In this study, the fractal dimensions and shape properties of particles were determined using image analysis. Exponential relationships between the fractal dimension and roundness, sphericity, angularity, convexity were described. A set of empirical correlations were also presented which clearly demonstrated the link between fractal dimension and shape properties of particles. Additionally, a new classification chart proposed for use in describing and comparing particle shape and fractal dimension.

HAIRIS: A Method for Automatic Image Registration Through Histogram-Based Image Segmentation

Abstract: Automatic image registration is still an actual challenge in several fields. Although several methods for automatic image registration have been proposed in the last few years, it is still far from a broad use in several applications, such as in remote sensing. In this paper, a method for automatic image registration through histogram-based image segmentation (HAIRIS) is proposed. This new approach mainly consists in combining several segmentations of the pair of images to be registered, according to a relaxation parameter on the histogram modes delineation (which itself is a new approach), followed by a consistent characterization of the extracted objects-through the objects area, ratio between the axis of the adjust ellipse, perimeter and fractal dimension-and a robust statistical based procedure for objects matching. The application of the proposed methodology is illustrated to simulated rotation and translation. The first dataset consists in a photograph and a rotated and shifted version of the same photograph, with different levels of added noise. It was also applied to a pair of satellite images with different spectral content and simulated translation, and to real remote sensing examples comprising different viewing angles, different acquisition dates and different sensors. An accuracy below 1° for rotation and at the subpixel level for translation were obtained, for the most part of the considered situations. HAIRIS allows for the registration of pairs of images (multitemporal and multisensor) with differences in rotation and translation, with small differences in the spectral content, leading to a subpixel accuracy.

Horizontal visibility graphs transformed from fractional Brownian motions: Topological properties versus the Hurst index

Abstract: Nonlinear time series analysis aims at understanding the dynamics of stochastic or chaotic processes. In recent years, quite a few methods have been proposed to transform a single time series to a complex network so that the dynamics of the process can be understood by investigating the topological properties of the network. We study the topological properties of horizontal visibility graphs constructed from fractional Brownian motions with different Hurst indexes H ∈ ( 0 , 1 ) . Special attention has been paid to the impact of the Hurst index on topological properties. It is found that the clustering coefficient C decreases when H increases. We also found that the mean length L of the shortest paths increases exponentially with H for fixed length N of the original time series. In addition, L increases linearly with respect to N when H is close to 1 and in a logarithmic form when H is close to 0. Although the occurrence of different motifs changes with H , the motif rank pattern remains unchanged for different H . Adopting the node-covering box-counting method, the horizontal visibility graphs are found to be fractals and the fractal dimension d B decreases with H . Furthermore, the Pearson coefficients of the networks are positive and the degree–degree correlations increase with degree, which indicate that the horizontal visibility graphs are assortative. With the increase of H , the Pearson coefficient decreases first and then increases, in which the turning point is around H = 0.6 . The presence of both fractality and assortativity in the horizontal visibility graphs converted from fractional Brownian motions is different from many cases where fractal networks are usually disassortative.

A quantitative approach to scar analysis.

Abstract: Analysis of collagen architecture is essential to wound healing research. However, to date no consistent methodologies exist for quantitatively assessing dermal collagen architecture in scars. In this study, we developed a standardized approach for quantitative analysis of scar collagen morphology by confocal microscopy using fractal dimension and lacunarity analysis. Full-thickness wounds were created on adult mice, closed by primary intention, and harvested at 14 days after wounding for morphometrics and standard Fourier transform-based scar analysis as well as fractal dimension and lacunarity analysis. In addition, transmission electron microscopy was used to evaluate collagen ultrastructure. We demonstrated that fractal dimension and lacunarity analysis were superior to Fourier transform analysis in discriminating scar versus unwounded tissue in a wild-type mouse model. To fully test the robustness of this scar analysis approach, a fibromodulin-null mouse model that heals with increased scar was also used. Fractal dimension and lacunarity analysis effectively discriminated unwounded fibromodulin-null versus wild-type skin as well as healing fibromodulin-null versus wild-type wounds, whereas Fourier transform analysis failed to do so. Furthermore, fractal dimension and lacunarity data also correlated well with transmission electron microscopy collagen ultrastructure analysis, adding to their validity. These results demonstrate that fractal dimension and lacunarity are more sensitive than Fourier transform analysis for quantification of scar morphology.

Fractal Characteristics of Coal Pores Based on Classic Geometry and Thermodynamics Models

Abstract: To better understand the characteristics of coal pores and their influence on coal reservoirs, coal pores in eight main coalfields of North China were analyzed by mercury porosimetry and scanning electron microscopy (SEM). Fractal characteristics of coal pores (size distribution and structure) were researched using two fractal models: classic geometry and thermodynamics. These two models establish the relationship between fractal dimensions and coal pores characteristics. New results include: (1) SEM imaging and fractal analysis show that coal reservoirs generally have very high heterogeneity; (2) coal pore structures have fractal characteristics and fractal dimensions characteristic of pore structures are controlled by the composition (e.g., ash, moisture, volatile component) and pore parameters (e.g., pore diameter, micro pores content) of coals; (3) the fractal dimensions (D1 and D2) of coal pores have good correlations with the heterogeneity of coal pore structures. Larger fractal dimensions correlate to higher heterogeneity of pore structures. The fractal dimensions (D1 and D2) have strong negative linear correlations with the sorted coefficient of coals (R2=0.719 and 0.639, respectively) that shows the heterogeneity of coal pores; (4) fractal dimension D1 and petrologic permeability of coals have a strong negative exponential correlation (R2=0.82). However, fractal dimension D2 and petrologic permeability of coals have no obvious correlation; and (5) the model of classic geometry is more accurate for fractal characterization of coal pores in coal reservoirs than that of thermodynamics by optimization.

Black carbon fractal morphology and short-wave radiative impact: a modelling study

Abstract: . We investigate the impact of the morphological properties of freshly emitted black carbon aerosols on optical properties and on radiative forcing. To this end, we model the optical properties of fractal black carbon aggregates by use of numerically exact solutions to Maxwell's equations within a spectral range from the UVC to the mid-IR. The results are coupled to radiative transfer computations, in which we consider six realistic case studies representing different atmospheric pollution conditions and surface albedos. The spectrally integrated radiative impacts of black carbon are compared for two different fractal morphologies, which brace the range of recently reported experimental observations of black carbon fractal structures. We also gauge our results by performing corresponding calculations based on the homogeneous sphere approximation, which is commonly employed in climate models. We find that at top of atmosphere the aggregate models yield radiative impacts that can be as much as 2 times higher than those based on the homogeneous sphere approximation. An aggregate model with a low fractal dimension can predict a radiative impact that is higher than that obtained with a high fractal dimension by a factor ranging between 1.1–1.6. Although the lower end of this scale seems like a rather small effect, a closer analysis reveals that the single scattering optical properties of more compact and more lacy aggregates differ considerably. In radiative flux computations there can be a partial cancellation due to the opposing effects of different error sources. However, this cancellation effect can strongly depend on atmospheric conditions and is therefore quite unpredictable. We conclude that the fractal morphology of black carbon aerosols and their fractal parameters can have a profound impact on their radiative forcing effect, and that the use of the homogeneous sphere model introduces unacceptably high biases in radiative impact studies. We emphasise that there are other potentially important morphological features that have not been addressed in the present study, such as sintering and coating of freshly emitted black carbon by films of organic material. Finally, we found that the spectral variation of the absorption cross section of black carbon significantly deviates from a simple 1/λ scaling law. We therefore discourage the use of single-wavelength absorption measurements in conjunction with a 1/λ scaling relation in broadband radiative forcing simulations of black carbon.

Random soups, carpets and fractal dimensions

Abstract: We study some properties of a class of random connected planar fractal sets induced by a Poissonian scale-invariant and translation-invariant point process. Using the second-moment method, we show that their Hausdorff dimensions are deterministic and equal to their expectation dimension. We also estimate their low-intensity limiting behavior. This applies in particular to the “conformal loop ensembles” defined via Poissonian clouds of Brownian loops for which the expectation dimension has been computed by Schramm, Sheffield and Wilson.

Waves in Fractal Media

Abstract: The term fractal was coined by Benoit Mandelbrot to denote an object that is broken or fractured in space or time. Fractals provide appropriate models for many media for some finite range of length scales with lower and upper cutoffs. Fractal geometric structures with cutoffs are called pre-fractals. By fractal media, we mean media with pre-fractal geometric structures. The basis of this study is the recently formulated extension of continuum thermomechanics to such media. The continuum theory is based on dimensional regularization, in which we employ fractional integrals to state global balance laws. The global forms of governing equations are cast in forms involving conventional (integer-order) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order. Using Hamilton’s principle, we derive the equations of motion of a fractal elastic solid under finite strains. Next, we consider one-dimensional models and obtain equations governing nonlinear waves in such a solid. Finally, we study shock fronts in linear viscoelastic solids under small strains. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions and reduce to conventional forms for continuous media with Euclidean geometries upon setting the dimensions to integers.

Deterministic fractals: extracting additional information from small-angle scattering data.

Abstract: The small-angle scattering curves of deterministic mass fractals are studied and analyzed in momentum space. In the fractal region, the curve $I(q){q}^{D}$ is found to be log-periodic with good accuracy, and the period is equal to the scaling factor of the fractal. Here, $D$ and $I(q)$ are the fractal dimension and the scattering intensity, respectively. The number of periods of this curve coincides with the number of fractal iterations. We show that the log-periodicity of $I(q){q}^{D}$ in the momentum space is related to the log-periodicity of the quantity $g(r){r}^{3\ensuremath{-}D}$ in the real space, where $g(r)$ is the pair distribution function. The minima and maxima positions of the scattering intensity are estimated explicitly by relating them to the pair distance distribution in real space. It is shown that the minima and maxima are damped with increasing polydispersity of the fractal sets; however, they remain quite pronounced even at sufficiently large values of polydispersity. A generalized self-similar Vicsek fractal with controllable fractal dimension is introduced, and its scattering properties are studied to illustrate the above findings. In contrast with the usual methods, the present analysis allows us to obtain not only the fractal dimension and the edges of the fractal region, but also the fractal iteration number, the scaling factor, and the number of structural units from which the fractal is composed.