Fractal Maps Research Articles

Fractal Spherical Harmonics

This paper tackles the construction of fractal maps on the unit sphere. The functions defined are a generalization of the classical spherical harmonics. The methodology used involves an iterated function system and a linear and bounded operator of functions on the sphere. For a suitable choice of the coefficients of the system, one obtains classical maps on the sphere. The different values of the system parameters provide Bessel sequences, frames, and Riesz fractal bases for the Lebesgue space of the square integrable functions on the sphere. The Laplace series expansion is generalized to a sum in terms of the new fractal mappings.

A Map of the Systems Thinking Forest: 48 Systems Thinking Genres plus 256 System Effects as the Trees of that Forest

Main Point: This paper examines the hypothesis that one's ability to spot and handle various types of non-linear system effects within minds and among minds (society) constitutes a very general capability factor in analogy to G-factor in intelligence research, to which one adds domain specific less comprehensive capabilities. Much of what we require as “experience” and “maturity” in choosing leaders appears to encompass precisely the 48 genres and 256 system effects in them mapped herein. Question 1 (from practice) What system effects do top performers, leaders, designers recognize that lesser mortals do not? Question 2 (from theory) What types of system effects in total are there, as they appear in human affairs, and what bodies of knowledge in the world embody them? Question 3 (from interventions) What that is faster than “experience” and maturity” would train people to spot and handle system effects so all, not just top people, mastered them? Question 4 (from generalizing intervening) What would such general spread of capability in non-linear systems effects handling do to improve wanted outcomes: production, innovation, adaptation, design, sales, revenue? Method Half the 150 expert people interviewed in earlier research to produce the fractal map of 256 system effect here, were re-interviewed about what fields “taught” spotting and handling all or subsets of them. Result 48 genres of systems effects, that organize the 256 and somewhat go beyond them.

On Hölder maps of cubes

A map of metric spaces f: X → Y satisfying the inequality $$ \left| {f(x) - f(y)} \right| \leqslant C\left| {x - y} \right|^\alpha $$ for some C and α and all x, y ∈ X is called a Holder map with exponent α. V. I. Arnold posed the following problem: Does there exist a Holdermap from the square onto the cube with exponent 2/3? The firstmain theorem of this paper gives a general method for constructing Holdermaps of compact metric spaces. This construction yields, in particular, a dimension-raising map f: In → Im with Holder exponent arbitrarily close to m/n for m > n > 1 and a map I1 → Im with Holder exponent 1/m. The second main theorem states the nonexistence of a regular fractal map f: In → Im with Holder exponent n/m from the n-cube onto the m-cube for m < 2n.

지형 고도 맵으로부터 기울기와 거칠기 추출 방법

Recently, the interests in the development and application of unmaned robots are increasing in various fields including surveillance and reconnaissance, planet exploration, and disaster relief Unmaned robots are usually controlled from distance using radio communications but they should be equipped with an autonomous travelling function to cope with unexpected terrains and obstacles. This means that they should be able to evaluate terrain’s characteristics quantitatively using mounted sensors so as to traverse harsh natural terrains autonomously. For this purpose, this paper presents a method for extracting terrain information, that is, slope and roughness from elevation maps as a prior step of traversability analysis. Slope is extracted using the curve fitting based on the least squares method and roughness using three metrics and their weighted average. The effectiveness of the proposed method is verified on both a fractal map and the world model map of a real terrain.

Fractal Mapping of Pore and Solid Attributes

Soil structure depends on its genesis and consists of highly variable pore and solid networks. Several internal and external factors affect the attributes of these networks, with water being the most aggressive agent. In this study, we used selected fractal parameters (called descriptors) to quantify the basic topological attributes—compactness and connectedness—as well as lacunarity and roughness of porous materials, with special attention to sampling error and population variance dynamics. Four microhorizons were sampled from a 1.0‐ by 1.0‐ by 1.6‐m monolith during a long‐term drying period under controlled conditions. A comparative fractal analysis of selected multitemporal (11 mo) and multiscale (50×, 500×, 1000×, and 5000×) scanning electron microscopy (SEM) images was accomplished for the following layers with contrasting texture or structure: loam topsoil, pure basaltic compacted sand, and two amorphous lacustrine clays. The scale invariance (self‐similarity) of pore and solid networks was documented. A complex relationship was found between the descriptor mean values and corresponding variances. Simple power‐law relations were established between the standard deviation (and variance) and layer depth, as well as scale and sampling time. Pearson's r correlation criterion was applied to measure the degree of association between multiobjective fractal descriptors. Systematic and standardized fractal analysis of multiscale images is recommended for the statistical quantitative description of complex and dynamic pore and solid networks.

Fractal dimension of metallic fracture surface

In this study, a complete method of determination of the fractal dimension for fracture surfaces of ferrous alloys has been proposed. This dimension is determined for the vertical profile obtained by the profile technique cross-section. The image of the profile, seen through the microscope coupled with a camera, is recorded in a computer, where numerical processing is performed. For calculation of the same fractal dimension, the fd3 program has been used, which is available through the Internet. The essential element of the method is optimisation concerning microscopic magnification (scale of a length), resolution of the recorded image and selection of the grey level threshold at binarization. The tests for the stability of discretization, which enable minimization of the error of the measurement, have also been carried out. These tests consist in checking the difference in fractal dimensions for the same profile obtained in two different methods of contouring as well as the difference between capacitive, informative and correlative dimensions. In both cases, too big difference suggests that the determined dimension is not reliable. This method allows determination of the fractal dimension with an absolute accuracy of 0.05 in non-dimensional units. The method has been employed in many studies. In this paper the following tests have been presented: a “fractal map” of the fracture surface was made, an influence of the mechanical notch radius in a compact specimen on the fractal dimension of the fracture surface, an influence of the distortion rate on the fractal dimension, an effect of fatigue crack propagation rate on the fractal dimension and influence of the stress-intensity factor on the fractal dimension of the fracture surface. The following materials were examined: Armco iron, P355N steel and 41Cr4 steel in different states after the heat treatment. The measurements have been made for the specimens of the compact type. There was considered an influence of location of the place of measurement on the fractal dimension being determined. The dimension was determined on the profiles lying longwise and crosswise the crack propagation direction. It has been found that the fractal dimension of the fracture surface does not depend on a place of measurement. This suggests, among other things, that a distinction between the places, which were created under conditions of the plane stress, and the places, which were created under conditions of the plane strain state, cannot be made with the help of the fractal dimension. When testing an influence of the radius of the mechanical tip notch on the fractal dimension of a fracture surface, this dimension was determined in the places located at different distances from the tip of the mechanical notch. With respect to the radii up to 1.0 mm, no significant differences in fractal dimensions have been found. The fractal dimensions of the fracture surface for all examined materials were practically the same and they ranged from 2.02 to 2.10. However in some ranges of da/dN rate the dimension was changing inversely proportional to da/dN. Obtained results confirm that fractal dimension do not depend on the investigated material.

Surface Laplacian and fractal brain mapping

The electric potential recorded on the brain cortex results from the generation of sources of current. However, it depends on the reference electrode chosen and devices used. We propose here explicit formulae to compute the fractal dimension of experimental recordings by means of fractal interpolation. Another way of avoiding the reference dependence is to compute the scalp current density as surface Laplacian of the electroencephalogram. In practice, the value of the potential is known in a given number of electrodes, but the information about the density is not explicit. By using different interpolation methods by multivariate splines, formulae for the approximation of the density are obtained. These procedures are tested on a theoretical model of brain electrical potential given by the current of a single dipole inside the skull. By computing relative errors for different values of the eccentricity of the dipole it can be observed that errors decrease with increasing dipole eccentricities. The smallest errors are computed in the case of a pseudocubic spline. This method is used to perform two and three-dimensional brain mapping representations and to locate epileptic peaks.

Dust and gas distribution in molecular clouds: an observational approach

The interstellar medium (ISM), gas and dust, appears to be arranged in clouds, whose dimensions, masses and densities span a large range of scales: from giant molecular clouds to small isolated globules. The structure of these objects show a high degree of complexity appearing, in the range of the observed scales, as a non-homogeneous ("clumpy") distribution of matter. The arrangement of the ISM is clearly relevant for the study of the fragmentation of the clouds and then of the star formation processes. To quantify observationally the ISM structure, many methods have been developed and our study is focused on some of them, exploiting multiwavelength observations of IS objects. The investigations presented here have been carried out by considering both the dust absorption (in optical and near IR wavelengths) and the gas emission (in the submm-radio spectral range). We present the maps obtained from the reduction of raw data and a first tentative analysis by means of methods as the structure function, the autocorrelation, and the Δ-variance. These are appropriate tools to highlight the complex structure of the ISM with reference to the paradigm given by the supersonic turbulence. Three observational cases are briefly discussed. In order to analyse the structure of objects characterized by different sizes, we applied the above-mentioned algorithms to the extinction map of the dark globule CB 107 and to the CO(J = 1-0) integrated intensity map of Vela Molecular Ridge, D Cloud. Finally we compare the results obtained with synthetic fractal maps known as "fractional Brownian motion" fBm images.

Simulating dispersal of reintroduced species within heterogeneous landscapes

This paper describes the development and application of a spatially explicit, individual based model of animal dispersal (J-walk) to determine the relative effects of landscape heterogeneity, prey availability, predation risk, and the energy requirements and behavior of dispersing organisms on dispersal success. Significant unknowns exist for the simulation of complex movement behavior within heterogeneous landscapes. Therefore, initial simulations with J-walk examined the relative effect of landscape patterns and species-specific characteristics on dispersal success. Differences in landscape pattern were simulated by random generation of fractal maps with average available energy (i.e. prey) and predation risk expressed as a function of habitat type. Variation in species-specific patterns were then simulated by a series of scenarios that varied the response of dispersing individuals to habitat heterogeneity, including: habitat selection to maximize energy intake, habitat selection to minimize predation risk, or habitat selection contingent on energy reserves. Results showed that significant shifts in dispersal could be related to (1) the unique spatial arrangement of habitat within each map, (2) changes in relative prey abundance, and (3) variation in the relationship between energy availability and predation risk. Hypothetical management scenarios were used to identify critical data needed to assure the persistence of reintroduced populations of American martens ( Martes americana).

Generation of volumetric escape time fractals

Aesthetically appealing two-dimensional escape time fractals are often based on complex-function transformations. The same methods of analysis and rendering can be extended to create fractal solids. Methods for generation of these objects very often use quaternion representations for transforming points in three or four dimensions. In this tutorial and artistic statement, we describe a simple yet effective approach for visualizing escape times of general quadratic fractal maps in the third dimension. The approach allows us to automatically generate diverse and beautiful volumetric shapes in real time on currently available personal computers.

Equidimensional Fractal Maps for Indirect Estimation of Deformation Damage in Nonuniform Aircraft Alloys

This article describes the surface relief formation on Al single-crystal plates (sensors) with orientation {100}〈001〉 that were rigidly attached to weld specimens of aircraft alloy 2024 T351 during fatigue loading. Taking into account the nonuniform structure of the weld alloy, we located sensors in different zones of the weld specimens to receive information about the sites of strain localization and probability of destruction. The qualitative and quantitative information about the sensor relief was extracted by the online method automated on the basis of a stereomicroscope with a charge-coupled device (CCD) camera attached to a personal computer by a video adapter and frame-grabber card. The quantitative and qualitative relief parameters are shown to correlate with the deformation prehistory of the weld specimens. Panoramic views were created and used for fractal analysis of deformation relief. Spatial distributions of information fractal dimension on the sensor surface for different numbers of cycles were plotted in the shape of contour lines with equal values (equidimensional maps). Equidimensional maps for sufficiently large sensors allow us to find strain localization sites in specimens and monitor their evolution in an online regimen.

Hybrid image coding based on partial fractal mapping

The fractal image compression technique models a natural image using a contractive mapping called fractal mapping in the image space. In this paper, we demonstrate that the fractal image coding algorithm is compatible with other image coding methods. In other words, we can encode only part of the image using fractal technique and model the remaining part using other algorithms. According to such an idea, a new mapping in the image space called partial fractal mapping is proposed. Furthermore, a general framework of fractal-based hybrid image coding encoding/decoding systems is presented. The framework provides us with much flexibility for real implementations. Many different hybrid image coding schemes can be derived from it. Finally, a new hybrid image coding scheme is proposed where non-fractal coded regions are used to help the encoding of fractal coded regions. Experiments show that the proposed system performs better than the quadtree-based fractal image coding algorithm and the JPEG image compression standard at high compression ratios larger than 30.

Research on some theoretical problems of map data handling using fractal approach

Some theoretical problems of fractal geographical map data handling are discussed and some new methods about fractal dimension introducing, developing, comparing and estimating are proposed in this paper.

An object-oriented coder using block-based motion vectors and residual image compensation

This paper proposes a two-stage global motion parameter estimation method using block-based motion vectors and a model failure (MF) object compensation algorithm by object-oriented fractal mapping of the residual image for an object-oriented coder. In the first stage of the two-stage motion parameter estimation algorithm, coarse motion parameters are estimated by fitting block-based motion vectors computed hierarchically to a six-parameter model, and in the second stage, the estimated motion parameters are refined by the gradient method using an image reconstructed by motion parameters detected in the first stage. The local prediction error by the six-parameter method is locally reduced by blockwise motion parameter correction using the residual image. Finally, the MF object is compensated by object-oriented fractal mapping of the previous residual image into the current one, in which geometric affine mapping is followed by the massic transformation. For MF object compensation, the selection between motion parameter compensation and fractal mapping is achieved by the validity test. Computer simulation results show that the proposed method gives better performance than the conventional ones in terms of the peak signal-to-noise ratio (PSNR) and compression ratio (CR).

An eigenmode solution algorithm based on high-order power iteration with fractally ordered shifts

This paper presents a technique which allows the high-order power method with shifts to solve reliably for mid-spectrum eigenmodes. Normally, the method fails at high orders because undesirable modes can never be reduced below round-off error levels, even though analytically those modes have been reduced to much lower levels. This paper describes a technique of reordering the shifts which drastically reduces the round-off error effect. The ordering technique is based on a fractal mapping of the eigenmode spectrum which is self-similar at all scales along the spectrum. The technique is rather stunning in its capability. For example, an eigenmode problem of 10 000 unknowns based on the finite difference electromagnetic wave equations can be solved for the first 50 modes in a few hours on a PC, using single precision arithmetic. Furthermore, any given interval of the spectrum can be searched for an eigenmode without the need to find lower modes. And because the fractal mapping technique treats all regions and all scales of the spectrum identically, mid-spectrum modes pose no more difficult a problem than do the lowest or highest modes.

Landscape Connectivity and Population Distributions in Heterogeneous Environments

Landscape connectivity refers to the functional relationship among habitat patches, owing to the spatial contagion of habitat and the movement responses of organisms to landscape structure. Heterogeneous landscapes provide a particular challenge for modelling population-level responses to habitat fragmentation, because individuals may be utilizing multiple habitats to varying degrees across the landscape. We apply neutral landscape models to understand how species' habitat affinities interacted with landscape structure (i.e., habitat abundance, distribution, and quality as measured by carrying capacity) to affect the redistribution of individuals. Two types of neutral models are presented: random maps, in which the distribution of habitat is spatially independent, and fractal maps, in which habitat exhibits an intermediate level of spatial dependence. The neutral landscapes comprised varying proportions of three habitat types, for which species exhibited a preference gradient (high, medium, low). We performed a series of simulation experiments as a factorial design of parameter states to tease apart the underlying factors responsible for population distributional patterns (random vs clumped) in spatially complex mosaics. Landscape connectivity is a threshold phenomenon, in which even a minimal loss of habitat near the critical threshold (p c ) is likely to disconnect the landscape, and which may have consequences for population distributions. The exact value of p c depends upon the spatial arrangement of habitat; fractal landscapes exhibited connectivity across a greater range of habitat abundance (p) than random maps (fractal p c = 0.29-0.50. random p c = 0.59). Although the spatial arrangement of habitat (random vs fractal) was the most important determinant of population distributional patterns, different landscape factors were important in structuring populations in the two types of maps. The relative abundance of habitat had the greatest effect on populations in random landscapes, whereas scale-dependent patterns were evident in fractal landscapes. At fine scales, population dispersion was determined by habitat abundance in both random and fractal maps, although populations were more aggregated (as measured by Morisita's Index, l m ) at this scale in random landscapes. But at coarse scales on fractal maps, population distribution was primarily influenced by species' habitat affinities. Assessment of the independent effects of habitat affinity and habitat carrying capacity on population distributions revealed that the differential interaction of species with landscape structure (i.e., different residence probabilities in each habitat type) was the primary determinant of distributional patterns. Neutral landscape models thus provide a useful tool for determining the relative importance of various components of landscape structure that affect population distributions.

Still image coding based on vector quantization and fractal approximation

In this paper, we propose a coding algorithm for still images using vector quantization (VQ) and fractal approximation, in which low-frequency components of an input image are approximated by VQ, and its residual is coded by fractal mapping. The conventional fractal coding algorithms indirectly used the gray patterns of an original image with contraction mapping, whereas the proposed fractal coding method employs an approximated and then decimated image as a domain pool and uses its gray patterns. Thus, the proposed algorithm utilizes fractal approximation without the constraint of contraction mapping. For approximation of an original image, we employ the discrete cosine transform (DCT) rather than conventional polynomial-based transforms. In addition, for variable blocksize segmentation, we use the fractal dimension of a block that represents the roughness of the gray surface of a region. Computer simulations with several test images show that the proposed method shows better performance than the conventional fractal coding methods for encoding still pictures.

A note on quantum mechanics, diffusional interference and informions

Following some ideas initially advanced by Eddington, we consider first the mathematical possibility of ‘self’ interference in diffusional processes. Subsequently, we consider the results of computer simulation of the two-slit experiment using iterated fractal maps and finally some real experiments suggested by Walter and Scully using quantum eraser. In the light of this discussion it is concluded that the hypothetical particles, cantorions, which are supposed somehow to carry information in a cantorian micro space-time may be ascribed real existence. These so called informions could then be drawn upon to explain the new experimental insights besides or as an alternative to Higgs bosons.

Comment on “Fractal mapping of digitized images: Application to the topography of arizona and comparison with synthetic images” by J. Huang and D. L. Turcotte

Comment on “Fractal mapping of digitized images: Application to the topography of arizona and comparison with synthetic images” by J. Huang and D. L. Turcotte

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