Fractals in physical geography

1996 ◽  
Vol 20 (2) ◽  
pp. 178-191 ◽  
Author(s):  
Jay Gao ◽  
Zong-guo Xia
Keyword(s):  
Fractal Analysis ◽  
Fractal Dimensions ◽  
Great Depth ◽  
Physical Geography ◽  
Topographic Features ◽  
Straight Line ◽  
Mathematical Algorithms ◽  
Determination Methods ◽  
Landform Regions ◽  
Geomorphic Processes

Since the fractal concept was introduced to measuring coastline length over three decades ago, fractal analysis has been prolifically applied to many topographic studies. A number of mathematical algorithms are now available to determine the fractal dimension for both linear and areal features. These determination methods require one or more straight-line segments to fit the Richardson's plot. A close examination of the literature shows that not all topographic features are fractal at all scales studied. While the multifractal nature of some geographical phenomena has been explored in great depth, it is not completely understood why some terrains are better modelled with fractal geometry than others. Fractal analysis has been successfully used to measure and characterize irregular linear features such as coastlines and shorelines, to describe and characterize landforms, and to delineate landform regions statistically. Fractal analysis can also be used to produce terrain simulations with a known dimension against which hypotheses can be tested. These studies fail to link fractal dimensions to the underlying geomorphic processes. The failure stems from the fact that there is no one-to-one relationship between geomorphic processes and the landforms they shape.

Structure characterization of ground calcium carbonate flocs by fractal analysis and their effects on handsheet properties

TAPPI Journal ◽  
2013 ◽  
Vol 12 (3) ◽  
pp. 17-23 ◽  
Author(s):  
WANHEE IM ◽  
HAK LAE LEE ◽  
HYE JUNG YOUN ◽  
DONGIL SEO
Keyword(s):  
Fractal Analysis ◽  
Structural Characteristics ◽  
Fractal Dimensions ◽  
Strength Loss ◽  
Structure Characterization ◽  
Uniform Size ◽  
Cross Sectional ◽  
Floc Size ◽  
Mass Fractal ◽  
Handsheet Properties

Preflocculation of filler particles before their addition to pulp stock provides the most viable and practical solution to increase filler content while minimizing strength loss. The characteristics of filler flocs, such as floc size and structure, have a strong influence on preflocculation efficiency. The influence of flocculant systems on the structural characteristics of filler flocs was examined using a mass fractal analysis method. Mass fractal dimensions of filler flocs under high shear conditions were obtained using light diffraction spectroscopy for three different flocculants. A single polymer (C-PAM), a dual cationic polymer (p-DADMAC/C-PAM) and a C-PAM/micropolymer system were used as flocculants, and their effects on handsheet properties were investigated. The C-PAM/micropolymer system gave the greatest improvement in tensile index. The mass fractal analysis showed that this can be attributed to the formation of highly dense and spherical flocs by this flocculant. A cross-sectional analysis of the handsheets showed that filler flocs with more uniform size were formed when a C-PAM/micropolymer was used. The results suggest that a better understanding of the characteristics of preflocculated fillers and their influence on the properties of paper can be gained based on a fractal analysis.

Architectural Bias in Recurrent Neural Networks: Fractal Analysis

2003 ◽  
Vol 15 (8) ◽  
pp. 1931-1957 ◽  
Author(s):  
Peter Tiňo ◽  
Barbara Hammer
Keyword(s):  
Neural Networks ◽  
Fractal Analysis ◽  
Recurrent Neural Networks ◽  
Markov Models ◽  
Fractal Dimensions ◽  
Transition Diagram ◽  
Activation Patterns ◽  
Box Counting ◽  
Fractal Clusters ◽  
Finite State

We have recently shown that when initialized with “small” weights, recurrent neural networks (RNNs) with standard sigmoid-type activation functions are inherently biased toward Markov models; even prior to any training, RNN dynamics can be readily used to extract finite memory machines (Hammer & Tiňo, 2002; Tiňo, Čerňanský, &Beňušková, 2002a, 2002b). Following Christiansen and Chater (1999), we refer to this phenomenon as the architectural bias of RNNs. In this article, we extend our work on the architectural bias in RNNs by performing a rigorous fractal analysis of recurrent activation patterns. We assume the network is driven by sequences obtained by traversing an underlying finite-state transition diagram&a scenario that has been frequently considered in the past, for example, when studying RNN-based learning and implementation of regular grammars and finite-state transducers. We obtain lower and upper bounds on various types of fractal dimensions, such as box counting and Hausdorff dimensions. It turns out that not only can the recurrent activations inside RNNs with small initial weights be explored to build Markovian predictive models, but also the activations form fractal clusters, the dimension of which can be bounded by the scaled entropy of the underlying driving source. The scaling factors are fixed and are given by the RNN parameters.

scholarly journals Fractal Analysis of Morphological Parameters of the Structure Materials

2019 ◽  
Author(s):  
Константин Макаренко ◽  
Konstantin Makarenko ◽  
Александр Никитин ◽  
Alexander Nikitin
Keyword(s):  
Elementary Particle ◽  
Fractal Analysis ◽  
Ductile Iron ◽  
Crystallization Process ◽  
Morphological Characteristics ◽  
Fractal Dimensions ◽  
Structural Materials ◽  
Morphological Parameters ◽  
Graphite Cell ◽  
Mesh Method

It is proposed to use the methods of fractal analysis to determine the morphological characteristics of the structure of structural materials. The questions of fractal aggregation of particles in the process of crystallization of ductile iron are considered, an austenitic-graphite cell is used as an elementary particle. Based on the mesh method, images of the microstructure of ductile irons are analysed and conclusions are drawn about the similarity of the nature of the process of their crystallization and fractal aggregation of particles. Based on the calculated fractal dimensions, a theory is proposed to explain the features of the crystallization process of ductile irons.

scholarly journals PECULIARITIES OF THE RADIO SIGNALS AND HINDRANCES IN THE NAVIGATION SYSTEM OF THE REMOTE-PILOTED VEHICLES

2018 ◽  
Vol 8 (1) ◽  
pp. 40-43 ◽  
Author(s):  
Mykola Mykyjchuk ◽  
Volodymyr Markiv
Keyword(s):  
Control System ◽  
Fractal Analysis ◽  
Navigation System ◽  
Fractal Dimensions ◽  
Vehicle Control ◽  
Vital Role ◽  
Environment Research ◽  
Radio Signals

The article dwells upon the peculiarities of radio signals concerning the use of remote-piloted vehicles. It is highlighted that it is important take into consideration the fractal analysis of remote-piloted vehicles based on diverse fractal dimensions. The significance of remote-piloted vehicle control system investigation based on radio signals is presented. Also it is highlighted that there are many hindrances during the remote-piloted vehicle flight and it is important to take them into consideration and develop methods in order to omit them. Also the vital role of remote-piloted vehicles in different spheres of life, for example, in environment research is depicted.

scholarly journals Postnatal Development of the Retina in Rats Exposed to Hyperoxia: A Fractal Analysis

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Anne Claudia Ştefănuţ ◽  
Ştefan Ţălu ◽  
Viorel Miclăuş ◽  
Adriana Mureşan ◽  
Remus Moldovan ◽  
...  
Keyword(s):  
Fractal Analysis ◽  
Fractal Dimensions ◽  
Control Group ◽  
Newborn Rats ◽  
Vascular Abnormalities ◽  
Retinal Layers ◽  
Fractal Analyses ◽  
Group 10 ◽  
Box Counting Method ◽  
Experimental Group

Purpose. The aim of this study was to investigate and quantify changes in the newborn rats retinal layers during the hyperoxia (80% O2) exposure using fractal analysis. Materials and Methods. This study was conducted on two groups of 20 newborn rats: a control (normal) group (10 rats) and an experimental group (10 rats). The control group was composed of 10 newborn rats, which were placed at 12 hours after birth, in a pediatric incubator, together with their mother, in conditions of normoxia for 21 days. The experimental group consisted of 10 newborn rats, which were placed at 12 hours after birth, in a pediatric incubator with their mother, in conditions of normoxia for 7 days, then 7 days of hyperoxia (80% O2) for 22.5 hours/day, and then 7 days in conditions of normoxia. Slaughtering of the rats was performed on day 21 and the eye globes were harvested in order to perform histopathological examinations. The fractal analyses of the retinal digital images were performed using the fractal analysis software Image J, and the fractal dimensions were calculated using the standard box-counting method. Results. Microscopic examination revealed a normal development of the retina in the control group. In the experimental group, all the animals exposed to hyperoxia revealed both structural and vascular abnormalities on entire retina. Conclusions. The results showed that the fractal analysis is a valuable tool to quantify histoarchitectural changes in the newborn rats retinal layers during the hyperoxia (80% O2).

Cake porosity analysis using 1D–3D fractal dimensions in coagulation–microfiltration of NOM

2015 ◽  
Vol 71 (5) ◽  
pp. 740-746
Author(s):  
G. S. Raspati ◽  
T. O. Leiknes
Keyword(s):  
Natural Organic Matter ◽  
Fractal Analysis ◽  
Membrane Fouling ◽  
Fractal Dimensions ◽  
The Other ◽  
Transmembrane Pressure ◽  
Process Conditions ◽  
Iron Based ◽  
Cake Porosity ◽  
Porosity Analysis

Fouling during coagulation–ceramic microfiltration of natural organic matter was investigated. Two process configurations (inline coagulation (IC) and tank coagulation (TC)) and two process conditions (types of coagulants–aluminum-based PAX and iron-based PIX–and G-values) were studied. The rate of irreversible fouling corresponding to the increase of initial transmembrane pressure after backwash of IC-PAX was lowest followed by TC-PAX and TC-PIX, while the performance of IC-PIX was found worst. The 1D and 2D fractal analysis revealed that flocs from IC were morphologically different from those of TC, leading to different filtration characteristics. The 3D fractal analysis revealed two groups of morphologically similar flocs: one led to successful filtration experiments, whereas the other led to unsuccessful ones. Cake porosity was found dependent on the floc morphology. Thus, such an approach was found complementary with fouling analysis by means of a membrane fouling model and minimization of fouling phenomenon was achieved by combining the two approaches.

scholarly journals FRACTAL ANALYSIS OF TRABECULAR BONE: A STANDARDISED METHODOLOGY

2011 ◽  
Vol 19 (1) ◽  
pp. 45 ◽  
Author(s):  
Ian Parkinson ◽  
Nick Fazzalari
Keyword(s):  
Fractal Dimension ◽  
Trabecular Bone ◽  
Fractal Analysis ◽  
Fractal Dimensions ◽  
Box Counting ◽  
Cell Tissue ◽  
Image Analyser ◽  
Model Independent ◽  
Box Counting Method ◽  
Image Orientation

A standardised methodology for the fractal analysis of histological sections of trabecular bone has been established. A modified box counting method has been developed for use on a PC based image analyser (Quantimet 500MC, Leica Cambridge). The effect of image analyser settings, magnification, image orientation and threshold levels, was determined. Also, the range of scale over which trabecular bone is effectively fractal was determined and a method formulated to objectively calculate more than one fractal dimension from the modified Richardson plot. The results show that magnification, image orientation and threshold settings have little effect on the estimate of fractal dimension. Trabecular bone has a lower limit below which it is not fractal (λ<25 μm) and the upper limit is 4250 μm. There are three distinct fractal dimensions for trabecular bone (sectional fractals), with magnitudes greater than 1.0 and less than 2.0. It has been shown that trabecular bone is effectively fractal over a defined range of scale. Also, within this range, there is more than 1 fractal dimension, describing spatial structural entities. Fractal analysis is a model independent method for describing a complex multifaceted structure, which can be adapted for the study of other biological systems. This may be at the cell, tissue or organ level and compliments conventional histomorphometric and stereological techniques.

scholarly journals Exploring Geomorphic Processes and Martian Gale Crater Topography on Mars using CTX and HiRISE Express Image Dataset

2019 ◽  
Author(s):  
Pavan Kumar ◽  
Jiwan Singh Rawat ◽  
Sufia Rehman ◽  
Haroon Sajjad ◽  
Meenu Rani ◽  
...  
Keyword(s):  
Alluvial Fan ◽  
Martian Surface ◽  
Laser Altimeter ◽  
Gale Crater ◽  
Topographic Features ◽  
Geomorphic Features ◽  
Imaging Science ◽  
Resolution Imaging ◽  
Orbital Data ◽  
Geomorphic Processes

Abstract. Exploration of Martian surface and the crater deposition has recently attracted scientific community. We hypothesized the existence of momentous topographic features of different origin on the Martian surface. It was observed that Gale Crater has a thick sediment deposition mainly of fluvial and aeolian origin. This study has utilized spatially referenced topographic dataset Context Camera images acquired from Mars reconnaissance orbiter for exploring the geomorphic processes and topography of Gale Crater. A base map was prepared by mosaicking all imagesfor preparing geomorphologic map of the crater. Surface map of the topography of the Crater was prepared using Mars Orbiter Laser Altimeter (MOLA) data. HiRISE images were used to examine the identified geomorphic features. Results revealed that the presence of both fluvial and aeolian processes and their respective associated landforms in the Crater. Depositional landforms such as alluvial fan, inverted channel and mound and erosional landform namely canyons were distinctly identified on the image. Yardangs produced by erosional and ripple and dunes formed by the depositional work of wind were also identified in the Crater. The fluvial channel, inverted channels and fan shaped deposits further signifies ongoing aqueous activity on the Gale Crater. Sinuous ridges are the common features present on the floor of Crater. Rock particles, sand and silt in the crater were found to have been transported and deposited by flowing water on its floor. Presence of different sedimentary structures and valley revealed ancient sedimentary deposition due to water action. Significant ejecta morphologies were also identified on Martian surface. Our study confirms the previous studies that presence of H2O sub surface volatiles. Concentrations of volatiles have produced double layer ejecta morphology. The concept and findings of this study will escalate knowledge about the surface features on Gale Crater. However, more coherent investigation is needed for modelling and understanding the processes and landforms of the crater. Context Camera (CTX) and High-Resolution Imaging Science Experiment (HiRISE) images have proved useful for geomorphic and topographic mapping of the Mars planet. The dataset used in this study can be accessed on Mars Orbital Data Explorer (https://ode.rsl.wustl.edu/mars/indexproductsearch.aspx).

scholarly journals Fractal analysis of urban catchments and their representation in semi-distributed models: imperviousness and sewer system

2016 ◽  
Author(s):  
Auguste Gires ◽  
Ioulia Tchiguirinskaia ◽  
Daniel Schertzer ◽  
Susana Ochoa Rodriguez ◽  
Patrick Willems ◽  
...  
Keyword(s):  
Fractal Dimension ◽  
Fractal Analysis ◽  
Fractal Dimensions ◽  
Hydrological Models ◽  
Scale Invariant ◽  
Sewer System ◽  
Distributed Models ◽  
Urban Catchments ◽  
Distributed Hydrological Models ◽  
Dimension Characteristic

Abstract. Fractal analysis relies on scale invariance and the concept of fractal dimension enables to characterise and quantify the space filled by a geometrical set exhibiting complex and tortuous patterns. Fractal tools have been widely used in hydrology but seldom in the specific context of urban hydrology. In this paper fractal tools are used to analyse surface and sewer data from 10 urban or peri-urban catchments located in 5 European countries. The aim was to characterise urban catchment properties accounting for the complexity and inhomogeneity typical of urban water systems. Sewer system density and imperviousness (roads or buildings), represented in rasterized maps of 2 m × 2 m pixels, were analysed to quantify their fractal dimension, characteristic of scaling invariance. The results showed that both sewer density and imperviousness exhibit scale invariant features and can be characterized with the help of fractal dimensions ranging from 1.6 to 2, depending on the catchment. In a given area consistent results were found for the two geometrical features, yielding a robust and innovative way of quantifying the level of urbanization. The representation of imperviousness in operational semi-distributed hydrological models for these catchments was also investigated by computing fractal dimensions of the geometrical sets made up of the sub-catchments with coefficients of imperviousness greater than a range of thresholds. It enabled to quantify how well spatial structures of imperviousness were represented in the urban hydrological models.

Preliminary results of fractal analysis of the poligonal survey from cave: case study of Małołączniak area (Tatra Mts.)

2013 ◽  
Vol 2 (1) ◽  
pp. 95-100
Author(s):  
Jacek Szczygieł ◽  
Maciej J. Mendecki
Keyword(s):  
Fractal Analysis ◽  
Fractal Geometry ◽  
Close Relation ◽  
Cave System ◽  
Box Counting ◽  
Straight Line ◽  
Tatra Mts ◽  
Hydrological System ◽  
Geological Constraints

Abstract Traverses (polygons) from two caves have been tested: Wielka Śnieżna Cave System (2858 vectors) and Śnieżna Studnia cave (742). The box counting technique was applied to evaluate the fractal analysis of spatial orientation of traverses. The polygonal survey of Wielka Śnieżna Cave, Śnieżna Studnia and both caves merged together have a fractal geometry. It may be concluded that these caves are close to the full recognition of passages forming by a hydrological system and they could have close relation with geological structures. The usual explanation of fractal dimension D higher than 1 indicates that caves with such dimension fill more space than those with ideal dimension of 1.00 (for example a straight line), and the geological constraints limit the dimension to be lower than 2 (Verbovsek 2007). It may suggest that systems can be developped into more complicated passages in future. The fact that both caves merged together also have showed fractal geometry indicates that they are belonging to the same hydrological system. It was noticed that D-value of merged caves is slightly larger than individual cave. It can be explained by “occupying” more space in rock mass.

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