A Unified Approach to Fractal Dimensions

2011 ◽  
pp. 304-325 ◽  
Author(s):  
Witold Kinsner
Keyword(s):  
Fractal Dimension ◽  
Fractal Dimensions ◽  
Unified Approach ◽  
Feature Maps ◽  
Progressive Development ◽  
Practical Algorithms ◽  
Diffusion Limited ◽  
Radio Transmitters ◽  
The Individual ◽  
New Applications

Many scientific chapters treat the diversity of fractal dimensions as mere variations on either the same theme or a single definition. There is a need for a unified approach to fractal dimensions for there are fundamental differences between their definitions. This chapter presents a new description of three essential classes of fractal dimensions based on: (a) morphology, (b) entropy, and (c) transforms, all unified through the generalized-entropy-based Rényi fractal dimension spectrum. It discusses practical algorithms for computing 15 different fractal dimensions representing the classes. Although the individual dimensions have already been described in the literature, the unified approach presented in this chapter is unique in terms of its progressive development of the fractal dimension concept, similarity in the definitions and expressions, analysis of the relation between the dimensions, and their taxonomy. As a result, a number of new observations have been made, and new applications discovered. Of particular interest are behavioral processes (such as dishabituation), irreversible and birth-death growth phenomena (e.g., diffusion-limited aggregates, DLAs, dielectric discharges, and cellular automata), as well as dynamical nonstationary transient processes (such as speech and transients in radio transmitters), multifractal optimization of image compression using learned vector quantization with Kohonen’s self-organizing feature maps (SOFMs), and multifractal-based signal denoising.

scholarly journals Projection of compact fractal sets: application to diffusion-limited and cluster-cluster aggregates

2008 ◽  
Vol 15 (4) ◽  
pp. 695-699 ◽  
Author(s):  
F. Maggi
Keyword(s):  
Fractal Dimension ◽  
Three Dimensional ◽  
Fractal Dimensions ◽  
Cohesive Sediment ◽  
Mathematical Approach ◽  
Two Dimensional ◽  
Atmospheric Sciences ◽  
Practical Applications ◽  
Fractal Sets ◽  
Diffusion Limited

Abstract. The need to assess the three-dimensional fractal dimension of fractal aggregates from the fractal dimension of two-dimensional projections is very frequent in geophysics, soil, and atmospheric sciences. However, a generally valid approach to relate the two- and three-dimensional fractal dimensions is missing, thus questioning the accuracy of the method used until now in practical applications. A mathematical approach developed for application to suspended aggregates made of cohesive sediment is investigated and applied here more generally to Diffusion-Limited Aggregates (DLA) and Cluster-Cluster Aggregates (CCA), showing higher accuracy in determining the three-dimensional fractal dimension compared to the method currently used.

ПСИХОЛОГІЧНИЙ СУПРОВІД СТАНОВЛЕННЯ УСПІШНОЇ ОСОБИСТОСТІ У ПРОФЕСІЙНОМУ СЕРЕДОВИЩІ В УМОВАХ СОЦІАЛЬНИХ ТА ОСВІТНІХ РЕФОРМ

1970 ◽  
Vol 8 (1) ◽  
pp. 221-226
Author(s):  
Олена Горова
Keyword(s):  
Social Activity ◽  
Professional Activity ◽  
Subjective Feeling ◽  
Support Worker ◽  
Gainful Employment ◽  
Progressive Development ◽  
Professional Success ◽  
The Social ◽  
The Individual ◽  
Favorable Conditions

Професійне   становлення   особистості   супроводжує   всі   етапи  соціально-вікового   розвитку  особистості.  Трудова  діяльність  є  основним  видом  суспільної  активності,  який  дозволяє  працівнику  задовольняти  основні  потреби,  особливо  у  процесі  постійних  соціальних,  освітніх  реформ.  Важливим  завданням психологічного супроводу працівника у процесі виконання професійної діяльності є забезпечення  сприятливих  умов  формування  професійно  важливих  якостей.  Соціальна  успішність  є  результатом  ефективного  розв’язання  виробничих  завдань, які  мають  суспільно корисну  важливість  та  пов’язані  з  потребами інших людей. Якісний прогресивний розвиток працівника можливий лише за умови збереження  стійкого  позитивного  ставлення  до  професії.  Позитивна  професійна  самоідентифікація  пов’язана  з  ототожненням  та  персоналізацією  працівником  особистісних  рис  працівників,  які  досягли  успіху  у  професії,  мають  суспільно  визнані  результати  діяльності.  Таким  чином,  професійна  успішність  як  суб’єктне  новоутворення  у  якості  відчуття  гордості  за  власні  результати  діяльності  забезпечує  реалізацію традиції наставництва і  передачі позитивного професійного досвіду.    Професійно  успішний  працівник  усвідомлює  необхідність  та  важливість  результатів  своєї  діяльності  для  інших,  що  вимагає,  відповідно,  від  соціального  середовища  усвідомлення  необхідності  визнання  результатів  діяльності  фахівців.  Знехтуваний  суспільством  працівник,  або  той,  результати  діяльності  якого  позиціонуються  як  меншовартісні,  дистанціюється  від  професії  та  має  негативний  потенціал розвитку. Professional formation of the person accompanies all phases of social and age of the individual. Gainful  employment is the main form of social activity that allows the employee to realize the basic needs. An important task  of psychological support worker in the course of professional activity is to provide favorable conditions for the  formation  of  professionally  important  qualities.  Professional  success  is  the  result  of  an  effective  solution  of  industrial jobs that are socially useful and important related to the needs of others. High-quality progressive  development of an employee is only possible while maintaining a stable positive attitude towards the profession.  Positive  professional  identity  associated  with  the  identification  and  personalization  of  employee  personality traits of employees who have been successful in the profession, who have publicly acknowledged  performance. Thus professional success as the subjective feeling of a lump in the pride of their own results of  operations  ensures  the  implementation  of  the  tradition  of  mentoring  and  of  positive  transfer  of  professional  experience.  Professionally successful employees aware of the need and the importance of the results of its operations  for the other, which requires, respectively, from the social environment - awareness of the need to recognize the  performance of specialists. Unclaimed society worker, or the results of operations, which are positioned as less  important, is moving away from the profession and has a negative potential. 

THE SUBJECT FIELD OF INSTITUTIONAL EDUCATION: THE SENSE AND PROSPECTS OF DEVELOPMENT IN UKRAINE

2020 ◽  
pp. 25-38
Author(s):  
Irina Mordous
Keyword(s):  
Decisive Role ◽  
Subject Field ◽  
National Education ◽  
Progressive Development ◽  
Separate Area ◽  
Economic Transformations ◽  
The Difference ◽  
The Individual ◽  
Political Economic

The development of modern civilization attests to its decisive role in the progressive development of institutions. They identified the difference between Western civilization and the rest of the world. Confirmation of the institutional advantages of the West was its early industrialization. The genesis and formation of institutionalism in its ideological and conceptualmethodological orientation occurs as a process alternative to neoclassic in the context of world heterodoxia, which quickly spread in social science. Highlighting institutional education as a separate area of sociocultural activity is determined by the factor of differentiation of institutional theory as a whole. A feature of institutional education is its orientation toward the individual and his/her transformation into a personality. The content of institutional education is revealed through the analysis of the institution, which includes a set of established customs, traditions, ways of thinking, behavioral stereotypes of individuals and social groups. The dynamics of socio-political, economic transformations in Ukraine requires a review of the foundations of national education and determination of the prospects for its development in the 21st century in the context of institutionalism.

FRACTAL DIMENSION, PRIMES, AND THE PERSISTENCE OF MEMORY

2003 ◽  
Vol 06 (02) ◽  
pp. 241-249
Author(s):  
JOSEPH L. PE
Keyword(s):  
Number Theory ◽  
Fractal Dimension ◽  
Fractal Dimensions ◽  
Distinguishing Characteristic ◽  
Local Behavior ◽  
Remarkable Similarity ◽  
Recursive Procedures

Many sequences from number theory, such as the primes, are defined by recursive procedures, often leading to complex local behavior, but also to graphical similarity on different scales — a property that can be analyzed by fractal dimension. This paper computes sample fractal dimensions from the graphs of some number-theoretic functions. It argues for the usefulness of empirical fractal dimension as a distinguishing characteristic of the graph. Also, it notes a remarkable similarity between two apparently unrelated sequences: the persistence of a number, and the memory of a prime. This similarity is quantified using fractal dimension.

scholarly journals Supramolecular Fractal Growth of Self-Assembled Fibrillar Networks

Gels ◽  
2021 ◽  
Vol 7 (2) ◽  
pp. 46
Author(s):  
Pedram Nasr ◽  
Hannah Leung ◽  
France-Isabelle Auzanneau ◽  
Michael A. Rogers
Keyword(s):  
Fractal Dimension ◽  
Crystallization Temperature ◽  
Cayley Tree ◽  
Fractal Dimensions ◽  
Self Similarity ◽  
Avrami Model ◽  
Cluster Aggregation ◽  
Box Counting ◽  
Self Assembled ◽  
Limited Diffusion

Complex morphologies, as is the case in self-assembled fibrillar networks (SAFiNs) of 1,3:2,4-Dibenzylidene sorbitol (DBS), are often characterized by their Fractal dimension and not Euclidean. Self-similarity presents for DBS-polyethylene glycol (PEG) SAFiNs in the Cayley Tree branching pattern, similar box-counting fractal dimensions across length scales, and fractals derived from the Avrami model. Irrespective of the crystallization temperature, fractal values corresponded to limited diffusion aggregation and not ballistic particle–cluster aggregation. Additionally, the fractal dimension of the SAFiN was affected more by changes in solvent viscosity (e.g., PEG200 compared to PEG600) than crystallization temperature. Most surprising was the evidence of Cayley branching not only for the radial fibers within the spherulitic but also on the fiber surfaces.

scholarly journals Dynamic characteristics and fractal representations of crack propagation of rock with different fissures under multiple impact loadings

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Bing Sun ◽  
Shun Liu ◽  
Sheng Zeng ◽  
Shanyong Wang ◽  
Shaoping Wang
Keyword(s):  
Fractal Dimension ◽  
Crack Propagation ◽  
Crack Growth ◽  
Impact Test ◽  
Fractal Theory ◽  
Dynamic Change ◽  
Fractal Dimensions ◽  
Peak Stress ◽  
Dynamic Mechanical Properties ◽  
Gravitational Potential Energy

AbstractTo investigate the influence of the fissure morphology on the dynamic mechanical properties of the rock and the crack propagation, a drop hammer impact test device was used to conduct impact failure tests on sandstones with different fissure numbers and fissure dips, simultaneously recorded the crack growth after each impact. The box fractal dimension is used to quantitatively analyze the dynamic change in the sandstone cracks and a fractal model of crack growth over time is established based on fractal theory. The results demonstrate that under impact test conditions of the same mass and different heights, the energy absorbed by sandstone accounts for about 26.7% of the gravitational potential energy. But at the same height and different mass, the energy absorbed by the sandstone accounts for about 68.6% of the total energy. As the fissure dip increases and the number of fissures increases, the dynamic peak stress and dynamic elastic modulus of the fractured sandstone gradually decrease. The fractal dimensions of crack evolution tend to increase with time as a whole and assume as a parabolic. Except for one fissure, 60° and 90° specimens, with the extension of time, the increase rate of fractal dimension is decreasing correspondingly.

LATTICE DYNAMICS OF THE BINARY APERIODIC CHAINS OF ATOMS I: FRACTAL DIMENSION OF PHONON SPECTRA

1995 ◽  
Vol 09 (12) ◽  
pp. 1429-1451 ◽  
Author(s):  
WŁODZIMIERZ SALEJDA
Keyword(s):  
Fractal Dimension ◽  
Lattice Dynamics ◽  
Nearest Neighbor ◽  
Average Distance ◽  
Local Maximum ◽  
Fractal Dimensions ◽  
Dimension Formula ◽  
Phonon Spectra ◽  
Wide Range ◽  
Silver Mean

The microscopic harmonic model of lattice dynamics of the binary chains of atoms is formulated and studied numerically. The dependence of spring constants of the nearest-neighbor (NN) interactions on the average distance between atoms are taken into account. The covering fractal dimensions [Formula: see text] of the Cantor-set-like phonon spec-tra (PS) of generalized Fibonacci and non-Fibonaccian aperiodic chains containing of 16384≤N≤33461 atoms are determined numerically. The dependence of [Formula: see text] on the strength Q of NN interactions and on R=mH/mL, where mH and mL denotes the mass of heavy and light atoms, respectively, are calculated for a wide range of Q and R. In particular we found: (1) The fractal dimension [Formula: see text] of the PS for the so-called goldenmean, silver-mean, bronze-mean, dodecagonal and Severin chain shows a local maximum at increasing magnitude of Q and R>1; (2) At sufficiently large Q we observe power-like diminishing of [Formula: see text] i.e. [Formula: see text], where α=−0.14±0.02 and α=−0.10±0.02 for the above specified chains and so-called octagonal, copper-mean, nickel-mean, Thue-Morse, Rudin-Shapiro chain, respectively.

Universality of fractal aggregates as probed by light scattering

1989 ◽  
Vol 423 (1864) ◽  
pp. 71-87 ◽  
Keyword(s):  
Light Scattering ◽  
Dynamic Light Scattering ◽  
Master Curve ◽  
Rotational Diffusion ◽  
Fractal Dimensions ◽  
Scattering Data ◽  
Diffusion Limited ◽  
Single Master Curve ◽  
Using Data ◽  
Dynamic Light Scattering Data

Fractal colloid aggregates are studied with both static and dynamic light scattering. The dynamic light scattering data are scaled onto a single master curve, whose shape is sensitive to the structure of the aggregates and their mass distribution. By using the structure factor determined from computer-simulated aggregates, and including the effects of rotational diffusion, we predict the shape of the master curve for different cluster distributions. Excellent agreement is found between our predictions and the data for the two limiting régimes, diffusion-limited and reaction-limited colloid aggregation. Furthermore, using data from several completely different colloids, we find that the shapes of the master curves are identical for each régime. In addition, the cluster fractal dimensions and the aggregation kinetics are identical in each régime. This provides convincing experimental evidence of the universality of these two régimes of colloid aggregation.

Fractal dimension based features are useful descriptors of leaf complexity and shape

1999 ◽  
Vol 29 (9) ◽  
pp. 1301-1310 ◽  
Author(s):  
Wojciech Borkowski
Keyword(s):  
Fractal Dimension ◽  
Tree Species ◽  
Fractal Dimensions ◽  
Plant Identification ◽  
Simple Method ◽  
Mass Dimension ◽  
Scale Range ◽  
Computer Identification

An application of fractal dimensions as measures of leaf complexity to morphometric studies and automated plant identification is presented. Detailed algorithms for the calculation of compass dimension and averaged mass dimension together with a simple method of grasping the scale range related variability are given. An analysis of complexity of more than 300 leaves from 10 tree species is reported. Several classical biometric descriptors as well as 16 fractal dimension features were computed on digitized leaf silhouettes. It is demonstrated that properly defined fractal dimension based features may be used to discriminate between species with more than 90% accuracy, especially when used together with other measures. It seems, therefore, that they can be utilized in computer identification systems and for purely taxonomical purposes.

A Unified Approach of Object-Level Saliency Detection in Images and Videos

2013 ◽  
Vol 765-767 ◽  
pp. 1401-1405
Author(s):  
Chi Zhang ◽  
Wei Qiang Wang
Keyword(s):  
Quality Evaluation ◽  
Saliency Detection ◽  
Visual Saliency ◽  
Unified Approach ◽  
Feature Maps ◽  
Still Images ◽  
Saliency Maps ◽  
Novel Method ◽  
Object Level ◽  
Important Branch

Object-level saliency detection is an important branch of visual saliency. In this paper, we propose a novel method which can conduct object-level saliency detection in both images and videos in a unified way. We employ a more effective spatial compactness assumption to measure saliency instead of the popular contrast assumption. In addition, we present a combination framework which integrates multiple saliency maps generated in different feature maps. The proposed algorithm can automatically select saliency maps of high quality according to the quality evaluation score we define. The experimental results demonstrate that the proposed method outperforms all state-of-the-art methods on both of the datasets of still images and video sequences.

Sign in / Sign up

Export Citation Format

Share Document