Related papers: Algorithmic interpretations of fractal dimension
Least box number coverage problem for calculating dimension of fractal networks is a NP-hard problem. Meanwhile, the time complexity of random ball coverage for calculating dimension is very low. In this paper we strictly present the upper…
It is shown that fractal dimension can be estimated seeking a solution of functional equation defined for areas of coverages of different scales. The method proposed is compared with widely known way to estimate fractal dimension via linear…
Estimates of the fractal dimension $D$ of the set of galaxies in the universe, based on ever improving data sets, tend to settle on $D\approx 2$. This result raised a raging debate due to its glaring contradiction with astrophysical models…
We introduce fractional flat space, described by a continuous geometry with constant non-integer Hausdorff and spectral dimensions. This is the analogue of Euclidean space, but with anomalous scaling and diffusion properties. The basic tool…
Urban form has been empirically demonstrated to be of scaling invariance and can be described with fractal geometry. However, the rational range of fractal dimension value and the relationships between various fractal indicators of cities…
We discuss a number of techniques for determining the Minkowski dimension of bounded subsets of some Euclidean space of any dimension, including: the box-counting dimension and equivalent definitions based on various box-counting functions;…
A fractal is in essence a hierarchy with cascade structure, which can be described with a set of exponential functions. From these exponential functions, a set of power laws indicative of scaling can be derived. Hierarchy structure and…
This chapter explores the notion of "dimension" of a set. Various power laws by which an Euclidean space can be characterized are used to define dimensions, which then explore different aspects of the set. Also discussed are the…
Complexity measures are designed to capture complex behavior and quantify *how* complex, according to that measure, that particular behavior is. It can be expected that different complexity measures from possibly entirely different fields…
We study the porosity properties of fractal percolation sets $E\subset\mathbb{R}^d$. Among other things, for all $0<\varepsilon<\tfrac12$, we obtain dimension bounds for the set of exceptional points where the upper porosity of $E$ is less…
Randomized dimensionality reduction is a widely-used algorithmic technique for speeding up large-scale Euclidean optimization problems. In this paper, we study dimension reduction for a variety of maximization problems, including…
Irregular boundary lines can be characterized by fractal dimension, which provides important information for spatial analysis of complex geographical phenomena such as cities. However, it is difficult to calculate fractal dimension of…
In this report we present experimental results using \emph{Haussdorf-Besicovich} fractal dimension for performing morphological galaxy classification. The fractal dimension is a topological, structural and spatial property that give us…
Analysis on fractals is a growing field, with hints of potential for widespread applicability across all of STEM. One of the most heavily researched type of fractals are the nested fractals, fractal shapes defined by virtue of being made of…
The fractal structure of real world objects is often analyzed using digital images. In this context, the compression fractal dimension is put forward. It provides a simple method for the direct estimation of the dimension of fractals stored…
It is wellknown that the ordinary calculus is inadequate to handle fractal structures and processes and another suitable calculus needs to be developed for this purpose. Recently it was realized that fractional calculus with suitable…
The purpose of this paper is to study the fractal phenomena in large data sets and the associated questions of dimension reduction. We examine situations where the classical Principal Component Analysis is not effective in identifying the…
In this article a definition of optimal cover for fractal structures is proposed. Expression for Minkowsky dimension is rewritten in terms of functional equation on areas of covers that constructed for different scales.Given the definition,…
The present work shows a novel fractal dimension method for shape analysis. The proposed technique extracts descriptors from the shape by applying a multiscale approach to the calculus of the fractal dimension of that shape. The fractal…
This paper surveys work on the relation between fractal dimensions and algorithmic information theory over the past thirty years. It covers the basic development of prefix-free Kolmogorov complexity from an information theoretic point of…