Dissertations / Theses on the topic 'Box-counting dimension'

O objetivo deste trabalho é o estudo da dimensão fractal, nomeadamente a dimensão de Hausdorff, dimensão de capacidade e dimensão de correlação, relacionando-as e efetuando o cálculo em alguns exemplos. Sempre que se considera indispensável, são apresentadas noções introdutórias para uma melhor compreensão dos conceitos analisados. O Capítulo 2 é dedicado ao estudo da dimensão de Hausdorff, introduzindo, previamente, uma noção de medida, de Hausdorff. No Capítulo 3 analisamos a dimensão de capacidade, suas propriedades e inconvenientes, relacionando, no final, esta dimensão com a dimensão de Hausdorff. O Capítulo 4 estuda técnicas para calcular dimensões. São estudados subconjuntos de medida. Finita, sistemas de funções iteradas, conjuntos auto-semelhantes e auto-afins e dimensões de gráficos. O Capítulo 5 é dedicado à dimensão de correlação. Estuda o expoente de correlação  Introduzido por Grassberger e Procaccia. São analisadas funções de dimensão 1 e no plano. Terminamos com o estudo de séries temporais de variável única. ABSTRACT: The aim of this work is the study of the fractal dimension, namely the Hausdorff dimension, the box-counting dimension and the correlation dimension, relating and computing them in some examples. Everytime it is necessary we introduce the basic concepts to a better understanding of the concepts analysed in this work. Chapter 2 is dedicated to the study of the Hausdorff dimension, introducing first the notion of Hausdorff measure. Chapter 3 is concerned with the box-counting dimension, its properties and problems. Then we relate this dimension With Hausdorff dimension studied in Chapter 2. Chapter 4 is dedicated to the techniques for calculating dimensions. We study subsets of finite measure, iterated function schemes, self-similar and self-affine sets and dimensions of graphs. Finally, in Chapter 5 we present the correlation dimension. We study the correlation exponent, introduced by Grassberger and Procaccia. We finish this Chapter with a study of single-variable time series.

We consider random fractals generated by random recursive constructions, prove zero-one laws concerning their dimensions and find their packing and Minkowski dimensions. Also we investigate the packing measure in corresponding dimension. For a class of random distribution functions we prove that their packing and Hausdorff dimensions coincide.

Tato diplomová práce se zabývá teorií fraktálů a popisuje patričné potíže při zavedení pojmu fraktál. Dále se v práci navrhuje několik metod, které se použijí na aproximaci fraktálních dimenzí různých množin zobrazených na zařízeních s konečným rozlišením. Tyto metody se otestují na takových množinách, jejichž dimenze známe, a na závěr se výsledky porovnávají.

A city consists of many elements such as humans, buildings, and roads. The complexity of cities is difficult to measure using Euclidean geometry. In this study, we use fractal geometry (scaling analysis) to measure the complexity of urban areas. We observe urban development from different perspectives using the bottom-up approach. In a bottom-up approach, we observe an urban region from a basic to higher level from our daily life perspective to an overall view. Furthermore, an urban environment is not constant, but it is complex; cities with greater complexity are more prosperous. There are many disciplines that analyze changes in the Earth’s surface, such as urban planning, detection of melting ice, and deforestation management. Moreover, these disciplines can take advantage of remote sensing for research. This study not only uses satellite imaging to analyze urban areas but also uses check-in and points of interest (POI) data. It uses straightforward means to observe an urban environment using the bottom-up approach and measure its complexity using fractal geometry.   Web 2.0, which has many volunteers who share their information on different platforms, was one of the most important tools in this study. We can easily obtain rough data from various platforms such as the Stanford Large Network Dataset Collection (SLNDC), the Earth Observation Group (EOG), and CloudMade. The check-in data in this thesis were downloaded from SLNDC, the POI data were obtained from CloudMade, and the nighttime lights imaging data were collected from EOG. In this study, we used these three types of data to derive natural cities representing city regions using a bottom-up approach. Natural cities were derived from open geographic data without human manipulation. After refining data, we used rough data to derive natural cities. This study used a triangulated irregular network to derive natural cities from check-in and POI data.   In this study, we focus on the four largest US natural cities regions: Chicago, New York, San Francisco, and Los Angeles. The result is that the New York City region is the most complex area in the United States. Box-counting fractal dimension, lacunarity, and ht-index (head/tail breaks index) can be used to explain this. Box-counting fractal dimension is used to represent the New York City region as the most prosperous of the four city regions. Lacunarity indicates the New York City region as the most compact area in the United States. Ht-index shows the New York City region having the highest hierarchy of the four city regions. This conforms to central place theory: higher-level cities have better service than lower-level cities. In addition, ht-index cannot represent hierarchy clearly when data distribution does not fit a long-tail distribution exactly. However, the ht-index is the only method that can analyze the complexity of natural cities without using images.

La geometria euclidea risulta spesso inadeguata a descrivere le forme della natura. I Frattali, oggetti interrotti e irregolari, come indica il nome stesso, sono più adatti a rappresentare la forma frastagliata delle linee costiere o altri elementi naturali. Lo strumento necessario per studiare rigorosamente i frattali sono i teoremi riguardanti la misura di Hausdorff, con i quali possono definirsi gli s-sets, dove s è la dimensione di Hausdorff. Se s non è intero, l'insieme in gioco può riconoscersi come frattale e non presenta tangenti e densità in quasi nessun punto. I frattali più classici, come gli insiemi di Cantor, Koch e Sierpinski, presentano anche la proprietà di auto-similarità e la dimensione di similitudine viene a coincidere con quella di Hausdorff. Una tecnica basata sulla dimensione frattale, detta box-counting, interviene in applicazioni bio-mediche e risulta utile per studiare le placche senili di varie specie di mammiferi tra cui l'uomo o anche per distinguere un melanoma maligno da una diversa lesione della cute.

We study the geometric and topological properties of strange non-chaotic attractors created in non-smooth saddle-node bifurcations of quasiperiodically forced interval maps. By interpreting the attractors as limit objects of the iterates of a continuous curve and controlling the geometry of the latter, we determine their Hausdorff and box-counting dimension and show that these take distinct values. Moreover, the same approach allows us to describe the topological structure of the attractors and to prove their minimality.

Viene studiata una nuova metodologia di diagnostica, basata sull’analisi di dati derivanti dall’Emissione Acustica (EA), per lo studio della nucleazione e della propagazione di difetti durante prove di laboratorio su provino sotto carico e serbatoi in pressione. L’applicazione dell’analisi frattale al segnale di EA risulta particolarmente efficace e permette di identificare la distribuzione spaziale delle sorgenti stesse, esplicitandone la correlazione tra gli eventi. E’ così possibile ottenere molte informazioni associate al danneggiamento dei differenti casi studiati. L’intensità della sollecitazione, il danneggiamento del pezzo, o lo stato di fatica del materiale, sono parametri strettamente correlati con l’EA. La dimensione frattale (Dt) evolve con il carico (sigma) o la pressione (p) o con il numero di cicli (N). Le curve Dt-sigma, Dt-p o Dt-N risultano utili per l’individuazione della nucleazione e propagazione del difetto e per l’identificazione di una condizione di incipiente collasso della struttura. I risultati ottenuti mediante questa tecnica suggeriscono la possibilità d’individuare con anticipo la formazione della cricca, rispetto a tecniche sperimentali e/o teoriche.
A new experimental methodology was investigated for the evaluation of material damage by analyzing the behavior of several specimens under stress. The application of fractal analysis to Acoustic Emission (AE) signal resulted particularly effective it is possible to characterize the spatial distribution of the prime AE sources, and the relationship between different event of AE. In fact, it is possible to obtain several information, associated with the damage of the different tested materials. The intensity of the prime stress, or the state of fatigue, of the material, i.e. of the flaws that damaged the rheology of the material during its previous stress history, is closely related to AE. The fractal dimension (D) evolves altogether with the stress (sigma) or the pressure (p) or the number of fatigue cycles (N). D-sigma, D-p and D-N curves resulted useful for identifying the condition of incipient collapse or nucleation and propagation of the fatigue cracks. The results of such experimental technique suggest that it is possible anticipating the detection of the crack onset, relating to other theoretical and/or experimental techniques.

Questa tesi ha lo scopo di presentare i frattali, per prima formalizzati dal matematico Benoit Mandelbrot, descrivendo a livello matematico le due principali proprietà che li caratterizzano: la dimensione frattale (dimensione di Hausdorff) e l'autosimilarità. Si pone inoltre l'attenzione alla dimensione di Box-counting, analizzandone la relazione che questa ha con la dimensione di Hausdorff.

In questa tesi vengono presentati i frattali, strutture matematiche accumunate da dimensione di Hausdorff non intera e proprietà di autosimilarità. Dopo alcuni concetti di base della teoria della misura, si studieranno la misura e la dimensione di Hausdorff, ponendo attenzione ad alcune loro proprietà e fornendo esempi tra cui l'insieme di Cantor e la funzione di Weierstrass. Si analizzerà inoltre la proprietà di autosimilarità descritta tramite contrazioni e punti fissi. Infine verrà presentato il metodo del box counting e il suo utilizzo in campo medico per l’analisi di immagini frattali riguardanti la vascolarizzazione tumorale.

L'obiettivo della tesi è quello di mostrare la costruzione della dimensione di Hausdorff, per poi presentarne alcune tecniche di calcolo e stima.Nel primo capitolo enunciamo i principali risultati di teoria della misura, utili per definire la premisura di Hausdorff da cui si ricava la misura alpha-dimensionale di Hausdorff, che si verifica essere una misura esterna. Dopo aver dimostrato alcune proprietà fondamentali di questa potremo, quindi,definire la dimensione di Hausdorff e descriverne il comportamento in relazione a isometrie e a mappe lipschitziane. Il primo capitolo termina con il calcolo della dimensione di alcuni fra i più noti e semplici frattali. Il secondo capitolo si pone, invece, come obiettivo quello di esporre tecniche di calcolo della dimensione di Hausdorff. In prima istanza, viene definita la dimensione di box-counting, utilizzata per stimare dall'alto la dimensione di Hausdorff, mentre, a fine capitolo, si presenta il metodo del potenziale teorico, utile per la stima dal basso. Nell'ultimo capitolo si prendono in considerazione gli insiemi autosimilari, per i quali è possibile stimare la dimensione di Hausdorff sotto certe ipotesi. Il principale scopo, quindi,di questo elaborato è quello di presentare alcune tecniche di stima e di calcolo della dimensione di Hausdorff. Il filo conduttore della tesi è dato da alcuni esempi noti di frattali come l'insieme di Cantor e la curva di Koch, di cui viene calcolata la dimensione prima manualmente, per poi farlo con le tecniche mostrate.

碩士
國立中興大學
資訊管理學系所
104
Cancer has been the leading first of - first ten causes of death in humans in the past ten years. Many researchers have invested in the cancer experiments in order to find the cure for cancer. Because cancer cells have to rely on oxygen and nutrients in biological vessel to survive, researchers usually need small animals for experimental sample (for example: rabbits, mouses, etc ….), put cancer cells into the small animals’ bodies to make them infected, inject the experimental drug and observe the changes in blood vessels to determine whether the experimental drug against cancer effectively. In recent years, the cost of animal samples becomes increase. Moreover, the animals must be dissected in the experiments, The processing of animal experiments is very troublesome. However, the chicken chorioallantoic membrane grows faster, the price is cheaper, and it is easy to observe the results immediately. Chicken chorioallantoic membrane which has vascular structure is suited to replace other animal samples. Due to the large quantity of samples, the researchers needs to takes a lot of time on viewing the results of the reaction to judge good or bad results. Therefore we proposed a technology to determine the results of the chicken chorioallantoic membrane. This technology can be divided into two parts: non-yolk and yolk region segmentation and vessel segmentation in yolk region. In the first part, we use R, G, B three kinds of gray scale to let the boundary between non-yolk and yolk region more obvious. Then, Local Cross Thresholding is used to segment non-yolk and yolk region. In the second part, Run-Length is used to make blood vessels in the yolk region more obvious and Local Cross Thresholding is used to segment vessels in yolk region. Because there is noise after blood vessels were segmented, we use Opening to divide them, and Thinning and Region Labeling to remove the noise. We use Box Counting Dimension (BCD) to determine the density of blood vessels. Then, BCD values of Ground Truth, propose methods and artificial blood vessels judgment method are calculated. According to the experimental results, BCD values of the proposed method are close to those of Ground Truth. The proposed method has better results.

Poniższa rozprawa składa się z dwóch części. Obie z nich badają geometryczne własności miar występujących w skończenie wymiarowych układach dynamicznych, głównie z punktu widzenia teorii wymiaru. Część pierwsza dotyczy probabilistycznych aspektów twierdzenia Takensa o zanurzaniu, zajmującego się zagadnieniem rekonstrukcji układu dynamicznego z ciągu pomiarów wykonanych za pomocą jednowymiarowej obserwabli. Klasyczne wyniki z tej dziedziny orzekają, że dla typowej obserwabli, dowolny stan początkowy układu jest jednoznacznie wyznaczony przez ciąg pomiarów, o ile ich ilość przekracza dwukrotnie wymiar przestrzeni fazowej. Główny wynik tej części rozprawy stwierdza, że w kontekście probabilistycznym liczba pomiarów może być dwukrotnie zmniejszona, tzn. prawie każdy stan początkowy układu jest wyznaczony jednoznacznie, o ile ilość pomiarów jest większa od wymiaru Hausdorffa przestrzeni fazowej. Powyższy wynik dowodzi częściowo hipotezy Shroera, Sauera, Otta oraz Yorka z 1998 roku. Przedstawiamy także niedynamiczną wersję probabilistycznego twierdzenia o zanurzaniu oraz szereg przykładów. W drugiej części rozprawy rozważamy rodzinę stacjonarnych miar probabilistycznych dla pewnych losowych układów dynamicznych na odcinku jednostkowym oraz badamy ich własności geometryczne. Rozważane miary mogą być traktowane jako miary stacjonarne dla procesu Markowa na odcinku, otrzymanego przez losowe iterowanie dwóch kawałkami afinicznych homeomorfizmów odcinka. Układy tej postaci nazywamy układami Alsedy-Misiurewicza (albo AM-układami), gdyż badania nad nimi rozpoczęli Alseda oraz Misiurewicz, którzy postawili w 2014 roku hipotezę, że typowe miary stacjonarne dla takich układów są singularne względem miary Lebesgue'a. Głównym celem naszej pracy jest scharakteryzowanie parametrów posiadających tę własność. Naszym głównym wynikiem jest znalezienie pewnych zbiorów parametrów dla których odpowiednie miary są singularne, co dowodzi powyższą hipotezę dla tych zbiorów. Przedstawiamy dwa różne podejścia do dowodzenia singularności - jedno oparte na znajdowaniu minimalnych niezmienniczych zbiorów Cantora oraz drugie, wykorzystujące szacowanie oczekiwanego czasu powrotu do odpowiednio dobranego przedziału. W pierwszym przypadku wyliczamy wymiar Hausdorffa miary stacjonarnej dla pewnych parametrów. Przedstawiamy również kilka dodatkowych wyników dotyczących AM-układów.
This dissertation consists of two parts, both studying geometric properties of measures occuring in finite-dimensional dynamical systems, mainly from the point of view of the dimension theory. The first part concerns probabilistic aspects of the Takens embedding theorem, dealing with the problem of reconstructing a dynamical system from a sequence of measurements performed via a one-dimensional observable. Classical results of that type state that for a typical observable, every initial state of the system is uniquely determined by a sequence of measurements as long as the number of measurements is greater than twice the dimension of the phase space. The main result of this part of the dissertation states that in the probabilistic setting the number of measurements can be reduced by half, i.e. almost every initial state of the system can be uniquely determined provided that the number of measurements is greater than the Hausdorff dimension of the phase space. This result partially proves a conjecture of Shroer, Sauer, Ott and Yorke from 1998. We provide also a non-dynamical probabilistic embedding theorem and several examples. In the second part of the dissertation we consider a family of stationary probability measures for certain random dynamical systems on the unit interval and study their geometric properties. The measures we are interested in can be seen as stationary measures for Markov processes on the unit interval, which arise from random iterations of two piecewise-affine homeomorphisms of the interval. We call such random systems Alseda-Misiurewicz systems (or AM-systems), as they were introduced and studied by Alseda and Misiurewicz, who conjectured in 2014 that typically measures of that type should be singular with respect to the Lebesgue measure. We work towards characterization of parameters exhibiting this property. Our main result is establishing singularity of the corresponding stationary measures for certain sets of parameters, hence confirming the conjecture on these sets. We present two different approaches to proving singularity - one based on constructing invariant minimal Cantor sets and one based on estimating the expected return time to a suitably chosen interval. In the first case we calculate the Hausdorff dimension of the measure for certain parameters. We present also several auxiliary results concerning AM-systems.