Ο όρος μορφοκλασματικό σύνολο παρουσιάστηκε για πρώτη φορά από τον Barnsley, το 1975. Ουσιαστικά ονόμασε έτσι το όριο μιας επαναληπτικής διαδικασίας, όταν ο αριθμός των επαναλήψεων τείνει στο άπειρο. Μέσα από τη μηχανή πολλαπλών αντιγράφων ορίζουμε το μορφοκλασματικό σύνολο και τις ιδιότητές του. Τα μορφοκλασματικά σύνολα έχουν εφαρμογές σε πολλά επιστημονικά πεδία, αλλά η κυριότερη είναι ως μέσο στη συμπίεση εικόνων. Ξεκινώντας από τη συμπίεση εικόνων οδηγούμαστε στον ορισμό του επαναληπτικού συστήματος συναρτήσεων και την ανάλυση της θεωρίας του, μέσω των οποίων επιτυγχάνουμε συμπίεση εικόνων. Η ανάγκη για εξοικονόμηση μνήμης, λόγω του μεγάλου όγκου δεδομένων, έκανε τον Barnsley να σκεφτεί αντί να αποθηκεύουμε ολόκληρη την εικόνα να αποθηκεύουμε μόνο το κατάλληλο επαναληπτικό σύστημα συναρτήσεων. Η διαδικασία της συμπίεσης εικόνων με μορφοκλασματικά σύνολα συντίθεται από δύο φάσεις. Την κωδικοποίηση και την αποκωδικοποίηση. Στην πρώτη φάση, προσπαθούμε να λύσουμε ένα αντίστροφο πρόβλημα και συγκεκριμένα να βρούμε το κατάλληλο επαναληπτικό σύστημα συναρτήσεων που αν το εφαρμόσουμε σε μια αρχική εικόνα θα συγκλίνουμε στον ελκυστή. Στη δεύετρη φάση, τη φάση της αποκωδικοποίησης, με χρήση του επαναληπτικού συστήματος συναρτήσεων, προσεγγίζουμε τον ελκυστή μετά από έναν πεπερασμένο αριθμό επαναλήψεων.Η υλοποίηση αυτών των φάσεων καθώς και οι δυνατοί τρόποι μείωσης της πολυπλοκότητας που απαιτείται κατά τη φάση της κωδικοποίησης είναι και το αντικείμενο της παρούσας εργασίας, στα πλαίσια της οποίας παρουσιάζονται και υλοποιούνται παραλλαγές γνωστών τεχνικών οι οποίες παρουσιάζουν βελτίωση στο PSNR, το λόγο συμπίεσης και το χρόνο κωδικοποίησης. Τέλος, προτείνεται μια νέα τεχνική για τον προσδιορισμό του κατάλληλου επαναληπτικού συστήματος συναρτήσεων κατά τη φάση της κωδικοποίησης, η οποία βασίζεται στην ελαχιστοποίηση μιας μη γραμμικής συνάρτησης κόστους. Αποτιμάται η απόδοσή της και συγκρίνεται με αυτή άλλων γνωστών τεχνικών κωδικοποίησης. / The term ‘fractal’ was first introduced by B. Mandelbrot, in 1975. It denotes the limit of an iterative process, when the number of iterations is infinite. Through the multiple photocopy machine, we define fractal and its properties. Fractals have many applications, one of which is in image compression. This is the object of the present research. Starting with data compression, we aim at the definition of an iterative function system and the IFS theory, through which we can achieve image compression. The need of massive storage in memory made Barnsley think that instead of storing the whole image in memory we can store only the suitable IFS. The process consists of two phases. The encoding and decoding. In encoding we try to solve the inverse problem, i.e. finding the best IFS that when we apply it to a starting image we will get the attractor. In the second phase, after a number of iterations we approximate the attractor. Several fractal image compression methods that already exist, together with their techniques and variations are presented and implemented in this research, in order to achieve improvements in PSNR, compression ratio and encoding time. A new method for fractal image compression on grayscale images is proposed at the end of this research, based on the minimization of a cost function. The experimental as well as the comparative results are shown.

This thesis is structured as follows. Chapter 1 introduces fractal sets before recalling basic mathematical concepts from dynamical systems, measure theory, dimension theory and probability theory. In Chapter 2 we give an overview of both deterministic and stochastic sets obtained from iterated function systems. We summarise classical results and set most of the basic notation. This is followed by the introduction of random graph directed systems in Chapter 3, based on the single authored paper [T1] to be published in Journal of Fractal Geometry. We prove that these attractors have equal Hausdorff and upper box-counting dimension irrespective of overlaps. It follows that the same holds for the classical models introduced in Chapter 2. This chapter also contains results about the Assouad dimensions for these random sets. Chapter 4 is based on the single authored paper [T2] and establishes the box-counting dimension for random box-like self-affine sets using some of the results and the notation developed in Chapter 3. We give some examples to illustrate the results. In Chapter 5 we consider the Hausdorff and packing measure of random attractors and show that for reasonable random systems the Hausdorff measure is zero almost surely. We further establish bounds on the gauge functions necessary to obtain positive or finite Hausdorff measure for random homogeneous systems. Chapter 6 is based on a joint article with J. M. Fraser and J.-J. Miao [FMT] to appear in Ergodic Theory and Dynamical Systems. It is chronologically the first and contains results that were extended in the paper on which Chapter 3 is based. However, we will give some simpler, alternative proofs in this section and crucially also find the Assouad dimension of some random self-affine carpets and show that the Assouad dimension is always `maximal' in both measure theoretic and topological meanings.

Four functional magnetic resonance imaging (fMRI) studies of pitch processing in auditory cortex were designed to reduce the impact of a number of methodological issues that have hitherto limited previous research findings. Due to adaptation effects, it is necessary to repeatedly present short stimulus bursts rather than long-duration stimuli. Thus, conventionally, in neuroimaging studies of pitch perception, a number of short bursts of the pitch stimulus, separated by silent intervals, are compared to a Gaussian noise presented in the same way. The results of the first experiment indicate that replacing the silent intervals with an energetically matched noise context increases the pitch-specific response by removing the 'energy-onset response' that saturates the overall response if silent intervals are used. In the second experiment, a particular pitch-evoking stimulus, iterated ripple noise (IRN), which is commonly used in neuroimaging studies of pitch perception, was examined. Hall and Plack (Cerebral Cortex 2009;19:576-585) showed that IRN contains slowly varying spectro-temporal features unrelated to pitch, and suggested that these features could account for at least some of the cortical activation produced by IRN. The results support this hypothesis, but also suggest that there is an additional pitch-dependent effect in the same region of auditory cortex.The third experiment assessed the effect of using a different control stimulus to the usual Gaussian noise. The new matched controls were a pulse train with randomly jittered inter-pulse intervals and a random-phase unresolved harmonic complex tone. These low-pitch-salience controls were compared to a regular interval pulse train, which is identical to a cosine-phase unresolved harmonic complex tone. The third experiment did not provide evidence for sensitivity to pitch-salience in pitch-responsive regions of auditory cortex. The fourth and final experiment was a factorial design seeking to answer two main questions: 1) Is the pitch-sensitive region of auditory cortex responsive to the salience of other sound features (e.g. modulation)? 2) Are the responses to pitch and to modulation within this region co-located? Two different pitch-evoking stimuli with different levels of pitch salience were used, presented in a noise context. Results indicate that the pitch-sensitive region contains representations for both pitch and modulation. Furthermore, there was no evidence for an interaction between pitch and modulation, suggesting that the two responses are independent. Overall, the results suggest that careful stimulus design, and appropriate experimental control, is necessary to obtain reliable information on the cortical response to pitch. In addition, the results have shed further light on the likely neural substrates of pitch processing in the cortex.

Indiana University-Purdue University Indianapolis (IUPUI) / Given a parametrized family of Iterated Function System (IFS) we give sufficient conditions for a parameter on the boundary of the connectedness locus, M, to be accessible from the complement of M. Moreover, we provide a few examples of such parameters and describe how they are connected to Misiurewicz parameter in the Mandelbrot set, i.e. the connectedness locus of the quadratic family z^2+c.

No / This paper presents a new class of heuristics which embed an exact algorithm within the framework of a local search heuristic. This approach was inspired by related heuristics which we developed for a practical problem arising in electronics manufacture. The basic idea of this heuristic is to break the original problem into small subproblems having similar properties to the original problem. These subproblems are then solved using time intensive heuristic approaches or exact algorithms and the solution is re-embedded into the original problem. The electronics manufacturing problem where we originally used the embedded local search approach, contains the Travelling Salesman Problem (TSP) as a major subproblem. In this paper we further develop our embedded search heuristic, HyperOpt, and investigate its performance for the TSP in comparison to other local search based approaches. We introduce an interesting hybrid of HyperOpt and 3-opt for asymmetric TSPs which proves more efficient than HyperOpt or 3-opt alone. Since pure local search seldom yields solutions of high quality we also investigate the performance of the approaches in an iterated local search framework. We examine iterated approaches of Large-Step Markov Chain and Variable Neighbourhood Search type and investigate their performance when used in combination with HyperOpt. We report extensive computational results to investigate the performance of our heuristic approaches for asymmetric and Euclidean Travelling Salesman Problems. While for the symmetric TSP our approaches yield solutions of comparable quality to 2-opt heuristic, the hybrid methods proposed for asymmetric problems seem capable of compensating for the time intensive embedded heuristic by finding tours of better average quality than iterated 3-opt in many less iterations and providing the best heuristic solutions known, for some instance classes.

The increasing availability of richer, high-dimensional, home sales data-sets, as well as spatially geocoded data, allows for the use of new econometric and computational methods to explore novel research questions. This dissertation consists of three separate research papers which aim to leverage this trend to answer empirical inferential questions, propose new computational approaches in environmental valuation, and address future challenges. The first research chapter estimates the effect on home values of 10 large-scale urban stream restoration projects situated near the project sites. The study area is the Johnson Creek Watershed in Portland, Oregon. The research design incorporates four matching model approaches that vary based on the temporal bands' width, a narrow and a wider band, and two spatial zoning buffers, a smaller and larger that account for the affected homes' distances. Estimated effects tend to be positive for six projects when the restoration projects' distance is smaller, and the temporal bands are narrow, while two restoration projects have positive effects on home values across all four modeling approaches. The second research chapter focuses on the underlying statistical and computational properties of matching methods for causal treatment effects. The prevailing notion in the literature is that there is a tradeoff between bias and variance linked to the number of matched control observations for each treatment unit. In addition, in the era of Big Data, there is a paucity of research addressing the tradeoffs between inferential accuracy and computational time across different matching methods. Is it worth employing computationally costly matching methods if the gains in bias reduction and efficiency are negligible? We revisit the notion of bias-variance tradeoff and address the subject of computational time considerations. We conduct a simulation study and evaluate 160 models and 320 estimands. The results suggest that the conventional notion of a bias-variance tradeoff, with bias increasing and variance decreasing with the number of matched controls, does not hold under the bias-corrected matching estimator (BCME), developed by Abadie and Imbens (2011). Specifically, for the BCME, the trend of bias decreases as the number of matches per treated unit increases. Moreover, when the pre-matching balance's quality is already good, choosing only one match results in a significantly larger bias under all methods and estimators. In addition, the genetic search matching algorithm, GenMatch, is superior compared to the baseline Greedy Method by achieving a better balance between the observed covariate distributions of the treated and matched control groups. On the down side, GenMatch is 408 times slower compared to a greedy matching method. However, when we employ the BCME on matched data, there is a negligible difference in bias reduction between the two matching methods. Traditionally, environmental valuation methods using residential property transactions follow two approaches, hedonic price functions and Random Utility sorting models. An alternative approach is the Iterated Bidding Algorithm (IBA), introduced by Kuminoff and Jarrah (2010). This third chapter aims to improve the IBA approach to property and environmental valuation compared to its early applications. We implement this approach in an artificially simulated residential housing market, maintaining full control over the data generating mechanism. We implement the Mesh Adaptive Direct Search Algorithm (MADS) and introduce a convergence criterion that leverages the knowledge of individuals' actual pairing to homes. We proceed to estimate the preference parameters of the distribution of an underlying artificially simulated housing market. We estimate with significantly higher precision than the original baseline Nelder-Mead optimization that relied only on a price discrepancy convergence criterion, as implemented during the IBAs earlier applications. / Doctor of Philosophy / The increasing availability of richer, high-dimensional, home sales data sets enables us to employ new methods to explore novel research questions involving housing markets. This dissertation consists of three separate research papers which leverage this trend. The first research paper estimates the effects on home values of 10 large-scale urban stream restoration projects in Portland, Oregon. These homes are located near the project sites. The results show that the distance of the homes from the project sites and the duration of the construction cause different effects on home values. However, two restorations have positive effects regardless of the distance and the duration period. The second research study is focused on the issue of causality. The study demonstrates that a traditional notion concerning causality known as the ``bias-variance tradeoff" is not always valid. In addition, the research shows that sophisticated but time-consuming algorithms have negligible effects in improving the accuracy of estimating the causal effects when we account for the required computational time. The third research study improves an environmental evaluation method that relies on residential property transactions. The methodology leverages the features of more informative residential data sets in conjunction with a more efficient optimization method, leading to significant improvements. The study concludes that due to these improvements, this alternative method can be employed to elicit the true preferences of homeowners over housing and locational characteristics by avoiding the shortcomings of existing techniques.

In this paper, we begin in Chapter 1 by giving a brief overview of the history of fractal geometry, focusing on six of the most important mathematicians in this field. Chapter 2 explains the main definitions needed for the remainder of the paper. In Chapter 3, we clarify the process of creating fractals from iterated function systems. Chapter 4 consists of an examination of the properties of traditional fractals. Next, in Chapter 5, we examine the newly discovered V-variable fractals and their properties. Finally we consider applications and future research in the field of fractal geometry.

<p>This thesis consists of four papers, all concerning random iteration of isometries. The papers are:</p><p>I. Ambroladze A, Ådahl M, Random iteration of isometries in unbounded metric spaces. Nonlinearity 16 (2003) 1107-1117.</p><p>II. Ådahl M, Random iteration of isometries controlled by a Markov chain. Manuscript.</p><p>III. Ådahl M, Melbourne I, Nicol M, Random iteration of Euclidean isometries. Nonlinearity 16 (2003) 977-987.</p><p>IV. Johansson A, Ådahl M, Recurrence of a perturbed random walk and an iterated function system depending on a parameter. Manuscript.</p><p>In the first paper we consider an iterated function system consisting of isometries on an unbounded metric space. Under suitable conditions it is proved that the random orbit {<i>Z</i>n} ^∞_n=0, of the iterations corresponding to an initial point Z₀, “escapes to infinity" in the sense that <i>P</i>(<i>Z</i>n Є <i>K)</i> → 0, as <i>n</i> → ∞ for every bounded set <i>K</i>. As an application we prove the corresponding result in the Euclidean and hyperbolic spaces under the condition that the isometries do not have a common fixed point.</p><p>In the second paper we let a Markov chain control the random orbit of an iterated function system of isometries on an unbounded metric space. We prove under necessary conditions that the random orbit \escapes to infinity" and we also give a simple geometric description of these conditions in the Euclidean and hyperbolic spaces. The results generalises the results of Paper I.</p><p>In the third paper we consider the statistical behaviour of the reversed random orbit corresponding to an iterated function system consisting of a finite number of Euclidean isometries of <b>R</b>n. We give a new proof of the central limit theorem and weak invariance principles, and we obtain the law of the iterated logarithm. Our results generalise immediately to Markov chains. Our proofs are based on dynamical systems theory rather than a purely probabilistic approach.</p><p>In the fourth paper we obtain a suficient condition for the recurrence of a perturbed (one-sided) random walk on the real line. We apply this result to the study of an iterated function system depending on a parameter and defined on the open unit disk in the complex plane. </p>

This thesis consists of four papers, all concerning random iteration of isometries. The papers are: I. Ambroladze A, Ådahl M, Random iteration of isometries in unbounded metric spaces. Nonlinearity 16 (2003) 1107-1117. II. Ådahl M, Random iteration of isometries controlled by a Markov chain. Manuscript. III. Ådahl M, Melbourne I, Nicol M, Random iteration of Euclidean isometries. Nonlinearity 16 (2003) 977-987. IV. Johansson A, Ådahl M, Recurrence of a perturbed random walk and an iterated function system depending on a parameter. Manuscript. In the first paper we consider an iterated function system consisting of isometries on an unbounded metric space. Under suitable conditions it is proved that the random orbit {Zn} ∞n=0, of the iterations corresponding to an initial point Z0, “escapes to infinity" in the sense that P(Zn Є K) → 0, as n → ∞ for every bounded set K. As an application we prove the corresponding result in the Euclidean and hyperbolic spaces under the condition that the isometries do not have a common fixed point. In the second paper we let a Markov chain control the random orbit of an iterated function system of isometries on an unbounded metric space. We prove under necessary conditions that the random orbit \escapes to infinity" and we also give a simple geometric description of these conditions in the Euclidean and hyperbolic spaces. The results generalises the results of Paper I. In the third paper we consider the statistical behaviour of the reversed random orbit corresponding to an iterated function system consisting of a finite number of Euclidean isometries of <b>R</b>n. We give a new proof of the central limit theorem and weak invariance principles, and we obtain the law of the iterated logarithm. Our results generalise immediately to Markov chains. Our proofs are based on dynamical systems theory rather than a purely probabilistic approach. In the fourth paper we obtain a suficient condition for the recurrence of a perturbed (one-sided) random walk on the real line. We apply this result to the study of an iterated function system depending on a parameter and defined on the open unit disk in the complex plane.

Made available in DSpace on 2016-03-15T19:37:38Z (GMT). No. of bitstreams: 1 Glauco dos Santos Reis.pdf: 1334999 bytes, checksum: d2d72d3f95a449c19482f55f82b7f61e (MD5) Previous issue date: 2011-08-22 / Fundo Mackenzie de Pesquisa / A new image compression technique is proposed, based on the affine transformations (ATs) that define an iterated function system (IFS). Previous related research in the field has shown that an image may be approximated by iteratively subjecting a set of sub-regions to a group of ATs. In this case, the original image should be partitioned in regions, and each one of the active pixels are transformed by the AT. The new transformed set should be approximated to other image regions. This iterated execution to find ATs for the best set of areas might result in smaller storage space since the similar areas might be replaced by AT coefficients. Despite this advantage, the technique is computationally intensive, because both the sub-regions and the corresponding ATs that have to be searched for. Here, a new form of similarity is proposed, based on the successive points generated by the iteration of affine transformations. By understanding an AT as a discrete dynamical system, with each image point represented by an iteration of the AT, the method captures similarities between these points, namely, those with the same color in the image; by saving the starting point and the transformations coefficients, the points can be iterated back, to reconstruct the original image. This results in lighter computational effort, since the comparison is made point by point, instead of region by region. Experiments were made on a group of 10 images, representing a broad set of distinct features and resolutions. The proposed algorithm competes in terms of storage size, when compared to JPEG, mainly when the image size is small, and the number of colors are reduced, as currently happens for most images used in the Internet. Although the proposed method is faster than the traditional method for IFS compression, it is slower than common file formats like JPEG. / Uma nova técnica para compressão de imagens é proposta, baseada em conjuntos de transformações afins (affine transformations - ATs), normalmente conhecidos como sistemas de funções iteradas (iterated function system -IFS). Pesquisas anteriores mostraram que uma imagem poderia ser aproximada pela aplicação de um grupo de ATs em conjuntos de sub-regiões da imagem, de forma iterativa. Através deste processo, a imagem original seria subdividida em regiões e sobre a coordenada de cada ponto habilitado de cada região seria aplicada uma transformação afim. O resultado representaria um novo conjunto de pontos similares a outras regiões da imagem. A execução de forma iterada deste processo de identificação das ATs para o maior conjunto de regiões similares de uma determinada imagem permitiria uma redução no armazenamento, já que as regiões similares poderiam ser armazenadas como os coeficientes das transformações afins. Apesar desta vantagem em termos de compressão, a técnica é computacionalmente intensiva, pela busca exaustiva de sub-regiões e das ATs geradoras, de forma a proporcionar o melhor preenchimento em outras regiões da imagem. Esta pesquisa propõe uma nova forma de compressão baseada em ATs, utilizando a sequência de pontos gerada pela iteração das ATs. Entendendo uma AT como um sistema dinâmico em tempo discreto, cada novo ponto identificado é consequência direta da iteração da AT sobre o ponto anterior, permitindo a captura de similaridades nesta sequência de pontos. Através do salvamento dos coeficientes das ATs e das coordenadas iniciais, é possível a reconstrução da imagem pela iteração da AT a partir do ponto inicial. Isto pode resultar em menor esforço computacional, pois apenas comparações simples de pontos são necessárias, ao invés de comparações entre os pontos de regiões da imagem. Foram feitos experimentos em um conjunto de 10 classes de imagens, representando um espectro de diferentes características gerais e resoluções. O algoritmo proposto rivaliza em termos de armazenamento quando comparado ao formato JPEG, principalmente para imagens de pequeno tamanho e com número de cores reduzidas, como as utilizadas com frequência na Internet. Apesar de ser mais rápido para a compressão do que outros métodos baseados em IFS, ele é mais lento do que métodos clássicos como o JPEG.