In this thesis, fixed point theory is used to construct a fractal type sets and to solve data classification problem. Fixed point method, which is a beautiful mixture of analysis, topology, and geometry has been revealed as a very powerful and important tool in the study of nonlinear phenomena. The existence of fixed points is therefore of paramount importance in several areas of mathematics and other sciences. In particular, fixed points techniques have been applied in such diverse fields as biology, chemistry, economics, engineering, game theory and physics. In Chapter 2 of this thesis it is demonstrated how to define and construct a fractal type sets with the help of iterations of a finite family of generalized F-contraction mappings, a class of mappings more general than contraction mappings, defined in the context of b-metric space. This leads to a variety of results for iterated function system satisfying a different set of contractive conditions. The results unify, generalize and extend various results in the existing literature. In Chapter 3, the theory of support vector machine for linear and nonlinear classification of data and the notion of generalized support vector machine is considered. In the thesis it is also shown that the problem of generalized support vector machine can be considered in the framework of generalized variation inequalities and results on the existence of solutions are established. / FUSION

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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.

京都大学 / 0048 / 新制・課程博士 / 博士(人間・環境学) / 甲第22534号 / 人博第937号 / 新制||人||223(附属図書館) / 2019||人博||937(吉田南総合図書館) / 京都大学大学院人間・環境学研究科共生人間学専攻 / (主査)教授 角 大輝, 教授 上木 直昌, 准教授 木坂 正史 / 学位規則第4条第1項該当 / Doctor of Human and Environmental Studies / Kyoto University / DFAM

京都大学 / 新制・課程博士 / 博士(人間・環境学) / 甲第23984号 / 人博第1036号 / 新制||人||243(附属図書館) / 2022||人博||1036(吉田南総合図書館) / 京都大学大学院人間・環境学研究科共生人間学専攻 / (主査)教授 角 大輝, 教授 上木 直昌, 准教授 木坂 正史, 教授 宍倉 光広 / 学位規則第4条第1項該当 / Doctor of Human and Environmental Studies / Kyoto University / DFAM

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.

<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.

The aim of this thesis is to develop the dimension theory of self-affine carpets in several directions. Self-affine carpets are an important class of planar self-affine sets which have received a great deal of attention in the literature on fractal geometry over the last 30 years. These constructions are important for several reasons. In particular, they provide a bridge between the relatively well-understood world of self-similar sets and the far from understood world of general self-affine sets. These carpets are designed in such a way as to facilitate the computation of their dimensions, and they display many interesting and surprising features which the simpler self-similar constructions do not have. For example, they can have distinct Hausdorff and packing dimensions and the Hausdorff and packing measures are typically infinite in the critical dimensions. Furthermore, they often provide exceptions to the seminal result of Falconer from 1988 which gives the `generic' dimensions of self-affine sets in a natural setting. The work in this thesis will be based on five research papers I wrote during my time as a PhD student. The first contribution of this thesis will be to introduce a new class of self-affine carpets, which we call box-like self-affine sets, and compute their box and packing dimensions via a modified singular value function. This not only generalises current results on self-affine carpets, but also helps to reconcile the `exceptional constructions' with Falconer's singular value function approach in the generic case. This will appear in Chapter 2 and is based on a paper which appeared in 'Nonlinearity' in 2012. In Chapter 3 we continue studying the dimension theory of self-affine sets by computing the Assouad and lower dimensions of certain classes. The Assouad and lower dimensions have not received much attention in the literature on fractals to date and their importance has been more related to quasi-conformal maps and embeddability problems. This appears to be changing, however, and so our results constitute a timely and important contribution to a growing body of literature on the subject. The material in this Chapter will be based on a paper which has been accepted for publication in 'Transactions of the American Mathematical Society'. In Chapters 4-6 we move away from the classical setting of iterated function systems to consider two more exotic constructions, namely, inhomogeneous attractors and random 1-variable attractors, with the aim of developing the dimension theory of self-affine carpets in these directions. In order to put our work into context, in Chapter 4 we consider inhomogeneous self-similar sets and significantly generalise the results on box dimensions obtained by Olsen and Snigireva, answering several questions posed in the literature in the process. We then move to the self-affine setting and, in Chapter 5, investigate the dimensions of inhomogeneous self-affine carpets and prove that new phenomena can occur in this setting which do not occur in the setting of self-similar sets. The material in Chapter 4 will be based on a paper which appeared in 'Studia Mathematica' in 2012, and the material in Chapter 5 is based on a paper, which is in preparation. Finally, in Chapter 6 we consider random self-affine sets. The traditional approach to random iterated function systems is probabilistic, but here we allow the randomness in the construction to be provided by the topological structure of the sample space, employing ideas from Baire category. We are able to obtain very general results in this setting, relaxing the conditions on the maps from `affine' to `bi-Lipschitz'. In order to get precise results on the Hausdorff and packing measures of typical attractors, we need to specialise to the setting of random self-similar sets and we show again that several interesting and new phenomena can occur when we relax to the setting of random self-affine carpets. The material in this Chapter will be based on a paper which has been accepted for publication by 'Ergodic Theory and Dynamical Systems'.

O objetivo do presente trabalho foi propor métodos de processamento de sinais e reconhecimento de padrões dos sinais de respostas de sensores de gás, utilizando técnicas e modelos da geometria fractal. Foram analisados e estudados os sinais de resposta de dois tipos de sensores. O primeiro sensor foi um dispositivo de óxido de estanho, cujo princípio de funcionamento baseia-se na mudança da resistividade do filme. Este forneceu sinais de respostas com características ruidosas como resposta à interação com as moléculas de gás. O segundo sensor foi um dispositivo Metal-Óxido-Semicondutor (MOS) com princípio de funcionamento baseado na geração de foto corrente, fornecendo respostas imagens bidimensionais. Para as análises dos sinais ruidosos do sensor de óxido de estanho, foi proposto um método de processamento baseado no modelo do movimento Browniano fracionário. Com este método foi possível a discriminação de gases combustíveis com uma taxa de acerto igual a 100%. Para as análises das respostas do tipo imagem do sensor MOS, foram propostos dois diferentes métodos. O primeiro foi embasado no princípio de compressão fractal de imagens e o segundo método proposto, foi baseado na análise e determinação da dimensão fractal multiescala. Ambos os métodos propostos mostram-se eficazes para a determinação da assinatura, como o reconhecimento, de todos os gases que foram utilizados nos experimentos. Os resultados obtidos no presente trabalho abrem novas fronteiras e perspectivas nos paradigmas de processamento de sinais e reconhecimento de padrões, quando utilizada a teoria da geometria fractal. / The aim of the present work was to propose methods for signal possessing and pattern, recognition from the signals response of gas sensors using models and techniques from the fractal geometry. The data studied and analyzed were obtained from two kinds of sensors. The first sensor was the tin oxide device, which detection principle is based on the resistivity changes of the tin oxide film and it provides noisy signals as response to the gas interaction. The second sensor was a metal-oxide-semiconductor (MOS) device, which has as the working principle the photocurrent generation. This sensor provides two-dimensional images signals. A method using a fractional Brownian motion was proposed to analyze the noise signal from the tin oxide device. The fuel gases discrimination employing this model was 100% successful. Two different methods were proposed to analyze the signal response from the MOS device. The first method was based on the fractal image compression technique and the second one was based on the analysis and determination of the multiscale fractal dimension. Both proposed methods have shown to be efficient tools for signature determination as the pattern recognition of all gases that were used in the experiment. The results obtained in the present work open new frontiers and perspectives inside the paradigms of the signal processing and pattern recognition by using the fractal theory.