We introduce the notion of the modulus of fractality and consider the problem of approximation of functions with a restriction on the modulus of fractality by partial sums of trigonometric Fourier series (Fourier sums). The upper estimate of the difference between the function and the corresponding Fourier sum in terms of the modulus of continuity and the modulus of fractality is given. Examples of functions from the considered classes with trigonometric Fourier series diverging at some point are constructed. / Вводится понятие модуля фрактальности и рассматривается задача приближения функций с ограничением на модуль фрактальности частичными суммами тригонометрических рядов Фурье (суммами Фурье). Приведена оценка сверху модуля разности функции и соответствующей суммы Фурье, выраженная в терминах модуля непрерывности и модуля фрактальности. Построены примеры функций из рассматриваемых классов с расходящимся в некоторой точке тригонометрическим рядом Фурье.

Nanoparticles are used in many different applications because of their small size and unique properties. The usage is increasing rapidly, which will increase the nanoparticle exposure to the environment. Up till now, environmental behavior and ecotoxicology of nanoparticles have only been studied to a certain extent and because of the increasing usage, research should focus more on nanoparticle behavior and ecotoxicology. An effective way of studying nanoparticles in aqueous environments is to use mathematical models. In this study, the In vitro Sedimentation, Diffusion, and Dosimetry (ISDD) model was investigated and applied to copper, manganese, and zinc oxide nanoparticles to determine their sedimentation velocity in 1 mM NaClO4(aq).   The results show that the simulated sedimentation of nanoparticles in solution, i.e. the output from the ISDD model, can vary a lot depending on some of the input parameters in the model. The fact that some of these parameters have to be estimated increases the uncertainty of the ISDD model, although it is possible to yield results in great agreement with experimentally determined sedimentation velocities for the studied systems. The simulation results could always be explained by the theory behind it, which increases the reliability of the ISDD model.   The possibility of measuring the effective density of nanoparticle agglomerates using the volumetric centrifugation method was also investigated. This method makes it possible to avoid estimating the fractal dimension, an input parameter with great uncertainty in the ISDD model. The results look promising, although further investigation is needed.    The ISDD model seems to be a promising model for future simulation work. The model should be investigated further in order to minimize the uncertainties due to estimations. The possibility to predict nanoparticle sedimentation using a mathematical model will save a lot of time and money, and it can be a helpful tool in the extensive work of identifying the behavior of nanoparticles in aqueous environments.

Yes / The most recent face recognition systems are mainly dependent on feature representations obtained using either local handcrafted-descriptors, such as local binary patterns (LBP), or use a deep learning approach, such as deep belief network (DBN). However, the former usually suffers from the wide variations in face images, while the latter usually discards the local facial features, which are proven to be important for face recognition. In this paper, a novel framework based on merging the advantages of the local handcrafted feature descriptors with the DBN is proposed to address the face recognition problem in unconstrained conditions. Firstly, a novel multimodal local feature extraction approach based on merging the advantages of the Curvelet transform with Fractal dimension is proposed and termed the Curvelet–Fractal approach. The main motivation of this approach is that theCurvelet transform, a newanisotropic and multidirectional transform, can efficiently represent themain structure of the face (e.g., edges and curves), while the Fractal dimension is one of the most powerful texture descriptors for face images. Secondly, a novel framework is proposed, termed the multimodal deep face recognition (MDFR)framework, to add feature representations by training aDBNon top of the local feature representations instead of the pixel intensity representations. We demonstrate that representations acquired by the proposed MDFR framework are complementary to those acquired by the Curvelet–Fractal approach. Finally, the performance of the proposed approaches has been evaluated by conducting a number of extensive experiments on four large-scale face datasets: the SDUMLA-HMT, FERET, CAS-PEAL-R1, and LFW databases. The results obtained from the proposed approaches outperform other state-of-the-art of approaches (e.g., LBP, DBN, WPCA) by achieving new state-of-the-art results on all the employed datasets.

Yes / n this paper, a powerful face recognition system for authentication and identification tasks is presented and a new facial feature extraction approach is proposed. A novel feature extraction method based on combining the characteristics of the Curvelet transform and Fractal dimension transform is proposed. The proposed system consists of four stages. Firstly, a simple preprocessing algorithm based on a sigmoid function is applied to standardize the intensity dynamic range in the input image. Secondly, a face detection stage based on the Viola-Jones algorithm is used for detecting the face region in the input image. After that, the feature extraction stage using a combination of the Digital Curvelet via wrapping transform and a Fractal Dimension transform is implemented. Finally, the K-Nearest Neighbor (K-NN) and Correlation Coefficient (CC) Classifiers are used in the recognition task. Lastly, the performance of the proposed approach has been tested by carrying out a number of experiments on three well-known datasets with high diversity in the facial expressions: SDUMLA-HMT, Faces96 and UMIST datasets. All the experiments conducted indicate the robustness and the effectiveness of the proposed approach for both authentication and identification tasks compared to other established approaches.

In everyday situations individuals make decisions. For example, a tourist usually chooses a crowded or recommended restaurant to have dinner. Perhaps it is an individual decision, but the observed pattern of decision-making is a collective phenomenon. Collective behaviour emerges from the local interactions that give rise to a complex pattern at the group level. In our example, the recommendations or simple copying the choices of others make a crowded restaurant even more crowded. The rules of interaction between individuals are important to study. Such studies should be complemented by biological experiments. Recent studies of collective phenomena in animal groups help us to understand these rules and develop mathematical models of collective behaviour. The most important communication mechanism is positive feedback between group members, which we observe in our example. In this thesis, we use a generic experimentally validated model of positive feedback to study collective decision-making. The first part of the thesis is based on the modelling of decision-making associated to the selection of feeding sites. This has been extensively studied for ants and slime moulds. The main contribution of our research is to demonstrate how such aspects as "irrationality", speed and quality of decisions can be modelled using differential equations. We study bifurcation phenomena and describe collective patterns above critical values of a bifurcation points in mathematical and biological terms. In the second part, we demonstrate how the primitive unicellular slime mould Physarum Polycephalum provides an easy test-bed for theoretical assumptions and model predictions about decision-making. We study its searching strategies and model decision-making associated to the selection of food options. We also consider the aggregation model to investigate the fractal structure of Physarum Polycephalum plasmodia. / <p>Fel serie i tryckt bok /Wrong series in the printed book</p>

Texture is one of the primary visual attributes used to describe patterns found in nature. Several texture analysis methods have been used as powerful tools for real applications involving analysis and computer vision. However, existing methods do not successfully discriminate the complexity of texture patterns. Such methods disregard the possibility of describing image structures by means of measures such as the fractal dimension. Fractality-based measures allow a non-integer geometric interpretation with applications in areas such as mathematics, physics, and biology. With this gap in mind, the central hypothesis of this thesis is that textures can be described as irregular fractal surfaces due to their complex geometry; such geometry can be exploited for image analysis and computer vision. By exploring such possibilities, pushing the limits of the state-of-the-art, this thesis starts with an analysis of texture features achieved by means of agents on image surfaces. To do so, we used the Bouligand-Minkowski fractal dimension, swarm-system Artificial Crawlers, and non-linear diffusion of Perona-Malik, techniques that led to methodologies with efficacy and efficiency comparable to the state-of-the-art. Our first method combines fractal dimension with random walks on the surface of images. In a second approach, non-linear diffusion is used to represent texture images at different scales, which are described via their fractal dimension for image classification purposes. In a third proposal, we employ fractal dimension concepts over multiple scales derived from the same image for a richer texture description. One of the purposes is the automatic detection of diseases in soybean leaves. Finally, texture characteristics were exploited in a method based on complex networks used to analyze the agglomeration of particles in nanotechnology images. The results achieved in the four methodologies described in this thesis demonstrated the potential of using texture features in tasks of classification and pattern recognition. The contributions of this work shall support significant advances in materials engineering, computer vision, and agriculture. / A textura é um dos principais atributos visuais para a descrição de padrões encontrados na natureza. Diversos métodos de análise de textura têm sido usados como uma poderosa ferramenta para aplicações reais que envolvem análise de imagens e visão computacional. Entretanto, os métodos existentes não conseguem discriminar com sucesso a complexidade dos padrões de textura. Tais métodos desconsideram a possibilidade de se descrever estruturas de imagens por meio de medidas como a dimensão fractal. Medidas baseadas em fractalidade permitem uma interpretação geométrica não-inteira que possui aplicações encontradas em áreas como matemática, física, e biologia. Sobre esta lacuna metodológica, a hipótese central desta tese é que texturas presentes na natureza podem ser medidas como superfícies fractais irregulares devido à sua geometria complexa, o que pode ser explorado para fins de análise de imagens e visão computacional. Para superar tais limitações, avançando o estado da arte, esta tese se inicia com uma análise das características de texturas baseada em caminhadas aleatórias de agentes sobre superfícies de imagens. Esta primeira análise leva a um método que combina dimensão fractal com caminhadas de agentes sobre a superfície de imagens. Em uma segunda abordagem, usa-se a difusão não-linear para representar imagens de texturas em diferentes escalas, as quais são descritas via dimensão fractal para fins de classificação de imagens. Em uma terceira proposta, emprega-se a dimensão fractal sobre múltiplas escalas derivadas de uma mesma imagem com o propósito de se realizar a descrição multi-escala de texturas. Um dos propósitos específicos foi a detecção automática de doenças em folhas de soja. Por último, as características de textura foram exploradas segundo uma metodologia baseada em redes complexas para análise de aglomeração de partículas em imagens de nanotecnologia. Os resultados alcançados nesta tese demonstraram o potencial do uso de características de textura. Para tanto foram usadas técnicas de dimensão fractal de Bouligand-Minkowski, multiagentes Artificial Crawlerse difusão não-linear de Perona-Malik, os quais alcançaram eficácia e eficiência comparáveis ao do estado da arte. As contribuições obtidas devem suportar avanços significativos nas áreas de engenharia de materiais, visão computacional, e agricultura.

O sucesso das cirurgias de implantes depende de diversos fatores como a saúde sistêmica dos pacientes, qualidade e quantidades ósseas disponíveis e, o tipo de macro e microarquitetura dos implantes osseointegráveis. Este estudo propõe avaliar o tecido ósseo em radiografias panorâmicas antes e após cirurgias de implantes correlacionando com análises das Dimensões Fractais (DF), Intensidade de pixel (IP), Índice mentual (IM) e análises de estabilidade primária de implantes (AFR e TI). Foram analisadas 138 regiões de interesse de 50 pacientes, homens e mulheres com idade entre 22 e 77 anos cujas imagens pertencem ao banco de dados do curso de pós-graduação em Implantodontia da Universidade de Santos (Unimes). Foram obtidos dados cirúrgicos do ato da instalação dos implantes, análises de Frequência de Ressonância (AFR) e Torque de Inserção dos implantes (TI) e, com os dados quantitativos obtidos das radiografias panorâmicas antes e após a instalação dos implantes, DF, IP e IM foram realizados testes estatísticos de Correlação de Spearmam, Teste Kruskal Wallis e Wilcoxon para amostras relacionadas com auxílio do software IBM SPSS Advanced Statistics ® (IBM, USA) com nível de significância de 95% (p<0,05). Os resultados obtidos para as Correlações significantes foram entre a Idade e DF (r=-.647) e TI e DF (r=0.434). O teste de Wilcoxon identificou diferença significante (p>0,05) para as amostras relacionadas IP antes e IP depois (p=0,020 e 0,000) entre os grupos 1 e 2. Foram analisadas as possíveis diferenças entre as variáveis independentes para os grupos 1 e 2 em relação a Idade, com diferenças entre IP (p=0,046), TI (p=0,044), AFR_MD (p=0,040) e AFR_VL (p=0,010), em relação ao gênero encontrados diferenças entre as variáveis DF (p=0,039), IP_antes (p=0,001) e IP_depois (p=0,033). O estudo concluiu que a IP pode detectar alterações de mineralização óssea antes e após a instalação de implantes; DF é correlacionada com TI para pacientes jovens (até 45 anos) e, a análise da Idade e Gênero devem ser considerados na avaliação de exames por imagem para planejamento com implantes. / The success of dental implants depends on several factors as patient´s systemic health, bone quality and quantity, geometry of dental implants and their microarquitecture relationship to the alveolar bone. The aim of this study is to assess changes of alveolar bone of panoramic radiographs correlating with Fractal Dimension (FD), Pixel Intensity (PI), Mentual Index (MI) and Implant Stability Quocient (ISQ) measured with Insertion Torque (IT) and Resonance Frequency (RF). 138 regions of interest (ROI) of 50 participants (22 to 77 years old) have been analyzed, all data belonged to the Implantology graduate course of the Universidade de Santos (Unimes). Retrospective surgical data from dental implant surgeries were obtained (RF and IT) and statistically tested with ROI values achieved from DF, PI and MI of panoramic radiographs taken before and after all dental implants placement. Spearman correlation, Kruskal Wallis and Wilcoxon tests were done with the IBM SPSS Advanced Statistics ® software (IBM, USA), level of significance of 95% (p<0,05). Significant correlation was obtained between Age and DF (r=-.647) and TI and DF (r=0.434). Wilcoxon test identified significant difference (p>0,05) for related samples PI before and PI after implant (p=0,020 e 0,000) between groups 1 and 2. Possible differences were analyzed among the independent variables for groups 1 and 2 related to Age, the results showed differences between PI (p=0,046), IT (p=0,044), RF_MD (p=0,040) e RF_VL (p=0,010), between Gender the results showed differences between the variables FD (p=0,039), PI_before (p=0,001) e PI_after (p=0,033). The study concluded that PI can detect bone mineralization changes before and after dental implant placement. FD correlates with the variable IT for younger participants (22 to 45 years old). Age and Gender should be considered regarding imaging exams tests (FD and PI) for dental implant planning.

Neste trabalho, estudamos uma generalização dos semigrupos gradientes, os semigrupos gradiente-like, algumas de suas propriedades e a sua invariância por pequenas perturbações; isto é, pequenas perturbações de sistemas gradiente-like continuam sendo gradiente-like. Como consequência da caracterização dos atratores para este tipo de sistema, estudamos a atração exponencial de atratores. Por fim, estudamos o concetio de dimensão de Hausdorff e dimensão fractal de atratores e apresentamos alguns resultados sobre este assunto, e estudamos a construção de uma nova classe de atratores, os atratores exponenciais fractais / In this work, we study a generalization of gradient discrete semigroups, the gradientlike semigroups, some of its properties and its invariance under small perturbations; that is, small perturbations of gradient-like semigroups are still gradient-like semigroups. As a consequence of the characterization of the attractors for this sort of semigroups, we study the exponential attraction of attractors. Finally, we study some concepts of Hausdorff dimension and fractal dimension and present some results about this subject, and we studied the construction of a new class of attractors, the exponential fractal attractors

A avaliação do estado nutricional das plantas de milho usualmente é feita através de análises químicas ou pela diagnose visual das folhas da planta, esta última, sujeita a erros de interpretação já que a ausência de algum nutriente na planta gera um padrão de mudança específico na superfície da folha que depende do nível de ausência do nutriente. As dificuldades que apresentam neste processo e sua importância na agricultura, criam a necessidade de pesquisar sistemas automáticos para a avaliação do estado nutricional de plantas. Desta forma, este mestrado teve como objetivo principal o desenvolvimento de um sistema de visão artificial para verificar a possibilidade de identificação de níveis dos macronutrientes Cálcio, Enxofre, Magnésio, Nitrogênio e Potássio em plantas de milho através da análise da superfície das folhas usando métodos de visão computacional. Este projeto realiza uma revisão bibliográfica do estado da arte dos métodos de extração de características de cor, textura em escala de cinza e textura colorida utilizadas em processamento de imagens. A alta similaridade entre os sintomas produzidos pelas deficiências e a pouca similaridade entre amostras de uma mesma deficiência motivou o desenvolvimento de novos métodos de extração de características que pudessem fornecer dados necessários para uma correta separação entre as classes. Os resultados obtidos demonstraram que o sistema desenvolvido possibilita a predição de deficiências nutricionais em estágios iniciais do crescimento da planta usando unicamente a textura da superfície da folha como fonte de informação / The evaluation of the nutritional status of corn plants is usually done through chemical analysis or by visual diagnosis of the plant leaves. Visual diagnosis is subject to misinterpretation as the lack of some nutrient in the plant generates a specific pattern of change in the leaf surface that depends on the degree on which the nutrient is absent on the plant. The difficulties present in this process and its importance in agriculture creates the necessity to search automated systems for the assessment of nutritional status of plants. Thus, this dissertation had as main objective the development of an artificial vision system to verify the possibility of identifying levels of macronutrients calcium, sulfur, magnesium, potassium and nitrogen in corn plants by analyzing the surface of the leaves using computer vision methods. This project performs a review of the literature of the state of the art methods for feature extraction of color, grayscale and colored texture used in image processing. The high similarity between the symptoms caused by deficiencies and low similarity between samples of the same deficiency motivated the development of new methods for extracting features that could provide the data needed for a correct separation between classes. The results showed that the system enables the prediction of nutritional deficiencies in an initial stage of plant growth using only texture of the leaf surface as a source of information

Este trabalho apresenta um estudo sobre a morfologia de filmes finos de polianilina, fabricados por diferentes processos e depositados sobre diferentes substratos pela técnica de microscopia de força atômica (AFM). Os processos usados para a fabricação dos filmes foram o de espalhamento fluido, o de centrifugação, o de auto montagem e o eletroquímico. Os substratos usados foram: vidro, ouro/cromo/vidro, quartzo, ITO, silício e mica. Além da análise visual da qualidade dos filmes, ou seja, do grau de recobrimento do substrato e da uniformidade dos filmes, foram feitas análises de parâmetros quantitativos através de medidas de rugosidade. Usando modelos de crescimento como o de deposição balística e de análise de rugosidade, como o da densidade do espectro de potência, (transformada de Fourier da rugosidade da imagem em função da freqüência espectral) em superfícies auto-afins, foram obtidos parâmetros como: expoente de rugosidade, comprimento característico de correlação e dimensão fractal. Apesar dos filmes de polianilina serem sistemas extremamente complexos, - compostos de macromoléculas formando uma estrutura semicristalina - os parâmetros obtidos por esses modelos, que foram elaborados para sistemas bem mais simples, deram resultados muito bons e auto-coerentes. Como conclusão, mostramos que a conjuminação da técnica de microscopia de força atômica com esses modelos compõem um método poderoso para o estudo de filmes finos orgânicos. / This work presents a detailed study about the characteristic of thin films of polyaniline, fabricated by different methods and deposited over different substrates using Atomic Force Microscopy technique (AFM). The processes used for film depositions were casting, spin-coating, self-assembly, and the electrochemistry methods. The substrata used were: glass plates, gold/chromium/glass, quartz, ITO, silicon, and mica. Besides the visual analysis, that observes the film uniformity and verifies the coating degree of the substratum, the application of some mathematical models was employed to obtain quantitative parameters about the roughness of the film surfaces. From the ballistic deposition model, developed to explain growing processes, and the power spectrum density that takes the Fourier transform of the roughness in function of the spectral frequency, both developed for self-affine surfaces, it was obtained parameters as: the roughness exponent, the correlation characteristic length, and the fractal dimension. Despite the highly complex structure of the polyaniline - composed by macromolecules forming a semicrystalline structure- the obtained parameters by such models developed for very simple systems, provided reasonable and self-coherent results. As a final result of this work we showed that the synergy between the AFM technique and the above mentioned models compose a powerful method for the study of thin organic films.