Made available in DSpace on 2016-04-27T16:57:17Z (GMT). No. of bitstreams: 1 Juliano de Paula Mineli.pdf: 2480017 bytes, checksum: e6777d50d1d15cbae78249914a247c2f (MD5) Previous issue date: 2012-06-15 / This research aimed to study the formation of skills of elementary school students to solve problems, or more specifically, problems whose solutions can be expressed by 1st order equations. My teaching experience has made me raise a hypothesis that the difficulties that students encountered in the organization of the equation, related to the situations of Generalization of Patterns. From then define the objectives of this research is to: investigate problems related to skills training for the Generalization of Patterns and teaching situations to facilitate the training. For data collection, drafted a sequence containing three Curriculum Activities supported on the elements of Fractal Geometry, with a view to generalization of Standards and their implementation based on the Theory of Didactic Situations of Brousseau - theoretical framework used in this research.This sequence was applied in seven sessions to a class of 7th grade (6th grade) of a private Institution located in the city of Ribeirão Preto.This is a qualitative research study using a methodology supported by the Didactic Engineering. The observations and analyzes showed that: the characteristics of fractal figures provide pattern recognition, pattern recognition encourages students to make generalization, the use of geometric shapes provides better organization of data from a mathematical problem, which favors their resolution; generalization of a given situation helps the development of algebraic thinking as well as their attitudes and autonomy in order to observe, raising conjectures, draw conclusions and justify their answers / Esta pesquisa objetivou estudar formação de habilidades de estudantes do Ensino Fundamental para resolver problemas, ou mais especificamente problemas cujas soluções podem ser expressas por equações do 1º grau. Minha experiência docente me fez levantar uma hipótese de que as dificuldades que os estudantes encontravam, na organização da equação, relacionavam-se à situações de Generalização de Padrões. A partir daí definimos os objetivos desta pesquisa que são: investigar dificuldades relacionadas à formação de habilidades para a Generalização de Padrões e de situações de ensino facilitadoras para essa formação. Para a coleta de dados, elaborei uma Sequência Didática contendo três Atividades apoiadas nos elementos da Geometria Fractal, tendo em vista a Generalização de Padrões e sua aplicação baseada na Teoria das Situações Didáticas de Brousseau referenciais teóricos utilizados nesta pesquisa. Essa Sequência foi aplicada em sete sessões a uma turma do 7° ano (6ª série) de uma instituição privada situada na cidade de Ribeirão Preto. Trata-se de uma pesquisa de caráter qualitativo com metodologia apoiada na Engenharia Didática. Como conclusão podemos apontar que: as características das figuras fractais propiciam o reconhecimento de padrões; o reconhecimento de padrões favorece ao aluno realizar generalização; a utilização de figuras geométricas propicia melhor organização dos dados de um problema matemático, o que favorece sua resolução; a generalização de uma dada situação auxilia o desenvolvimento do pensamento algébrico, bem como suas atitudes e autonomia no sentido de observar, levantar conjecturas, tirar conclusões e justificar suas respostas

Made available in DSpace on 2016-04-27T16:57:45Z (GMT). No. of bitstreams: 1 EDM - Arlete Ap O Almeida.pdf: 11456266 bytes, checksum: 26513687e3c57ad3afc46f05972c39ca (MD5) Previous issue date: 2006-10-31 / This work had as purpose to do research on the approach to the learning of geometry of the Fractals under the view of the teacher and your understanding to the development of a project didactic about the concept of Fractals. The research was done with a group of public school teachers and it was developed in two meeting. These short meetings to the formation of teachers was to show if short meetings could influence on pedagogical practice and to introduce new subsidies to teaching-learning, overcoming obstacles and offering a new possibility of appropriation of knowledge in mathematic. We have intention to make a review these works; we ll go to offer a new adventure to become the geometry more near the reality of our students and to create a real identification with the nature and the geometry, by the project didactic about the concept of Fractals that was created also to application subsequent at the students. Our work also showed that the teacher is open-minded to new suggestions, and he needs just opportunity and moments to reflect on your pedagogical practice. It s this that will try show in this work, giving a historical prospect and the importance to put in evidence the value of the geometry of the fractals to represent better the nature forms. In this first moment, we leave the calculation more complex out of the work, because the central idea is give information about this new field of mathematic, which is so interesting and dazzling / Este trabalho teve como objetivo investigar uma abordagem para o ensino da geometria dos Fractais sob o olhar do professor e sua receptividade no desenvolvimento de uma seqüência didática para abordar o conceito de Fractais. A pesquisa foi realizada junto a um grupo de professores da Rede Pública Estadual, desenvolvida em dois encontros. Tínhamos como hipótese do trabalho, se um mini-curso para professores poderia influenciar na prática docente sugerindo novas propostas de trabalho, passando por obstáculos da aceitação para chegar às salas de aula, proporcionando uma nova visão da aplicação da matemática. Com o intuito de rever estes estudos, é que iremos proporcionar uma nova aventura para tornar mais próxima a geometria da realidade do nosso aluno, promovendo uma identificação real com a natureza e a geometria, por meio de uma seqüência didática que foi elaborada visando também sua aplicação posterior aos alunos. É isto que tentaremos mostrar nesse trabalho, dando um panorama histórico e a necessidade do surgimento da geometria dos fractais para representar melhor as formas da natureza. Nesse primeiro momento, nos poupamos dos cálculos mais complexos, pois a idéia central foi fornecer informações sobre essa nova área da matemática que é tão envolvente e deslumbrante. Nossa análise aponta que o professor está receptivo às novas propostas, precisando apenas de oportunidade e espaço para reflexão sobre suas ações

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.

Due to the increasing amount and complexity of data processed in real world applications, similarity search became a vital task to store and retrieve such data. However, missing attribute values are very frequent and metric access methods (MAMs), designed to support similarity search, do not operate on datasets when attribute values are missing. Currently, the approach to use the existing indexing techniques on datasets with missing attribute values just use an indicator to identify the missing values and employ a traditional indexing technique. Although, this approach can be applied over multidimensional indexing techniques, it is impractical for metric access methods. This dissertation presents the results of a research conducted to identify and deal with the issues related to indexing and querying datasets with missing values in metric spaces. An empirical analysis of the metric access methods when applied on incomplete datasets leads us to identify two main issues: distortion of the internal structure of the index when data are missing at random and skew of the index structure when data are not missing at random. Based on those findings, a new variant of the Slim-tree access method, called Hollow-tree, is presented. It employs new techniques that are capable to handle missing data issues when missingness is ignorable. The first technique includes a set of indexing policies that allow to index objects with missing attribute values and prevent distortions to occur in the internal structure of the indexes. The second technique targets the similarity queries to improve the query performance over incomplete datasets. This technique employs the fractal dimension of the dataset and the local density around the query object to estimate an ideal radius able to achieve an accurate query answer, considering data with missing values as a potential response. Results from experiments with a variety of real and synthetic datasets show that Hollow-tree achieves nearly 100% of precision and recall for Range queries and more than 90% for k Nearest Neighbor queries, while Slim-tree access method deteriorates with the increasing amount of missing values. The results confirm that the indexing technique helps to establish consistency in the index structure and the searching technique achieves a remarkable performance. When combined, the new techniques allow to explore properly all the available data even with high amounts of missing attribute values. As they are independent of the underlying access method, they can be adopted by a broad range of metric access methods, allowing to extend the class of MAMs. / O crescimento em quantidade e complexidade dos dados processados e armazenados torna a busca por similaridade uma tarefa fundamental para tratar esses dados. No entanto, atributos faltantes ocorrem freqüentemente, inviabilizando os métodos de acesso métricos (MAMs) projetados para apoiar a busca por similaridade. Assim, técnicas de tratamento de dados faltantes precisam ser desenvolvidas. A abordagem mais comum para executar as técnicas de indexação existentes sobre conjuntos de dados com valores faltantes é usar um indicador de valores faltantes e usar as técnicas de indexação tradicionais. Embora, esta técnica seja útil para os métodos de indexação multidimensionais, é impraticável para os métodos de acesso métricos. Esta dissertação apresenta os resultados da pesquisa realizada para identificar e lidar com os problemas de indexação e recuperação de dados em espaços métricos com valores faltantes. Uma análise experimental dos MAMs aplicados a conjuntos de dados incompletos identificou dois problemas principais: distorção na estrutura interna do índice quando a falta é aleatória e busca tendenciosa na estrutura do índice quando o processo de falta não é aleatório. Uma variante do MAM Slim-tree, chamada Hollow-tree foi proposta com base nestes resultados. A Hollow-tree usa novas técnicas de indexação e de recuperação de dados com valores faltantes quando o processo de falta é aleatório. A técnica de indexação inclui um conjunto de políticas de indexação que visam a evitar distorções na estrutura interna dos índices. A técnica de recuperação de dados melhora o desempenho das consultas por similaridade sobre bases de dados incompletas. Essas técnicas utilizam o conceito de dimensão fractal do conjunto de dados e a densidade local da região de busca para estimar um raio de busca ideal para obter uma resposta mais correta, considerando os dados com valores faltantes como uma resposta potencial. As técnicas propostas foram avaliadas sobre diversos conjuntos de dados reais e sintéticos. Os resultados mostram que a Hollow-tree atinge quase 100% de precisão e revocação para consultas por abrangência e mais de 90% para k vizinhos mais próximos, enquanto a Slim-tree rapidamente deteriora com o aumento da quantidade de valores faltantes. Tais resultados indicam que a técnica de indexação proposta ajuda a estabelecer a consistência na estrutura do índice e a técnica de busca pode ser realizada com um desempenho notável. As técnicas propostas são independentes do MAM básico usado e podem ser aplicadas em uma grande variedade deles, permitindo estender a classe dos MAMs em geral para tratar dados faltantes.

A complexidade é uma característica de grande importância em processos de reconhecimento de padrões, especialmente naqueles que envolvem imagens biológicas. Este trabalho tem como objetivo estudar métodos que realizam a análise de imagens por meio da análise de sua complexidade. Os métodos a serem estudados foram selecionados com base na similaridade de seus algoritmos e metodologia: dimensão fractal, Caminhada Determinística do Turista e Redes Complexas. Estes métodos permitem realizar a análise e segmentação de formas ou texturas contidas em uma imagem com base na sua variação de complexidade. Dos três métodos considerados, dois deles fazem parte do estado da arte em análise de complexidade, enquanto que a dimensão fractal já é aplicada a mais tempo na análise de formas e texturas. Os trabalhos aqui desenvolvidos visam comparar e analisar os métodos selecionados por meio de experimentos com imagens de forma e texturas, sendo utilizadas texturas naturais e de Brodatz, freqüentemente utilizadas na literatura como benchmark para texturas. Com base no conhecimento adquirido, novas técnicas voltadas para a análise e segmentação de formas e texturas foram desenvolvidas, assim como foram analisadas as deficiências e propostas melhorias às técnicas estudadas. Além disso, diversos experimentos com estas metodologias foram realizados em aplicações de Bioinformática / Complexity is a feature of great importance in pattern recognition processes, especially those involving biological images. This work aims to study methods that perform image analysis by the analysis of its complexity. The methods to be studied were selected based on similarity of their algorithms and methodology: fractal dimension, Deterministic Tourist Walk and Complex Networks. These methods enable us to perform the analysis and segmentation of shapes and textures contained in an image based on the variation of its complexity. Of the three methods considered, two of them are part of the state of the art in complexity analysis, while the fractal dimension is already applied in shapes and textures analysis. The work developed here aims to compare and analyze the selected methods through experiments with shape and texture images, utilizing for this natural and Brodatz textures samples, often used in literature as benchmark for textures analysis. Based on the knowledge acquired, new techniques for analysis and segmentation of shapes and textures were developed, as also were analyzed the deficiencies and proposed improvements to the techniques studied. Moreover, several experiments with these methods were performed in bioinformatics applications

FundaÃÃo de Amparo à Pesquisa do Estado do Cearà / CoordenaÃÃo de AperfeiÃoamento de NÃvel Superior / Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Neste trabalho propomos dois modelos para a geraÃÃo de fraturas em substratos regulares. No primeiro modelo, empregamos iterativamente o conceito de caminho crÃtico para determinar sistematicamente o elemento de menor âcondutividadeâ da rede. Estes elementos sÃo entÃo identificados como âfalhasâ e removidos permanentemente da estrutura atà que uma fratura macroscÃpica destrua a conectividade global da rede. Uma vez detectada, esta fratura à caracterizada topologicamente como uma estrutura auto-similar de dimensÃo fractal Dp ≈ 1.21. No segundo modelo, empregamos iterativamente o algoritmo de Dijkstra para determinar o caminho mÃnimo em uma paisagem aleatÃria, retirando sistematicamente desta estrutura o elemento de maior energia. Como no modelo anterior, estes elementos sÃo identificados como âfalhasâ atà que um conjunto conecto deles resulte em uma fratura macroscÃpica. A mÃdia realizada sobre vÃrias amostras de fraturas em diferentes tamanhos de substratos revela a presenÃa de uma estrutura auto-similar de dimensÃo fractal Df ≈ 1.21. A semelhanÃa numÃrica entre os expoentes Dp e Df sugere que os dois modelos pertencem à mesma classe de universalidade. / In this work we propose two models for fracture generation in regular substrates. In the first model, we iteratively apply the concept of critical path to systematically determine the lower âconductivityâ element in the connected spanning network. At each iteration, once these elements are identified as local âcracks ́ ́, they are permanently removed from the structure up to the point in which a macroscopic fracture can destroy the global network connectivity. This fracture is then topologically characterized as self-similar with fractal dimension Dp ≈ 1.21. In the second model, we employ the algorithm of Dijkstra to determine the minimal path in a random energy landscape and remove its highest energy element. As in the previous model, these elements are considered to be local âcracks ́ ́ till a subset of them can be identified as a macroscopic fracture. The average over many samples of fractures calculated for different system sizes reveals the presence of a self-similar structure with fractal dimension Df ≈ 1.21. The resemblance between the two exponents Dp e Df suggests that the two models belong to the same universality class.

Made available in DSpace on 2017-07-10T19:24:55Z (GMT). No. of bitstreams: 1 Gilberto Comiran.pdf: 1846926 bytes, checksum: 1b96f9d3924516f3c3a1814a89cd0014 (MD5) Previous issue date: 2006-07-21 / In this work, it is shown an application of a model to obtain the hydraulic conductivity of soils no saturated using concepts Fractal Geometry and the laws of Laplace and of Poiseuille. For the obtaining of the soil water retention curve it was used the camera of pressure of Richards and the experimental data of hydraulic conductivity in function of the soil water content were obtained through the Method of Hillel with the tensiometry use, according to a potency model. The equation van Genuchten with their parameters was used for the obtaining of the hydraulic conductivity. The application of the models of the Geometric Average and of the Neutral Pore suggested by Fuentes was shown simple and capable of predicting the hydraulic conductivity satisfactorily. / Neste trabalho, apresenta-se a aplicação de um modelo para se estimar a condutividade hidráulica de solos não saturados, utilizando os conceitos da Geometria Fractal e das leis de Laplace e de Poiseuille. Para a obtenção da curva de retenção de água, foi utilizada a câmara de pressão de Richards e os dados experimentais de condutividade hidráulica, em função da umidade do solo, foram obtidos através do Método de Hillel com a utilização de tensiometria, conforme um modelo de potência. A equação de van Genuchten, com seus parâmetros, foi utilizada para a obtenção da condutividade hidráulica. A aplicação dos modelos da Média Geométrica e do Poro Neutro sugerido por Fuentes mostrou-se simples e capaz de predizer a condutividade hidráulica satisfatoriamente.

Made available in DSpace on 2017-05-12T14:48:17Z (GMT). No. of bitstreams: 1 Gilberto Comiran.pdf: 1846926 bytes, checksum: 1b96f9d3924516f3c3a1814a89cd0014 (MD5) Previous issue date: 2006-07-21 / In this work, it is shown an application of a model to obtain the hydraulic conductivity of soils no saturated using concepts Fractal Geometry and the laws of Laplace and of Poiseuille. For the obtaining of the soil water retention curve it was used the camera of pressure of Richards and the experimental data of hydraulic conductivity in function of the soil water content were obtained through the Method of Hillel with the tensiometry use, according to a potency model. The equation van Genuchten with their parameters was used for the obtaining of the hydraulic conductivity. The application of the models of the Geometric Average and of the Neutral Pore suggested by Fuentes was shown simple and capable of predicting the hydraulic conductivity satisfactorily. / Neste trabalho, apresenta-se a aplicação de um modelo para se estimar a condutividade hidráulica de solos não saturados, utilizando os conceitos da Geometria Fractal e das leis de Laplace e de Poiseuille. Para a obtenção da curva de retenção de água, foi utilizada a câmara de pressão de Richards e os dados experimentais de condutividade hidráulica, em função da umidade do solo, foram obtidos através do Método de Hillel com a utilização de tensiometria, conforme um modelo de potência. A equação de van Genuchten, com seus parâmetros, foi utilizada para a obtenção da condutividade hidráulica. A aplicação dos modelos da Média Geométrica e do Poro Neutro sugerido por Fuentes mostrou-se simples e capaz de predizer a condutividade hidráulica satisfatoriamente.

La génération de formes naturelles a été le sujet de nombreuses recherches depuis plusieurs années. Plusieurs méthodes ont été proposées afin de générer des objets naturels et réalistes tels que des terrains, des plantes et arbres, des nuages, etc. Les modèles itératifs sont très connus dans ce domaine de recherche grâce à leur capacité à générer des formes rugueuses et complexes qui sont adaptées à la représentation d’objets naturels. L’inconvénient majeur de tels modèles est le manque de contrôle sur le résultat final. Ce dernier peut venir de la méthode de construction stochastique interdisant tout contrôle par définition. Pour les modèles dont la construction est déterministe, les paramètres de générations sont souvent non intuitifs et limitent ainsi le contrôle. Pour ces raisons un grand nombre de recherches ont port sur le problème du contrôle de ces modèles ainsi que sur la possibilité d’utiliser des modèles non-itératifs (esquisses, basés exemples, etc.). Bien souvent, le contrôle introduit par ces modèles est un contrôle global, c’est-à-dire sur la totalité de l’objet final et ne prend donc pas en compte les détails locaux de ce dernier. Dans notre travail, nous nous attaquons au problème du contrôle sur les formes naturelles en tenant compte du contrôle local. À cette fin, nous introduisons deux modèles différents. Le premier repose sur un formalisme itératif avec notion de détail qui se dicline en deux sous-familles, l’une basée sur les IFS et l’autre basée sur les surfaces de subdivision. Le deuxième modèle permet l’édition de caractéristiques d’un terrain sous forme de primitives vectorielles puis la génération du terrain par une méthode de diffusion guidée. Cette dernière fait l’objet d’une implémentation parallèle sur la carte graphique (GPU). / The generation of natural shapes has been the subject of much research for many years. Several methods have been proposed to generate realistic natural objects such as terrain, plants and trees, clouds, etc... Iterative models are well known in this field of research due to their ability to generate complex and rough shapes that are adapted to the representation of natural objects. The major drawback of such models is the lack of control over the final result. The latter can come from the stochastic construction method which prevents any control by definition. For models whose construction is deterministic, the parameters of generation are often non-intuitive and thus limit control. For these reasons many studieshave focused on the problem of controlling these models as well as the possibility of using non-iterative models (sketches, based on examples, etc). Often, control introduced by these models is a global control, on the whole final object and therefore does not include local details of this object. In our work, we focus in the problem of control over natural shapes, taking into account local control. To this end, we introduce two different models. The first is based on an iterative formalism with detail concept which is divided into two subfamilies, one based on IFS and the other one based on subdivisionsurfaces. The second model allows the editing of terrain features under a form of vectorial primitives which one used to generate the terrain by guided diffusion method. The latter is the subject of a parallel implementation on graphics card (GPU).

Human face recognition is an important area in the field of biometrics. It has been an active area of research for several decades, but still remains a challenging problem because of the complexity of the human face. In this thesis we describe fully automatic solutions that can locate faces and then perform identification and verification. We present a solution for face localisation using eye locations. We derive an efficient representation for the decision hyperplane of linear and nonlinear Support Vector Machines (SVMs). For this we introduce the novel concept of $\rho$ and $\eta$ prototypes. The standard formulation for the decision hyperplane is reformulated and expressed in terms of the two prototypes. Different kernels are treated separately to achieve further classification efficiency and to facilitate its adaptation to operate with the fast Fourier transform to achieve fast eye detection. Using the eye locations, we extract and normalise the face for size and in-plane rotations. Our method produces a more efficient representation of the SVM decision hyperplane than the well-known reduced set methods. As a result, our eye detection subsystem is faster and more accurate. The use of fractals and fractal image coding for object recognition has been proposed and used by others. Fractal codes have been used as features for recognition, but we need to take into account the distance between codes, and to ensure the continuity of the parameters of the code. We use a method based on fractal image coding for recognition, which we call the Fractal Neighbour Distance (FND). The FND relies on the Euclidean metric and the uniqueness of the attractor of a fractal code. An advantage of using the FND over fractal codes as features is that we do not have to worry about the uniqueness of, and distance between, codes. We only require the uniqueness of the attractor, which is already an implied property of a properly generated fractal code. Similar methods to the FND have been proposed by others, but what distinguishes our work from the rest is that we investigate the FND in greater detail and use our findings to improve the recognition rate. Our investigations reveal that the FND has some inherent invariance to translation, scale, rotation and changes to illumination. These invariances are image dependent and are affected by fractal encoding parameters. The parameters that have the greatest effect on recognition accuracy are the contrast scaling factor, luminance shift factor and the type of range block partitioning. The contrast scaling factor affect the convergence and eventual convergence rate of a fractal decoding process. We propose a novel method of controlling the convergence rate by altering the contrast scaling factor in a controlled manner, which has not been possible before. This helped us improve the recognition rate because under certain conditions better results are achievable from using a slower rate of convergence. We also investigate the effects of varying the luminance shift factor, and examine three different types of range block partitioning schemes. They are Quad-tree, HV and uniform partitioning. We performed experiments using various face datasets, and the results show that our method indeed performs better than many accepted methods such as eigenfaces. The experiments also show that the FND based classifier increases the separation between classes. The standard FND is further improved by incorporating the use of localised weights. A local search algorithm is introduced to find a best matching local feature using this locally weighted FND. The scores from a set of these locally weighted FND operations are then combined to obtain a global score, which is used as a measure of the similarity between two face images. Each local FND operation possesses the distortion invariant properties described above. Combined with the search procedure, the method has the potential to be invariant to a larger class of non-linear distortions. We also present a set of locally weighted FNDs that concentrate around the upper part of the face encompassing the eyes and nose. This design was motivated by the fact that the region around the eyes has more information for discrimination. Better performance is achieved by using different sets of weights for identification and verification. For facial verification, performance is further improved by using normalised scores and client specific thresholding. In this case, our results are competitive with current state-of-the-art methods, and in some cases outperform all those to which they were compared. For facial identification, under some conditions the weighted FND performs better than the standard FND. However, the weighted FND still has its short comings when some datasets are used, where its performance is not much better than the standard FND. To alleviate this problem we introduce a voting scheme that operates with normalised versions of the weighted FND. Although there are no improvements at lower matching ranks using this method, there are significant improvements for larger matching ranks. Our methods offer advantages over some well-accepted approaches such as eigenfaces, neural networks and those that use statistical learning theory. Some of the advantages are: new faces can be enrolled without re-training involving the whole database; faces can be removed from the database without the need for re-training; there are inherent invariances to face distortions; it is relatively simple to implement; and it is not model-based so there are no model parameters that need to be tweaked.