Submitted by Izabel Franco (izabel-franco@ufscar.br) on 2016-09-23T18:54:56Z No. of bitstreams: 1 DissREN.pdf: 3941226 bytes, checksum: 8164c389310d09352cd0a12f8d23b722 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-09-26T20:43:52Z (GMT) No. of bitstreams: 1 DissREN.pdf: 3941226 bytes, checksum: 8164c389310d09352cd0a12f8d23b722 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-09-26T20:43:59Z (GMT) No. of bitstreams: 1 DissREN.pdf: 3941226 bytes, checksum: 8164c389310d09352cd0a12f8d23b722 (MD5) / Made available in DSpace on 2016-09-26T20:44:07Z (GMT). No. of bitstreams: 1 DissREN.pdf: 3941226 bytes, checksum: 8164c389310d09352cd0a12f8d23b722 (MD5) Previous issue date: 2015-08-21 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / This work handles topics involving geometric transformations and friezes for the study of symmetry. It aimed to present students from the Ninth Grade of a public school in São Paulo State activities, in order to enrich them scientifically, culturally; sharpening their curiosity and critical reasoning. For this, a series of activities related to the theme symmetry was developed: first of all, students solved questions on a diagnosing sheet of paper about translation, reflection and rotation movements on the plan, followed by a slide presentation of these same topics. Then, they watched a video about Alhambra Palace in Spain to motivate them to study the symmetries in decorative friezes. Students solved various problems of OBMEP involving symmetrical situations and they ended up the work building seven patterns of friezes. The analysis of these solutions is presented in order to verify whether there was a contribution to the learning of geometric transformations related to symmetries. Such analysis is based on the researcher’s experience and also on testimonials from the participants. / Este trabalho aborda questões que envolvem transformações geométricas e frisos para o estudo de simetrias. A intenção principal foi a de apresentar aos alunos do nono ano de uma escola pública do interior paulista atividades com o intuito de enriquecê-los cientificamente e culturalmente, aguçando-lhes a curiosidade e o espírito crítico. Para isto, foi desenvolvida uma série de atividades ligadas ao tema simetria: inicialmente os alunos resolveram questões de uma folha atividade diagnóstica sobre os movimentos de translação, reflexão e rotação no plano, seguido da apresentação de slides sobre esses mesmos assuntos. Em continuidade, assistiram a um vídeo sobre o palácio da Alhambra na Espanha para motivá-los ao estudo das simetrias em frisos decorativos. Os alunos resolveram vários problemas da OBMEP (Olimpíada Brasileira de Matemática das Escolas Públicas) que envolvem situações simétricas e terminaram o trabalho construindo os sete tipos de padrões de frisos. A análise dessas soluções é apresentada com vistas a verificar se houve uma contribuição na aprendizagem das transformações geométricas ligadas às simetrias. Esta análise é fundamentada na experiência do pesquisador e também em depoimentos dos alunos participantes.

Submitted by Ana Paula Inforsato (ana.inforsato@gmail.com) on 2018-05-21T19:31:12Z No. of bitstreams: 1 Ana Paula final.pdf: 2233592 bytes, checksum: 79e4da8a53734795b00c77eb03daa0af (MD5) / Approved for entry into archive by Ana Paula Santulo Custódio de Medeiros null (asantulo@rc.unesp.br) on 2018-05-22T13:00:36Z (GMT) No. of bitstreams: 1 inforsato_ap_me_rcla.pdf: 2232239 bytes, checksum: 6e5ef796e47228af1164f8da713ea8de (MD5) / Made available in DSpace on 2018-05-22T13:00:36Z (GMT). No. of bitstreams: 1 inforsato_ap_me_rcla.pdf: 2232239 bytes, checksum: 6e5ef796e47228af1164f8da713ea8de (MD5) Previous issue date: 2018-05-04 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Neste trabalho tratamos da classificação dos grupos de friso. Para realizar este objetivo abordamos elementos básicos da estrutura algébrica de grupo bem como apresentamos transformações geométricas, entre estas destacamos: translações, reflexões, rotações e reflexão com deslizamento. Além disso, localizamos este assunto como tópico da estrutura curricular do Ensino Fundamental e executamos uma atividade em sala de aula em que os alunos criaram frisos ornamentais. / In this work we deal with the classification of frieze groups. In order to accomplish this objective we approach basic elements of the algebraic group structure as well as present geometric transformations, among which we highlight: translations, reflections, rotations and glide reflection. In addition, we locate this subject as a topic of the curricular structure of Elementary School and perform a classroom activity in which the students created ornamental friezes.

Este estudo tem como objetivo examinar as possibilidades e potencialidades do ensino das Transformações Geométricas no Ensino Fundamental, assunto este pouco abordado nos currículos deste nível de ensino. Realiza um resgate histórico sobre o ensino da Geometria e constata que a introdução do tema das transformações se dá a partir do Movimento da Matemática Moderna. Identifica os Parâmetros Curriculares Nacionais como incentivador deste estudo, a partir do momento em que, devido a essas orientações, autores de livros didáticos passam a tratar do tema em suas coleções, mesmo que, em alguns casos, de forma tímida. Para verificar as potencialidades do estudo das Transformações Geométricas foram elaborados dois conjuntos de atividades, sendo um desenvolvido com professoras atuantes nos Anos Iniciais do Ensino Fundamental e outro com uma turma de alunos do 6º ano desse nível de ensino, ambas em uma escola municipal da Rede Pública de Ensino da cidade de São Leopoldo, RS. Esta pesquisa de caráter qualitativo segue princípios da pesquisa-ação, adotando procedimentos de acompanhamento e controle da intervenção produzida. Da análise da aplicação da proposta, foi produzido um livro do tipo paradidático intitulado Matemática das Transformações, o qual trata o tema com aspectos lúdicos, de caráter artístico e que, pela linguagem nele utilizada, tem a pretensão de incentivar a leitura. / This study intents to examine the possibilities and potential of Geometric Transformations in teaching Elementary School, a few discussed subject in the curriculum of this educational level. Performs a historical survey about the teaching of geometry and notes that the introduction of the theme of change is given from the Movement of Modern Mathematics. Identifies how the National Curriculum Parameters for this study, from the time when, due to these guidelines, authors of textbooks come to address the issue in their collections, even though in some cases, so shy. To verify the potential of the study of Geometric Transformations were prepared two sets of activities, one developed with teachers working in early years of elementary school and another with a group of students from the 6th year of this level of education, both in a municipal school of the Network public Schools in the city of São Leopoldo, RS. This qualitative study follows the principles of action-research, adopting procedures for monitoring and control of the intervention produced. An analysis of the implementation of the proposal, was produced a textbook titled Transformation’s Mathematics , which treats the subject with playful aspects of artistic character, and that the language used therein, intend to encourage reading.

Este trabalho de pesquisa investigou o processo de aprendizagem de geometria em uma turma do sexto ano do Ensino Fundamental de uma escola da rede municipal de Porto Alegre. Durante dois meses do ano de 2016, foram desenvolvidos os conceitos de quadriláteros, triângulos e de Transformações Geométricas (translação, rotação e reflexão) sob a perspectiva da Investigação Matemática em sala de aula, metodologia de ensino que possui potencial para desencadear o processo de construção do conhecimento. Durante este período, os estudantes realizaram a investigação de quadriláteros e de triângulos, utilizando o software GeoGebra como recurso das Tecnologias da Informação e Comunicação (TIC). Os alunos construíram estes polígonos no GeoGebra, através de orientações passo-a-passo que foram disponibilizadas através de formulários online. Ao longo destas construções, os estudantes responderam a questionamentos também contidos nestes formulários online, de forma a identificar as propriedades contidas em cada construção, referentes a cada figura geométrica. Posteriormente, registraram as propriedades de cada polígono em uma tabela de características, de forma a organizar as propriedades de cada quadrilátero e de cada triângulo estudado. Para a investigação das Transformações Geométricas, desenvolveu-se um trabalho fazendo-se uso de tesselações no plano (coberturas para o plano). Para esta etapa da investigação, utilizou-se o applet “Design a Tessellation”, que é um recurso online e gratuito no qual o usuário pode criar diferentes coberturas para o plano através de uma unidade de tesselação quadrada. Os alunos fizeram uso de formulários online para responder aos questionamentos sobre as Transformações Geométricas estudadas, assim como folhas com atividades e malhas impressas para a criação de tesselações. Para a análise do processo de aprendizagem dos estudantes, foi utilizada a perspectiva dos níveis de Van Hiele, que classifica os níveis de pensamento geométrico, utilizando também uma abordagem que admite a existência de níveis intermediários. Além disso, este trabalho também formulou uma complementação para os níveis de Van Hiele quanto às Transformações Geométricas, de forma a analisar os dados obtidos com a pesquisa de uma maneira mais detalhada. Com a pesquisa finalizada, conclui-se que houve progresso dos níveis de Van Hiele para os estudantes analisados. / This research investigated the learning process of geometry in a class of the sixth grade of Elementary School, of a municipal school in Porto Alegre. During two months of 2016, the concepts of quadrilaterals, triangles and Geometric Transformations (translation, rotation and reflection) were developed from the perspective of Math Investigation in the classroom, teaching methodology which has the potential to develop the process of knowledge construction. During this period, students performed the investigation of quadrilaterals and triangles, using GeoGebra software as a resource of Information and Communication Technologies (TIC). The students constructed these polygons in GeoGebra, through step-by-step guidelines, which were made available through online forms. Throughout these constructions, the students answered the questions also contained in these online forms, in order to identify the properties contained in each construction, referring to each geometric figure. Later, they registered the properties of each polygon in a table of characteristics, in order to organize the properties of each quadrilateral and of each triangle studied. For the investigation of the Geometric Transformations, a work was developed making use of tessellations in the plane (covers for the plane). For this stage of the research, the "Design a Tessellation" applet was used, which is an online and free resource, where the user can create different covers for the plane, through a square tessellation unit. Students used online forms to answer questions about Geometric Transformations studied, as well as sheets with activities, and printed meshes for the creation of tessellations. For the analysis of the students' learning process, the Van Hiele levels perspective was also used, which classifies the levels of geometric thinking, using an approach that admits the existence of intermediate levels. In addition, this work also formulated a complementation for the Van Hiele levels, regarding Geometric Transformations, in order to analyze the data obtained with the research in a more detailed way. With the research completed, it is concluded that there was progress of the levels of Van Hiele for the analyzed students.

Este estudo tem como objetivo examinar as possibilidades e potencialidades do ensino das Transformações Geométricas no Ensino Fundamental, assunto este pouco abordado nos currículos deste nível de ensino. Realiza um resgate histórico sobre o ensino da Geometria e constata que a introdução do tema das transformações se dá a partir do Movimento da Matemática Moderna. Identifica os Parâmetros Curriculares Nacionais como incentivador deste estudo, a partir do momento em que, devido a essas orientações, autores de livros didáticos passam a tratar do tema em suas coleções, mesmo que, em alguns casos, de forma tímida. Para verificar as potencialidades do estudo das Transformações Geométricas foram elaborados dois conjuntos de atividades, sendo um desenvolvido com professoras atuantes nos Anos Iniciais do Ensino Fundamental e outro com uma turma de alunos do 6º ano desse nível de ensino, ambas em uma escola municipal da Rede Pública de Ensino da cidade de São Leopoldo, RS. Esta pesquisa de caráter qualitativo segue princípios da pesquisa-ação, adotando procedimentos de acompanhamento e controle da intervenção produzida. Da análise da aplicação da proposta, foi produzido um livro do tipo paradidático intitulado Matemática das Transformações, o qual trata o tema com aspectos lúdicos, de caráter artístico e que, pela linguagem nele utilizada, tem a pretensão de incentivar a leitura. / This study intents to examine the possibilities and potential of Geometric Transformations in teaching Elementary School, a few discussed subject in the curriculum of this educational level. Performs a historical survey about the teaching of geometry and notes that the introduction of the theme of change is given from the Movement of Modern Mathematics. Identifies how the National Curriculum Parameters for this study, from the time when, due to these guidelines, authors of textbooks come to address the issue in their collections, even though in some cases, so shy. To verify the potential of the study of Geometric Transformations were prepared two sets of activities, one developed with teachers working in early years of elementary school and another with a group of students from the 6th year of this level of education, both in a municipal school of the Network public Schools in the city of São Leopoldo, RS. This qualitative study follows the principles of action-research, adopting procedures for monitoring and control of the intervention produced. An analysis of the implementation of the proposal, was produced a textbook titled Transformation’s Mathematics , which treats the subject with playful aspects of artistic character, and that the language used therein, intend to encourage reading.

Neste estudo é proposto um método para resolução de problemas de tangências, especificamente para o décimo caso do problema de Apolônio. Este método é baseado na transformação geométrica inversão e no uso do conceito de feixes de circunferências. Além de permitir a resolução de todas as configurações do problema, ele é aplicável também à resolução de outros problemas. Através do trabalho indicamos a importância do tema Desenho Geométrico no ensino de Desenho que há muito tempo enfatiza apenas o desenvolvimento da visualização espacial. Destacamos ao longo do texto como o ensino de Desenho Geométrico pode ser utilizado eficazmente para fomentar o raciocínio lógico-dedutivo dos estudantes através da prática de demonstrações. / A new method to solve the tenth case of Appolonius problem is presented in this study. This method is based on the geometric transformation called inversion and the concept of coaxal circumferences. Besides allowing the resolution of all configurations of the problem, it can also be used to solve other problems. We indicate the importance of the subject about geometric constructions in teaching Drawing, which, since a long time ago has given attention only to the development of the spatial visualization ability. We detach along the text how the teaching of geometric construction can be used efficiently to foment the deductive logical reasoning of the students through the practice of demonstrations.

This master's thesis deals with implementation of a system for creating general image operations using an image processing chain. The text introduces the concept of image processing using chains and shows its advantages and possibilities. Furthermore, the text focuses on the detailed design of a system based on that concept. The system was successfully implemented and tested, while the results were described and discussed. Finally, the system is shown in a practical application of several image processing tasks.

For decades curriculum materials have been used as a means to reform the manner in which mathematics is taught. In an attempt to better understand what impact curriculum materials have on the teaching and learning of mathematics, researchers have begun to analyze the process in which teachers transform curriculum materials into instruction. Given that many of these studies have been broad in nature, I sought to study the specific decisions that three teachers made when planning lesson on geometric transformation and their reasoning for those decisions. In this study I found that while the participants in this study ignored the curriculum material I gave them as well as their district adopted materials in planning instruction, they made a wide variety of decisions with regards to other curriculum. In an attempt to describe this variety of decisions I expanded the previous methods of describing curriculum use decisions adding a self-create category as well as differentiating between the different types of adaptations teachers make. I also found that although teachers used different curriculum materials, they made similar decisions in how they planned the mathematics content of geometric transformations that seem problematic.

Dans son contexte spécifique, le terme « biométrie » est souvent associé à l'étude des caractéristiques physiques et comportementales des individus afin de parvenir à leur identification ou à leur vérification. Ainsi, le travail développé dans cette thèse nous a conduit à proposer un algorithme d'identification robuste, en considérant les caractéristiques intrinsèques des phalanges de la main. Considérée comme une biométrie cachée, cette nouvelle approche peut s'avérer intéressante, notamment lorsqu'il est question d'assurer un niveau de sécurité élevé, robuste aux différentes attaques qu'un système biométrique doit contrer. La base des techniques proposées requière trois phases, à savoir: (1) la segmentation des phalanges, (2) l'extraction de leurs caractéristiques par la génération d'une empreinte, appelée « Phalange-Code » et (3) l'identification basée sur la méthode du 1-plus proche voisin ou la vérification basée sur une métrique de similarité. Ces algorithmes opèrent sur des niveaux hiérarchiques permettant l'extraction de certains paramètres, invariants à des transformations géométriques telles que l'orientation et la translation. De plus, nous avons considéré des techniques robustes au bruit, pouvant opérer à différentes résolutions d'images. Plus précisément, nous avons élaboré trois approches de reconnaissance biométrique : la première approche utilise l'information spectrale des contours des phalanges de la main comme signature individuelle, alors que la deuxième approche nécessite l'utilisation des caractéristiques géométriques et morphologiques des phalanges (i.e. surface, périmètre, longueur, largeur, capacité). Enfin, la troisième approche requière la génération d'un nouveau rapport de vraisemblance entre les phalanges, utilisant la théorie de probabilités géométriques. En second lieu, la construction d'une base de données avec la plus faible dose de rayonnement a été l'un des grands défis de notre étude. Nous avons donc procédé par la collecte de 403 images radiographiques de la main, acquises en utilisant la machine Apollo EZ X-Ray. Ces images sont issues de 115 adultes volontaires (hommes et femmes), non pathologiques. L'âge moyen étant de 27.2 ans et l'écart-type est de 8.5. La base de données ainsi construite intègre des images de la main droite et gauche, acquises à des positions différentes et en considérant des résolutions différentes et des doses de rayonnement différentes (i.e. réduction jusqu'à 98 % de la dose standard recommandée par les radiologues « 1 µSv »).Nos expériences montrent que les individus peuvent être distingués par les caractéristiques de leurs phalanges, que ce soit celles de la main droite ou celles de la main gauche. Cette distinction est également valable pour le genre des individus (homme/femme). L'étude menée a montré que l'approche utilisant l'information spectrale des contours des phalanges permet une identification par seulement trois phalanges, à un taux EER (Equal Error Rate) inférieur à 0.24 %. Par ailleurs, il a été constaté « de manière surprenante » que la technique fondée sur les rapports de vraisemblance entre les phalanges permet d'atteindre un taux d'identification de 100 % et un taux d'EER de 0.37 %, avec une seule phalange. Hormis l'aspect identification/authentification, notre étude s'est penchée sur l'optimisation de la dose de rayonnement permettant une identification saine des individus. Ainsi, il a été démontré qu'il était possible d'acquérir plus de 12500/an d'images radiographiques de la main, sans pour autant dépasser le seuil administratif de 0.25 mSv / In its specific context, the term "biometrics" is often associated with the study of the physical and behavioral of individual's characteristics to achieve their identification or verification. Thus, the work developed in this thesis has led us to suggest a robust identification algorithm, taking into account the intrinsic characteristics of the hand phalanges. Considered as hidden biometrics, this new approach can be of high interest, particularly when it comes to ensure a high level of security, robust to various attacks that a biometric system must address. The basis of the proposed techniques requires three phases, namely: (1) the segmentation of the phalanges (2) extracting their characteristics by generating an imprint, called "Phalange-Code" and (3) the identification based on the method of 1-nearest neighbor or the verification based on a similarity metric. This algorithm operates on hierarchical levels allowing the extraction of certain parameters invariant to geometric transformations such as image orientation and translation. Furthermore, the considered algorithm is particularly robust to noise, and can function at different resolutions of images. Thus, we developed three approaches to biometric recognition: the first approach produces individual signature from the spectral information of the contours issued from the hand phalanges, whereas the second approach requires the use of geometric and morphological characteristics of the phalanges (i.e. surface, perimeter, length, width, and capacity). Finally, the third approach requires the generation of a new likelihood ratio between the phalanges, using the geometric probability theory. Furthermore, the construction of a database with the lowest radiation dose was one of the great challenges of our study. We therefore proceeded with the collection of 403 x-ray images of the hand, acquired using the Apollo EZ X-Ray machine. These images are from 115 non-pathological volunteering adult (men and women). The average age is 27.2 years and the standard deviation is 8.5. Thus, the constructed database incorporates images of the right and left hands, acquired at different positions and by considering different resolutions and different radiation doses (i.e. reduced till 98% of the standard dose recommended by radiologists "1 µSv").Our experiments show that individuals can be distinguished by the characteristics of their phalanges, whether those of the right hand or the left hand. This distinction also applies to the kind of individuals (male/female). The study has demonstrated that the approach using the spectral information of the phalanges' contours allows identification by only three phalanges, with an EER (Equal Error Rate) lower than 0.24 %. Furthermore, it was found “Surprisingly” that the technique based on the likelihood ratio between phalanges reaches an identification rate of 100% and an EER of 0.37% with a single phalanx. Apart from the identification/authentication aspect, our study focused on the optimization of the radiation dose in order to offer safe identification of individuals. Thus, it has been shown that it was possible to acquire more than 12,500/year radiographic hand images, without exceeding the administrative control of 0.25 mSv

Dans son contexte spécifique, le terme « biométrie » est souvent associé à l'étude des caractéristiques physiques et comportementales des individus afin de parvenir à leur identification ou à leur vérification. Ainsi, le travail développé dans cette thèse nous a conduit à proposer un algorithme d'identification robuste, en considérant les caractéristiques intrinsèques des phalanges de la main. Considérée comme une biométrie cachée, cette nouvelle approche peut s'avérer intéressante, notamment lorsqu'il est question d'assurer un niveau de sécurité élevé, robuste aux différentes attaques qu'un système biométrique doit contrer. La base des techniques proposées requière trois phases, à savoir: (1) la segmentation des phalanges, (2) l'extraction de leurs caractéristiques par la génération d'une empreinte, appelée « Phalange-Code » et (3) l'identification basée sur la méthode du 1-plus proche voisin ou la vérification basée sur une métrique de similarité. Ces algorithmes opèrent sur des niveaux hiérarchiques permettant l'extraction de certains paramètres, invariants à des transformations géométriques telles que l'orientation et la translation. De plus, nous avons considéré des techniques robustes au bruit, pouvant opérer à différentes résolutions d'images. Plus précisément, nous avons élaboré trois approches de reconnaissance biométrique : la première approche utilise l'information spectrale des contours des phalanges de la main comme signature individuelle, alors que la deuxième approche nécessite l'utilisation des caractéristiques géométriques et morphologiques des phalanges (i.e. surface, périmètre, longueur, largeur, capacité). Enfin, la troisième approche requière la génération d'un nouveau rapport de vraisemblance entre les phalanges, utilisant la théorie de probabilités géométriques. En second lieu, la construction d'une base de données avec la plus faible dose de rayonnement a été l'un des grands défis de notre étude. Nous avons donc procédé par la collecte de 403 images radiographiques de la main, acquises en utilisant la machine Apollo EZ X-Ray. Ces images sont issues de 115 adultes volontaires (hommes et femmes), non pathologiques. L'âge moyen étant de 27.2 ans et l'écart-type est de 8.5. La base de données ainsi construite intègre des images de la main droite et gauche, acquises à des positions différentes et en considérant des résolutions différentes et des doses de rayonnement différentes (i.e. réduction jusqu'à 98 % de la dose standard recommandée par les radiologues « 1 µSv »).Nos expériences montrent que les individus peuvent être distingués par les caractéristiques de leurs phalanges, que ce soit celles de la main droite ou celles de la main gauche. Cette distinction est également valable pour le genre des individus (homme/femme). L'étude menée a montré que l'approche utilisant l'information spectrale des contours des phalanges permet une identification par seulement trois phalanges, à un taux EER (Equal Error Rate) inférieur à 0.24 %. Par ailleurs, il a été constaté « de manière surprenante » que la technique fondée sur les rapports de vraisemblance entre les phalanges permet d'atteindre un taux d'identification de 100 % et un taux d'EER de 0.37 %, avec une seule phalange. Hormis l'aspect identification/authentification, notre étude s'est penchée sur l'optimisation de la dose de rayonnement permettant une identification saine des individus. Ainsi, il a été démontré qu'il était possible d'acquérir plus de 12500/an d'images radiographiques de la main, sans pour autant dépasser le seuil administratif de 0.25 mSv / In its specific context, the term "biometrics" is often associated with the study of the physical and behavioral of individual's characteristics to achieve their identification or verification. Thus, the work developed in this thesis has led us to suggest a robust identification algorithm, taking into account the intrinsic characteristics of the hand phalanges. Considered as hidden biometrics, this new approach can be of high interest, particularly when it comes to ensure a high level of security, robust to various attacks that a biometric system must address. The basis of the proposed techniques requires three phases, namely: (1) the segmentation of the phalanges (2) extracting their characteristics by generating an imprint, called "Phalange-Code" and (3) the identification based on the method of 1-nearest neighbor or the verification based on a similarity metric. This algorithm operates on hierarchical levels allowing the extraction of certain parameters invariant to geometric transformations such as image orientation and translation. Furthermore, the considered algorithm is particularly robust to noise, and can function at different resolutions of images. Thus, we developed three approaches to biometric recognition: the first approach produces individual signature from the spectral information of the contours issued from the hand phalanges, whereas the second approach requires the use of geometric and morphological characteristics of the phalanges (i.e. surface, perimeter, length, width, and capacity). Finally, the third approach requires the generation of a new likelihood ratio between the phalanges, using the geometric probability theory. Furthermore, the construction of a database with the lowest radiation dose was one of the great challenges of our study. We therefore proceeded with the collection of 403 x-ray images of the hand, acquired using the Apollo EZ X-Ray machine. These images are from 115 non-pathological volunteering adult (men and women). The average age is 27.2 years and the standard deviation is 8.5. Thus, the constructed database incorporates images of the right and left hands, acquired at different positions and by considering different resolutions and different radiation doses (i.e. reduced till 98% of the standard dose recommended by radiologists "1 µSv").Our experiments show that individuals can be distinguished by the characteristics of their phalanges, whether those of the right hand or the left hand. This distinction also applies to the kind of individuals (male/female). The study has demonstrated that the approach using the spectral information of the phalanges' contours allows identification by only three phalanges, with an EER (Equal Error Rate) lower than 0.24 %. Furthermore, it was found “Surprisingly” that the technique based on the likelihood ratio between phalanges reaches an identification rate of 100% and an EER of 0.37% with a single phalanx. Apart from the identification/authentication aspect, our study focused on the optimization of the radiation dose in order to offer safe identification of individuals. Thus, it has been shown that it was possible to acquire more than 12,500/year radiographic hand images, without exceeding the administrative control of 0.25 mSv