Isometrias: uma abordagem interdisciplinar no 8º ano de escolaridade

A sociedade atual requer indivíduos preparados para apresentar soluções inovadoras aos problemas encontrados nas mais variadas situações, sendo a criatividade considerada, na última década, uma competência essencial para o progresso. Neste contexto, emerge a necessidade da escola fomentar o seu desenvolvimento em qualquer área, e em particular, na Matemática. Por outro lado, não será alheio a este facto a promoção de um ensino, também ele, criativo. Tal ensino exige desde logo uma original, fluente e flexível gestão curricular envolvendo uma adequada seleção e ou (re) criação de tarefas relevantes, criteriosamente sequenciadas e autonomamente resolvidas pelos alunos, numa lógica de interdisciplinaridade e com recurso a tecnologia. No caso especifico das transformações geométricas isométricas, uma abordagem interdisciplinar reforçada com o recurso a ambientes de geometria dinâmica poderá constituir uma mais-valia nesse processo. Assim, desenvolveu-se este estudo, com o objetivo de avaliar o impacto de uma abordagem interdisciplinar, potenciada com o GeoGebra, no desenvolvimento de competências geométricas relacionadas com as isometrias, frisos e rosáceas, e em simultâneo, no desenvolvimento da criatividade e representações da mesma, de alunos do oitavo ano de escolaridade. Para isso desenvolveu-se um estudo de caso, centrado em três pares de alunos, que resolveram um conjunto de tarefas de natureza exploratória, com recurso ao GeoGebra e envolvendo a disciplina de Educação Visual. A análise dos dados recolhidos foi, essencialmente, de natureza qualitativa, tendo sido a observação, a inquirição e a análise documental,as principais técnicas de recolha de dados. A análise de conteúdo a que foram submetidos os dados, permitiu concluir que a implementação de uma abordagem interdisciplinar, centrada numa sequência de tarefas e aliada ao GeoGebra, potenciou a apropriação dos conhecimentos geométricos em causa e a sua aplicação. Contribuiu, também, para o desenvolvimento de atitudes favoráveis em relação à matemática e à geometria em particular. Por outro lado, os dados sugerem que tal abordagem permite obter indícios do desenvolvimento da criatividade nos alunos e de alterações a algumas das suas representações.

Análise de transformações geométricas para o georreferenciamento de imagens do satélite CBERS-1

Neste trabalho serão descritos métodos e técnicas na análise de transformações geométricas para o georreferenciamento de imagens do satélite CBERS I, utilizando o sensor CCD na região de Porto Alegre, com a utilização das transformações afim e equações projetivas e com o uso de pontos de apoio coletados com receptor GPS. Os resultados experimentais obtidos com as transformações afim e equações projetivas são animadores, recomendando-se então continuar os estudos para as imagens do CBERS-I que pode ser um atalho importante para atualizar a cartografia regional brasileira, pois neste caso do sensor CCD se vislumbra a possibilidade de se gerarem cartas imagens nas escalas de 1:100.000 e 1:50.000.

O uso do caleidoscópio no ensino de grupos de simetria e transformações geométricas

Pós-graduação em Educação Matemática - IGCE

Um estudo de fractais geométricos através de caleidoscópios e softwares de geometria dinâmica

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Reconhecimento de contorno de edifício em imagens de alta resolução usando os momentos complexos de Zernike

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Contributions to HEVC Prediction for Medical Image Compression

Medical imaging technology and applications are continuously evolving, dealing with images of increasing spatial and temporal resolutions, which allow easier and more accurate medical diagnosis. However, this increase in resolution demands a growing amount of data to be stored and transmitted. Despite the high coding efficiency achieved by the most recent image and video coding standards in lossy compression, they are not well suited for quality-critical medical image compression where either near-lossless or lossless coding is required. In this dissertation, two different approaches to improve lossless coding of volumetric medical images, such as Magnetic Resonance and Computed Tomography, were studied and implemented using the latest standard High Efficiency Video Encoder (HEVC). In a first approach, the use of geometric transformations to perform inter-slice prediction was investigated. For the second approach, a pixel-wise prediction technique, based on Least-Squares prediction, that exploits inter-slice redundancy was proposed to extend the current HEVC lossless tools. Experimental results show a bitrate reduction between 45% and 49%, when compared with DICOM recommended encoders, and 13.7% when compared with standard HEVC.

Optimization of Pattern Matching Algorithms for Multi- and Many-Core Platforms

Image and video compression play a major role in the world today, allowing the storage and transmission of large multimedia content volumes. However, the processing of this information requires high computational resources, hence the improvement of the computational performance of these compression algorithms is very important. The Multidimensional Multiscale Parser (MMP) is a pattern-matching-based compression algorithm for multimedia contents, namely images, achieving high compression ratios, maintaining good image quality, Rodrigues et al. [2008]. However, in comparison with other existing algorithms, this algorithm takes some time to execute. Therefore, two parallel implementations for GPUs were proposed by Ribeiro [2016] and Silva [2015] in CUDA and OpenCL-GPU, respectively. In this dissertation, to complement the referred work, we propose two parallel versions that run the MMP algorithm in CPU: one resorting to OpenMP and another that converts the existing OpenCL-GPU into OpenCL-CPU. The proposed solutions are able to improve the computational performance of MMP by 3 and 2:7 , respectively. The High Efficiency Video Coding (HEVC/H.265) is the most recent standard for compression of image and video. Its impressive compression performance, makes it a target for many adaptations, particularly for holoscopic image/video processing (or light field). Some of the proposed modifications to encode this new multimedia content are based on geometry-based disparity compensations (SS), developed by Conti et al. [2014], and a Geometric Transformations (GT) module, proposed by Monteiro et al. [2015]. These compression algorithms for holoscopic images based on HEVC present an implementation of specific search for similar micro-images that is more efficient than the one performed by HEVC, but its implementation is considerably slower than HEVC. In order to enable better execution times, we choose to use the OpenCL API as the GPU enabling language in order to increase the module performance. With its most costly setting, we are able to reduce the GT module execution time from 6.9 days to less then 4 hours, effectively attaining a speedup of 45 .

Polarisation diversity of conformal arrays based on geometric algebra via convex optimisation

A simple and general design procedure is presented for the polarisation diversity of arbitrary conformal arrays; this procedure is based on the mathematical framework of geometric algebra and can be solved optimally using convex optimisation. Aside from being simpler and more direct than other derivations in the literature, this derivation is also entirely general in that it expresses the transformations in terms of rotors in geometric algebra which can easily be formulated for any arbitrary conformal array geometry. Convex optimisation has a number of advantages; solvers are widespread and freely available, the process generally requires a small number of iterations and a wide variety of constraints can be readily incorporated. The study outlines a two-step approach for addressing polarisation diversity in arbitrary conformal arrays: first, the authors obtain the array polarisation patterns using geometric algebra and secondly use a convex optimisation approach to find the optimal weights for the polarisation diversity problem. The versatility of this approach is illustrated via simulations of a 7×10 cylindrical conformal array. © 2012 The Institution of Engineering and Technology.

Riemannian metric and geometric mean for positive semidefinite matrices of fixed rank

This paper introduces a new metric and mean on the set of positive semidefinite matrices of fixed-rank. The proposed metric is derived from a well-chosen Riemannian quotient geometry that generalizes the reductive geometry of the positive cone and the associated natural metric. The resulting Riemannian space has strong geometrical properties: it is geodesically complete, and the metric is invariant with respect to all transformations that preserve angles (orthogonal transformations, scalings, and pseudoinversion). A meaningful approximation of the associated Riemannian distance is proposed, that can be efficiently numerically computed via a simple algorithm based on SVD. The induced mean preserves the rank, possesses the most desirable characteristics of a geometric mean, and is easy to compute. © 2009 Society for Industrial and Applied Mathematics.

Geometric Distortion Insensitive Image Watermarking in Affine Covariant Regions

Feature-based image watermarking schemes, which aim to survive various geometric distortions, have attracted great attention in recent years. Existing schemes have shown robustness against rotation, scaling, and translation, but few are resistant to cropping, nonisotropic scaling, random bending attacks (RBAs), and affine transformations. Seo and Yoo present a geometrically invariant image watermarking based on affine covariant regions (ACRs) that provide a certain degree of robustness. To further enhance the robustness, we propose a new image watermarking scheme on the basis of Seo's work, which is insensitive to geometric distortions as well as common image processing operations. Our scheme is mainly composed of three components: 1) feature selection procedure based on graph theoretical clustering algorithm is applied to obtain a set of stable and nonoverlapped ACRs; 2) for each chosen ACR, local normalization, and orientation alignment are performed to generate a geometrically invariant region, which can obviously improve the robustness of the proposed watermarking scheme; and 3) in order to prevent the degradation in image quality caused by the normalization and inverse normalization, indirect inverse normalization is adopted to achieve a good compromise between the imperceptibility and robustness. Experiments are carried out on an image set of 100 images collected from Internet, and the preliminary results demonstrate that the developed method improves the performance over some representative image watermarking approaches in terms of robustness.

Affine Matching with Bounded Sensor Error: A Study of Geometric Hashing and Alignment

Affine transformations are often used in recognition systems, to approximate the effects of perspective projection. The underlying mathematics is for exact feature data, with no positional uncertainty. In practice, heuristics are added to handle uncertainty. We provide a precise analysis of affine point matching, obtaining an expression for the range of affine-invariant values consistent with bounded uncertainty. This analysis reveals that the range of affine-invariant values depends on the actual $x$-$y$-positions of the features, i.e. with uncertainty, affine representations are not invariant with respect to the Cartesian coordinate system. We analyze the effect of this on geometric hashing and alignment recognition methods.

Transformations quasi-conformes de maillages volumiques et applications en infographie

La modélisation géométrique est importante autant en infographie qu'en ingénierie. Notre capacité à représenter l'information géométrique fixe les limites et la facilité avec laquelle on manipule les objets 3D. Une de ces représentations géométriques est le maillage volumique, formé de polyèdres assemblés de sorte à approcher une forme désirée. Certaines applications, tels que le placage de textures et le remaillage, ont avantage à déformer le maillage vers un domaine plus régulier pour faciliter le traitement. On dit qu'une déformation est \emph{quasi-conforme} si elle borne la distorsion. Cette thèse porte sur l’étude et le développement d'algorithmes de déformation quasi-conforme de maillages volumiques. Nous étudions ces types de déformations parce qu’elles offrent de bonnes propriétés de préservation de l’aspect local d’un solide et qu’elles ont été peu étudiées dans le contexte de l’informatique graphique, contrairement à leurs pendants 2D. Cette recherche tente de généraliser aux volumes des concepts bien maitrisés pour la déformation de surfaces. Premièrement, nous présentons une approche linéaire de la quasi-conformité. Nous développons une méthode déformant l’objet vers son domaine paramétrique par une méthode des moindres carrés linéaires. Cette méthode est simple d'implémentation et rapide d'exécution, mais n'est qu'une approximation de la quasi-conformité car elle ne borne pas la distorsion. Deuxièmement, nous remédions à ce problème par une approche non linéaire basée sur les positions des sommets. Nous développons une technique déformant le domaine paramétrique vers le solide par une méthode des moindres carrés non linéaires. La non-linéarité permet l’inclusion de contraintes garantissant l’injectivité de la déformation. De plus, la déformation du domaine paramétrique au lieu de l’objet lui-même permet l’utilisation de domaines plus généraux. Troisièmement, nous présentons une approche non linéaire basée sur les angles dièdres. Cette méthode définit la déformation du solide par les angles dièdres au lieu des positions des sommets du maillage. Ce changement de variables permet une expression naturelle des bornes de distorsion de la déformation. Nous présentons quelques applications de cette nouvelle approche dont la paramétrisation, l'interpolation, l'optimisation et la compression de maillages tétraédriques.

The Noether theorem for geometric actions and area preserving diffeomorphisms on the torus

We find that within the formalism of coadjoint orbits of the infinite dimensional Lie group the Noether procedure leads, for a special class of transformations, to the constant of motion given by the fundamental group one-cocycle S. Use is made of the simplified formula giving the symplectic action in terms of S and the Maurer-Cartan one-form. The area preserving diffeomorphisms on the torus T2=S1⊗S1 constitute an algebra with central extension, given by the Floratos-Iliopoulos cocycle. We apply our general treatment based on the symplectic analysis of coadjoint orbits of Lie groups to write the symplectic action for this model and study its invariance. We find an interesting abelian symmetry structure of this non-linear problem.

Supervised variational relevance learning, an analytic geometric feature selection with applications to omic datasets

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Practical Transformations and Transformational Practices : Globalization, Postmodernism, and Early Childhood Education : Advances in Early Education and Day Care

Both traditional and progressive curricula are inadequate for the task of responding to the economic, political, social, and cultural changes that have occurred as a result of globalization. This book documents some of the ongoing work occurring in early childhood settings that is aimed at improving, and ultimately transforming, early childhood practice in these changed and changing times. The authors do not simply critique developmental approaches or the increasing standardization of the field. Instead, they describe how they are playing around with postmodern ideas in practice and developing unique approaches to the diverse educational circumstances that confront early childhood educators. Whether it is preparing teachers, using materials, or developing policies, each chapter provides readers with possibilities for enacting pedagogies that are responsive to the contemporary circumstances shaping the lives of young children.