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La morphologie mathématique est une théorie puissante pour l’analyse d’images 2 D. Elle se base sur la dilatation et l’érosion, qui correspondent à l’addition et la soustraction de Minkowski. Afin d’analyser des maillages 3D par morphologie mathématique, on doit disposer d’algorithmes performants et robustes pour le calcul exact de l’addition et de la soustraction pour ces maillages. Malheureusement, les travaux existants sont, soit approximés, soit non robustes ou limités par des contraintes. Aucun travail n’a traité la différence. Ces difficultés sont dues au fait qu’un maillage représente une surface linéaire par morceaux englobant un ensemble contenu et non dénombrable. Nous avons introduit la notion de sommets contributeurs et nous avons développé un algorithme efficace et robuste pour le calcul de la somme de polyèdres convexes. Nous l’avons par la suite adapté et proposé deux algorithmes performants pour la somme d’une paire de polyèdres non convexe/convexe, tout en gérant correctement les polyèdres complexes, les situations de non-variété ainsi que les changements topologiques. Nous avons également démontré la dualité des sommets contributeurs et nous l’avons exploité pour développer la première approche du calcul exact et efficace de la différence de polyèdres convexes. La dualité des sommets contributeurs ainsi que la robustesse et l’efficacité de nos approches motivent le développement d’une approche unifiée pour l’addition et la soustraction de polyèdres quelconques, ce qui permettra d’appliquer des traitements morphologiques à des maillages 3D. D’autres domaines tels que l’imagerie médicale, la robotique, la géométrie ou la chimie pourront en tirer profit / Mathematical morphology is a powerful theory for the analysis of 2D digital images. It is based on dilation and erosion, which correspond to Minkowski addition and subtraction. To be able to analyze 3D meshes using mathematical morphology, we must use efficient and robust algorithms for the exact computation of the addition and subtraction of meshes. Unfortunately, existing approaches are approximated, non-robust or limited by some constraints. No work has addressed the difference. These difficulties come from the the fact that a mesh represents a piecewise linear surface bounding a continuous and uncountable set. We introduced the concept of contributing vertices and developed an efficient and robust algorithm for the computation of the Minkowski sum of convex polyhedra. After that, we adapted and proposed two efficient algorithms for the computation of the Minkowski sum of a non-convex/convex pair of polyhedra, while properly handling complex polyhedra, non-manifold situations and topological changes. We also demonstrated the duality of the contributing vertices concept and exploited it to develop the first approach for the efficient and exact computation of the Minkowski difference of convex polyhedra. The duality of the contributing vertices concept as well as the robustness and efficiency of our approaches motivate the development of a unified approach for the Minkowski addition and subtraction of arbitrary polyhedral, which will permit the morphological analysis of 3D meshes. Other areas such as medical imaging, robotics, geometry or chemistry may benefit from our approaches

The image comparison operation ??sessing how well one image matches another ??rms a critical component of many image analysis systems and models of human visual processing. Two norms used commonly for this purpose are L1 and L2, which are specific instances of the Minkowski metric. However, there is often not a principled reason for selecting one norm over the other. One way to address this problem is by examining whether one metric better captures the perceptual notion of image similarity than the other. With this goal, we examined perceptual preferences for images retrieved on the basis of the L1 versus the L2 norm. These images were either small fragments without recognizable content, or larger patterns with recognizable content created via vector quantization. In both conditions the subjects showed a consistent preference for images matched using the L1 metric. These results suggest that, in the domain of natural images of the kind we have used, the L1 metric may better capture human notions of image similarity.

Thesis (M.S.)--West Virginia University, 2003. / Title from document title page. Document formatted into pages; contains v, 30 p. Includes abstract. Includes bibliographical references (p. 29-30).

Typescript (photocopy) / vi, 151 leaves ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Pure Mathematics, 1982

Thesis (B.A.)--Haverford College, Dept. of Mathematics, 2002. / Includes bibliographical references.

University, Diss., 2001--München.

University, Diss., 2004--München.

Dissertação (mestrado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2016. / Submitted by Fernanda Percia França (fernandafranca@bce.unb.br) on 2016-07-22T13:34:06Z No. of bitstreams: 1 2016_LumenaPauladeJesusBorges.pdf: 1020349 bytes, checksum: ccdbb2f9573ac7e8a5dfab408b8cdbf9 (MD5) / Approved for entry into archive by Raquel Viana(raquelviana@bce.unb.br) on 2016-08-19T21:12:40Z (GMT) No. of bitstreams: 1 2016_LumenaPauladeJesusBorges.pdf: 1020349 bytes, checksum: ccdbb2f9573ac7e8a5dfab408b8cdbf9 (MD5) / Made available in DSpace on 2016-08-19T21:12:40Z (GMT). No. of bitstreams: 1 2016_LumenaPauladeJesusBorges.pdf: 1020349 bytes, checksum: ccdbb2f9573ac7e8a5dfab408b8cdbf9 (MD5) / Os objetos de estudo nesta dissertação são as curvas de interseção entre duas superfícies no espaço Euclidiano e no espaço de Lorentz-Minkowski. As interseções podem ser do tipo transversal ou tangencial. As superfícies podem ser paramétricas ou implícitas e, portanto, os casos estudados são Paramétrica-Paramétrica, Paramétrica-Implícita e Implícita-Implícita. Quando as superfícies estão no espaço Euclidiano, o objetivo principal é apresentar algoritmos para se obter propriedades geométricas da curva de interseção, tais como curvatura, torção e vetor tangente, em cada caso das interseções. O propósito para o espaço de Lorentz-Minkowski é similar, no qual considera-se curvas de interseção transversal entre duas superfícies tipo espaço, bem como entre duas superfícies tipo tempo, apresentando-se expressões para a curvatura, torção e vetor tangente. Quando as superfícies são tipo espaço, a curva de interseção é também tipo espaço. Quando elas são tipo tempo, a curva pode ser tipo espaço, tipo tempo ou tipo luz. Uma análise para os casos tipo espaço e tipo tempo é feita neste trabalho. Além disso, para superfícies tipo espaço, são dadas condições para que a curva de interseção seja uma geodésica ou uma linha de curvatura das duas superfícies. Exemplos que ilustram esta teoria são acrescentados no final. ________________________________________________________________________________________________ ABSTRACT / The objects of study in this dissertation are the intersection curves of two surfaces in Euclidean space and Lorentz-Minkowski space. Intersections can be of transversal or tangential type. Surfaces can be parametric or implicit and, therefore, the cases studied are Parametric-Parametric, Parametric-Implicit and Implicit-Implicit. When the surfaces are in Euclidean space, the main objective is presenting algorithms to obtain geometrical properties of the intersection curve, such as curvature, torsion and tangent vector, in each case of the intersections. The purpose for Lorentz-Minkowski space is similar, in which is considered transversal intersection curves between two spacelike surfaces as well as between two timelike surfaces, presenting expressions for the curvature, torsion and tangent vector. When the surfaces are spacelike, the intersection curve is spacelike. When they are timelike, the curve can be spacelike, timelike or lightlike. An analysis for cases spacelike and timelike is considered in this work. Furthermore, for spacelike surfaces, conditions are given so that the intersection is a geodesic curve or line of curvature of both surfaces. Examples illustrating this theory are added at the end.

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