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Force fields of 3d objects are used for different purposes in computer graphics as skeletonization and collision detection. In this paper we present a novel method to approximate the force field of a discrete 3d object in linear time. Similar to the distance transformation we define a rule that describe how the forces associated with boundary points are propagated into the interior of the object. The result of this propagation depends on the order in which the points of the object are processed. Therefore we analyze how to obtain an order-invariant approximation formula. For a chosen iteration order (i, j, k) the set of boundary points that influence the force of a particular point p of the object can be described by a spatial region Rijk. The geometries of these regions are characterized both for the Cartesian and the body-centered cubic grid (bcc grid). We show that in the case of the bcc grid these regions can be combined in such a way that E3 is uniformly covered which basically means that each boundary point is contained in the same number of regions. Based on the covering an approximation formula for the force field is proposed that has linear complexity and gives good results for standard objects. We also show that such a uniform covering can not be built from the regions of influence of the Cartesian grid. With our method it becomes possible to use features of the force field for a fast and topology preserving skeletonization. We use a thinning strategy on the bcc grid to compute the skeleton and ensure that critical points of the force field are not removed. This leads to improved skeletons with respect to the properties of centeredness and rotational invariance.

Apesar de ser uma técnica muito difundida, a radioterapia pode causar danos ao reparo ósseo, como por exemplo, a diminuição da vascularização. Entretanto, a rede vascular óssea tem um papel importante na capacidade de regeneração dos ossos, pois fornece oxigênio e nutrientes essenciais, logo, ferramentas que auxiliem a análise dessas redes são importantes para o estudo de diversas terapias que têm influência sobre o tecido ósseo. Para analisar tais redes foi feita a reconstrução tridimensional de imagens coletadas a partir do seccionamento dos fêmures de um rato que recebeu doses de radiação em seu fêmur esquerdo, enquanto que o direito não foi irradiado sendo, portanto, utilizado para controle. Com o objetivo de auxiliar a análise desses volumes foi utilizada a técnica de esqueletização, que tem a finalidade de diminuir a quantidade de informação dos objetos e tornar a análise mais precisa e eficiente. Entretanto, existem diversos tipos de algoritmos esqueletização, sendo eles, de Afinamento, Geométricos, baseados na Transformada Distância, em Campo de Força e em Propagação de Ondas. Com o objetivo de analisar qual deles produz melhores resultados em volumes de redes vasculares foi escolhida uma implementação de cada tipo para ser testada e analisada em volumes pertencentes às redes vasculares. Além disso, o algoritmo escolhido para representar os métodos baseados em Propagação de Ondas foi desenvolvido e proposto neste trabalho exclusivamente para extrair canais de redes vasculares. Por fim, os esqueletos das redes vasculares conseguiram reproduzir com clareza a rede estudada e possibilitaram a conclusão de análises relacionadas ao impacto da radioterapia sobre a topologia vascular. Além disso, a comparação entre os tipos de algoritmos de esqueletização possibilitou um estudo aprofundado sobre o tema e sobre as diversas características de esqueletos de curva que podem ser usadas para classificar e comparar os métodos presentes na literatura. / Although a common technique, the radiotherapy can cause damage to bone repair, such as decrease in vascularization. However, the bone vascular network has an important role in capacity of bone regeneration because it provides oxygen and nutrients, therefore, tools that helps the analysis of vascular networks are important for the study of various therapies that have influence on the bone repair. In order to analyze such networks, we mande three-dimensional reconstructions of collected images from the sectioning of a mouse femurs that received radiation doses in the left femur, while the right was not irradiated and used for control. In order to aid the analysis of these volumes, skeletonization techniques were used to decrease the amount of the objects’s information and make the analysis more accurate and efficient. However, there are several types of skeletonization algorithms which uses different approachs as based on Forcefield, Thinning, based on Distance Transform, Geometrical and based on Wave Propagation. In order to analyze which of them produces the best results in vascular networks, an implementation of each type was chosen to be tested and analyzed in vascular network volumes. Furthermore, the algorithm chosen to represent the methods based on Wave Propagation was developed and proposed in this work exclusively to extract vascular networks. Finally, the skeletons of the vascular networks reproduced the network studied with clarity and enabled the conclusion of analysis related to the radiation impact on vascular topology. In addition, the comparison between the types of skeletonization algorithms allowed a deep study about the subject and on the various curve skeletons characteristics that can be used to classify and compare the methods in the literature. / Dissertação (Mestrado)

Force fields of 3d objects are used for different purposes in computer graphics as skeletonization and collision detection. In this paper we present a novel method to approximate the force field of a discrete 3d object in linear time. Similar to the distance transformation we define a rule that describe how the forces associated with boundary points are propagated into the interior of the object. The result of this propagation depends on the order in which the points of the object are processed. Therefore we analyze how to obtain an order-invariant approximation formula. For a chosen iteration order (i, j, k) the set of boundary points that influence the force of a particular point p of the object can be described by a spatial region Rijk. The geometries of these regions are characterized both for the Cartesian and the body-centered cubic grid (bcc grid). We show that in the case of the bcc grid these regions can be combined in such a way that E3 is uniformly covered which basically means that each boundary point is contained in the same number of regions. Based on the covering an approximation formula for the force field is proposed that has linear complexity and gives good results for standard objects. We also show that such a uniform covering can not be built from the regions of influence of the Cartesian grid. With our method it becomes possible to use features of the force field for a fast and topology preserving skeletonization. We use a thinning strategy on the bcc grid to compute the skeleton and ensure that critical points of the force field are not removed. This leads to improved skeletons with respect to the properties of centeredness and rotational invariance.