The Medial Axis Transform (MAT) is a powerful tool for shape analysis and manipulation. Traditional methods for working with shapes usually define shapes as boundaries between some “inside” and some “outside” region. While this definition is simple and intuitive, it does not lend itself well to the construction of algorithms for a number of seemingly simple tasks such as classification, deformation, and collision detection. The MAT is an alternative representation of shape that defines the “inside” region by its center and thickness. We present a method of constructing the MAT which overcomes a significant limitation of its use with real-world data: instability. As classically defined, the MAT is unstable with respect to the shape boundary that it represents. For data sources afflicted by noise this is a serious problem. We propose an algorithm, LSMAT, which constructs a stable least squares approximation to the MAT. / Graduate

Freehand sketching is a natural and crucial part of everyday humaninteraction, yet is almost totally unsupported by current user interfaces. With the increasing availability of tablet notebooks and pen based PDAs, sketchbased interaction has gained attention as a natural interaction modality.We are working to combine the flexibility and ease of use of paper and pencilwith the processing power of a computer, to produce a user interface fordesign that feels as natural as paper, yet is considerably smarter. One of themost basic tasks in accomplishing this is converting the original digitizedpen strokes in a sketch into the intended geometric objects. In this paper wedescribe an implemented system that combines multiple sources of knowledge toprovide robust early processing for freehand sketching. We also show how thisearly processing system can be used as part of a fast sketch recognition system with polynomial time segmentation and recognition algorithms.

[pt] Este trabalho busca reconstruir a geometria de uma malha partindo de sua conectividade e de um conjunto esparso de pontos com geometria conhecida, denominados pontos de controle. O problema é formulado como a maximização da suavidade da superfície fixando a posição dos pontos de controle. Nessa formulação, o método consiste em resolver um sistema linear esparso aplicando-se mínimos quadrados. Diferentes propostas para a seleção de pontos de controle, o método de minimização e a construção do sistema linear são apresentadas e comparadas. / [en] This work aims at reconstructing the geometry of a mesh from its connectivity and a small set of control points, whose geometry is known.The problem is formulated as a maximization of the surface smoothness restricting the position of the control points. With this formulation, the method reduces to solving a sparse linear system using least squares minimization. Several proposals for the selection of the control points, the minimization method and the linear system construction are presented and compared .

<p>New methods for global shape description of three-dimensional digital objects are presented. The shape of an object is first represented by a digital surface where the faces are either triangles or quadrilaterals. Techniques for computing a high-quality parameterization of the surface are developed and this parameterization is used to approximate the shape of the object. Spherical harmonics are used as basis functions for approximations of the coordinate functions. Information about the global shape is then captured by the coefficients in the spherical harmonics expansions.</p><p>For a starshaped object it is shown how a parameterization can be computed by a projection from its surface onto the unit sphere. An algorithm for computing the position at which the centre of the sphere should be placed, is presented. This algorithm is suited for digital voxel objects. Most of the work is concerned with digital objects whose surfaces are homeomorphic to the sphere. The standard method for computing parameterizations of such surfaces is shown to fail on many objects. This is due to the large distortions of the geometric properties of the surface that often occur with this method. Algorithms to handle this problem are suggested. Non-linear optimization methods are used to find a mapping between a surface and the sphere that minimizes geometric distortion and is useful as a parameterization of the surface. </p><p>The methods can be applied, for example, in medical imaging for shape recognition, detection of shape deformations and shape comparisons of three-dimensional objects.</p>

New methods for global shape description of three-dimensional digital objects are presented. The shape of an object is first represented by a digital surface where the faces are either triangles or quadrilaterals. Techniques for computing a high-quality parameterization of the surface are developed and this parameterization is used to approximate the shape of the object. Spherical harmonics are used as basis functions for approximations of the coordinate functions. Information about the global shape is then captured by the coefficients in the spherical harmonics expansions. For a starshaped object it is shown how a parameterization can be computed by a projection from its surface onto the unit sphere. An algorithm for computing the position at which the centre of the sphere should be placed, is presented. This algorithm is suited for digital voxel objects. Most of the work is concerned with digital objects whose surfaces are homeomorphic to the sphere. The standard method for computing parameterizations of such surfaces is shown to fail on many objects. This is due to the large distortions of the geometric properties of the surface that often occur with this method. Algorithms to handle this problem are suggested. Non-linear optimization methods are used to find a mapping between a surface and the sphere that minimizes geometric distortion and is useful as a parameterization of the surface. The methods can be applied, for example, in medical imaging for shape recognition, detection of shape deformations and shape comparisons of three-dimensional objects.

Représenter un objet géométrique complexe par un ensemble de primitives simples est une tâche souvent fondamentale, que ce soit pour la reconstruction et la réparation de données, ou encore pour faciliter la visualisation ou la manipulation des données. Le choix de la ou les primitives, ainsi que celui de la méthode d'approximation, impactent fortement les propriétés de la représentation de forme qui sera obtenue.Dans cette thèse, nous utilisons les boules comme seule primitive. Nous prenons ainsi un grand soin à décrire les unions finies de boules et leur structure. Pour cela, nous nous reposons sur les faisceaux de boules. En particulier, nous aboutissons à une description valide en toute dimension, sans hypothèse de position générale. En chemin, nous obtenons également plusieurs résultats portant sur les tests d'inclusion locale et globale dans une union de boules.Nous proposons également une nouvelle méthode d'approximation par union finie de boules, l'approximation par boules à (delta,epsilon)-près. Cette approche contraint l'union de boules à couvrir un sous-ensemble de la forme d'origine (précisément, un epsilon-érodé), tout en étant contenu dans un sur-ensemble de la forme (un delta-dilaté). En nous appuyant sur nos précédents résultats portant sur les unions de boules, nous démontrons plusieurs propriétés de ces approximations. Nous verrons ainsi que calculer une approximation par boules à (delta,epsilon)-près qui soit de cardinal minimum est un problème NP-complet. Pour des formes simples dans le plan, nous présentons un algorithme polynomial en temps et en espace qui permet de calculer ces approximations de cardinal minimum. Nous concluons par une généralisation de notre méthode d'approximation pour une plus large variété de sous-ensembles et sur-ensembles. / Describing a complex geometric shape with a set of simple primitives is often a fundamental task for shape reconstruction, visualization, analysis and manipulation. The type of primitives, as well as the choice of approximation scheme, both greatly impact the properties of the resulting shape representation.In this PhD, we focus on balls as primitives. Using pencils of balls, we carefully describe finite unions of balls and their structure. In particular, our description holds in all dimension without assuming general position. On our way, we also establish various results and tools to test local and global inclusions within these unions.We also propose a new approximation scheme by union of balls, the (delta,epsilon)-ball approximation. This scheme constrains the approximation to cover a core subset of the original shape (specifically, an epsilon-erosion), while being contained within a superset of the shape (a delta-dilation). Using our earlier results regarding finite unions of balls, we prove several properties of these approximations. We show that computing a cardinal minimum (delta,epsilon)-ball approximation is an NP-complete problem. For simple planar shapes however, we present a polynomial time and space algorithm that outputs a cardinal minimum approximation. We then conclude by generalizing the approximation scheme to a wider range of core subsets and bounding supersets.