Catálogo de publicaciones - libros

Access to the shape by its exterior is solved using convex hull. Many algorithms have been proposed in that way. This contribution addresses the open problem of the access of the shape by its interior also called convex skull. More precisely, we present approaches in discrete case. Furthermore, a simple algorithm to approximate the maximum convex subset of star-shaped polygons is described.

We say that a metric space is regular if a straight-line (in the metric space sense) passing through the center of a sphere has at least two diametrically opposite points. The normed vector spaces have this property. Nevertheless, this property might not be satisfied in some metric spaces. In this work, we give a characterization of an integer-valued translation-invariant regular metric defined on the discrete plane, in terms of a symmetric subset that induces through a recursive Minkowski sum, a chain of subsets that are morphologically closed with respect to .

A general algorithm for computing Euclidean skeletons of 3D data sets in linear time is presented. These skeletons are defined in terms of a new concept, called the () transform. The algorithm is based upon the computation of 3D feature transforms, using a modification of an algorithm for Euclidean distance transforms. The skeletonization algorithm has a time complexity which is linear in the amount of voxels, and can be easily parallelized. The relation of the skeleton to the usual definition in terms of centers of maximal disks is discussed.

Only the very restricted class of γ-regular shapes is proven not to change topology during digitization. Such shapes have a limited boundary curvature and cannot have corners. In this paper it is shown, how a much wider class of shapes, for which the morphological open-close and the close-open-operator with an -disc lead to the same result, can be digitized correctly in a topological sense by using an additional repairing step. It is also shown that this class is very general and includes several commonly used shape descriptions. The repairing step is easy to compute and does not change as much pixels as a preprocessing regularization step. The results are applicable for arbitrary, even irregular, sampling grids.

A downsampling method for binary images is presented, which aims at preserving the topology of the image. It uses a general reference sampling structure. The reference image is computed through the analysis of the connected components of the neighborhood of each pixel. The resulting downsampling operator is auto-dual, which ensures that white and black structures are treated in the same way. Experiments show that the image topology is indeed preserved, when there is enough space, satisfactorily.

Shortest distances, grey weighted distances and ultrametric distance are classically used in mathematical morphology. We introduce a lexicographic distance, for which any segmentation with markers becomes a Voronoï tessellation.

The spatial relationship “between” is a notion which is intrinsically both fuzzy and contextual, and depends in particular on the shape of the objects. The few existing definitions do not take into account these aspects. We propose here definitions which are based on morphological operators and a fuzzy notion of visibility in order to model the main intuitive acceptions of the relation. We distinguish between cases where objects have similar spatial extensions and cases where one object is much more extended than the other. These definitions are illustrated on real data from brain images.

While shock filters are popular morphological image enhancement methods, no well-posedness theory is available for their corresponding partial differential equations (PDEs). By analysing the dynamical system of ordinary differential equations that results from a space discretisation of a PDE for 1-D shock filtering, we derive an analytical solution and prove well-posedness. Finally we show that the results carry over to the fully discrete case when an explicit time discretisation is applied.

Partial differential equations (PDEs) have become very useful modeling and computational tools for many problems in image processing and computer vision related to multiscale analysis and optimization using variational calculus. In previous works, the basic continuous-scale morphological operators have been modeled by nonlinear geometric evolution PDEs. However, these lacked a variational interpretation. In this paper we contribute such a variational formulation and show that the PDEs generating multiscale dilations and erosions can be derived as gradient flows of variational problems with nonlinear constraints. We also extend the variational approach to more advanced object-oriented morphological filters by showing that levelings and the PDE that generates them result from minimizing a mean absolute error functional with local sup-inf constraints.

We present a constrained shape optimisation problem solved via metaheuristic stochastic techniques. Genetic Algorithms are briefly reviewed and their adaptation to surface topography optimisation is studied. An application to flow optimisation issues is presented.