Topology-preserving thinning in 2-D pseudomanifolds

International audiencePreserving topological properties of objects during thinning procedures is an important issue in the field of image analysis. In the case of 2-D digital images (i.e. images defined on Z^2) such procedures are usually based on the notion of simple point. By opposition to the case of spaces of higher dimensions (i.e. Z^n, n ≥ 3), it was proved in the 80’s that the exclusive use of simple points in Z^2 was indeed sufficient to develop thinning procedures providing an output that is minimal with respect to the topological characteristics of the object. Based on the recently introduced notion of minimal simple set (generalising the notion of simple point), we establish new properties related to topology-preserving thinning in 2-D spaces which extend, in particular, this classical result to more general spaces (the 2-D pseudomanifolds) and objects (the 2-D cubical complexes)

Topological properties of thinning in 2-D pseudomanifolds

International audiencePreserving topological properties of objects during thinning procedures is an important issue in the field of image analysis. In the case of 2-D digital images (i.e. images defined on Z^2) such procedures are usually based on the notion of simple point. In contrast to the situation in Z^n , n>=3, it was proved in the 80s that the exclusive use of simple points in Z^2 was indeed sufficient to develop thinning procedures providing an output that is minimal with respect to the topological characteristics of the object. Based on the recently introduced notion of minimal simple set (generalising the notion of simple point), we establish new properties related to topology-preserving thinning in 2-D spaces which extend, in particular, this classical result to cubical complexes in 2-D pseudomanifolds

Writing Reusable Digital Geometry Algorithms in a Generic Image Processing Framework

Digital Geometry software should reflect the generality of the underlying mathe- matics: mapping the latter to the former requires genericity. By designing generic solutions, one can effectively reuse digital geometry data structures and algorithms. We propose an image processing framework focused on the Generic Programming paradigm in which an algorithm on the paper can be turned into a single code, written once and usable with various input types. This approach enables users to design and implement new methods at a lower cost, try cross-domain experiments and help generalize resultsComment: Workshop on Applications of Discrete Geometry and Mathematical Morphology, Istanb : France (2010

Transformaciones geométricas para facilitar la identificación de objetos en imágenes digitales

La visión artificial por reconocimiento de patrones es un concepto que permite dar visión de mayor calidad, ver en tiempo real o bien dar visión artificial a dispositivos de menor capacidad tecnológica. Dependiendo de su posición, orientación o tamaño, un objeto puede generar millones de imágenes diferentes, lo que dificulta su identificación. Las características distintivas de un objeto suelen encontrarse en los bordes del mismo. Allí se buscan detalles de sectores del borde que estén en la menor cantidad de objetos, de modo de hacer rápida la obtención de resultados. Pero estos patrones del borde tienen relación con su ubicación dentro del borde del objeto, y también con la inclinación y escala del mismo. Dotando a una imagen digital de una topología y aprovechando las propiedades invariantes de la topología, se estudiará un algoritmo que mediante transformaciones geométricas específicas, obtenga una curva factible de ser comparada con formas estándares a fin de reconocer el objeto cuyo borde es tal curva.Eje: Computación gráfica, imágenes y visualizaciónRed de Universidades con Carreras en Informática (RedUNCI

Transformaciones geométricas para facilitar la identificación de objetos en imágenes digitales

La visión artificial por reconocimiento de patrones es un concepto que permite dar visión de mayor calidad, ver en tiempo real y seguir objetos. Dependiendo de su posición, orientación o tamaño, un objeto puede generar millones de imágenes diferentes, lo que dificulta su identificación. Las características topológicas de un objeto pueden estudiarse por medio de un análisis de los bordes. Convexidad, concavidad, curvatura, puntos extremos pueden detectarse siguiendo los puntos frontera del objeto, y en su conjunto permiten identificar el mismo. Estos patrones del borde tienen relación con su ubicación dentro del borde del objeto, y también con la inclinación y escala del mismo. Dotando a una imagen digital de una topología y aprovechando las propiedades invariantes, se estudian estrategias de reconocimiento de formas.Eje: Computación Gráfica, Imágenes y VisualizaciónRed de Universidades con Carreras en Informática (RedUNCI

On the equivalence between hierarchical segmentations and ultrametric watersheds

We study hierarchical segmentation in the framework of edge-weighted graphs. We define ultrametric watersheds as topological watersheds null on the minima. We prove that there exists a bijection between the set of ultrametric watersheds and the set of hierarchical segmentations. We end this paper by showing how to use the proposed framework in practice in the example of constrained connectivity; in particular it allows to compute such a hierarchy following a classical watershed-based morphological scheme, which provides an efficient algorithm to compute the whole hierarchy.Comment: 19 pages, double-colum

A graph-based mathematical morphology reader

This survey paper aims at providing a "literary" anthology of mathematical morphology on graphs. It describes in the English language many ideas stemming from a large number of different papers, hence providing a unified view of an active and diverse field of research

Skeletal representations of orthogonal shapes

Skeletal representations are important shape descriptors which encode topological and geometrical properties of shapes and reduce their dimension. Skeletons are used in several fields of science and attract the attention of many researchers. In the biocad field, the analysis of structural properties such as porosity of biomaterials requires the previous computation of a skeleton. As the size of three-dimensional images become larger, efficient and robust algorithms that extract simple skeletal structures are required. The most popular and prominent skeletal representation is the medial axis, defined as the shape points which have at least two closest points on the shape boundary. Unfortunately, the medial axis is highly sensitive to noise and perturbations of the shape boundary. That is, a small change of the shape boundary may involve a considerable change of its medial axis. Moreover, the exact computation of the medial axis is only possible for a few classes of shapes. For example, the medial axis of polyhedra is composed of non planar surfaces, and its accurate and robust computation is difficult. These problems led to the emergence of approximate medial axis representations. There exists two main approximation methods: the shape is approximated with another shape class or the Euclidean metric is approximated with another metric. The main contribution of this thesis is the combination of a specific shape and metric simplification. The input shape is approximated with an orthogonal shape, which are polygons or polyhedra enclosed by axis-aligned edges or faces, respectively. In the same vein, the Euclidean metric is replaced by the L infinity or Chebyshev metric. Despite the simpler structure of orthogonal shapes, there are few works on skeletal representations applied to orthogonal shapes. Much of the efforts have been devoted to binary images and volumes, which are a subset of orthogonal shapes. Two new skeletal representations based on this paradigm are introduced: the cube skeleton and the scale cube skeleton. The cube skeleton is shown to be composed of straight line segments or planar faces and to be homotopical equivalent to the input shape. The scale cube skeleton is based upon the cube skeleton, and introduces a family of skeletons that are more stable to shape noise and perturbations. In addition, the necessary algorithms to compute the cube skeleton of polygons and polyhedra and the scale cube skeleton of polygons are presented. Several experimental results confirm the efficiency, robustness and practical use of all the presented methods

Skeletal representations of orthogonal shapes

In this paper we present two skeletal representations applied to orthogonal shapes of R^n : the cube axis and a family of skeletal representations provided by the scale cube axis. Orthogonal shapes are a subset of polytopes, where the hyperplanes of the bounding facets are restricted to be axis aligned. Both skeletal representations rely on the L∞ metric and are proven to be homotopically equivalent to its shape. The resulting skeleton is composed of n − 1 dimensional facets. We also provide an efficient and robust algorithm to compute the scale cube axis in the plane and compare the resulting skeleton with other skeletal representations.Postprint (published version