Shape representation and analysis of 2D compact sets by shape diagrams

Shape diagrams are shape representations in the Euclidean plane introduced for studying 3D and 2D compact sets. A compact set is represented by a point within a shape diagram whose coordinates are morphological functionals defined from geometrical functionals and inequalities. Classically, the geometrical functionals for 2D sets are the area, the perimeter, the radii of the inscribed and circumscribed circles, and the minimum and maximum Feret diameters. The purpose of this paper is to present a particular shape diagram for which mathematical properties have been well-defined and to analyse the shape of several families of 2D sets for the discrimination of convex and non convex sets as well as the discrimination of similar sets

Shape diagrams for 2D compact sets - Part III: convexity discrimination for analytic and discretized simply connected sets.

International audienceShape diagrams are representations in the Euclidean plane introduced to study 3-dimensional and 2-dimensional compact convex sets. However, they can also been applied to more general compact sets than compact convex sets. A compact set is represented by a point within a shape diagram whose coordinates are morphometrical functionals defined as normalized ratios of geometrical functionals. Classically, the geometrical functionals are the area, the perimeter, the radii of the inscribed and circumscribed circles, and the minimum and maximum Feret diameters. They allow twenty-two shape diagrams to be built. Starting from these six classical geometrical functionals, a detailed comparative study has been performed in order to analyze the representation relevance and discrimination power of these twenty-two shape diagrams. The two first parts of this study are published in previous papers [8, 9]. They focus on analytic compact convex sets and analytic simply connected compact sets, respectively. The purpose of this paper is to present the third part, by focusing on the convexity discrimination for analytic and discretized simply connected compact sets

Shape diagrams for 2D compact sets - Part I: analytic convex sets. Australian Journal of

International audienceShape diagrams are representations in the Euclidean plane introduced to study 3-dimensional and 2-dimensional compact convex sets. Such a set is represented by a point within a shape diagram whose coordinates are morphometrical functionals defined as normalized ratios of geometrical functionals. Classically, the geometrical functionals are the area, the perimeter, the radii of the inscribed and circumscribed circles, and the minimum and maximum Feret diameters. They allow thirty-one shape diagrams to be built. Most of these shape diagrams can also been applied to more general compact sets than compact convex sets. Starting from these six classical geometrical functionals, a detailed comparative study has been performed in order to analyze the representation relevance and discrimination power of these thirty-one shape diagrams. The purpose of this paper is to present the first part of this study, by focusing on analytic compact convex sets. A set will be called analytic if its boundary is piecewise defined by explicit functions in such a way that the six geometrical functionals can be straightforwardly calculated. The second and third part of the comparative study are published in two following papers [19, 20]. They are focused on analytic simply connected sets and convexity discrimination for analytic and discretized simply connected sets, respectively

Shape diagrams for 2D compact sets - Part II: analytic simply connected sets.

International audienceShape diagrams are representations in the Euclidean plane introduced to study 3-dimensional and 2-dimensional compact convex sets. However, they can also been applied to more general compact sets than compact convex sets. A compact set is represented by a point within a shape diagram whose coordinates are morphometrical functionals defined as normalized ratios of geometrical functionals. Classically, the geometrical functionals are the area, the perimeter, the radii of the inscribed and circumscribed circles, and the minimum and maximum Feret diameters. They allow twenty-two shape diagrams to be built. Starting from these six classical geometrical functionals, a detailed comparative study has been performed in order to analyze the representation relevance and discrimination power of these twenty-two shape diagrams. The first part of this study is published in a previous paper [16]. It focused on analytic compact convex sets. A set will be called analytic if its boundary is piecewise defined by explicit functions in such a way that the six geometrical functionals can be straightforwardly calculated. The purpose of this paper is to present the second part, by focusing on analytic simply connected compact sets. The third part of the comparative study is published in a following paper [17]. It is focused on convexity discrimination for analytic and discretized simply connected compact sets

Fonctionnelles et fonctions de Minkowski à voisinages adaptatifs généraux pour l'analyse des images à tons de gris

http://documents.irevues.inist.fr/handle/2042/29071 http://hdl.handle.net/2042/29071 | DOI : 10.4267/2042/29071International audienceEn analyse d'image, les fonctionnelles de Minkowski sont des paramètres standards de mesures topologiques et géométriques d'objets. Néanmoins, elles sont souvent limitées aux images binaires, et déterminées de manière globale et mono-échelle. L'utilisation des Voisinages Adaptatifs Généraux (VAG), simultanément adaptatifs avec les échelles d'analyse, les structures spatiales et les intensités des images, permet de pallier ces limites. Une analyse locale, adaptative et multi-échelle des images à tons de gris est donc proposée sous forme de cartographies des fonctionnelles de Minkowski à VAG. Les VAG sont des voisinages définis en chaque point du support spatial d'une image à tons de gris, homogènes par rapport à un critère d'analyse suivant une tolérance d'homogénéité. Les fonctionnelles de Minkowski sont calculées pour chaque VAG de l'image, permettant de définir les cartographies des fonctionnelles de Minkowski adaptatives. A chaque point de l'image est donc attribuée une fonctionnelle topologique ou géométrique de la structure locale de l'image. Par ailleurs, l'évolution de la tolérance d'homogénéité permet de définir des cartographies multi-échelles. De plus, les cartographies sont analysées de manière qualitative sous l'influence d'une transformation morphologique multi-échelle (dilatation/érosion, ouverture/fermeture par reconstruction,...) ================================== In quantitative image analysis, Minkowski functionals are standard parameters for topological and geometrical measurements. Nevertheless, they are often limited to binary images and achieved in a global and monoscale way. The use of General Adaptive Neighborhoods (GANs), simultaneously adaptive with the analyzing scales, the spatial structures and the image intensities, enables to overcome these limitations. The GAN-based Minkowski functionals are introduced, which allow a gray-tone image analysis to be realized in a local, adaptive and multiscale way. The GANs are spatial neighborhoods defined around each point of the spatial support of a gray-tone image, homogeneous with respect to a criterion function according to an homogeneity tolerance. The Minkowski functionals are computed on the GAN of each point of the image, enabling to define the so-called Minkowski maps which assign the geometrical or the topological functional to each point. Also, the evolution of the GAN homogeneity tolerance allows the multiscale Minkowski maps to be defined. Futhermore, the impact of multiscale morphological transformations (dilation/erosion, opening/closing by reconstruction, . . . ), is analyzed in a qualitative way through these maps

General adaptive neighborhood-based Minkowski maps for gray-tone image analysis.

In quantitative image analysis, Minkowski functionals are standard parameters for topological and geometrical measurements. Nevertheless, they are often limited to binary images and achieved in a global and monoscale way. The use of General Adaptive Neighborhoods (GANs) enables to overcome these limitations. The GANs are spatial neighborhoods defined around each point of the spatial support of a gray-tone image, according to three (GAN) axiomatic criteria: a criterion function (luminance, contrast, . . . ), an homogeneity tolerance with respect to this criterion, and an algebraic model for the image space. Thus, the GANs are simultaneously adaptive with the analyzing scales, the spatial structures and the image intensities. The aim of this paper is to introduce the GAN-based Minkowski functionals, which allow a gray-tone image analysis to be realized in a local, adaptive and multiscale way. The Minkowski functionals are computed on the GAN of each point of the image, enabling to define the so-called Minkowski maps which assign the geometrical or the topological functional to each point. The impact of the GAN characteristics, as well as the impact of multiscale morphological transformations, is analyzed in a qualitative way through these maps. The GAN-based Minkowski maps are illustrated on the test image 'Lena' and also applied in the biomedical and materials areas

Analyse morphométrique d'images à tons de gris par diagrammes de forme

Session "Posters"National audienceLes diagrammes de forme sont des représentations introduites pour étudier les ensembles connexes compacts: un tel ensemble est représenté par un point dans le plan euclidien dont les coordonnées sont deux fonctionnelles morphométriques. Néanmoins, les diagrammes de forme sont souvent limités à l'analyse globale et mono-échelle d'image binaire. L'utilisation des Voisinages Adaptatifs Généraux (VAG) permet de pallier ces limites. Les VAG sont des voisinages spatiaux, définis autour de chaque point du support spatial d'une image à niveaux de gris, simultanément adatatifs avec les échelles d'analyse, les structures spatiales, et les intensités de l'image. Ce papier vise à introduire les diagrammes de forme à VAG, qui permettent d'analyser les images à niveaux de gris de manière locale, adaptative et multi-échelle. Les fonctionnelles morphométriques sont calculées pour le VAG de chaque point du support spatial de l'image, définissant les diagrammes de forme à VAG. Des distributions morphométriques des structures locales de l'image sont obtenues, permettant par exemple de les classifie

Integral Geometry and General Adaptive Neighborhoods for Multiscale Image Analysis

International audienceIn quantitative image analysis, Minkowski functionals are becoming standard parameters for topological and geometrical measurements. Nevertheless, they are limited to binary images or to sections of gray-tone images and are achieved in a global and monoscale way. The use of General Adaptive Neighborhoods (GANs) enables to overcome these limitations. The GANs are spatial neighborhoods defined around each point of the spatial support of a gray-tone image, according to three (GAN) axiomatic criteria: a criterion function (luminance, contrast, ...), an homogeneity tolerance with respect to this criterion, and an algebraic model for the image space. Thus, the GANs are simultaneously adaptive with the analyzing scales, the spatial structures and the image intensities. This paper aims to introduce the GAN-based Minkowski functionals, which allow a gray-tone image analysis to be realized in a local, adaptive and multiscale way. The Minkowski functionals are computed on the GAN of each point of the spatial support of a gray-tone image, enabling to define the so-called Minkowski maps by assigning the Minkowski functional value to each point. The histograms of these maps provide a statistical distribution of the topology and geometry of the gray-tone image structures, and not only of the image intensities. The impact of the GAN characteristics, as well as the impact of multiscale transformations, are analyzed in a qualitative global and local way through these GAN-based Minkowski maps and histograms. This multiscale image analysis is illustrated on the test image 'Lena' and also applied in both the biomedical and materials areas

Geometric and morphometric image analysis by shape diagrams and general adaptive neighborhoods

Les fonctionnelles de Minkowski définissent des mesures topologiques et géométriques d'ensembles, insuffisantes pour la caractérisation, des ensembles différents pouvant avoir les mêmes fonctionnelles. D'autres fonctionnelles de forme, géométriques et morphométriques, sont donc utilisées. Un diagramme de forme, défini grâce à deux fonctionnelles morphométriques, donne une représentation permettant d'étudier les formes d'ensembles. En analyse d'image, ces fonctionnelles et diagrammes sont souvent limités aux images binaires et déterminés de manière globale et mono-échelle. Les Voisinages Adaptatifs Généraux (VAG) simultanément adaptatifs avec les échelles d'analyse, structures spatiales et intensités des images, permettent de pallier ces limites. Une analyse locale, adaptative et multi-échelle des images à tons de gris est proposée sous forme de cartographies des fonctionnelles de forme à VAG.Les VAG, définis en tout point du support spatial d'une image à tons de gris, sont homogènes par rapport à un critère d'analyse représenté dans un modèle vectoriel, suivant une tolérance d'homogénéité. Les fonctionnelles de forme calculées pour chaque VAG de l'image définissent les cartographies des fonctionnelles de forme à VAG. Les histogrammes et diagrammes de ces cartographies donnent des distributions statistiques des formes des structures locales de l'image contrairement aux histogrammes classiques qui donnent une distribution globale des intensités de l'image. L'impact de la variation des critères axiomatiques des VAG est analysé à travers ces cartographies, histogrammes et diagrammes. Des cartographies multi-échelles sont construites, définissant des fonctions de forme à VAG.Minkowski functionals define set topological and geometrical measurements, insufficient for the characterization, because different sets may have the same functionals. Thus, other shape functionals, geometrical and morphometrical are used. A shape diagram, defined thanks to two morphometrical functionals, provides a representation allowing the study of set shapes. In quantitative image analysis, these functionals and diagrams are often limited to binary images and achieved in a global and monoscale way. The General Adaptive Neighborhoods (GANs) simultaneously adaptive with the analyzing scales, the spatial structures and the image intensities, enable to overcome these limitations. The GAN-based Minkowski functionals are introduced, which allow a gray-tone image analysis to be realized in a local, adaptive and multiscale way.The GANs, defined around each point of the spatial support of a gray-tone image, are homogeneous with respect to an analyzing criterion function represented in an algebraic model, according to an homogeneity tolerance. The shape functionals computed on the GAN of each point of the spatial support of the image, define the so-called GAN-based shape maps. The map histograms and diagrams provide statistical distributions of the shape of the gray-tone image local structures, contrary to the classical histogram that provides a global distribution of image intensities. The impact of axiomatic criteria variations is analyzed through these maps, histograms and diagrams. Thus, multiscale maps are built, defining GAN-based shape functions

Analyse d’image geometrique et morphometrique par diagrammes de forme et voisinages adaptatifs generaux

Minkowski functionals define set topological and geometrical measurements, insufficient for the characterization, because different sets may have the same functionals. Thus, other shape functionals, geometrical and morphometrical are used. A shape diagram, defined thanks to two morphometrical functionals, provides a representation allowing the study of set shapes. In quantitative image analysis, these functionals and diagrams are often limited to binary images and achieved in a global and monoscale way. The General Adaptive Neighborhoods (GANs) simultaneously adaptive with the analyzing scales, the spatial structures and the image intensities, enable to overcome these limitations. The GAN-based Minkowski functionals are introduced, which allow a gray-tone image analysis to be realized in a local, adaptive and multiscale way.The GANs, defined around each point of the spatial support of a gray-tone image, are homogeneous with respect to an analyzing criterion function represented in an algebraic model, according to an homogeneity tolerance. The shape functionals computed on the GAN of each point of the spatial support of the image, define the so-called GAN-based shape maps. The map histograms and diagrams provide statistical distributions of the shape of the gray-tone image local structures, contrary to the classical histogram that provides a global distribution of image intensities. The impact of axiomatic criteria variations is analyzed through these maps, histograms and diagrams. Thus, multiscale maps are built, defining GAN-based shape functions.Les fonctionnelles de Minkowski définissent des mesures topologiques et géométriques d'ensembles, insuffisantes pour la caractérisation, des ensembles différents pouvant avoir les mêmes fonctionnelles. D'autres fonctionnelles de forme, géométriques et morphométriques, sont donc utilisées. Un diagramme de forme, défini grâce à deux fonctionnelles morphométriques, donne une représentation permettant d'étudier les formes d'ensembles. En analyse d'image, ces fonctionnelles et diagrammes sont souvent limités aux images binaires et déterminés de manière globale et mono-échelle. Les Voisinages Adaptatifs Généraux (VAG) simultanément adaptatifs avec les échelles d'analyse, structures spatiales et intensités des images, permettent de pallier ces limites. Une analyse locale, adaptative et multi-échelle des images à tons de gris est proposée sous forme de cartographies des fonctionnelles de forme à VAG.Les VAG, définis en tout point du support spatial d'une image à tons de gris, sont homogènes par rapport à un critère d'analyse représenté dans un modèle vectoriel, suivant une tolérance d'homogénéité. Les fonctionnelles de forme calculées pour chaque VAG de l'image définissent les cartographies des fonctionnelles de forme à VAG. Les histogrammes et diagrammes de ces cartographies donnent des distributions statistiques des formes des structures locales de l'image contrairement aux histogrammes classiques qui donnent une distribution globale des intensités de l'image. L'impact de la variation des critères axiomatiques des VAG est analysé à travers ces cartographies, histogrammes et diagrammes. Des cartographies multi-échelles sont construites, définissant des fonctions de forme à VAG