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