Integral Geometry and General Adaptive Neighborhoods for Multiscale Image Analysis

Overview & Synopsis

In 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.