Probabilistic pseudo-morphology for grayscale and color images

• We propose a pseudo-morphology based on probabilistic tools.
• We use Chebyshev inequality and PCA for estimating pseudo-extrema of a set.
• Our operators are linear or non-linear depending on the choice of a parameter.
• We extend the approach to multivariate images, particularly for the color domain.
• Validation is performed through texture feature extraction.

Mathematical morphology offers popular image processing tools, successfully used for binary and grayscale images. Recently, its extension to color images has become of interest and several approaches were proposed. Due to various issues arising from the vectorial nature of the data, none of them imposed as a generally valid solution. We propose a probabilistic pseudo-morphological approach, by estimating two pseudo-extrema based on Chebyshev inequality. The framework embeds a parameter which allows controlling the linear versus non-linear behavior of the probabilistic pseudo-morphological operators. We compare our approach for grayscale images with the classical morphology and we emphasize the impact of this parameter on the results. Then, we extend the approach to color images, using principal component analysis. As validation criteria, we use the estimation of the color fractal dimension, color textured image segmentation and color texture classification. Furthermore, we compare our proposed method against two widely used approaches, one morphological and one pseudo-morphological.