Multivariate Max-Min Sampling Operators in Mathematical Morphology

Abstract.

We build a conceptual bridge between mathematical morphology and max-min sampling operators, offering new insights into the convergence behavior of discrete morphological transformations on multivariate grids as the spatial resolution of the data increases. To this end, we introduce and analyze nonlinear sampling operators rooted in the arithmetic operations of the max-min algebra. These new sampling operators can be interpreted as approximate morphological operations relying on the function values on the sampling grid and a specific kernel. We investigate the convergence behavior of these operators, alongside with the respective properties of their morphological operations as dilation, erosion, opening, and closing. In the two-dimensional setting, we show how these morphological sampling operators can be effectively applied to various image processing tasks, as upscaling, denoising, or segmentation.