Morphological Bilateral Filtering

A current challenging topic in mathematical morphology is the construction of locally adaptive operators; i.e., structuring functions are dependent on the input image itself at each position. Development of spatially-variant filtering is well established in the theory and practice of Gaussian filtering. The aim of the first part of the paper is to study how to generalize these convolutionbased approaches in order to introduce adaptive nonlinear filters which asymptotically correspond to spatially-variant morphological dilation and erosion. In particular, starting from the bilateral filtering framework and using the notion counter-harmonic mean, our goal is to propose a new low complexity approach to define spatially-variant bilateral structuring functions. Then, in the second part of the paper, an original formulation of spatially-variant flat morphological filters is proposed, where the adaptive structuring elements are obtained by thresholding the bilateral structuring functions. The methodological results of the paper are illustrated with various comparative examples.