Image adapted total ordering for mathematical morphology on multivariate images

In this paper, we discuss the problem of total ordering for mathematical morphology. We restrain the study to color images, but results are valid in any metric space. The discontinuities issue of total orders has already been evoked in mathematical morphology, but remains rarely studied. This phenomenon is highlighted and formalised in this paper. We propose a new approach to avoid discontinuities based on a recursive algorithm. The key point of the proposed method is to adapt the order to the studied image. The proposed framework presents invariance to isometric transformations of the color space. Promising results are presented.