Generalized Morphology using Sponges

Mathematical morphology has traditionally been grounded in lattice theory. For non-scalar data lattices often prove too restrictive, however. In this paper we present a more general alternative, sponges, that still allows useful definitions of various properties and concepts from morphological theory. It turns out that some of the existing work on “pseudo-morphology” for non-scalar data can in fact be considered “proper” mathematical morphology in this new framework, while other work cannot, and that this correlates with how useful/intuitive some of the resulting operators are

Median filtering and its extensions for color images

The construction of structure-preserving denoising filters for color images is a challenging task. A new approach is presented that is based on a recently proposed transformation from the RGB color space to the space of symmetric 2x2 matrices (Burgeth, B., Kleefeld A. (2014) An approach to color-morphology based on Einstein addition and Loewner order, Pattern Recognition Letters, 47, 29-39.). This new framework coupled with spatial adaptivity via morphological amoebas offers excellent capabilities for structure-preserving filtering of color images. Additionally, a generalization of the median-based concept is proposed leading to color-valued amoeba M-smoothers. Numerical experiments confirm the applicability and the potential of this novel approach (Kleefeld, A. et al. (2015) Adaptive Filters for Color Images: Median Filtering and its Extensions, Lecture Notes in Computer Science, Springer, Berlin, 9016, 149-158.)

Adaptive Filters for Color Images: Median Filtering and its Extensions

In this talk, the construction of structure-preserving denoising filters for color images is explained. This is based on a recently proposed transformation from the RGB color space to the space of symmetric $2\times2$ matrices that has already been used to transfer morphological operations such as dilation and erosion from matrix-valued data to color images (see [1]).The applicability of this framework is shown for the construction ofcolor-valued median filters. Additionally, spatial adaptivity is introducedinto this approach by morphological amoebas that offer excellent capabilities for structure-preserving filtering. Furthermore, color-valued amoeba M-smoothers as a generalization of the median-basedconcepts are defined. The experiments confirm that all these methods work wellwith color images. They demonstrate the potential of the new approach todefine color processing tools based on matrix field techniques (refer to [2]).[1] Burgeth, B., Kleefeld A. (2014) An approach to color-morphology based on Einstein addition and Loewner order, Pattern Recognition Letters, 47, 29-39.[2] Kleefeld, A. et al. (2015) Adaptive Filters for Color Images: Median Filtering and its Extensions, Lecture Notes in Computer Science, Springer, Berlin, 9016, 149-158