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Mathematical morphology (MM) is a powerful non-linear theory that can be used for signal and image processing and analysis. Although MM can be very well defined on complete lattices, which are partially ordered sets with well defined extrema operations, there is no natural ordering for multivalued images such as hyper-spectral and color images. Thus, a great deal of effort has been devoted to ordering schemes for multivalued MM. In a reduced ordering, in particular, elements are ranked according to the so-called ordering mapping. Despite successful applications, morphological operators based on reduced orderings are usually too reliant on the ordering mapping. In many practical situations, however, the ordering mapping may be subject to uncertainties such as measurement errors or the arbitrariness in the choice of the mapping. In view of this remark, in this paper we present two approaches to multivalued MM based on an uncertain reduced ordering. The new operators are formulated as the solution of an optimization problem which, apart from the uncertainty, can circumvent the false value problem and deal with irregularity issues.

Symmetry is present in many tasks in computer vision, where the same class of objects can appear transformed, e.g. rotated due to different camera orientations, or scaled due to perspective. The knowledge of such symmetries in data coupled with equivariance of neural networks can improve their generalization to new samples. Differential invariants are equivariant operators computed from the partial derivatives of a function. In this paper we use differential invariants to define equivariant operators that form the layers of an equivariant neural network. Specifically, we derive invariants of the Special Euclidean Group SE(2), composed of rotations and translations, and apply them to construct a SE(2)-equivariant network, called SE(2) Differential Invariants Network (SE2DINNet). The network is subsequently tested in classification tasks which require a degree of equivariance or invariance to rotations. The results compare positively with the state-of-the-art, even though the proposed SE2DINNet has far less parameters than the compared models.