Learning of structuring elements for morphological image model with a sparsity prior

A Learning Framework for Morphological Operators Using Counter–Harmonic Mean

“…To the best of our knowledge, we are the first to do this in a flexible and gradient-based framework without any prior. For instance, in classical approaches [13] or more recent ones [11], the operator needs to be fixed a priori. Figure 4-top shows an example of a closing with a line of length 10 and an orientation of 45 • , whereas Figure 4-bottom shows an example of an opening with a square of size 5.…”

“…However, most of the proposed approaches do not cover all operators. More importantly, they cannot learn both the structuring element and the operator, e.g., [11]. This is obviously a quite important limitation as it makes very hard or even impossible the composition of complex filtering pipelines.…”

Geometric Back-propagation in Morphological Neural Networks

“…Morphological operations are not fully differentiable however; therefore MNNs cannot readily be trained using linear back-propagation. As a consequence, three different approaches to back-propagating errors over sequences of morphological filters are established in the literature: (1) approximate (linearly) the min, max-operations to make them differentiable [10], [19], [20];…”

Geometric Back-Propagation in Morphological Neural Networks

“…From the chain rule, the term ∂E ∂f+(x) is all the information that is needed to obtain the derivatives of the error with respect to the input f − and the parameterized probe p (or kernel) of ⊗. a consequence, three different approaches to back-propagating errors over sequences of morphological filters are established in the literature: (1) approximate (linearly) the min, max-operations to make them differentiable [10,19,20];…”