Morphological neural networks, or layers, can be a powerful tool to boost the progress in mathematical morphology, either on theoretical aspects such as the representation of complete lattice operators, or in the development of image processing pipelines. However, these architectures turn out to be difficult to train when they count more than a few morphological layers, at least within popular machine learning frameworks which use gradient descent based optimization algorithms. In this paper we investigate the potential and limitations of differentiation based approaches and back-propagation applied to morphological networks, in light of the non-smooth optimization concept of Bouligand derivative. We provide insights and first theoretical guidelines, in particular regarding initialization and learning rates.

翻译：形态学神经网络（或形态学层）可成为推动数学形态学发展的重要工具，无论是在理论层面（如完备格算子的表示），还是在图像处理流程的开发中。然而，当这类架构包含多于少数几个形态学层时，其训练过程将变得尤为困难——至少在采用基于梯度下降优化算法的流行机器学习框架中如此。本文结合Bouligand导数这一非光滑优化概念，探究基于微分的方法及反向传播在形态学神经网络中的应用潜力与局限。我们提供了关键洞见与初步理论指导，尤其涉及参数初始化与学习率设置等方面。