A classical approach to designing binary image operators is Mathematical Morphology (MM). We propose the Discrete Morphological Neural Networks (DMNN) for binary image analysis to represent W-operators and estimate them via machine learning. A DMNN architecture, which is represented by a Morphological Computational Graph, is designed as in the classical heuristic design of morphological operators, in which the designer should combine a set of MM operators and Boolean operations based on prior information and theoretical knowledge. Then, once the architecture is fixed, instead of adjusting its parameters (i.e., structural elements or maximal intervals) by hand, we propose a lattice gradient descent algorithm (LGDA) to train these parameters based on a sample of input and output images under the usual machine learning approach. We also propose a stochastic version of the LGDA that is more efficient, is scalable and can obtain small error in practical problems. The class represented by a DMNN can be quite general or specialized according to expected properties of the target operator, i.e., prior information, and the semantic expressed by algebraic properties of classes of operators is a differential relative to other methods. The main contribution of this paper is the merger of the two main paradigms for designing morphological operators: classical heuristic design and automatic design via machine learning. Thus, conciliating classical heuristic morphological operator design with machine learning. We apply the DMNN to recognize the boundary of digits with noise, and we discuss many topics for future research.

翻译：设计二值图像算子的经典方法是数学形态学（MM）。我们提出用于二值图像分析的离散形态学神经网络（DMNN），以表示W算子并通过机器学习对其估计。DMNN架构由形态学计算图表示，其设计遵循经典形态学算子的启发式设计思路：设计者需基于先验信息和理论知识，组合一组MM算子与布尔运算。架构固定后，我们提出格梯度下降算法（LGDA），而非手动调整参数（即结构元素或最大区间），基于输入输出图像的样本在常规机器学习框架下训练这些参数。我们还提出LGDA的随机版本，该版本更高效、可扩展，并能解决实际问题中误差较小的问题。DMNN所表示的类别可根据目标算子的预期属性（即先验信息）具有通用性或专化性，而由算子类代数性质所表达的语义相对于其他方法具有区别性。本文的主要贡献在于融合了设计形态学算子的两大范式：经典启发式设计与基于机器学习的自动设计，从而调和了经典启发式形态学算子设计与机器学习。我们将DMNN应用于含噪数字边界的识别，并讨论了诸多未来研究方向。