The coincidence similarity index, based on a combination of the Jaccard and overlap similarity indices, has noticeable properties in comparing and classifying data, including enhanced selectivity and sensitivity, intrinsic normalization, and robustness to data perturbations and outliers. These features allow multiset neurons, which are based on the coincidence similarity operation, to perform effective pattern recognition applications, including the challenging task of image segmentation. A few prototype points have been used in previous related approaches to represent each pattern to be identified, each of them being associated with respective multiset neurons. The segmentation of the regions can then proceed by taking into account the outputs of these neurons. The present work describes multilayer multiset neuronal networks incorporating two or more layers of coincidence similarity neurons. In addition, as a means to improve performance, this work also explores the utilization of counter-prototype points, which are assigned to the image regions to be avoided. This approach is shown to allow effective segmentation of complex regions despite considering only one prototype and one counter-prototype point. As reported here, the balanced accuracy landscapes to be optimized in order to identify the weight of the neurons in subsequent layers have been found to be relatively smooth, while typically involving more than one attraction basin. The use of a simple gradient-based optimization methodology has been demonstrated to effectively train the considered neural networks with several architectures, at least for the given data type, configuration of parameters, and network architecture.

翻译：基于雅卡尔德相似性指数与重叠相似性指数相结合的巧合相似性指数，在数据比较与分类中具有显著特性，包括增强的选择性和敏感性、固有无量纲化、以及对数据扰动和离群值的鲁棒性。这些特性使得基于巧合相似性运算的多集合神经元能够执行有效的模式识别任务，包括极具挑战性的图像分割。在以往的相关方法中，通常使用少量原型点来表征待识别的每个模式，每个原型点对应一个多集合神经元。通过分析这些神经元的输出，即可进行区域分割。本文描述了包含两层或更多层巧合相似性神经元的多层多集合神经元网络。此外，为提升性能，本研究还探索了反原型点的利用——这些点被分配给需要避开的图像区域。结果表明，尽管仅使用一个原型点和一个反原型点，该方法仍能有效分割复杂区域。如本文所述，为确定后续各层神经元权重而需优化的平衡准确率景观通常较为平滑，但往往包含多个吸引域。基于简单梯度的优化方法已被证明能够有效训练所考虑的多种架构的神经网络，至少在给定数据类型、参数配置及网络架构的条件下如此。