Learning continuous activation fields from microscopic SEM images of lycoperdioid fungi via CNN-guided neural operator modeling.

Accurate interpretation of microscopic morphological features is critical for resolving taxonomic boundaries in morphologically similar fungal taxa, yet conventional image analysis approaches largely rely on discrete classification or pixel-level segmentation, which fail to capture the inherently continuous nature of biological microstructures. In this work, a continuous spatial representation learning framework is introduced for the analysis of scanning electron microscopy (SEM) images of lycoperdioid fungal spores by integrating convolutional neural network (CNN)-based local feature encoding with a coordinate-aware neural operator architecture. Microscopic images are reformulated as functions defined over a two-dimensional spatial domain, and a Deep Operator Network (DeepONet) is employed to learn mappings from image functions to continuous activation fields under weak supervision. The proposed framework was evaluated on high-resolution SEM images from five lycoperdioid species, acquired under standardized imaging conditions. Quantitative analyses within the evaluated model configuration indicate that the operator-learning-based approach achieves a 28-35% reduction in mean squared error relative to proxy-based activation references and an 18-22% improvement over patch-level CNN representations without operator coupling. Residual norm and line-profile analyses further reveal a ~ 30% suppression of high-frequency spatial fluctuations, particularly in boundary-dense regions, indicating enhanced spatial regularization. Explainability analysis based on continuous activation fields shows that over 70% of high-response regions consistently align with diagnostically relevant micromorphological features, while same-class activation variance is reduced by 20-25%, confirming improved representational stability without over-smoothing. By moving beyond discrete decision-making toward operator-based continuous modeling, this study establishes a scalable and interpretable framework for microscopic SEM image analysis. The proposed methodology provides a transferable foundation for explainable artificial intelligence applications across fungal taxonomy, biomedical imaging, and other domains requiring high-resolution spatial interpretation of complex microstructures.