Beyond Spatial Correspondence: Optical-Density-Guided Marginal and Relational Learning for Weakly-Paired H&E-to-IHC Virtual Staining

Weakly paired H&E-to-IHC virtual staining learns from adjacent tissue sections, where deformation, tissue loss, and expression heterogeneity weaken local correspondence. Spatial co-location and visual similarity do not necessarily imply compatible staining states, making them insufficient on their own for organizing stain-consistent supervision. Optical density (OD), widely used to quantify immunohistochemical (IHC) staining intensity, provides complementary evidence of stain expression. We propose PGLStain, which uses continuous OD compatibility not merely as an intensity-matching target, but as a signal for conditioning feature relations and allocating local supervision. Protein-Expression-Aware Correspondence Calibration (PECC) combines feature-context and OD-calibrated graphs, assigns relative weights to spatially proposed candidate pairs, and aligns permutation-invariant OD-state graph summaries. To complement candidate-based supervision with image-level expression constraints, Marginal–Relational Stain Alignment (MRSA) aligns unordered OD marginals and uses unbalanced Gromov–Wasserstein transport to compare intra-image expression-state relations under relaxed node-mass constraints, without fixed cross-section node assignments. Together, these objectives integrate stain-calibrated local supervision with correspondence-free expression-state learning, without additional inference-time computation. Experiments on public H&E-to-IHC benchmarks demonstrate competitive perceptual quality and quantitative staining agreement. Code are avaliable at https://anonymous.4open.science/r/PGLStain-anonymous-AC00anonymous repository