Relaxed Distribution-wise Consistency: Learning Affinity Order for Semi-supervised Medical Image Segmentation

Semi-supervised medical image segmentation learns from pseudo-labels produced by an imperfect teacher, and its supervision faces a tension between reliability and richness: pseudo-label filtering pursues reliability but discards the boundary pixels where supervision is most needed, whereas inter-pixel supervision pursues richness and reaches these pixels but aligns the teacher's noisy affinity values in a value-alignment manner. Since noisy values may still preserve the correct order, we propose Relaxed Distribution-wise Consistency (RDC), which relaxes inter-pixel alignment from values to order and combines it with filtered pixel-level pseudo-labels. RDC is a simple and lightweight addition to the standard teacher-student framework: each pixel is represented by affinities to well-separated reference pixels (agents), and the affinity order is aligned between teacher and student through a differentiable Kendall's $\tau$. Our analysis shows that, under symmetric teacher noise, value-alignment targets drift and can even reverse the order of two agents, whereas order alignment preserves the direction of the expected update for each agent pair and only weakens it; we further bound the expected fraction of inverted agent pairs. On ACDC, LA, Pancreas-NIH, and Synapse, RDC achieves state-of-the-art performance with the largest gains on the most difficult structures, and consistently improves GapMatch, GraphCL, VQ-Seg, and the SAM-based ESDE as a plug-in. With the agents fixed, every tested order-based objective outperforms every value-based one.