PACER Curriculum Self-Training with Spatial Uncertainty for Pseudo-Label Selection in Few-Shot Anatomical Landmark Detection

We study the semi-supervised setting of a small labeled (oracle) set alongside a larger unlabeled pool from the same distribution. Self-training is the natural framework, but its failure mode, confirmation bias, makes the selection rule the crux. PACER finetunes a pre-trained heatmap detector that predicts one probability heatmap per landmark; trained on one-hot targets, the network learns its own free-form heatmap, so the resulting dispersion reflects the model's confidence and is readable as uncertainty. PACER couples two components.

Expected Radial Error (ERE) as selection signal

For each predicted heatmap q_k and its refined mode û_k, the per-landmark ERE is the expected physical distance of heatmap mass from the predicted location, with s the pixel spacing in millimeters:

ERE_k(x; θ) = ∑_u∈Ω q_k(u)  ‖s ⊙ (u − û_k)‖₂,
e(x; θ) = (1/K) ∑_k ERE_k(x; θ)

ERE is label-free (a function of the prediction only), geometry-aware (close to zero for a sharp unimodal map, large for a diffuse or multimodal one), and physically interpretable (in millimeters, comparable across landmarks, images, and spacings). We gate at the image level, averaging per-landmark ERE into one image score e, which keeps targets image-wise consistent and stabilizes gradients.

Percentile-adaptive curriculum gate

A fixed millimeter threshold is fragile, since the right value depends on anatomy, pixel spacing, landmark count, and the current model. PACER instead sorts the pool of M images by ERE and, at gate level π, admits the m_π = round(π/100 · M) lowest-ERE images. The induced threshold ρ_π = e_(mπ) is read off the current pool rather than chosen in advance. The gate level grows linearly over T stages up to the cap Π:

π_t = (t / T) · Π,    t = 1, …, T

This realizes an easy-to-hard ordering grounded in the model's own millimeter-scaled uncertainty. By default PACER uses the 4-stage curriculum with Π = 100, so the whole pool is eventually admitted and nothing is discarded permanently. At the start of each stage the pool is re-scored and the pseudo-targets of the admitted images are frozen; they are refreshed once per stage, not per epoch, which gives a stable objective and curbs confirmation bias. The only extra work over vanilla self-training is one ERE scoring pass over the pool per stage.

Self-tightening

As training sharpens the heatmaps and the pool ERE shifts toward zero, the same π maps to a smaller ρ_π at each stage, so the gate grows stricter in absolute terms as the model grows more reliable, the opposite of a fixed threshold (Figure 1); the exception is the final stage under Π = 100, which readmits the whole pool by construction. Because ERE is physically scaled rather than bounded, this is a property a class-probability curriculum cannot have. Percentile admission is also calibration-invariant: a global change of s rescales every ERE equally, leaving ranking and admitted set unchanged, so the gate behaves identically on the uncalibrated Shenzhen benchmark.