Proximal Mapping Loss for Crowd Counting

We propose the Proximal Mapping Loss (PML), a unified and theoretically grounded framework for loss design in crowd counting and localization that operates without the intersection hypothesis, a common but unrealistic assumption in prior density-to-point methods that object regions do not overlap. By removing this constraint, our approach better reflects real-world crowd scenes where severe occlusions and dense overlaps are prevalent. Instead of relying on heuristics that break down under such conditions, PML interprets loss minimization through the lens of proximal operators from convex optimization, linking the predicted density map to the ground-truth point annotations via a well-defined regularized inversion process.

In this formulation, minimizing a loss corresponds to computing the proximal mapping of the target point measure under a chosen regularizer (e.g., entropy, total variation, or ℓ₁/ℓ₂ penalties). This perspective reveals that classic losses like MAE, MSE, or recent density-to-point losses (e.g., BL, GL) are special cases of PML under specific regularizers in a point-neighbor
case. In contrast, PML generalizes these losses to the more realistic setting where object supports may arbitrarily overlap, enabling accurate density recovery even in highly congested scenes. The framework is fully differentiable and allows flexible incorporation of prior knowledge through the choice of regularizer.

Experiments on standard benchmarks (ShanghaiTech, UCF-QNRF, and JHU++) demonstrate that PML achieves state-of-the-art performance in both counting accuracy and localization quality—especially in high-density regimes where overlapping is dominant. Notably, PML outperforms previous methods in metrics like AP and F1-score for localization, precisely because it does not enforce artificial non-overlap constraints. By discarding the intersection hypothesis and grounding loss design in optimization theory, our work provides a more faithful, robust, and interpretable foundation for crowd analysis.

Selected Publications