Marginalized Generalized IoU (MGIoU): A Unified Objective Function for Optimizing Any Convex Parametric Shapes

Optimizing the similarity between parametric shapes is crucial for numerous computer vision tasks, where Intersection over Union (IoU) stands as the canonical measure. However, existing optimization methods exhibit significant shortcomings: regression-based losses like L1/L2 lack correlation with IoU, IoU-based losses are unstable and limited to simple shapes, and task-specific methods are computationally intensive and not generalizable accross domains. As a result, the current landscape of parametric shape objective functions has become scattered, with each domain proposing distinct IoU approximations. To address this, we unify the parametric shape optimization objective functions by introducing Marginalized Generalized IoU (MGIoU), a novel loss function that overcomes these challenges by projecting structured convex shapes onto their unique shape Normals to compute one-dimensional normalized GIoU. MGIoU offers a simple, efficient, fully differentiable approximation strongly correlated with IoU. We then extend MGIoU to MGIoU+ that supports optimizing unstructured convex shapes. Together, MGIoU and MGIoU+ unify parametric shape optimization across diverse applications. Experiments on standard benchmarks demonstrate that MGIoU and MGIoU+ consistently outperform existing losses while reducing loss computation latency by 10-40x. Additionally, MGIoU and MGIoU+ satisfy metric properties and scale-invariance, ensuring robustness as an objective function. We further propose MGIoU- for minimizing overlaps in tasks like collision-free trajectory prediction. Code is available at https://ldtho.github.io/MGIoU