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

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