Optimizing the similarity between parametric shapes is crucial for numerous
computer vision tasks, où Intersection sur Union (IdO) stands as the
canonical measure. Cependant, existing optimization methods exhibit significant
shortcomings: les pertes basées sur la régression comme L1/L2 manquent de corrélation avec 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. À
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, efficace, fully differentiable approximation strongly correlated with
IdO. 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. En plus, 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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