Standard convolutions are prevalent in image processing and deep learning,
but their fixed kernel design limits adaptability. Several deformation
strategies of the reference kernel grid have been proposed. Yet, they lack a
unified theoretical framework. By returning to a metric perspective for images,
now seen as two-dimensional manifolds equipped with notions of local and
geodesic distances, either symmetric (Riemannian metrics) or not (Finsler
metrics), we provide a unifying principle: the kernel positions are samples of
unit balls of implicit metrics. With this new perspective, we also propose
metric convolutions, a novel approach that samples unit balls from explicit
signal-dependent metrics, providing interpretable operators with geometric
regularisation. This framework, compatible with gradient-based optimisation,
can directly replace existing convolutions applied to either input images or
deep features of neural networks. Metric convolutions typically require fewer
parameters and provide better generalisation. Our approach shows competitive
performance in standard denoising and classification tasks.

使用隐式度量的球体来采样，提出了可解释的具有几何正则化的几何卷积方法，可以替换现有的卷积操作并在标准去噪和分类任务中表现出竞争性能。