CRE: Convolutional basis-free rotation-equivariant operator on grids

Scientific and medical imaging often suffers from limited labeled data, making it important to exploit structure already present in the data. These images often contain geometric symmetries. In particular, rotation is a common nuisance factor: tissues, cells, and objects can appear at arbitrary orientations. Standard CNNs exploit translation symmetry but lack built-in rotation handling, while existing equivariant methods often rely on discrete rotation groups, predefined steerable bases, lifted orientation spaces, or non-standard image representations. We propose CRE, a local, basis-free convolutional block that models rotations directly on the native image grid without explicit orientation channels. CRE combines center- relative pixel interactions with invariant neighborhood attention, replacing fixed Cartesian kernel weights with geometry-conditioned local relations. We evaluate CRE on six benchmarks: three classification datasets and three segmentation datasets. In classification, CRE achieves the best mean accuracy across continuous rotations on all three datasets, while maintaining competitive variance over different angles. In segmentation, CRE remains competitive with baselines showing that it can be used as a practical simile for CNN-based pipelines.