# Scale Equivariant Neural Networks with Morphological Scale-Spaces

arXiv: https://arxiv.org/abs/2105.01335 · HTML: https://mlsanity.com/paper/2105.01335/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 May 2021.

## Built on by (most cited first, 8 of 8)

- [Scale Equivariant U-Net](https://mlsanity.com/paper/2210.04508/) (2022)
- [Riesz networks: scale invariant neural networks in a single forward pass](https://mlsanity.com/paper/2305.04665/) (2023)
- [MorphPool: Efficient Non-linear Pooling & Unpooling in CNNs](https://mlsanity.com/paper/2211.14037/) (2022)
- [PDE-CNNs: Axiomatic Derivations and Applications](https://mlsanity.com/paper/2403.15182/) (2024)
- [Current Symmetry Group Equivariant Convolution Frameworks for Representation Learning](https://mlsanity.com/paper/2409.07327/) (2024)
- [Adaptive aggregation of Monte Carlo augmented decomposed filters for efficient group-equivariant convolutional neural network](https://mlsanity.com/paper/2305.10110/) (2023)
- [Efficient 3D affinely equivariant CNNs with adaptive fusion of augmented spherical Fourier-Bessel bases](https://mlsanity.com/paper/2402.16825/) (2024)
- [On the representation of stack operators by mathematical morphology](https://mlsanity.com/paper/2504.09766/) (2025)

## Abstract (arXiv)

The translation equivariance of convolutions can make convolutional neural networks translation equivariant or invariant. Equivariance to other transformations (e.g. rotations, affine transformations, scalings) may also be desirable as soon as we know a priori that transformed versions of the same objects appear in the data. The semigroup cross-correlation, which is a linear operator equivariant to semigroup actions, was recently proposed and applied in conjunction with the Gaussian scale-space to create architectures which are equivariant to discrete scalings. In this paper, a generalization using a broad class of liftings, including morphological scale-spaces, is proposed. The architectures obtained from different scale-spaces are tested and compared in supervised classification and semantic segmentation tasks where objects in test images appear at different scales compared to training images. In both classification and segmentation tasks, the scale-equivariant architectures improve dramatically the generalization to unseen scales compared to a convolutional baseline. Besides, in our experiments morphological scale-spaces outperformed the Gaussian scale-space in geometrical tasks.