  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.
