  Encoding the scale information explicitly into the representation learned by
a convolutional neural network (CNN) is beneficial for many computer vision
tasks especially when dealing with multiscale inputs. We study, in this paper,
a scaling-translation-equivariant (ST-equivariant) CNN with joint convolutions
across the space and the scaling group, which is shown to be both sufficient
and necessary to achieve equivariance for the regular representation of the
scaling-translation group ST . To reduce the model complexity and computational
burden, we decompose the convolutional filters under two pre-fixed separable
bases and truncate the expansion to low-frequency components. A further benefit
of the truncated filter expansion is the improved deformation robustness of the
equivariant representation, a property which is theoretically analyzed and
empirically verified. Numerical experiments demonstrate that the proposed
scaling-translation-equivariant network with decomposed convolutional filters
(ScDCFNet) achieves significantly improved performance in multiscale image
classification and better interpretability than regular CNNs at a reduced model
size.
