  The convolutional layers of standard convolutional neural networks (CNNs) are
equivariant to translation. However, the convolution and fully-connected layers
are not equivariant or invariant to other affine geometric transformations.
Recently, a new class of CNNs is proposed in which the conventional layers of
CNNs are replaced with equivariant convolution, pooling, and
batch-normalization layers. The final classification layer in equivariant
neural networks is invariant to different affine geometric transformations such
as rotation, reflection and translation, and the scalar value is obtained by
either eliminating the spatial dimensions of filter responses using convolution
and down-sampling throughout the network or average is taken over the filter
responses. In this work, we propose to integrate the orthogonal moments which
gives the high-order statistics of the function as an effective means for
encoding global invariance with respect to rotation, reflection and translation
in fully-connected layers. As a result, the intermediate layers of the network
become equivariant while the classification layer becomes invariant. The most
widely used Zernike, pseudo-Zernike and orthogonal Fourier-Mellin moments are
considered for this purpose. The effectiveness of the proposed work is
evaluated by integrating the invariant transition and fully-connected layer in
the architecture of group-equivariant CNNs (G-CNNs) on rotated MNIST and
CIFAR10 datasets.
