  Symmetry is present in nature and science. In image processing, kernels for
spatial filtering possess some symmetry (e.g. Sobel operators, Gaussian,
Laplacian). Convolutional layers in artificial feed-forward neural networks
have typically considered the kernel weights without any constraint. In this
paper, we propose to investigate the impact of a symmetry constraint in
convolutional layers for image classification tasks, taking our inspiration
from the processes involved in the primary visual cortex and common image
processing techniques. The goal is to assess the extent to which it is possible
to enforce symmetrical constraints on the filters throughout the training
process of a convolutional neural network (CNN) by modifying the weight update
preformed during the backpropagation algorithm and to evaluate the change in
performance. The main hypothesis of this paper is that the symmetrical
constraint reduces the number of free parameters in the network, and it is able
to achieve near identical performance to the modern methodology of training. In
particular, we address the following cases: x/y-axis symmetry, point
reflection, and anti-point reflection. The performance has been evaluated on
four databases of images. The results support the conclusion that while random
weights offer more freedom to the model, the symmetry constraint provides a
similar level of performance while decreasing substantially the number of free
parameters in the model. Such an approach can be valuable in phase-sensitive
applications that require a linear phase property throughout the feature
extraction process.
