Convolutional neural networks have shown great success on feature extraction from raw input data such as images. Although convolutional neural networks are invariant to translations on the inputs, they are not invariant to other transformations, including rotation and flip. Recent attempts have been made to incorporate more invariance in image recognition applications, but they are not applicable to dense prediction tasks, such as image segmentation. In this paper, we propose a set of methods based on kernel rotation and flip to enable rotation and flip invariance in convolutional neural networks. The kernel rotation can be achieved on kernels of 3 @math 3, while kernel flip can be applied on kernels of any size. By rotating in eight or four angles, the convolutional layers could produce the corresponding number of feature maps based on eight or four different kernels. By using flip, the convolution layer can produce three feature maps. By combining produced feature maps using maxout, the resource requirement could be significantly reduced while still retain the invariance properties. Experimental results demonstrate that the proposed methods can achieve various invariance at reasonable resource requirements in terms of both memory and time.
Cohen and Welling @cite_2 studied rotation invariance and equivariance of convolutional neural networks from the perspective of group theory. They show that by operating on feature maps or kernels, the network can achieve invariance and equivariance. Both Dieleman @cite_9 and Cohen and Welling @cite_2 show that rotating feature maps and rotating kernels within convolutional layers have the same effect.
@cite_9 @cite_2
2951770173 2952054889
Many classes of images exhibit rotational symmetry. Convolutional neural networks are sometimes trained using data augmentation to exploit this, but they are still required to learn the rotation equivariance properties from the data. Encoding these properties into the network architecture, as we are already used to doing for translation equivariance by using convolutional layers, could result in a more efficient use of the parameter budget by relieving the model from learning them. We introduce four operations which can be inserted into neural network models as layers, and which can be combined to make these models partially equivariant to rotations. They also enable parameter sharing across different orientations. We evaluate the effect of these architectural modifications on three datasets which exhibit rotational symmetry and demonstrate improved performance with smaller models. We introduce Group equivariant Convolutional Neural Networks (G-CNNs), a natural generalization of convolutional neural networks that reduces sample complexity by exploiting symmetries. G-CNNs use G-convolutions, a new type of layer that enjoys a substantially higher degree of weight sharing than regular convolution layers. G-convolutions increase the expressive capacity of the network without increasing the number of parameters. Group convolution layers are easy to use and can be implemented with negligible computational overhead for discrete groups generated by translations, reflections and rotations. G-CNNs achieve state of the art results on CIFAR10 and rotated MNIST.