We propose a method to encode rotation equivariance or invariance into convolutional neural networks (CNNs). Each convolutional filter is applied with several orientations and returns a vector field that represents the magnitude and angle of the highest scoring rotation at the given spatial location. To propagate information about the main orientation of the different features to each layer in the network, we propose an enriched orientation pooling, i.e. max and argmax operators over the orientation space, allowing to keep the dimensionality of the feature maps low and to propagate only useful information. We name this approach RotEqNet. We apply RotEqNet to three datasets: first, a rotation invariant classification problem, the MNIST-rot benchmark, in which we improve over the state-of-the-art results. Then, a neuron membrane segmentation benchmark, where we show that RotEqNet can be applied successfully to obtain equivariance to rotation with a simple fully convolutional architecture. Finally, we improve significantly the state-of-the-art on the problem of estimating cars' absolute orientation in aerial images, a problem where the output is required to be covariant with respect to the object's orientation.

本文中，我们提出了旋转等变向量场网络（RotEqNet），这是一种卷积神经网络（CNN）体系结构，可以编码旋转等变性、不变性和协方差性，并在图像分类、生物医学图像分割、方向估计和拼接匹配等多个问题上进行了测试。结果表明，相较于许多数量级更大的网络，RotEqNet 可以提供非常紧凑的模型，并提供与后者类似的结果。