Learning geometric invariants through neural networks

Convolution neural networks have become a fundamental model for solving various computer vision tasks. However, these operations are only invariant to translations of objects and their performance suffer under rotation and other affine transformations. This work proposes a novel neural network that leverages geometric invariants, including curvature, higher-order differentials of curves extracted from object boundaries at multiple scales, and the relative orientations of edges. These features are invariant to affine transformation and can improve the robustness of shape recognition in neural networks. Our experiments on the smallNORB dataset with a 2-layer network operating over these geometric invariants outperforms a 3-layer convolutional network by 9.69% while being more robust to affine transformations, even when trained without any data augmentations. Notably, our network exhibits a mere 6% degradation in test accuracy when test images are rotated by 40\(^{\circ }\), in contrast to significant drops of 51.7 and 69% observed in VGG networks and convolution networks, respectively, under the same transformations. Additionally, our models show superior robustness than invariant feature descriptors such as the SIFT-based bag-of-words classifier, and its rotation invariant extension, the RIFT descriptor that suffer drops of 35 and 14.1% respectively, under similar image transformations. Our experimental results further show improved robustness against scale and shear transformations. Furthermore, the multi-scale extension of our geometric invariant network, that extracts curve differentials of higher orders, show enhanced robustness to scaling and shearing transformations.