Warped Convolutions: Efficient Invariance to Spatial Transformations

TL;DR

This paper introduces a simple, efficient method for achieving invariance to various spatial transformations in CNNs by using a constant image warp followed by standard convolution, enabling broad applicability with minimal computational overhead.

Contribution

It proposes a novel, exact construction that achieves invariance to multiple spatial transformations with the same complexity as standard convolutions, using a specific image warp.

Findings

Effective in vehicle pose estimation with rotation and scale invariance

Successful in face pose estimation with 3D rotations under perspective

Maintains computational efficiency comparable to standard convolutions

Abstract

Convolutional Neural Networks (CNNs) are extremely efficient, since they exploit the inherent translation-invariance of natural images. However, translation is just one of a myriad of useful spatial transformations. Can the same efficiency be attained when considering other spatial invariances? Such generalized convolutions have been considered in the past, but at a high computational cost. We present a construction that is simple and exact, yet has the same computational complexity that standard convolutions enjoy. It consists of a constant image warp followed by a simple convolution, which are standard blocks in deep learning toolboxes. With a carefully crafted warp, the resulting architecture can be made equivariant to a wide range of two-parameter spatial transformations. We show encouraging results in realistic scenarios, including the estimation of vehicle poses in the Google Earth dataset (rotation and scale), and face poses in Annotated Facial Landmarks in the Wild (3D rotations under perspective).