CARL: A Framework for Equivariant Image Registration

Image registration estimates spatial correspondences between image pairs. These estimates are typically obtained via numerical optimization or regression by a deep network. A desirable property is that a correspondence estimate (e.g., the true oracle correspondence) for an image pair is maintained under deformations of the input images. Formally, the estimator should be equivariant to a desired class of image transformations. In this work, we present careful analyses of equivariance properties in the context of multistep deep registration networks. Based on these analyses we 1) introduce the notions of [U, U ] equivariance (network equivariance to the same deformations of the input images) and [W, U ] equivariance (where input images can undergo different deformations); we 2) show that in a suitable multistep registration setup it is sufficient for overall [W, U ] equivariance if the first step has [W, U ] equivariance and all others have [U, U ] equivariance; we 3) show that common displacement-predicting networks only exhibit [U, U ] equivariance to translations instead of the more powerful [W, U ] equivariance; and we 4) show how to achieve multistep [W, U ] equivariance via a coordinate-attention mechanism combined with displacement-predicting networks. Our approach obtains excellent practical performance for 3D abdomen, lung, and brain medical image registration. We match or outperform state-of-the-art (SOTA) registration approaches on all the datasets with a particularly strong performance for the challenging abdomen registration.