Image registration estimates spatial correspondences between a pair of images. These estimates are typically obtained via numerical optimization or regression by a deep network. A desirable property of such estimators 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 the desired equivariance properties in the context of multi-step 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 multi-step 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 multi-step $[W,U]$ equivariance via a coordinate-attention mechanism combined with displacement-predicting refinement layers (CARL). Overall, our approach obtains excellent practical registration performance on several 3D medical image registration tasks and outperforms existing unsupervised approaches for the challenging problem of abdomen registration.

翻译：图像配准旨在估计一对图像之间的空间对应关系。此类估计通常通过数值优化或深度网络回归获得。此类估计器的一个理想特性是：对于给定的图像对，其对应关系估计（例如真实的理论对应关系）在输入图像发生形变时保持不变。从形式上看，估计器应对特定类别的图像变换具有等变性。本研究对多阶段深度配准网络中所需的等变性特性进行了细致分析。基于这些分析，我们：1）提出了 $[U,U]$ 等变性（网络对输入图像相同形变的等变性）与 $[W,U]$ 等变性（输入图像可经历不同形变）的概念；2）证明了在适当的多阶段配准框架中，若第一阶段具有 $[W,U]$ 等变性且后续所有阶段均具有 $[U,U]$ 等变性，则整体可实现 $[W,U]$ 等变性；3）指出常见的位移预测网络仅对平移变换具有 $[U,U]$ 等变性，而非更强大的 $[W,U]$ 等变性；4）提出通过坐标注意力机制结合位移预测优化层（CARL）实现多阶段 $[W,U]$ 等变性。整体而言，我们的方法在多个3D医学图像配准任务中取得了优异的实际配准性能，并在具有挑战性的腹部图像配准问题上超越了现有无监督方法。