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Deformable image registration is a crucial step in medical image analysis for finding a non-linear spatial transformation between a pair of fixed and moving images. Deep registration methods based on Convolutional Neural Networks (CNNs) have been widely used as they can perform image registration in a fast and end-to-end manner. However, these methods usually have limited performance for image pairs with large deformations. Recently, iterative deep registration methods have been used to alleviate this limitation, where the transformations are iteratively learned in a coarse-to-fine manner. However, iterative methods inevitably prolong the registration runtime, and tend to learn separate image features for each iteration, which hinders the features from being leveraged to facilitate the registration at later iterations. In this study, we propose a Non-Iterative Coarse-to-finE registration Network (NICE-Net) for deformable image registration. In the NICE-Net, we propose: (i) a Single-pass Deep Cumulative Learning (SDCL) decoder that can cumulatively learn coarse-to-fine transformations within a single pass (iteration) of the network, and (ii) a Selectively-propagated Feature Learning (SFL) encoder that can learn common image features for the whole coarse-to-fine registration process and selectively propagate the features as needed. Extensive experiments on six public datasets of 3D brain Magnetic Resonance Imaging (MRI) show that our proposed NICE-Net can outperform state-of-the-art iterative deep registration methods while only requiring similar runtime to non-iterative methods.