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Convolutional Neural Networks (CNNs) work very well for supervised learning problems when the training dataset is representative of the variations expected to be encountered at test time. In medical image segmentation, this premise is violated when there is a mismatch between training and test images in terms of their acquisition details, such as the scanner model or the protocol. Remarkable performance degradation of CNNs in this scenario is well documented in the literature. To address this problem, we design the segmentation CNN as a concatenation of two sub-networks: a relatively shallow image normalization CNN, followed by a deep CNN that segments the normalized image. We train both these sub-networks using a training dataset, consisting of annotated images from a particular scanner and protocol setting. Now, at test time, we adapt the image normalization sub-network for \emph{each test image}, guided by an implicit prior on the predicted segmentation labels. We employ an independently trained denoising autoencoder (DAE) in order to model such an implicit prior on plausible anatomical segmentation labels. We validate the proposed idea on multi-center Magnetic Resonance imaging datasets of three anatomies: brain, heart and prostate. The proposed test-time adaptation consistently provides performance improvement, demonstrating the promise and generality of the approach. Being agnostic to the architecture of the deep CNN, the second sub-network, the proposed design can be utilized with any segmentation network to increase robustness to variations in imaging scanners and protocols. Our code is available at: \url{https://github.com/neerakara/test-time-adaptable-neural-networks-for-domain-generalization}.