ENSURE: Ensemble Stein's Unbiased Risk Estimator for Unsupervised Learning

Deep learning accelerates the MR image reconstruction process after offline training of a deep neural network from a large volume of clean and fully sampled data. Unfortunately, fully sampled images may not be available or are difficult to acquire in several application areas such as high-resolution imaging. Previous studies have utilized Stein's Unbiased Risk Estimator (SURE) as a mean square error (MSE) estimate for the image denoising problem. Unrolled reconstruction algorithms, where the denoiser at each iteration is trained using SURE, has also been introduced. Unfortunately, the end-to-end training of a network using SURE remains challenging since the projected SURE loss is a poor approximation to the MSE, especially in the heavily undersampled setting. We propose an ENsemble SURE (ENSURE) approach to train a deep network only from undersampled measurements. In particular, we show that training a network using an ensemble of images, each acquired with a different sampling pattern, can closely approximate the MSE. Our preliminary experimental results show that the proposed ENSURE approach gives comparable reconstruction quality to supervised learning and a recent unsupervised learning method.