Deep equilibrium models (DEQ) have emerged as a powerful alternative to deep unfolding (DU) for image reconstruction. DEQ models-implicit neural networks with effectively infinite number of layers-were shown to achieve state-of-the-art image reconstruction without the memory complexity associated with DU. While the performance of DEQ has been widely investigated, the existing work has primarily focused on the settings where groundtruth data is available for training. We present self-supervised deep equilibrium model (SelfDEQ) as the first self-supervised reconstruction framework for training model-based implicit networks from undersampled and noisy MRI measurements. Our theoretical results show that SelfDEQ can compensate for unbalanced sampling across multiple acquisitions and match the performance of fully supervised DEQ. Our numerical results on in-vivo MRI data show that SelfDEQ leads to state-of-the-art performance using only undersampled and noisy training data.
Deep model-based architectures (DMBAs) integrating physical measurement models and learned image regularizers are widely used in parallel magnetic resonance imaging (PMRI). Traditional DMBAs for PMRI rely on pre-estimated coil sensitivity maps (CSMs) as a component of the measurement model. However, estimation of accurate CSMs is a challenging problem when measurements are highly undersampled. Additionally, traditional training of DMBAs requires high-quality groundtruth images, limiting their use in applications where groundtruth is difficult to obtain. This paper addresses these issues by presenting SPICE as a new method that integrates self-supervised learning and automatic coil sensitivity estimation. Instead of using pre-estimated CSMs, SPICE simultaneously reconstructs accurate MR images and estimates high-quality CSMs. SPICE also enables learning from undersampled noisy measurements without any groundtruth. We validate SPICE on experimentally collected data, showing that it can achieve state-of-the-art performance in highly accelerated data acquisition settings (up to 10x).