Self-Supervised Deep Equilibrium Models With Theoretical Guarantees and Applications to MRI Reconstruction
Weijie Gan, Chunwei Ying, Parna Eshraghi Boroojeni, Tongyao Wang, Cihat Eldeniz, Yuyang Hu, Jiaming Liu, Yasheng Chen, Hongyu An, Ulugbek S. Kamilov
Abstract
Deep equilibrium models (DEQ) have emerged as a powerful alternative to deep unfolding (DU) for image reconstruction. DEQ models— <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">implicit neural networks</i> 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 ground truth data is available for training. We present <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">self-supervised deep equilibrium model (SelfDEQ)</i> 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in-vivo</i> MRI data show that SelfDEQ leads to state-of-the-art performance using only undersampled and noisy training data.