Sketched Equivariant Imaging Regularization and Deep Internal Learning for Inverse Problems

Equivariant Imaging (EI) regularization has become the de-facto technique for unsupervised training of deep imaging networks, without any need of ground-truth data. Observing that the EI-based unsupervised training paradigm currently has significant computational redundancy leading to inefficiency in high-dimensional applications, we propose a sketched EI regularization which leverages the randomized sketching techniques for acceleration. We then extend our sketched EI regularization to develop an accelerated deep internal learning framework -- Sketched Equivariant Deep Image Prior (Sk-EI-DIP), which can be efficiently applied for single-image and task-adapted reconstruction. Additionally, for network adaptation tasks, we propose a parameter-efficient approach for accelerating both EI-DIP and Sk-EI-DIP via optimizing only the normalization layers. Our numerical study on X-ray CT image reconstruction tasks demonstrate that our approach can achieve order-of-magnitude computational acceleration over standard EI-based counterpart in single-input setting, and network adaptation at test time.