Provably Bounded Dynamic Sparsifying Transform Network for Compressive Imaging

Compressive imaging (CI) aims to recover the underlying image from the under-sampled observations. Recently, deep unfolded CI (DUCI) algorithms, which unfold the iterative algorithms into deep neural networks (DNNs), have achieved remarkable results. Theoretically, unfolding a convergent iterative algorithm could ensure a stable DUCI algorithm, i.e., its performance increases as the increasing stage. However, ensuring convergence often involves imposing constraints, such as bounded spectral norm or tight property, on the filter weights or sparsifying transform. Unfortunately, these constraints may compromise algorithm performance. To address this challenge, we present a provably bounded dynamic sparsifying transform network (BSTNet), which can be explicitly proven to be a bounded network without imposing constraints on the analysis sparsifying transform. Leveraging this advantage, the analysis sparsifying transform can be adaptively generated via a trainable DNN. Specifically, we elaborate a dynamic sparsifying transform generator capable of extracting multiple feature information from input instances, facilitating the creation of a faithful content-adaptive sparsifying transform. We explicitly demonstrate that the proposed BSTNet is a bounded network, and further embed it as the prior network into a DUCI framework to evaluate its performance on two CI tasks, i.e., spectral snapshot CI (SCI) and compressed sensing magnetic resonance imaging (CSMRI). Experimental results showcase that our DUCI algorithms can achieve competitive recovery quality compared to benchmark algorithms. Theoretically, we explicitly prove that the proposed BSTNet is bounded, and we provide a comprehensive theoretical convergence analysis of the proposed iteration algorithms.