Trainable Nonexpansive Denoisers for Contractive Image Reconstruction

Trainable denoisers with Lipschitz control have become central to convergent image reconstruction. However, training neural networks that simultaneously offer strong denoising performance and global Lipschitz guarantees is challenging. Existing approaches enforce Lipschitz control only empirically, providing no guarantees beyond the training data. In this work, we show that by exploiting the action of permutations on the image lattice, we can constrain a neural architecture that is globally nonexpansive (Lipschitz bound $\leqslant 1$). We integrate the proposed denoiser with forward imaging operators to develop a reconstruction mechanism that is provably contractive and therefore globally convergent. Experiments on standard inverse problems, such as superresolution and deblurring, demonstrate that our reconstruction performance is competitive with softly constrained baselines while providing Lipschitz guarantees.

By repeatedly applying an image denoiser as an implicit regularizer, one can reconstruct a sharp image from an observed blurred or low-resolution image. However, repeatedly applying an arbitrary denoiser may lead to unstable behavior, and suitable safeguards are therefore needed to ensure reliable reconstruction. In this work, we design a trainable denoiser that is stable by construction. The key idea is to compare an image with permuted versions of itself and learn which pixels or regions should contribute to denoising each part of the image. The neural network is left largely unconstrained, while stability is enforced externally through a controlled aggregation mechanism. When this denoiser is incorporated into a reconstruction algorithm, the resulting iterative process is guaranteed to be stable and reliable. This work represents a step toward more trustworthy AI-based image reconstruction systems, particularly in applications where stability is essential.