Deep Learning. DL has emerged as a powerful tool for addressing inverse problems (McCann et al., 2017; Ongie et al., 2020; Wen et al., 2023). Instead of explicitly defining a regularizer, DL methods use deep neural networks (DNNs) to map the measurements to the desired images (Wang et al., 2016; Jin et al., 2017; Kang et al., 2017; Chen et al., 2017; Delbracio et al., 2021; Delbracio & Milanfar, 2023). Model-based DL (MBDL) is a widely-used sub-family of DL algorithms that integrate physical measurement models with priors specified using CNNs (see reviews by Ongie et al. (2020); Monga et al. (2021)). The literature of 2 MBDLisvast, but some well-known examples include plug-and-play priors (PnP), regularization by denoising (RED), deep unfolding (DU), compressed sensing using generative models (CSGM), and deep equilibrium models (DEQ) (Bora et al., 2017; Romano et al., 2017; Zhang & Ghanem, 2018; Hauptmann et al., 2018; Gilton et al., 2021a; Liu et al., 2022; Hu et al., 2024d). These approaches come with different trade-offs in terms of imaging performance, computational and memory complexity, flexibility, need for supervision, and theoretical understanding. 

Denoisers as Priors. Score-based models (SBMs) are a powerful subset of DL methods for solving inverse problems that use deep Gaussian denoisers as imaging priors. Plug-and-Play (PnP) methods can be viewed as SBMs that incorporate denoisers within iterative optimization algorithms (see recent reviews (Ah mad et al., 2020; Kamilov et al., 2023)). These approaches construct a cost function by combining an explicit likelihood with a score function implicitly defined by the denoiser prior. Over the past few years, numerous variants of PnP have been developed (Venkatakrishnan et al., 2013; Romano et al., 2017; Metzler et al., 2018; Zhang et al., 2017; Meinhardt et al., 2017; Dong et al., 2019; Zhang et al., 2019; Wei et al., 2020; Hurault et al., 2022a), which has motivated an extensive research into their theoretical properties and empirical effectiveness (Chan et al., 2017; Buzzard et al., 2018; Ryu et al., 2019; Sun et al., 2019; Tirer & Giryes, 2019; Teodoro et al., 2019; Xu et al., 2020; Sun et al., 2021; Cohen et al., 2021b; Hu et al., 2022; Laumont et al., 2022; Hurault et al., 2022b; Gan et al., 2023a; Cohen et al., 2021a; Fang et al., 2024; Renaud et al., 2024b; Hu et al., 2024a; Renaud et al., 2024a; Terris et al., 2024). Diffusion Models (DMs) represent another category of SBMs; they are trained to learn the score function of the underlying probability distribution governed by stochastic differential equations (SDEs) (Ho et al., 2020; Song et al., 2021). Once trained, these models can be used as powerful priors for inverse problems by leveraging their learned score functions. Specifically, pre-trained DMs facilitate posterior sampling by guiding the denoising process to generate data consistent with observed measurements. This approach enables DMs to address inverse problems, often achieving im pressive perceptual performance even for highly ill-posed inverse problems (Chung et al., 2023; Zhu et al., 2023; Wang et al., 2023; Feng et al., 2023; Sun et al., 2024; Wu et al., 2024; Song et al., 2024; Hu et al., 2024b; Al¸calar & Ak¸ cakaya, 2024; Zhao et al., 2024; Rout et al., 2024; Bai et al., 2024). 

Restoration Networks as Priors. In addition to denoiser-based methods, recent work has also con sidered using restoration models as implicit priors for solving inverse problems (Zhang et al., 2019; Liu et al., 2020; Gilton et al., 2021b; Hu et al., 2024c). It has been observed that pre-trained restoration models can be effective priors for addressing unseen inverse problems, sometimes surpassing traditional denoiser-based approaches (Hu et al., 2024c). However, existing methods present two main limitations. First, restoration models considered so far have relied on a fixed prior tailored to a specific degradation. Although deep restoration models can be trained in various settings—such as different blur kernels for image deblurring or diverse undersampling masks for MRI reconstruction—current approaches do not leverage this capability, limiting their robustness to diverse artifacts. Second, prior work has not explored the potential of learning restoration priors directly from undersampled measurements, without access to fully sampled data. Unlike Gaussian denoisers, training without fully sampled data is natural for restoration models (Yaman et al., 2020; Liu et al., 2020; Tachella et al., 2022; Chen et al., 2022; Millard & Chiew, 2023; Gan et al., 2023b). It is also worth highlighting the related work that has explored using corrupt measurements for training Ambient DMs (Daras et al., 2023; Aali et al., 2024). Ambient DMs seek to sample from px using DMs trained directly on undersampled measurements. Thus, during inference Ambient DMs assume access to the image prior px, while ShaRP only assumes access to the ensemble of likelihoods of multiple degraded observations. 

Our contribution. (1) We propose ShaRP, a new framework for solving inverse problems leveraging a set of priors implicit in a pre-trained deep restoration network. ShaRP generalizes Regularization by Denoising (RED) (Romano et al., 2017) and Stochastic Denoising Regularization (SNORE) (Renaud et al., 2024b) by using more flexible restoration operators and extends Deep Restoration Priors (DRP) (Hu et al., 2024c) by using multiple restoration priors instead of relying on a single one. (2) We introduce a novel regularization concept for inverse problems that encourages solutions that produce degraded versions closely resembling real degraded images. For example, our regularizer favors an MR image solution only if its various degraded versions are consistent with the characteristics of actual degraded MR images. (3) We show that ShaRP minimizes a composite objective that incorporates our proposed regularizer. We provide a theoretical analysis of its convergence for both exact and approximate MMSE restoration operators. (4) We implement ShaRP with both supervised and self-supervised restoration models as priors and test it on two inverse problems: compressed sensing MRI (CS-MRI) and single-image super-resolution (SISR). Our results highlight the capability of restoration models to achieve state-of-the-art performance. Notably, in the MRI context, we show that restoration networks trained directly on subsampled and noisy MRI data can serve as effective priors, a scenario where training traditional Gaussian denoisers is infeasible.
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