# Multi-Scale Energy (MuSE) plug and play framework for inverse problems

arXiv: https://arxiv.org/abs/2305.04775 · HTML: https://mlsanity.com/paper/2305.04775/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 May 2023.

## Built on by (most cited first, 8 of 8)

- [MRI Super-Resolution with Deep Learning: A Comprehensive Survey](https://mlsanity.com/paper/2511.16854/) (2025)
- [Robust plug-and-play methods for highly accelerated non-Cartesian MRI reconstruction](https://mlsanity.com/paper/2411.01955/) (2024)
- [Diffusion Probabilistic Generative Models for Accelerated, in-NICU Permanent Magnet Neonatal MRI](https://mlsanity.com/paper/2505.15984/) (2025)
- [Three-Dimensional Diffusion-Weighted Multi-Slab MRI With Slice Profile Compensation Using Deep Energy Model](https://mlsanity.com/paper/2501.17152/) (2025)
- [MAP Image Recovery with Guarantees using Locally Convex Multi-Scale Energy (LC-MUSE) Model](https://mlsanity.com/paper/2502.03302/) (2025)
- [Deep End-to-End Posterior ENergy (DEEPEN) for image recovery](https://mlsanity.com/paper/2503.17244/) (2025)
- [Motion-compensated cardiac MRI using low-rank diffeomorphic flow (DMoCo)](https://mlsanity.com/paper/2505.03149/) (2025)
- [Annealed Langevin Posterior Sampling (ALPS): A Rapid Algorithm for Image Restoration with Multiscale Energy Models](https://mlsanity.com/paper/2601.02594/) (2026)

## Abstract (arXiv)

We introduce multi-scale energy models to learn the prior distribution of images, which can be used in inverse problems to derive the Maximum A Posteriori (MAP) estimate and to sample from the posterior distribution. Compared to the traditional single-scale energy models, the multi-scale strategy improves the estimation accuracy and convergence of the MAP algorithm, even when it is initialized far away from the solution. We propose two kinds of multi-scale strategies: a) the explicit (e-MuSE) framework, where we use a sequence of explicit energies, each corresponding to a smooth approximation of the original negative log-prior, and b) the implicit (i-MuSE), where we rely on a single energy function whose gradients at different scales closely match the corresponding e-MuSE gradients. Although both schemes improve convergence and accuracy, the e-MuSE MAP solution depends on the scheduling strategy, including the choice of intermediate scales and exit conditions. In contrast, the i-MuSE formulation is significantly simpler, resulting in faster convergence and improved performance. We compare the performance of the proposed MuSE models in the context of Magnetic Resonance (MR) image recovery. The results demonstrate that the multi-scale framework yields a MAP reconstruction comparable in quality to the End-to-End (E2E) trained models, while being relatively unaffected by the changes in the forward model. In addition, the i-MuSE scheme also allows the generation of samples from the posterior distribution, enabling us to estimate the uncertainty maps.