MAP Image Recovery with Guarantees using Locally Convex Multi-Scale Energy (LC-MUSE) Model

TL;DR

This paper introduces a novel multi-scale deep energy model called LC-MUSE for inverse problems, providing guarantees of uniqueness, convergence, and robustness, and demonstrating superior performance in MRI image reconstruction.

Contribution

The paper presents a locally convex multi-scale energy model with theoretical guarantees, improving inverse problem solutions and MRI reconstruction performance.

Findings

Outperforms state-of-the-art convex regularizers in MRI reconstruction

Provides convergence guarantees and robustness to perturbations

Achieves comparable results to plug-and-play and end-to-end methods

Abstract

We propose a multi-scale deep energy model that is strongly convex in the local neighbourhood around the data manifold to represent its probability density, with application in inverse problems. In particular, we represent the negative log-prior as a multi-scale energy model parameterized by a Convolutional Neural Network (CNN). We restrict the gradient of the CNN to be locally monotone, which constrains the model as a Locally Convex Multi-Scale Energy (LC-MuSE). We use the learned energy model in image-based inverse problems, where the formulation offers several desirable properties: i) uniqueness of the solution, ii) convergence guarantees to a minimum of the inverse problem, and iii) robustness to input perturbations. In the context of parallel Magnetic Resonance (MR) image reconstruction, we show that the proposed method performs better than the state-of-the-art convex regularizers, while the performance is comparable to plug-and-play regularizers and end-to-end trained methods.