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

Jyothi Rikhab Chand, Mathews Jacob

TL;DR

The paper addresses robust MAP reconstruction from undersampled MR data by introducing LC-MuSE, a locally convex multi-scale energy model parameterized by a CNN. By enforcing $m$-strong local convexity via a local monotonicity constraint on the energy gradient, the approach guarantees uniqueness and convergence within a neighborhood while remaining compatible with practical architectures. The energy is learned through a DSM-based objective with a Lipschitz regularization, and the MAP solution is obtained via a Majorization-Minimization update, yielding robust and reliable reconstructions. Empirical results on parallel MR imaging show LC-MuSE outperforms globally convex regularizers and matches or closely approaches state-of-the-art non-convex and end-to-end methods, demonstrating both theoretical guarantees and competitive practical performance across acquisition settings. The framework is broadly applicable to computational imaging problems beyond MR, offering a principled way to combine energy-based priors with deep networks without imposing overly restrictive architectural constraints.

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.

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

TL;DR

The paper addresses robust MAP reconstruction from undersampled MR data by introducing LC-MuSE, a locally convex multi-scale energy model parameterized by a CNN. By enforcing -strong local convexity via a local monotonicity constraint on the energy gradient, the approach guarantees uniqueness and convergence within a neighborhood while remaining compatible with practical architectures. The energy is learned through a DSM-based objective with a Lipschitz regularization, and the MAP solution is obtained via a Majorization-Minimization update, yielding robust and reliable reconstructions. Empirical results on parallel MR imaging show LC-MuSE outperforms globally convex regularizers and matches or closely approaches state-of-the-art non-convex and end-to-end methods, demonstrating both theoretical guarantees and competitive practical performance across acquisition settings. The framework is broadly applicable to computational imaging problems beyond MR, offering a principled way to combine energy-based priors with deep networks without imposing overly restrictive architectural constraints.

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.

Paper Structure

This paper contains 14 sections, 7 theorems, 12 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Proposed method
  4. Training Procedure
  5. MAP estimation
  6. Theoretical Guarantees
  7. Results
  8. Data set
  9. Architecture and implementation
  10. State-of-the-art (SOTA) methods for comparison
  11. Experimental results
  12. Conclusion
  13. Compliance with ethical standards
  14. Acknowledgments

Key Result

Lemma 2.1

${\boldsymbol{E}}_{\boldsymbol{\boldsymbol \theta}}(\cdot)$ is m-strongly convex iff ${\boldsymbol{H}}_{\boldsymbol{\boldsymbol \theta}} ({\boldsymbol{x}})=\nabla_{\boldsymbol{x}} {\boldsymbol{E}}_{\boldsymbol{\boldsymbol \theta}} ({\boldsymbol{x}})$ is m-strongly monotone monotone_prop:

Figures (1)

  • Figure 1: Comparison of LC-MuSE with MuSE, ICNN and E2E-trained MoL at a) two-fold and b) six-fold acceleration using 1D and 2D undersampling mask, respectively. The top, middle, and last rows show the reconstructed, enlarged, and error images, respectively.

Theorems & Definitions (7)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3: Uniqueness
  • Lemma 3.4
  • Lemma 3.5