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Progressive Divide-and-Conquer via Subsampling Decomposition for Accelerated MRI

Chong Wang, Lanqing Guo, Yufei Wang, Hao Cheng, Yi Yu, Bihan Wen

TL;DR

This work presents a rigorous derivation of the pro-posed PDAC framework, which could be further unfolded into an end-to-end trainable network and achieves superior performance on the pub-licly available fastMRI and Stanford2D FSE datasets in both multi-coil and single-coil settings.

Abstract

Deep unfolding networks (DUN) have emerged as a popular iterative framework for accelerated magnetic resonance imaging (MRI) reconstruction. However, conventional DUN aims to reconstruct all the missing information within the entire null space in each iteration. Thus it could be challenging when dealing with highly ill-posed degradation, usually leading to unsatisfactory reconstruction. In this work, we propose a Progressive Divide-And-Conquer (PDAC) strategy, aiming to break down the subsampling process in the actual severe degradation and thus perform reconstruction sequentially. Starting from decomposing the original maximum-a-posteriori problem of accelerated MRI, we present a rigorous derivation of the proposed PDAC framework, which could be further unfolded into an end-to-end trainable network. Specifically, each iterative stage in PDAC focuses on recovering a distinct moderate degradation according to the decomposition. Furthermore, as part of the PDAC iteration, such decomposition is adaptively learned as an auxiliary task through a degradation predictor which provides an estimation of the decomposed sampling mask. Following this prediction, the sampling mask is further integrated via a severity conditioning module to ensure awareness of the degradation severity at each stage. Extensive experiments demonstrate that our proposed method achieves superior performance on the publicly available fastMRI and Stanford2D FSE datasets in both multi-coil and single-coil settings.

Progressive Divide-and-Conquer via Subsampling Decomposition for Accelerated MRI

TL;DR

This work presents a rigorous derivation of the pro-posed PDAC framework, which could be further unfolded into an end-to-end trainable network and achieves superior performance on the pub-licly available fastMRI and Stanford2D FSE datasets in both multi-coil and single-coil settings.

Abstract

Deep unfolding networks (DUN) have emerged as a popular iterative framework for accelerated magnetic resonance imaging (MRI) reconstruction. However, conventional DUN aims to reconstruct all the missing information within the entire null space in each iteration. Thus it could be challenging when dealing with highly ill-posed degradation, usually leading to unsatisfactory reconstruction. In this work, we propose a Progressive Divide-And-Conquer (PDAC) strategy, aiming to break down the subsampling process in the actual severe degradation and thus perform reconstruction sequentially. Starting from decomposing the original maximum-a-posteriori problem of accelerated MRI, we present a rigorous derivation of the proposed PDAC framework, which could be further unfolded into an end-to-end trainable network. Specifically, each iterative stage in PDAC focuses on recovering a distinct moderate degradation according to the decomposition. Furthermore, as part of the PDAC iteration, such decomposition is adaptively learned as an auxiliary task through a degradation predictor which provides an estimation of the decomposed sampling mask. Following this prediction, the sampling mask is further integrated via a severity conditioning module to ensure awareness of the degradation severity at each stage. Extensive experiments demonstrate that our proposed method achieves superior performance on the publicly available fastMRI and Stanford2D FSE datasets in both multi-coil and single-coil settings.
Paper Structure (17 sections, 19 equations, 4 figures, 4 tables, 1 algorithm)

This paper contains 17 sections, 19 equations, 4 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Accelerated MRI
  4. Deep Unfolding Networks
  5. Preliminary
  6. MRI Acquisition
  7. Half Quadratic Splitting
  8. Method
  9. Decomposing the Subsampling
  10. Progressive Divide-and-Conquer
  11. Learning to Decompose
  12. Training Objectives
  13. Experiments
  14. Experimental Setups
  15. Comparison with State-of-the-Art
  16. ...and 2 more sections

Figures (4)

  • Figure 1: In each iteration: (a) Existing deep unfolding networks recover the information in the entire null space; (b) Our proposed method decomposes the entire null space and selectively retrieves information within specific segments of the null space, progressing from those that are easier to recover to more challenging ones.
  • Figure 2: Illustration of the proposed progressive divide-and-conquer (PDAC) framework, where the iterative process is detailed in Algorithm \ref{['alg1']}. Each iteration consists of data consistency, network reconstruction, and degradation using $M_t$. Besides, the network learns the decomposed degradation, characterized by $M_t$, as an auxiliary task along iterations via (1) Degradation Prediction: we adaptive learn a decomposed sampling mask $M_t$ which indicates the frequency components to preserve in $\tilde{\bm{z}}_t$; (2) Severity Conditioning: we adopt a severity embedding module $E_{\theta_t}$ to guarantee awareness of the degradation pattern in $M_{t-1}$.
  • Figure 3: Examples of multi-coil accelerated MRI reconstruction results of zero-filled input, U-Net ronneberger2015u, E2E-VarNet sriram2020end, HUMUSNet fabian2022humus, Ours and ground truth on the fastMRI zbontar2018fastMRI knee and Stanford2D FSE stanford2d datasets. Please zoom in to see the details.
  • Figure 4: Ablation study on different sampling budget schedules.

Theorems & Definitions (1)

  • Definition 1