Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

T2LR-Net: An unrolling network learning transformed tensor low-rank prior for dynamic MR image reconstruction

Yinghao Zhang, Peng Li, Yue Hu

TL;DR

Dynamic MR reconstruction from undersampled data is improved by learning a transformed tensor low-rank prior. The authors extend t-SVD to arbitrary unitary transforms and introduce the UTNN, a convex envelope of the transformed tensor sum rank, then solve via an ADMM framework that is unfolded into the T$^2$LR-Net. Each iteration module learns a CNN-based transform and a per-slice SVT, enabling both explicit low-rank modeling and implicit deep priors. Experiments on open cardiac datasets show superior performance over state-of-the-art optimization-based and unrolling methods, with robust results across single- and multi-coil and prospective settings. This approach offers a principled way to combine tensor structure with data-driven transform learning for fast, high-quality dynamic MR reconstruction."

Abstract

The tensor low-rank prior has attracted considerable attention in dynamic MR reconstruction. Tensor low-rank methods preserve the inherent high-dimensional structure of data, allowing for improved extraction and utilization of intrinsic low-rank characteristics. However, most current methods are still confined to utilizing low-rank structures either in the image domain or predefined transformed domains. Designing an optimal transformation adaptable to dynamic MRI reconstruction through manual efforts is inherently challenging. In this paper, we propose a deep unrolling network that utilizes the convolutional neural network (CNN) to adaptively learn the transformed domain for leveraging tensor low-rank priors. Under the supervised mechanism, the learning of the tensor low-rank domain is directly guided by the reconstruction accuracy. Specifically, we generalize the traditional t-SVD to a transformed version based on arbitrary high-dimensional unitary transformations and introduce a novel unitary transformed tensor nuclear norm (UTNN). Subsequently, we present a dynamic MRI reconstruction model based on UTNN and devise an efficient iterative optimization algorithm using ADMM, which is finally unfolded into the proposed T2LR-Net. Experiments on two dynamic cardiac MRI datasets demonstrate that T2LR-Net outperforms the state-of-the-art optimization-based and unrolling network-based methods.

T2LR-Net: An unrolling network learning transformed tensor low-rank prior for dynamic MR image reconstruction

TL;DR

Dynamic MR reconstruction from undersampled data is improved by learning a transformed tensor low-rank prior. The authors extend t-SVD to arbitrary unitary transforms and introduce the UTNN, a convex envelope of the transformed tensor sum rank, then solve via an ADMM framework that is unfolded into the TLR-Net. Each iteration module learns a CNN-based transform and a per-slice SVT, enabling both explicit low-rank modeling and implicit deep priors. Experiments on open cardiac datasets show superior performance over state-of-the-art optimization-based and unrolling methods, with robust results across single- and multi-coil and prospective settings. This approach offers a principled way to combine tensor structure with data-driven transform learning for fast, high-quality dynamic MR reconstruction."

Abstract

The tensor low-rank prior has attracted considerable attention in dynamic MR reconstruction. Tensor low-rank methods preserve the inherent high-dimensional structure of data, allowing for improved extraction and utilization of intrinsic low-rank characteristics. However, most current methods are still confined to utilizing low-rank structures either in the image domain or predefined transformed domains. Designing an optimal transformation adaptable to dynamic MRI reconstruction through manual efforts is inherently challenging. In this paper, we propose a deep unrolling network that utilizes the convolutional neural network (CNN) to adaptively learn the transformed domain for leveraging tensor low-rank priors. Under the supervised mechanism, the learning of the tensor low-rank domain is directly guided by the reconstruction accuracy. Specifically, we generalize the traditional t-SVD to a transformed version based on arbitrary high-dimensional unitary transformations and introduce a novel unitary transformed tensor nuclear norm (UTNN). Subsequently, we present a dynamic MRI reconstruction model based on UTNN and devise an efficient iterative optimization algorithm using ADMM, which is finally unfolded into the proposed T2LR-Net. Experiments on two dynamic cardiac MRI datasets demonstrate that T2LR-Net outperforms the state-of-the-art optimization-based and unrolling network-based methods.
Paper Structure (30 sections, 2 theorems, 39 equations, 9 figures, 8 tables, 1 algorithm)

This paper contains 30 sections, 2 theorems, 39 equations, 9 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Notations and Preliminaries
  4. Problem Formulation
  5. Methodology
  6. Transformed t-SVD and UTNN Based on Arbitrary Unitary Transform
  7. UTNN-based Iterative Optimization Algorithm
  8. $\mathcal{Z}$ Subproblem
  9. $\mathcal{X}$ Subproblem
  10. The Proposed Unrolling Network: T$^2$LR-Net
  11. Transformed Tensor Low-rank Prior Block $\mathbf{Z}_n$
  12. Reconstruction Block $\mathbf{X}_n$
  13. Multiplier Update Block $\mathbf{L}_n$
  14. Loss Function
  15. Implementation Details
  16. ...and 15 more sections

Key Result

Lemma 1

A transform $\mathsf{T}$ is the unitary transform only if it preserves the Frobenius norm and inner product horn2012matrix, i.e., where $\mathcal{X} \in \mathbb{C}^{n_1 \times n_2 \times n_3}$, $\mathcal{B} \in \mathbb{C}^{n_2 \times n_4 \times n_3}$, $\mathsf{T}:\mathbb{C}^{n_1 \times n_2 \times n_3} \rightarrow \mathbb{C}^{m_1 \times m_2 \times m_3}$, and $\hat{\mathcal{X}}_\mathsf{T} = \mathsf

Figures (9)

  • Figure 1: An illustration of the proposed transformed t-SVD factorization of a tensor with dimensions $n_1 \times n_2 \times n_3$.
  • Figure 2: The proposed T$^2$LR-Net framework. The T$^2$LR-Net is an unrolling neural network that unrolls N (fixed) iteration of the algorithm \ref{['iter_alg2']} into N iteration modules. Each iteration module contains three blocks: the transformed tensor low-rank prior block $\mathbf{Z}_n$, the reconstruction block $\mathbf{X}_n$, and the multiplier update block $\mathbf{L}_n$. The transformed tensor low-rank prior block is incorporated with the CNN and the hyperparameters are learned through the training process. The number above each color block represents the current channel count. The first row of the figure shows the T$^2$LR-Net framework, and the second row shows the detail of the three blocks.
  • Figure 3: The training and test loss curves of the proposed T$^2$LR-Net.
  • Figure 4: The OCMR reconstruction results of different methods under the pseudo-radial sampling pattern ref_ktslr with 16 lines in the single-coil scenario. The first row shows the reconstruction images of the different methods, and the second row shows the enlarged view of the heart regions marked by the orange box. The first image in the third row displays the sampling mask, while the other images show the reconstruction error maps w.r.t. the different methods. The fourth row and the fifth row show the x-t images indicated by the blue dot line and their reconstruction error maps. The reconstruction SNRs of different methods are listed in parentheses.
  • Figure 6: The OCMR reconstruction results of different methods under the Vds pattern with 10 acceleration rate in multi-coil scenario.
  • ...and 4 more figures

Theorems & Definitions (5)

  • Lemma 1: Unitary transform
  • Definition 1: $\mathsf{T}$-product
  • Theorem 1: Unitary Transformed t-SVD
  • Definition 2: Transformed tensor sum rank
  • Definition 3: UTNN