Introduction

Physics-driven deep learning (PD-DL) reconstruction is a powerful tool for fast MRI^1,2. Early methods used supervised learning with fully-sampled k-space as reference labels^3,4. To address the challenges of obtaining fully-sampled data in various scenarios, unsupervised learning methods have been proposed⁵, including self-supervised learning^6,7 and generative models^8,9. In the former, e.g. SSDU⁶, parts of acquired k-space are masked out, and the network learns to predict these from remaining k-space. In the latter, a generative model is used to estimate the prior distribution of images, which is then used in conjunction with a log-likelihood data term during test-time. In our work, we propose an alternative approach for unsupervised training of PD-DL networks, drawing inspiration from statistical image processing and compressed sensing (CS)^10,11. In particular, while we still rely on the power of non-linear low-dimensional representations afforded by PD-DL networks for signal recovery, we use compressibility to evaluate reconstruction quality. Results show that our approach performs similar to SSDU⁶ and supervised learning, while outperforming conventional CS.

Methods

PD-DL Reconstruction:
Regularized MRI reconstruction solves:$$\arg\min_{\mathbf{x}}\left\|\mathbf{y}_{\Omega}-\mathbf{E}_{\Omega}\mathbf{x}\right\|_2^2+\mathcal{R}(\mathbf{x}),$$where $$$y_\Omega$$$ is acquired k-space data at locations $$$\Omega$$$, $$$E_\Omega$$$ is the multi-coil encoding operator, and $$$x$$$ is the image of interest. A popular PD-DL approach is to unroll an iterative algorithm for solving this objective function, e.g. proximal gradient descent, for a fixed number of steps, and train the network end-to-end, learning both the proximal operator for $$$\mathcal{R(\cdot)}$$$, implemented implicitly with a neural network, and any weights used for data fidelity.

Proposed Compressibility-Inspired Loss Function:
We propose a loss function aimed at encouraging sparsity of the reconstruction in a transform domain. We first consider a supervised scenario, where the reference image is available. In this setting, we use the reweighted $$$\ell_1$$$-norm, which corresponds to the $$$\ell_0$$$-norm of a sparse signal¹². Let $$$\mathbf{x}^{out}=f(\mathbf{y}_\Omega, \mathbf{E}_\Omega; \mathbf{\theta})$$$ be the output of the PD-DL network parametrized by learnable $$$\mathbf{\theta}$$$. The proposed loss is given as $$\mathcal{L}_{rew-\ell_1}\left(\mathbf{x}^{ref},\mathbf{x}^{\text{out}}\right)=\frac{1}{N}\cdot\sum_{i=1}^N\left(\frac{\left|\left(\mathbf{W}\mathbf{x}^{\text{out}}\right)_i\right|}{\left|\left(\mathbf{W}\mathbf{x}^{\text{ref}}\right)_i\right|+\epsilon}\right)+\lambda\cdot\frac{\left\|\mathbf{y}_{\Omega}-\mathbf{E}_{\Omega}\mathbf{x}^{\text{out}}\right\|_2}{\left\|\mathbf{y}_{\Omega}\right\|_2},$$where $$$\mathbf{x}^{\text{ref}}$$$ is the reference image, $$$\mathbf{W}$$$ denotes the transform under which $$$\mathbf{x}$$$ is sparsely represented, and $$$(\mathbf{Wx})_i$$$ denotes the $$$i^{th}$$$ (out of N) transform-domain coefficients, and $$$\epsilon$$$ is for numerical stability. Both terms in the loss are normalized to be unitless. We note the second term is needed to ensure that the network output is not driven to zero. Even in the presence of the data fidelity term within the PD-DL network, excluding the second term and training solely on the reweighted $$$\ell_1$$$ norm will push the regularizer output to $$$\mathbf{0}$$$, while simultaneously driving the weighting assigned to the regularizer to infinity since this combination forces the final PD-DL network output to be $$$\mathbf{0}$$$, minimizing the first term alone.

For the unsupervised setting, this motivates a reweighted $$$\ell_1$$$ approach that first starts with an initial estimate as the denominator weights, and then updates these weights progressively after each reweighting (Fig. 1). In particular, we propose:$$\mathcal{L}\left(\mathbf{x}^{(k)},\mathbf{x}^{\text{out}}\right)=\frac{1}{N}\cdot\sum_{i=1}^N\left(\frac{\left|\left(\mathbf{W}\mathbf{x}^{\text{out}}\right)_i\right|}{\left|\left(\mathbf{W}\mathbf{x}^{\text{(k)}}\right)_i\right|+\epsilon}\right)+\lambda\cdot\frac{\left\|\mathbf{y}_{\Omega}-\mathbf{E}_{\Omega}\mathbf{x}^{\text{out}}\right\|_2}{\left\|\mathbf{y}_{\Omega}\right\|_2},$$where $$$\mathbf{x}^{\text{(k)}}$$$ represents the signal estimate at the $$$k^{th}$$$ reweighting step. It is also worth noting that this method can be applied in a zero-shot manner to perform scan-specific accelerated MRI¹³.

Implementation Details:
Coronal proton-density knee fastMRI datasets¹⁴ were uniformly undersampled with R=4, 24 ACS. Training was performed using 300 knee slices and testing was performed on 10 distinct subjects. PD-DL network used variable splitting with data fidelity using conjugate-gradient². The proximal operator for the regularizer was implemented with a ResNet¹⁵.

Biorthogonal 1.5 was used as $$$\mathbf{W}$$$. $$$\lambda$$$ was selected as 100 for both supervised and unsupervised proposed loss functions. 5 reweightings were used in the unsupervised case with initial $$$\mathbf{x}^{(0)}$$$ estimated using conventional $$$\ell_1$$$-wavelet-regularized least-squares minimization. Comparisons were made to supervised learning with an $$$\ell_1-\ell_2$$$ loss¹³, and SSDU⁶. Conventional CS reconstruction was also performed to highlight the differences between the use of PD-DL with the proposed loss and convex reconstruction with linear representations. Finally, proposed unsupervised loss was compared to ZS-SSDU¹⁶ for subject-specific learning. Results were quantitatively assessed using SSIM and PSNR.

Results

Fig. 2 shows representative reconstructions for supervised and unsupervised learning. Proposed loss formulation improves upon CS, as expected, while showing similar quality to well-established PD-DL methods.
Fig. 3 summarizes the PSNR and SSIM metrics, matching these observations.
Fig. 4 showcases the zero-shot setting, where our proposed loss function performs similar to ZS-SSDU, while outperforming CS.

Discussion and Conclusions:

In this study, we proposed a new unsupervised loss formulation for PD-DL networks. Results show similar performance to self-supervised learning with SSDU. This provides an alternative approach that does not require masking of k-space, but instead focuses on compressibility of the output image.

Acknowledgements

Grant support: NIH R01HL153146, NIH R01EB032830, NIH P41EB027061.