Machine Learning for Quantitative MR Image Reconstruction

TL;DR

In the following chapter, the reader is provided with an extensive overview of methods that can be employed for (quantitative) MR image reconstruction, highlighting their advantages and limitations both from a theoretical and computational point of view.

Abstract

In the last years, the design of image reconstruction methods in the field of quantitative Magnetic Resonance Imaging (qMRI) has experienced a paradigm shift. Often, when dealing with (quantitative) MR image reconstruction problems, one is concerned with solving one or a couple of ill-posed inverse problems which require the use of advanced regularization methods. An increasing amount of attention is nowadays put on the development of data-driven methods using Neural Networks (NNs) to learn meaningful prior information without the need to explicitly model hand-crafted priors. In addition, the available hardware and computational resources nowadays offer the possibility to learn regularization models in a so-called model-aware fashion, which is a unique key feature that distinguishes these models from regularization methods learned in a more classical, model-agnostic manner. Model-aware methods are not only tailored to the considered data, but also to the class of considered imaging problems and nowadays constitute the state-of-the-art in image reconstruction methods. In the following chapter, we provide the reader with an extensive overview of methods that can be employed for (quantitative) MR image reconstruction, also highlighting their advantages and limitations both from a theoretical and computational point of view.

Figures (12)

• Figure 1: An example for the reconstruction of three different quantitative parameters $\mathbf{p}=[\mathbf{M}_0, \boldsymbol{\alpha}, \mathbf{T}_1]^{ \boldsymbol{\mathsf{T}}}$ from undersampled $k$-space data (shown for a single-coil acquisition for simplicity). Thereby, the set $\mathcal{T}=\{t_1, \ldots, t_{Q}\}$ denotes different inversion times and the quantitative parameter vector $\mathbf{p}$ contains the equilibrium magnetization $\mathbf{M}_0$, the flip-angle $\boldsymbol{\alpha}$ and the longitudinal relaxation parameter $\mathbf{T}_1$.
• Figure 2: A comparison of examples for the reconstruction of $\mathbf{M}_0$ and $\mathbf{T}_1$ from simulated 4-fold and 8-fold Cartesian undersampled saturation recovery data. Without any regularization, the results obtained at 4-fold undersampling (top row) are severely degraded by artifacts. TV-regularization, i.e.ṡetting $\mathcal{R}(\mathbf{x})$ according to \ref{['def:TV']} in \ref{['eq:qmri_variational_problem_splitted2']}, improves the results visibly at 4-fold (second row), but fails at 8-fold undersampling (third row). This is the setting that is used to demonstrate different NN-based approaches in Figure \ref{['fig:model_aware_vs_model_unaware_vs_pinqi_qmri']}. The last row shows the ground-truth target labels of $\mathbf{M}_0$ and $\mathbf{T}_1$ used in the simulation (generated from the BrainWeb datasetaubert2006twenty).
• Figure 3: An example of cardiac cine MR images reconstructed from undersampled radially acquired $k$-space data. From left to right: approximation of the Moore-Penrose pseudo-inverse, TV-minimization block2007undersampled, $kt$-SENSE tsao2003k and a dictionary learning-based approach pali2021adaptive. Additionally, the target image from which the $k$-space data was retrospectively simulated as well as the initial reconstruction, given by applying the adjoint operator to the density-compensated $k$-space data, are also shown. The data-driven dictionary learning approach improves the images compared to the other shown methods both in terms of PSNR as well as SSIM.
• Figure 4: An example of a U-Net ronneberger2015u for reducing artifacts and noise from the zero-filled reconstruction of undersampled $k$-space data. This particular example of a U-Net consists of three resolution scales. The numbers over the convolution arrows signify varying numbers of convolution filters used. Here, each convolutional block consists of two convolutional layers followed by a rectified linear unit (ReLU) activation function. The last layer maps to the real and imaginary part of the complex-valued output image.
• Figure 5: An example of a model-agnostic NN given by a learned direct inversion of the forward model $\mathbf{A}_I$ is shown for a brain MRI example. For qualitative MR image reconstruction, the method named AUTOMAP (AUtomated TransfOrm by Manifold APproximation) can be found in zhu2018image.