## Highlights

- •Machine learning reconstruction of MRI data is becoming increasingly popular in research.
- •Many methods exist to perform machine learning reconstruction of MRI data.
- •The limited availability of publicly available training data sets, restricts current development and comparison of existing methods.
- •There is currently very limited clinical validation of MRI images reconstructed using machine learning.

## Abstract

*k*-space data. However, clinical uptake of vastly accelerated acquisitions has been limited, in particular in compressed sensing, due to the time-consuming nature of the reconstructions and unnatural looking images. Following the success of machine learning in a wide range of imaging tasks, there has been a recent explosion in the use of machine learning in the field of MRI image reconstruction.

*k*-space and/or image-space.

## Keywords

## 1. Introduction

### 1.1 The image reconstruction problem

*k*-space. In the simple case, the inverse Fourier transform (iFT) can then be used to reconstruct the

*k*-space data into clinically interpretable images.

*k*-space samples collected. Therefore, it is desirable to collect as few samples as possible. However, if the sampling rate is reduced below that required by the Nyquist criterion, aliasing artefacts will appear in the image.

*k*-space data, $A$ is the system matrix, $x$ is the image and $\u220a$ is a random noise term. When

*k*-space data is undersampled and noise corrupted, the inverse problem in Eq. (1) is ill-posed: a solution might not exist, infinite solutions might exist, and it may be unstable with respect to measurement errors. As a result, direct inversion of $A$ is generally not possible. Instead, an optimal solution in the least-squares sense may be obtained by recasting the problem as the following minimization:

*k*-space over the last few decades. Two broad technologies stand out for their importance and deserve a brief overview here, namely parallel imaging and compressed sensing. These enable substantial reductions in acquisition time while preserving image quality.

*k*-space. This is enabled by the fact that receiver coils exhibit spatially varying responses, which can be leveraged to unfold aliased images or estimate missing

*k*-space samples. Parallel imaging techniques, such as SENSE (Sensitivity Encoding) [

### 1.2 Deep learning

*learns*, the network approximates the mapping from inputs to outputs.

*hidden state*, which acts like a memory about previous inputs in the sequence.

*k*-space, those which operate in different domains, those that learn the direct mapping from

*k*-space to image-space, and unrolled optimization methods (Fig. 1).

## 2. Supervised machine learning

*gold-standard*fully sampled data set (the desired output), with paired undersampled data (the input). These approaches require a qualitative metric, or

*loss function*, which is used to evaluate how close the current output of the network is to the target image. The most commonly used loss functions are pixel-wise Mean Squared Error (MSE, ${\mathcal{l}}_{2}$-loss) and Mean Absolute Error (MAE, ${\mathcal{l}}_{1}$-loss). However, these metrics do not reflect a radiologists’ perspective well [

### 2.1 Image restoration methods

*U-Net*[

*k*-space results in aliasing artefacts in the reconstructed images, which are dependent on the trajectory and undersampling pattern. Where the undersampling is performed in a non-uniform manner, the resultant artefacts are incoherent and noise-like. Therefore, it is possible to train a machine learning network to remove such artefacts in a similar manner to image de-noising. It has been shown that it is possible to perform de-aliasing from data acquired using a random undersampling scheme in the phase direction of 2D images [

### 2.2 *k*-space methods

*k*-space enhancement (see Fig. 1-b), in a supervised manner, similarly to GRAPPA. Some approaches use large training databases without the need for explicit coil-sensitivity information, whereas others learn the relationship between coil elements from a small amount of fully sampled reference data (the auto-calibration signal, ACS).

*k*-space data. It is based on the SPIRiT (iterative self‐consistent parallel imaging reconstruction) algorithm [

*k*-space are trained separately in a multi-resolution approach, using a large database without the need for explicit coil sensitivity maps or reference data. Where multiple contiguous slices are available, spatially adjacent slices can be used as multi-channel input to improve the accuracy; adaptive convolutional neural networks for

*k*-space data interpolation (ACNN-

*k*-Space) [

*k*-space [

*k*-space neighborhoods) [

### 2.3 Direct mapping

*k*-space data and the uncorrupted images (see Fig. 1-c). These end-to-end reconstructions have the potential to mitigate against errors caused by field inhomogeneity, eddy current effects, phase distortions, and re-gridding.

*k*-space data (input), and reconstructed images (desired output). The network architecture consists of a feedforward deep neural network consisting of fully connected layers with hyperbolic tangent activations (which learns the transform), followed by convolutional layers with rectifier nonlinearity activations that form a convolutional autoencoder (which performs image domain refinement). Unfortunately, this results in a large number of parameters, which grows quadratically with the number of image pixels, which limited the use of AUTOMAP to small images (up to 128 × 128).

### 2.4 Cross-domain methods

*k*-space and images exhibit different properties; therefore, a combination of them might outperform them separately. Typically, frequency domain subnetworks attempt to estimate the missing

*k*-space samples, while image domain subnetworks attempt to remove residual artefacts.

*k*-space completion step, followed by an image restoration step. This is the case of W-Net [

*k*-space subnetwork and a U-Net for the image subnetwork in a dynamic imaging context.

*k*-space and image deep CNN’s are applied, separated by the Fourier transform (the network architecture operates on

*k*-space, image-space,

*k*-space, and then image-space sequentially). Another proposal, the hybrid cascade [

- Souza R.
- Lebel R.M.
- Hybrid F.R.A.

*k*-space and image CNN’s. The W-Net method was extended to WW-Net [

*k*-space correlations between coils can be efficiently exploited by

*k*-space domain networks.

*k*-space, while the other is fed the zero-filled reconstructed images. The features from both paths are combined via concatenation and fed into a single decoder path, which produces the reconstructed image.

*k*-space domain. This is the case of IKWI-Net [

*k*-space, wavelet domain and image domain).

### 2.5 Unrolled optimization

*k*-space to the corresponding reconstructed image. Then image transforms, sparsity-promoting functions, regularization parameters and update rates can be treated as either explicitly or implicitly trainable and fitted to a training dataset using back-propagation. This has three advantages with respect to classic optimization. First, learned parameters may be better adapted to image characteristics than hand-engineered ones. Second, it avoids the need for manual tuning, which is not a trivial process. Finally, reconstruction is faster, because such learned iterative schemes are trained to produce results with fewer iterations.

where $f\left(Ax,y\right)$ is a generic data consistency term, which ensures that the solution $x$ agrees with the observations $y$, and $g\left(x\right)$ is a generic regularization term which incorporates prior information. The definitions of $f$ and $g$, together with the optimization strategy, determine the fundamental structure of the resulting neural network. Several approaches are outlined hereafter. A summary of the techniques described is presented in Table 1, which the reader is encouraged to use for reference.

Ref. | Name | Algorithm | $f$ | $g$ | Learned parameters |
---|---|---|---|---|---|

63 , 65 | ADMM-Net | ADMM | $\frac{1}{2}{\Vert Ax-y\Vert}_{2}^{2}$ | ${\sum}_{l=1}^{L}{\lambda}_{l}\mathcal{R}\left({D}_{l}x\right)$ | ${D}_{l}$(Conv), $\mathcal{R}$ (implicit, proximal operator, piecewise linear function) |

[55] | VarNet | GD | ${D}_{l}$(Conv), $g$ (implicit, first order derivative, radial basis functions) | ||

[58] | ISTA-Net | PGD (ISTA) | $\lambda {\Vert D\left(x\right)\Vert}_{1}$ | $D$(Conv-ReLU-Conv) | |

[57] | R-GANCS | PGD | $\mathcal{R}\left(x\right)$ | $\mathcal{R}$(implicit, proximal operator, GAN). | |

[56] | HC-PGD | PGD | $\mathcal{R}\left(x\right)$ | $\mathcal{R}$(implicit, proximal operator, CNN). | |

[50] | DC-CNN | PGD | $\lambda {\Vert x-\mathcal{C}\left(x\right)\Vert}_{2}^{2}$ | $\mathcal{C}$(CNN) | |

[59] | MoDL | AMA | $\mathcal{C}$(CNN) | ||

[66] | MoDL-SToRM | AMA | ${\lambda}_{1}{\Vert x-\mathcal{C}\left(x\right)\Vert}_{2}^{2}+{\lambda}_{2}\mathrm{tr}\left({x}^{T}Lx\right)$ | $\mathcal{C}$(CNN) | |

[60] | VS-Net | AMA | $\mathcal{R}\left(x\right)$ | $\mathcal{R}$(implicit, proximal operator, CNN). | |

[61] | CRNN-MRI | AMA | $\mathcal{R}\left(x\right)$ | $\mathcal{R}$(implicit, proximal operator, CRNN). | |

[64] | TVINet | PDHG | $\mathcal{R}\left(Dx\right)$ | $D$(conv), $\mathcal{R}$ (CNN) | |

[67] | PDHG-CSNet | PDHG | $\mathcal{R}\left(x\right)$ | $\mathcal{R}$(implicit, proximal operator, CNN). | |

[67] | CP-Net, PD-Net | PDHG | $\mathcal{F}\left(Ax,y\right)$ | $\mathcal{R}\left(x\right)$ | $\mathcal{F}$, $\mathcal{R}$ (implicit, proximal operators, CNN’s) |

## 3. Unsupervised machine learning

*ground-truth*data or user guidance. This is particularly challenging in the field of MRI reconstruction. It has been shown that state-of-the-art unsupervised learning techniques are currently unable to achieve as good image quality as supervised learning techniques [

*ground-truth*fully sampled datasets are unavailable and difficult or impossible to acquire (e.g. 4D flow), unsupervised learning techniques provide a promising alternative.

*blind compressed sensing*) [

*k*-space to image domain [

*k*-space data), and the discriminator network tries to differentiate between the original

*k*-space and a randomly undersampled

*k*-space created from the generated image. GAN’s have also been used to learn the probability distribution of uncorrupted MRI data in an unsupervised manor, and provide implicit priors for iterative reconstruction approaches [

## 4. Clinical implications

*k*-space interpolation ML reconstruction (see section 2.2) [

## 5. Current limitations

*k*-space data sets; mridata.org, NYU fastMRI [

## 6. Conclusion

## Declaration of Competing Interest

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