Magnetic Resonance Imaging (MRI) is a widely used medical imaging modality known for its ability to produce high soft tissue contrast, high-resolution images without the use of ionizing radiation. Its applications range from diagnosis and treatment planning to real-time MR-guided tasks such as surgery or radiotherapy. However, the long acquisition times of the MRI measurements, known as k-space, hinder the full potential of MRI-guidance. Reducing these times could make MRI-guidance cheaper and extend its functionality in real-time settings.

Over the past two decades, several methods have been put to use in clinical practice for accelerating the MRI acquisition. The two most conventionally applied methods to-date are Parallel Imaging (PI) and Compressed Sensing (CS), which are typically incorporated into modern, state-of-the-art MRI scanners.

Compressed Sensing seeks to reconstruct images from incoherently subsampled k-space measurements through mathematical optimization techniques [[1], [2], [3], [4]]. Subsampling the k-space is, in general, a violation of the Nyquist-Shannon sampling criterion [5] and thus prone to producing aliasing artifacts. CS reconstruction algorithms employ optimization methods like Total Variation (TV) [6] to obtain faithful images from sparse input signals.

Parallel Imaging on the other hand, employs an array of multiple - instead of one - radio-frequency receiver coils which measure sets of spatially localised k-space frequencies while maintaining the same spatial resolution [[7], [8], [9]]. Each independent receiver coil receives distinct measurements corresponding to their spatial location in relation to the scanned object.

With the recent advancements in Deep Learning (DL) and Computer Vision (CV), a plethora of algorithms have emerged targeting to solve imaging inverse problems, with Accelerated MRI Reconstruction being a par excellence example. By utilizing CS optimization approaches and PI, numerous DL-based methods involving convolutional neural networks (CNNs) have been proposed in the literature [10,11] applied to the task of Accelerated MRI Reconstruction. These methods are usually trained in a supervised manner using retrospectively subsampled (from available fully-sampled) k-space datasets and their target is to make a prediction of the fully-sampled k-space or its image reconstruction.

In clinical 2D parallel MRI acquisitions, rectilinear Cartesian patterns are the most commonly used subsampling techniques due to their ease of implementation [7]. Consequently, DL-based applications for Accelerated MRI Reconstruction use rectilinear subsampling masks to retrospectively subsample fully-sampled data. Notable MRI reconstruction challenges such as fastMRI [12] and CMRxRecon challenges [13] have provided the public with knee, brain (fastMRI), and cardiac (CMRxRecon) MRI data subsampled using random and/or equispaced rectilinear subsampling schemes. Additionally, DL-based MRI reconstruction research has widely incorporated such patterns into their experiments [[14], [15], [16], [17]].

However, a variety of prospective and retrospective non-rectilinear Cartesian sampling and subsampling patterns exist. For instance, non-Cartesian patterns such as radial or spiral are being applied in real-time MRI acquisitions due to the fact that they are less susceptible to motion compared to Cartesian ones [18].In the DL Reconstruction community, the Multi-Coil MRI (MC-MRI) reconstruction challenge [19] included reconstructing brain k-space data subsampled with a variable density Poisson Cartesian scheme. Moreover, multiple studies have employed non-rectilinear Cartesian or pseudo non-Cartesian schemes in their DL-based reconstruction experiments for MRIs from various organs and contrasts [[20], [21], [22]]. Some works have also experimented with a mixture of subsampling schemes on various datasets [[23], [24], [25], [26]], although the direct comparison of models trained using these schemes was not the focus of these studies. The authors in [27] by employing a deep neural network architecture, namely the Recurrent Inference Machine (RIM) [20], explored the effects of training RIMs by applying either rectilinear or radial retrospective subsampling, and concluded that the RIM trained using the latter could produce higher-fidelity reconstructions.

While our paper primarily investigates retrospective undersampling schemes and their impact on DL-based MRI reconstruction, it's worth noting that other studies have delved into the realm of adaptive k-space sampling optimization. In addition to conventional undersampling techniques, recent studies have explored the optimization of k-space sampling strategies by harnessing traditional mathematical approaches and/or the power of deep learning. Some studies have designed sampling patterns based on constrained optimization problems, aiming to simultaneously learn the optimal sampling pattern and regularization parameters, essentially focusing on optimizing Cartesian sampling within the compressed sensing framework [28]. In [29], the authors optimized non-Cartesian k-space filling using gradient descent optimization.

In the domain of deep learning, researchers have also explored the use of neural networks to optimize k-space acquisition. Some have proposed deep reinforcement learning to select optimal k-space locations [30], while in [31], this is achieved using phase-encoding lines to reduce uncertainty in reconstructed MR images. Another work utilized a policy DL-based network to perform a greedy policy search for 2D Cartesian rectilinear acquisition in the k-space [32]. Moreover, a neural network-based approach was introduced in [33] for learning k-space sampling point locations in non-Cartesian acquisitions, although it is sensitive to the selection of initial trajectory points, resulting in trajectories with rapidly changing gradient waveforms. Furthermore, in [34], the authors proposed an adaptive deep learning model to learn optimal policies for undersampling and simultaneously reconstructing the acquired data. A recent approach involved formulating the problem as a solution to an ordinary differential equation and applying it to both Cartesian and non-Cartesian scenarios [35].

In this work, we aim to investigate and compare the effects of employing various retrospective subsampling schemes on the quality of DL-based learned image reconstructions. To this end, we trained and tested Recurrent Variational Networks [21] (RecurrentVarNets) on retrospectively subsampled k-space measurements. We performed experiments under either scheme-specific or multi-scheme setups, in which models were trained and evaluated on data subsampled with either individual or multiple, respectively, subsampling schemes.

The contributions and findings of our work can be summarized as follows:

• We provide a review of eight currently employed (retrospective) subsampling techniques.

• We experimentally demonstrate that DL models trained and evaluated on non-rectilinearly, compared to rectilinearly, subsampled data output superior reconstructions, especially for high acceleration factors.

• We illustrate that models trained on data subsampled with multiple patterns, rather than individual ones, can reconstruct rectilinearly subsampled data with higher fidelity.