# RIP Sensing Matrices Construction for Sparsifying Dictionaries with Application to MRI Imaging
## 2 Compressed Sensing Applications  
Compressed Sensing (CS) has found numerous applications across various fields. One significant area is addressing the future data storage challenges [18]. For example, advancements in communication and signal processing are producing vast amounts of data; 5G networks can reach peak data rates of about 300 Mb per second, while an image of a black hole needs several petabytes for storage. In medical signal processing, the volume of required data is also substantial. CS can enhance wireless communication capabilities in cognitive radio networks and enable spatial compression and projection in wireless sensor networks [19]. Additionally, modern imaging techniques rely on arrays of millions of detector pixels. In contrast, CS is foundational to single-pixel cameras, which utilize mask patterns to filter scenes and capture transmitted intensity measurements with a one-pixel detector. This technology is well-suited for imaging beyond the visible spectrum. It allows for precise timing or depth resolution, making it ideal for applications such as infrared imaging and 3D situational awareness in autonomous vehicles [20].  
An important CS application, in which the sensing mechanisms is dictated by hardware constraints, is magnetic resonance imaging (MRI). In MRI a scanner collects Fourier domain measurements of a target medical image. Accelerating the speed of MRI data acquisition by reducing the number of required measurements remains of great interest to the medical community. Inference of the true underlying spatial image can be achieved via several CS strategies when non-uniformly under-sampling below the requirements of the Shannon-Nyquist theory within the maximum frequency spectrum, by incorporating the a priori knowledge about sparsity of medical images in a dictionary transform domain or of its spatial gradients [21]. While deep learning approaches have been a recent center of attention to recover data under-sampled in this way [22], recovery guarantees and interpretability set the CS approach apart from such machine learning techniques [23]. Though state-of-the-art machine learning models for MRI have been validated for clinical interchangeability in four-fold data acquisition acceleration [24], they rely on empirical studies and present challenges concerning, for instance, reconstruction hallucinations that can be problematic for interpreting radiologists [25], unknown training data biases [26], or robustness to distribution shifts between training and application data (e.g., scanning technology [27], target anatomy [28], or acceleration factors [29]). Significant issues concerning the stability of distribution shifts are highlighted in [30]. The authors stress the need to apply a method across different datasets featuring anatomical regions and images collected through different scanning protocols. This approach is crucial for showcasing the method's versatility and evaluating the model's resilience to distribution shifts. In contrast, the sparsity-driven CS approach can be fine-tuned to perform close to deep learning methods for MRI acceleration, via unrolling algorithms which consist of a small fraction of the parameters employed by deep learning approaches [23].