# RIP Sensing Matrices Construction for Sparsifying Dictionaries with Application to MRI Imaging
## 6 Conclusions  
This article describes a novel approach to CS involving the construction a sensing matrix for a prescribed sparsifying dictionary, such that their composition has the RIP. This property theoretically guarantees the accurate recovery of a fixed number of sparse coefficients. While previous theoretical work has demonstrated RIP assurances only for orthonormal matrices, our study extends these insights to include general over-complete dictionaries. We assert that our sensing matrix can recover sparse coefficients from an over-complete dictionary as effectively as a random sensing matrix for an orthonormal matrix. This advancement is significant, as many practical applications that employ compressive sensing rely on over-complete dictionaries. We further apply the factorization approach to accelerated MR imaging--deriving an embedding for under-sampled \(k\)-space data that facilitates RIP recovery guarantees. In this application one of the factors in our proposed decomposition is adapted to the physically prescribed sensing mechanism, which otherwise does not allow for direct RIP based recovery guarantees in the \(\ell_{1}\)-synthesis approach to CS.  
In addition to the achievements mentioned, it is notable to recognize that the analysis could be further improved if the factorization structure in \(G\) has a specific arrangement. A structured sensing matrix can significantly enhance storage and computational efficiency in large systems. A reviewer proposes a promising avenue for future exploration for MRI acceleration. Traditional compressive sensing (CS) recovery techniques and modern neural network methods generate different artifacts. While the artifacts from conventional algorithms can usually be traced back to the algorithm and the regularization penalty, those produced by neural networks tend to be more unpredictable. Different neural networks tend to produce distinct model-shifting artifacts. Additionally, the unpredictable nature of the artifacts generated by fixed neural networks presents a significant challenge. Experienced clinicians can quickly identify the artifacts associated with compressive sensing (CS) recovery methods, which helps reduce false positives. However, neural network methods offer a significant advantage, achieving better recovery accuracy and inference speed than traditional techniques. In line with the reviewer's suggestions, we recommend developing a hybrid approach that combines the strengths and weaknesses of both neural networks and CS techniques to optimize overall performance.  
Conceptualization, W.-L.H.; Methodology, J.H., W.-L.H. and A.H.; Software, J.H.; Validation, J.H., W.-L.H. and A.H.; Formal analysis, J.H., W.-L.H. and A.H.; Data curation, J.H.; Writing--original draft, W.-L.H.; Writing--review & editing, A.H.; Visualization, J.H.; Supervision, W.-L.H. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
All MRI data in Figure 6(a0), Figure 7(a0), Figure 8(a0) and Figure 9(a0) have been made publicly available at the fastMRI dataset and can be accessed at [[https://fastmri.med.nyue.edu/](https://fastmri.med.nyue.edu/)]([https://fastmri.med.nyue.edu/](https://fastmri.med.nyue.edu/)) (accessed date: 1 July 2022).  
The authors declare no conflicts of interest.