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The low-rank idea was further ex- tended to structured low-rank approach for parallel imaging [ 55 ] and single coil imaging with the finite support [ 56 ]. The duality between the compressed sensing and low-rank Hankel matrix approaches were discovered by Jin and his colleagues [ 57 , 58 , 59 , 60 ] and Ongie et al. In particular, the unified framework for compressed sensing and parallel MRI in terms of low-rank Hankel matrix approaches was presented by Jin et al. In the following, we provide more detailed reviews of these historical milestones in compressed sensing MRI.

Basic formulation of compressed sensing MRI Although some MR images such as angiograms are already sparse in the pixel representation, more complicated images are rarely sparse, but only have a sparse representation in some transform domain, for example, in terms of spatial finite- differences or their wavelet coefficients. Based on this observation, Lustig et al. The resulting optimization algorithms are, however, computational expensive. For cartesian sampling trajectory, this problem can be overcome as follows.

The ones are at those diagonal entries that corresponds to the sampled locations in the k-space. The first experimental demonstration by Lustig et al. For example, to deal with several hyperparameters to trade off between sparsity and data fidelity terms. Many research efforts have been made to quantity such trade- off, provide different forms of reconstruction, even to propose hyperparameter free reconstruction method [ 63 , 64 , 65 , 66 , 67 , 68 , 69 , 70 ].

Advanced formulation for compressed sensing MRI Non-Cartesian compressed sensing MRI Compressed sensing with non Cartesian sampling has been also extensively stud- ied, since they are really a great combination given the sampling behavior of non Cartesian sampling schemes and the incoherence requirement in compressed sens- ing reconstruction. Aside from the radial and spiral sampling patterns discussed before, the works in [ 71 , 72 , 73 , 74 , 75 , 76 , 77 , 78 , 79 , 80 , 81 , 82 , 83 ] have fo- cused on designing better sampling trajectories with good incoherent properties.

For example, Haldar et al. However, one of the main technical issues associated with non-cartesian compressed sensing MRI is that the fast reconstruction trick shown in 21 cannot be used, which increases the overall computational time. Combination of parallel imaging with CS Recall that parallel MRI pMRI [ 1 , 3 ] exploits the diversity in the receiver coil sen- sitivity maps that are multiplied by an unknown image.

This provides additional spatial information for the unknown image, resulting in accelerated MR data acquisition through k-space sample reduction. Because the aim of the parallel imaging and CS approaches is similar, extensive research efforts have been made to syner- gistically combine the two for further acceleration [ 18 , 20 , 38 , 40 , 84 ]. One of the most simplest approaches can be a SENSE type approach that explic- itly utilizes the estimated coil maps to obtain an augmented compressed sensing problem [ 18 , 20 , 38 , 84 ].

In both approaches, an accurate estimation of coil sensitivity maps or GRAPPA kernel is essential to fully exploit the coil sensitivity diversity. To address this problem, Uecker et al. Blind compressed sensing MR using dictionary learning Blind compressed sensing approaches attempted to simultaneously reconstruct the underlying image as well as the sparsifying transform from highly undersampled measurements.

Ravishankar and his colleagues pioneered two distinct approaches - synthesis dictionary learning [ 48 ] and analysis transform learning [ 86 ] - when the underlying sparsifying transform is unknown a priori. The dictionary, and the image patch, are assumed to be much smaller than the image. This model can be used as a signal model, and Ravishankar et al.

To address the optimization problem P0 , Ravishankar et al. The dictionary learning MRI have shown superior image reconstructions for MRI, as compared to non-adaptive compressed sensing schemes. Approximate iterative algorithms for P0 typically solve the synthesis sparse coding problem re- peatedly, which makes them computationally expensive. In order to overcome some of the aforementioned drawbacks of synthesis dictionary-based BCS, Ravishankar et al.

Sparsify- ing transform learning has been shown to be effective and efficient in applications, while also enjoying good convergence guarantees. One of the important advantages of P1 is that there exists a closed-form update for W, so the computationally expensive dictionary learning step can be avoided. In dynamic MRI, there exists significant redundancies along the temporal directions, which can be extensively studied in various compressed sensing approaches.

Note that p s, f is usually sparse because the periodic motions from heart or slow varying motions from fMRI can be easily sparsified using the temporal Fourier transform. In addition to the temporal Fourier transform, the authors in [ 88 ] used the wavelet transform in the spatial dimension to exploit the spatial redundancy. First, rather than directly enforcing the sparseness of the x-f image, the k-t FOCUSS further sparsifies the x-f image using the initial estimate.

More specifically, by using incoherence sampling patterns, multiple iterations and correct weighting factor for the diagonal matrix, the authors of k-t FOCUSS [ 18 , 20 ] clearly demonstrated the performance improvement. Another powerful aspect of k-t FOCUSS was that the idea can be easily extended to exploit the sparsity in other transform domains.

For example, the residual step in 36 can be interpreted as sparsity promoting step by subtracting the temporal mean images. Thus, Jung et al. More specifi- cally, rather than subtracting the temporal mean values, they subtracted the motion estimated frame. First, at least one reference frame is required. Spreads range from 1.

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