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requirement can be satisfied by employing a pseudorandom data acquisition pattern on Cartesian k-space grid. The idea of compressed sensing has received much attention thereafter and has been extended to the use of non-Cartesian trajectories such as radial [20] and spiral [22].

Another place where a high level of sparseness exists is the temporal domain of dynamic MRI. Not only does each individual temporal frame have inherent sparseness under the appropriate transform, and hence allows undersampling using compressed sensing, there also exists a high level of redundancy in the time domain due to the generally slow object variation over time. Such redundancy can be exploited in terms of its sparseness under appropriate transform to allow undersampling in the temporal domain. In fact, this idea has been well exploited prior to the use of compressed sensing: Fourier-encoding the overlaps using the temporal dimension (UNFOLD) [23], k-t broad-use linear acquisition speed-up technique (k-t BLAST) [24] and k-t focal underdetermined system solver (k-t FOCUSS, [25]) are all good examples of this. They all include an additional temporal frequency dimension in addition to the spatial frequency dimensions of the data set, and utilize the sparseness in the resulting spatial–temporal frequency domain. In fact, k-t FOCUSS was later shown to be a general implementation of compressed sensing by inherently utilizing the l1 norm constraint. In [26], compressed sensing is explicitly applied in the additional temporal dimension to in MR cardiac imaging, and it was shown to outperform the k-t BLAST method.

In general, the application of compressed sensing to MRI is a recently emerged and fast developing area. The use of the l1 norm in the appropriate domain, which is the core of compressed sensing, can be used as a general regularization constraint in many cases. One good example is in the conjoint use of parallel imaging, which normally faces the penalty of reduced SNR with reduced sampling density. The l1 norm regularization intrinsically suppresses noise and hence offers complementary characteristics to those of parallel imaging [27].

14.4.2 Results from Our Work

Our own work in applying sparse sampling in MRI has led to the development of two new algorithms: Prior estimate-based compressed sensing (PECS) and sensitivity encoding with compressed sensing (SENSECS). PECS has been demonstrated in both brain imaging, that is imaging of a static structure, and in contrast-enhanced angiography, that is dynamic imaging as part of a pilot study on normal volunteers [28, 29]. SENSECS has been demonstrated in brain imaging [27]. We briefly summarize the work in the remainder of this section.

14.4.2.1 PECS

The success of compressed sensing is determined by the sparsity of the underlying signal. In our experience to date, the sparsest representation of a typical MRI

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Fig. 14.6 Presentation of an argument for the approximate ordering process incorporating prior knowledge into PECS recovery. Figure (b) depicts the computation of an approximate data order from a low-resolution version of the signal. Figure (a) depicts the use of that sampling order on the original signal. Figure (c) depicts the effect of sorting on the discrepancies between the original signal and the prior knowledge. After transformation, the large amplitude coefficients retain information about the sorting prior, while the smaller coefficients are mainly associated with the discrepancies between the original and low-resolution signals

anatomical image is obtained by ordering the set of pixel (or voxel) amplitudes as described in Sect. 14.2.3. Assume that a set of undersampled k-space data have been collected with a particular MRI sequence with the purpose of forming a highresolution image. In addition, a prior estimate is available of the image, for example a low-resolution image. In PECS, the prior estimate of the image is first used to derive a data ordering, R ; compressed sensing is used to recover an image from the measured k-space data, incorporating the approximate ordering (to promote sparsity) and a TV minimization (to promote piecewise smoothness in the resulting image). The process is illustrated in Fig. 14.6 for the case of a 1D signal. Recall from Sect. 14.2.3 that ordering the amplitudes of a signal into a monotonic function allows that signal to be represented by fewer coefficients under an appropriate transform (DCT or DWT, say). In this case the approximate ordering, R , is derived from a low resolution (low pass filtered) version of the signal. When R is applied to the high resolution data, the result is a highly noisy signal which only approximates in form to a monotonic function. Under the transform the largest coefficients

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Fig. 14.7 Reconstructions comparing PECS with other methods: (a) top left quadrant of a reconstruction of an axial brain slice with full k-space sampling; (b–d) reconstructions at an acceleration factor of 4 (the sampling patterns used are shown in the insets). (b) CS with uniformly distributed randomly selected k-space samples; (c) CS with randomly selected k-space samples, but including the center 32 lines; and (d) PECS with ordering derived from a low-resolution approximation using just the center 32 lines of k-space. The arrows indicate areas where (d) shows better recovery than (c)

retain the form, while the errors tend to generate a noise-like spectrum with low amplitudes spread across many coefficients. After thresholding only the significant coefficients are nonzero and these retain the form of the true ordering. We argue that prior knowledge about the signal is thereby introduced by the application of the approximate data ordering [27].

A result for PECS is shown in Fig. 14.7. A 1.5T GE scanner equipped with an eight-channel head coil was used to obtain a 2D T2-weighted axial brain slice of a healthy adult volunteer. A fully sampled k-space data set (256 × 256) was obtained and then various forms of sampling patterns were applied in post processing to simulate the under-sampling required [29]. Note that in this case the under-sampling is applied in the single phase encoding direction (anterior–posterior). In Fig. 14.7a is shown the reconstruction for the slice utilizing the fully sampled k-space data; the top left quadrant is selected for more detailed study. To the right are three different reconstructions obtained from only one quarter of the k-space data, simulating an acceleration factor of 4 for the imaging process. In Fig. 14.7b, the reconstruction is by CS with a uniform sampling density, while in Fig. 14.7c an otherwise similar sampling pattern is altered to make the center 32 lines of k-space fully sampled. The improvement in (c) compared to (b) is obvious. A PECS reconstruction is shown in Fig. 14.7d, with the prior estimate used to generate R being a low-resolution image formed from the center 32 k-space lines. The arrows indicate particular areas, where PECS in (d) performs better that CS in (c).

14.4.2.2 SENSECS

As the name implies, SENSECS combines SENSE with CS. A regular sampling pattern is employed which promotes the performance of SENSE, except that several

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Fig. 14.8 Reconstructions comparing SENSECS with other methods at high acceleration factors (shown in the top left of the reconstructions): (a) reconstruction of an axial brain slice with full k-space sampling; in the lower section of the figure, the left column is SENSE, the center column is CS, and the right column is SENSECS. Note that the SENSECS reconstructions use the SENSE reconstructions to derive the data sorting order

additional lines are sampled at the center of k-space. SENSE is applied to achieve an intermediate reconstruction [11]. Because of the high acceleration factor being used, the image is likely to be quite noisy. The noise is in part at least due to the imperfect knowledge available of the sensitivities of the individual receiver coils. An approximate sorting order R is derived from the intermediate SENSE-derived image. Then PECS is performed with the same set of k-space data and using R .

Again, a single set of results is shown for the new algorithm in Fig. 14.8. The same set of data as described above in this section was used and the fully sampled k-space data was again undersampled in postprocessing [29]. High acceleration factors of 5.8 and 6.5 were simulated. In Fig. 14.8a is shown the reconstruction for the slice utilizing the fully sampled k-space data; the top left quadrant is selected for