14 Sparse Sampling in MRI

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coefficients that need to be retained to represent the image to within 1% relative mean square error (RMSE). It is clear that many fewer coefficients are required to represent the sorted images than the originals, with the ratio being around 10 to 1. Under the definition of sparsity we employ here, therefore, the sorting operation achieves a considerably sparser image.

14.3 Theory of Compressed Sensing

Recall that an image x can be called sparse if under the linear transformation y = Φ x just K N of the data values in y are enough to accurately represent the image, where N is the size of x (Sect. 14.1.2). Assume that y is of size N (under the sparsity assumption, N − K of the elements yi are close to zero). Therefore, Φ is N × N. If we knew a priori which of the elements yi may be neglected, we could reduce Φ to K × N and attempt to estimate y and recover an estimate of x. However, in general it is not known which may be neglected and the data measured may be in a different space.

In MRI, data are measured in k-space and can be represented by a vector d = Wx, where W is the Fourier transform matrix. It is desirable to minimize the number of measurements and so we seek to reduce the size of d as much as possible, to M elements say (dM ), while recovering an acceptable quality of estimate of x. A direct method would be to form a transformation Ψ of dimension K ×M such that y = Ψd, with y comprising only the important values of y being estimated. Again, however, such an approach requires prior knowledge of which of the elements yi may be neglected.

The alternative approach used by many proponents of compressed sensing is to pose the problem in terms of an optimization. Before looking at this in detail, however, let us consider the nature of the measurement process.

14.3.1 Data Acquisition

We have established that speeding up the MRI can be achieved primarily by making fewer measurements. However, there is inevitably a cost incurred from making fewer measurements. First, fewer measurements with other properties of the scanning apparatus unchanged means a lower SNR for the data [11]. Second, undersampling in k-space causes aliasing in the image domain, that is simply inverting the relationship dM = Wx to estimate an image x = W−1x produces a heavily aliased image. Even if a more sophisticated image recovery process is adopted, it is clear that the choice of the sampling locations plays an important role.

The effect of noise can be ameliorated to some extent in post-processing, particularly if that noise is random in nature and its distribution throughout the image. The effect of aliasing can be reduced to acceptable levels by the use of prior

330	P.J. Bones and B. Wu

Fig. 14.5 The transform point spread functions (TPSFs) corresponding to one 2D DWT coefficient with random and regular sampling patterns. (a) Original 128 × 128 axial image; (b) TPSF for random k-space sampling; (c) TPSF for regular k-space sampling (showing clear aliasing); (d) low-resolution image formed from 1/16 k-space data; (e) TPSF for random k-space sampling with data ordering; and (f) TPSF for regular k-space sampling with data ordering

information. However, the pattern of undersampling plays a particularly important role. In CS, random sampling patterns are often employed [3, 12]. In Fig. 14.5, we illustrate the important roles that both random sampling patterns and data ordering can play. Figure 14.5a is the original 128 × 128 axial brain image formed from a fully sampled k-space dataset. Two subsets of the k-space samples were taken by a random pattern and a regular pattern. A DWT was formed (Debauchies-4 wavelets) in each case and one coefficient was chosen to be estimated from the undersampled k-space data. Figure 14.5b, c shows the estimates in the DWT domain for random and regular undersampling patterns, respectively. Lustig, Donoho, and Pauly [13] refer to this type of plot as the “transform point spread function” (TPSF). In Fig. 14.5b, the coefficient is estimated with relatively little aliasing, whereas in Fig. 14.5c the process generates several aliases for the coefficient. Many authors who have written on CS describe this as an “incoherence” property [3, 12, 13].

The remainder of Fig. 14.5 illustrates what happens when data ordering is introduced into the process, with the ordering based on the low-resolution re-

construction shown	in	Fig. 14.5d (formed from the	center 1/16 of the k-space
data). Figure 14.5e,	f	shows the TPSF for random	and regular undersampling

patterns, respectively, with data ordering and reordering included. In this case, little difference is seen between random and regular k-space undersampling patterns, with relatively minor aliasing occurring in both cases. We believe this indicates that the data ordering itself introduces the incoherence property. In Sect. 14.4.2 below, we relate our experience with a number of different sampling strategies.