14 Sparse Sampling in MRI

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of pixels are being recorded and stored. Applying a linear transform, such as the discrete wavelet transform (DWT), to the image allows many of the DWT coefficients to be set to zero, resulting in a compressed form of storage. The compressed data set can be used to reconstruct a good likeness of the original image. The compressed sensing (CS) approach is illustrated in Fig. 14.1b: by some process many fewer samples are made of the original scene and the nonzero coefficients of the compressed image are directly estimated. Thus, the waste associated with full data measurement followed by compression is avoided in CS.

At the time of writing, the group at Rice University has established a very comprehensive bibliography of literature related to the theory and application of compressed sensing (see http://dsp.rice.edu).

14.1.3 The Role of Prior Knowledge

The term “prior knowledge” is most frequently associated with Bayesian inference, in which a posterior estimate based on gathered measurements is influenced by knowledge of prior distributions for the measurements and for the result. See Geman and Geman [5] and Hu et al. [6] for a full treatment of Bayesian estimation methods. Here, we use the term in a wider context, however. By prior knowledge is here meant any constraint which can be put on the estimate. For example if the interior of an object is being imaged that is known to exist within a given boundary, then that boundary, or possibly a conservative estimate of it, can be incorporated into the imaging process: this is an example of a “support constraint.” In the situation cited, those pixels (or voxels, if 3D images are being considered), which lie outside the support region do not need to be estimated, suggesting that incorporating the prior knowledge may make the estimation process easier.

The spatial support constraint represents only one from a rich set of forms of prior knowledge, which may be applied to imaging problems in general and MRI in particular. Other examples include:

1.Knowledge that the image is sparse in some sense is itself a form of prior knowledge

2.Knowledge that the object being imaged is approximately piecewise homogeneous

3.Knowledge that the object has a smooth boundary or boundaries

4.Knowledge that changes in time occur smoothly

5.Knowledge that only a relatively small portion of the object undergoes changes with time

Some authors may argue that all of the examples of prior knowledge listed above may be able to be labeled as “sparsity,” but here we use a more restricted definition: sparsity is the property that when the image is expressed in terms of some basis, many fewer coefficients are required to accurately represent the image than is implied by Nyquist sampling.