Abstract
Classic signal processing theory dictates that, in order to faithfully reconstruct a band-limited signal (e.g., an image), the sampling rate must be at least twice the maximum frequency contained within the signal, i.e., the Nyquist frequency. Recent developments in applied mathematics, however, have shown that it is often possible to reconstruct signals sampled below the Nyquist rate. This new method of compressed sensing (CS) requires that the signal have a concise and extremely dense representation in some mathematical basis. Magnetic resonance imaging (MRI) is particularly well suited for CS approaches, owing to the flexibility of data collection in the spatial frequency (Fourier) domain available in most MRI protocols. With custom CS acquisition and reconstruction strategies, one can quickly obtain a small subset of the full data and then iteratively reconstruct images that are consistent with the acquired data and sparse by some measure. Successful use of CS results in a substantial decrease in the time required to collect an individual image. This extra time can then be harnessed to increase spatial resolution, temporal resolution, signal-to-noise, or any combination of the three. In this article, we first review the salient features of CS theory and then discuss the specific barriers confronting CS before it can be readily incorporated into clinical quantitative MRI studies of cancer. We finally illustrate applications of the technique by describing examples of CS in dynamic contrast-enhanced MRI and dynamic susceptibility contrast MRI.