Compressed Sensing

Compressed sensing is a new rapidly growing research field emerging primarily in the USA, which investigates ways in which we can sample signals at roughly the "information rate" rather than the Nyquist rate. For example compressed sensing theory has already enabled a 4-fold reduction in acquisition time for MRI images by allowing the under-sampling of k-space (Lustig et al. 2007). It is potentially a very disruptive technology and provides a new way of thinking about how to acquire and code signals in the most efficient manner.

Its growing popularity is evident in the forthcoming special issue in IEEE Signal Processing Magazine (Eds: Donoho, Baranuik and Vetterli), the large number of research papers and the growing number of applications that are being explored across a range of disciplines, including:

  • Medical imaging
  • Seismic imaging
  • Distributed and remote sensing
  • Analogue to Information (Digital) Conversion

The foundations for compressed sensing emerged over the last two years from theoretical work developed within the field of sparse signal representations. A sparse representation is one which accounts for most or all of the information of a signal with a linear combination of a small number of elementary signals called atoms.

The Fourier transform, for example, can represent a signal containing a single frequency with a single non-zero frequency component. This sparseness is one of the reasons for the extensive use of popular transforms such as the discrete Fourier transform (DFT) and the wavelet transform in practical signal source coding schemes. The aim of these transforms is often to reveal certain structures of a signal and to represent these structures in a compact form. Sparse representations extend this idea by also considering more flexible redundant representations (called dictionaries) where the linear analysis transform is replaced by a nonlinear sparse representation operator.