Edinburgh Imaging

Compressed sensing

Compressed sensing is a generic method of reconstructing signals & images from fewer measurements (i.e. fewer samples) than are required by conventional theory.


A simple example of compressed sensing is the reproduction of a musical tone produced by a musical instrument:

  • We could record the tonal waveform at a high enough sampling rate, so as to capture all the nuances
  • This would give us several thousand samples - the ‘classical’ approach to compressed sensing
  • For some instruments, the tone can be adequately described by the sum of a number of sinusoidal waveforms - the ‘fundamental’ & its harmonics
  • Since each component requires only a few parameters to describe it (frequency, amplitude & phase), far fewer samples are needed in total

Very similar compression schemes (JPEG & MP3) are widely used to compress the sizes of image & sound files, for storage & transmission over the internet

Medical images also can be though of as ‘sparse’, meaning that they can be described by far fewer samples than the number of pixels. The logical consequence is that if we don’t need all those samples, why collect them in the first place?


  • We investigate how compressed sensing can applied to medical imaging



Compressed sensing in medical imaging has three requirements:

  1. The image must be intrinsically sparse
  2. The samples should be collected in a pseudorandom fashion from all possible samples, to avoid reconstruction artefacts
  3. An iterative reconstruction algorithm is required


MR is a suitable imaging candidate for compressed sensing:

  • MR is inherently a slow imaging technique (e.g. compared with ultrasound or CT)
  • Collecting fewer samples can reduce examination time


The images on this page illustrate the principle of compressed sensing:

  • Fig a) a fully sampled image generated by conventional Fourier based reconstruction
  • Fig b) a four times undersampled image (i.e. in which only 1/4 of the total number of samples are acquired thereby reducing scan time by a factor of 4) & reconstructed by conventional Fourier-based method
  • Fig c) a four times undersampled image generated by Compressed Sensing (CS) reconstruction


Compressed sensing example