AdvancedTopic 100 of 101Signals, Control & Comms
Compressed sensing
Recovers sparse signals from far fewer samples than Nyquist suggests.
In depth
If a signal is sparse in some basis, it can be recovered from far fewer random linear measurements than the Nyquist rate suggests, by solving an L1 minimisation (basis pursuit, LASSO). Requires incoherence and restricted isometry property. Applications: MRI acceleration, radar, single-pixel cameras.
Key points
- Sparsity
- L1 vs L0 norm
- Random sensing matrices
ExampleMRI scans can be done several times faster by under-sampling k-space and reconstructing with compressed sensing.
What is Compressed sensing?
Recovers sparse signals from far fewer samples than Nyquist suggests.
Why is it important to understand compressed sensing?
Compressed sensing recovers signals from fewer measurements than traditional sampling suggests, which can cut the cost of imaging and sensing.
What does a signal need to be sparse?
It must have few non-zero components in some basis, such as a wavelet or Fourier basis, so it can be recovered from fewer samples.
What algorithms recover the signal?
Optimisation methods such as basis pursuit or greedy algorithms find the sparsest solution that fits the measurements.
How is Compressed sensing used in practice?
Typical applications include reconstructing an MRI image from a reduced number of samples to shorten the scan, recording a sparse signal with a smaller number of sensors and reducing the data rate of a wideband spectrum monitor.
What should I learn before Compressed sensing?
Study "Network information theory and 6G" first (topic 99 of 101 in Signals, Control & Comms). It covers the groundwork this topic builds on.
What are the key points of Compressed sensing?
Sparsity. L1 vs L0 norm. Random sensing matrices.
Can you give an example of Compressed sensing?
MRI scans can be done several times faster by under-sampling k-space and reconstructing with compressed sensing.
What comes after Compressed sensing?
The next topic is "Extended Kalman filter", which builds directly on this one.
How difficult is Compressed sensing?
It is classed as Advanced in Signals, Control & Comms (topic 100 of 101). It is an advanced topic. It assumes you are comfortable with the intermediate topics in Signals, Control & Comms.