Compressed sensing in signal processing may help give an intuitive overview of what’s going on. Consider a stream of data samples S, collected periodically at a rate R samples per second. According to the sampling theorem, we can perfectly reconstruct a continuous signal from those samples up to a frequency of rate 1/(2R). So for samples taken every 1/100th of a second, we can perfectly reconstruct signals from 0 to 50 hz.
Now, take two different subsets of S and compare them:
if we take a subset of S at fixed intervals, we can only reconstruct part of the original signal perfectly. The part that we can reconstruct is at a lower frequency: for example, if we take every other sample from S, we can only reconstruct perfectly up to a limit of 25 hz. Frequencies above that cannot be uniquely determined.
if we take a completely random subset of S, we can reconstruct the entire frequency range, but we can not reconstruct it perfectly. This is similar to holography, where cutting the hologram in half still reproduces the whole image, but at lower resolution.
We use fixed subsampling when we know our signal is band limited, and that we can safely throw away some band of frequencies.
The use of random sampling (compressed sensing) is when we have a signal that is not band limited, but is sparse enough that we can still accurately describe using a smaller number of data points. The more frequency sparse the signal is, the more accurate our estimate will be. For most compressed sensing applications, the signal can be very sparse indeed, and the number of needed samples can be quite low.
Compressed sensing in signal processing may help give an intuitive overview of what’s going on. Consider a stream of data samples S, collected periodically at a rate R samples per second. According to the sampling theorem, we can perfectly reconstruct a continuous signal from those samples up to a frequency of rate 1/(2R). So for samples taken every 1/100th of a second, we can perfectly reconstruct signals from 0 to 50 hz.
Now, take two different subsets of S and compare them:
if we take a subset of S at fixed intervals, we can only reconstruct part of the original signal perfectly. The part that we can reconstruct is at a lower frequency: for example, if we take every other sample from S, we can only reconstruct perfectly up to a limit of 25 hz. Frequencies above that cannot be uniquely determined.
if we take a completely random subset of S, we can reconstruct the entire frequency range, but we can not reconstruct it perfectly. This is similar to holography, where cutting the hologram in half still reproduces the whole image, but at lower resolution.
We use fixed subsampling when we know our signal is band limited, and that we can safely throw away some band of frequencies.
The use of random sampling (compressed sensing) is when we have a signal that is not band limited, but is sparse enough that we can still accurately describe using a smaller number of data points. The more frequency sparse the signal is, the more accurate our estimate will be. For most compressed sensing applications, the signal can be very sparse indeed, and the number of needed samples can be quite low.