Compressed sensing with prior Information: Strategies, geometry, and bounds

We address the problem of compressed sensing (CS)
with prior information: reconstruct a target CS signal with the
aid of a similar signal that is known beforehand, our prior
information. We integrate the additional knowledge of the similar
signal into CS via ℓ1-ℓ1 and ℓ1-ℓ2 minimization. We then
establish bounds on the number of measurements required by
these problems to successfully reconstruct the original signal.
Our bounds and geometrical interpretations reveal that if the
prior information has good enough quality, ℓ1-ℓ1 minimization
improves the performance of CS dramatically. In contrast, ℓ1-
ℓ2 minimization has a performance very similar to classical
CS and brings no significant benefits. In addition, we use the
insight provided by our bounds to design practical schemes to
improve prior information. All our findings are illustrated with
experimental results.