This article considers constrained ℓ1 minimization methods for the
recovery of high dimensional sparse signals in three settings: noiseless,
bounded error and Gaussian noise. A unified and elementary treatment is given
in these noise settings for two ℓ1 minimization methods: the Dantzig
selector and ℓ1 minimization with an ℓ2 constraint. The results of
this paper improve the existing results in the literature by weakening the
conditions and tightening the error bounds. The improvement on the conditions
shows that signals with larger support can be recovered accurately. This paper
also establishes connections between restricted isometry property and the
mutual incoherence property. Some results of Candes, Romberg and Tao (2006) and
Donoho, Elad, and Temlyakov (2006) are extended