Performance Analysis of $\ell_1$-synthesis with Coherent Frames

Abstract: Signals with sparse frame representations comprise a much more realistic model of nature than that with orthonomal bases. Studies about the signal recovery associated with such sparsity models have been one of major focuses in compressed sensing. In such settings, one important and widely used signal recovery approach is known as $\ell_1$-synthesis (or Basis Pursuit). We present in this article a more effective performance analysis (than what are available) of this approach in which the dictionary $\Dbf$ may be highly, and even perfectly correlated. Under suitable conditions on the sensing matrix $\Phibf$, an error bound of the recovered signal $\hat{\fbf}$ (by the $\ell_1$-synthesis method) is established. Such an error bound is governed by the decaying property of $\tilde{\Dbf}{\text{o}}^*\fbf$, where $\fbf$ is the true signal and $\tilde{\Dbf}{\text{o}}$ denotes the optimal dual frame of $\Dbf$ in the sense that $|\tilde{\Dbf}_{\text{o}}^*\hat{\fbf}|_1$ produces the smallest $|\tilde{\Dbf}^*\tilde{\fbf}|_1$ in value among all dual frames $\tilde{\Dbf}$ of $\Dbf$ and all feasible signals $\tilde{\fbf}$. This new performance analysis departs from the usual description of the combo $\Phibf\Dbf$, and places the description on $\Phibf$. Examples are demonstrated to show that when the usual analysis fails to explain the working performance of the synthesis approach, the newly established results do.