Estimation of q for ℓ q $\ell _{q}$ -minimization in signal recovery with tight frame

Abstract This study aims to reconstruct signals that are sparse with a tight frame from undersampled data by using the ℓ q $\ell _{q}$ -minimization method. This problem can be cast as a ℓ q $\ell _{q}$ -minimization problem with a tight frame subjected to an undersampled measurement with a known noise bound. We proved that if the measurement matrix satisfies the restricted isometry property with δ 2 s ≤ 1 / 2 $\delta _{2s}\leq 1/2$ , there exists a value q 0 $q_{0}$ such that for any q ∈ ( 0 , q 0 ] $q\in (0,q_{0}]$ , any signal that is s-sparse with a tight frame can be robustly recovered to the true signal. We estimated q 0 $q_{0}$ as q 0 = 2 / 3 $q_{0} = 2/3$ in the case of δ 2 s ≤ 1 / 2 $\delta _{2s}\leq 1/2$ and discussed that the value of q 0 $q_{0}$ can be much higher. We also showed that when δ 2 s ≤ 0.3317 $\delta _{2s}\leq 0.3317$ , for any q ∈ ( 0 , 1 ] $q\in (0,1]$ , robust recovery for signals via ℓ q $\ell _{q}$ -minimization holds, which is consistent with the case of ℓ q $\ell _{q}$ -minimization without a tight frame.