  Our aim of this article is to reconstruct a signal from undersampled data in
the situation that the signal is sparse in terms of a tight frame. We present a
condition, which is independent of the coherence of the tight frame, to
guarantee accurate recovery of signals which are sparse in the tight frame,
from undersampled data with minimal $l_1$-norm of transform coefficients. This
improves the result in [1]. Also, the $l_q$-minimization $(0<q<1)$ approaches
are introduced. We show that under a suitable condition, there exists a value
$q_0\in(0,1]$ such that for any $q\in(0,q_0)$, each solution of the
$l_q$-minimization is approximately well to the true signal. In particular,
when the tight frame is an identity matrix or an orthonormal basis, all results
obtained in this paper appeared in [13] and [26].
