New Sufficient Conditions of Signal Recovery With Tight Frames via ${l}_1$ -Analysis Approach

This paper discusses the recovery of signals that are nearly sparse with respect to a tight frame
$D$
by means of the
$l_{1}$
-analysis approach. We establish several new sufficient conditions regarding the
$D$
-restricted isometry property to ensure stable reconstruction of signals that are approximately sparse with respect to
$D$
. It is shown that if the measurement matrix
$\Phi $
fulfills the condition
$\delta _{ts}<t/(4-t)$
for
$0<t<4/3$
, then signals which are approximately sparse with respect to
$D$
can be stably recovered by the
$l_{1}$
-analysis approach. In the case of
$D=I$
, the bound is sharp (see Cai and Zhang’s work). In addition, numerical simulations are conducted to indicate that the
$l_{1}$
-analysis method can stably reconstruct the sparse signal in terms of tight frames.