  Sparse recovery is widely applied in many fields, since many signals or
vectors can be sparsely represented under some frames or dictionaries. Most of
fast algorithms at present are based on solving $l^0$ or $l^1$ minimization
problems and they are efficient in sparse recovery. However, compared with the
practical results, the theoretical sufficient conditions on the sparsity of the
signal for $l^0$ or $l^1$ minimization problems and algorithms are too strict.
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  In many applications, there are signals with certain structures as piecewise
sparsity. Piecewise sparsity means that the sparse signal $\mathbf{x}$ is a
union of several sparse sub-signals, i.e.,
$\mathbf{x}=(\mathbf{x}_1^T,\ldots,\mathbf{x}_N^T)^T$, corresponding to the
matrix $A$ which is composed of union of bases $A=[A_1,\ldots,A_N]$. In this
paper, we consider the uniqueness and feasible conditions for piecewise sparse
recovery. We introduce the mutual coherence for the sub-matrices $A_i\
(i=1,\ldots,N)$ to study the new upper bounds of $\|\mathbf{x}\|_0$ (number of
nonzero entries of signal) recovered by $l^0$ or $l^1$ optimizations. The
structured information of measurement matrix $A$ is used to improve the
sufficient conditions for successful piecewise sparse recovery and also improve
the reliability of $l_0$ and $l_1$ optimization models on recovering global
sparse vectors.

Numerical Analysis;Numerical Analysis