  This paper discusses reconstruction of signals from few measurements in the
situation that signals are sparse or approximately sparse in terms of a general
frame via the $l_q$-analysis optimization with $0<q\leq 1$. We first introduce
a notion of restricted $q$-isometry property ($q$-RIP) adapted to a dictionary,
which is a natural extension of the standard $q$-RIP, and establish a
generalized $q$-RIP condition for approximate reconstruction of signals via the
$l_q$-analysis optimization. We then determine how many random, Gaussian
measurements are needed for the condition to hold with high probability. The
resulting sufficient condition is met by fewer measurements for smaller $q$
than when $q=1$.
  The introduced generalized $q$-RIP is also useful in compressed data
separation. In compressed data separation, one considers the problem of
reconstruction of signals' distinct subcomponents, which are (approximately)
sparse in morphologically different dictionaries, from few measurements. With
the notion of generalized $q$-RIP, we show that under an usual assumption that
the dictionaries satisfy a mutual coherence condition, the $l_q$ split analysis
with $0<q\leq1 $ can approximately reconstruct the distinct components from
fewer random Gaussian measurements with small $q$ than when $q=1$
