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

Restricted $q$-Isometry Properties Adapted to Frames for Nonconvex $l_q$-Analysis