  Compressed sensing (CS) is a powerful tool for reducing the amount of data to
be collected while maintaining high spatial resolution. Such techniques work
well in practice and at the same time are supported by solid theory. Standard
CS results assume measurements to be made directly on the targeted signal. In
many practical applications, however, CS information can only be taken from
indirect data $h_\star = W x_\star$ related to the original signal by an
additional forward operator. If inverting the forward operator is ill-posed,
then existing CS theory is not applicable. In this paper, we address this issue
and present two joint reconstruction approaches, namely relaxed $\ell^1$
co-regularization and strict $\ell^1$ co-regularization, for CS from indirect
data. As main results, we derive error estimates for recovering $x_\star$ and
$h_\star$. In particular, we derive a linear convergence rate in the norm for
the latter. To obtain these results, solutions are required to satisfy a source
condition and the CS measurement operator is required to satisfy a restricted
injectivity condition. We further show that these conditions are not only
sufficient but even necessary to obtain linear convergence.
