  We present an algorithm for finding sparse solutions of the system of linear
equations $\Phi\mathbf{x}=\mathbf{y}$ with rectangular matrices $\Phi$ of size
$n\times N$, where $n<N$, when measurement vector $\mathbf{y}$ is corrupted by
a sparse vector of errors $\mathbf e$. We call our algorithm the
$\ell^1$-greedy-generous (LGGA) since it combines both greedy and generous
strategies in decoding. Main advantage of LGGA over traditional error
correcting methods consists in its ability to work efficiently directly on
linear data measurements. It uses the natural residual redundancy of the
measurements and does not require any additional redundant channel encoding. We
show how to use this algorithm for encoding-decoding multichannel sources. This
algorithm has a significant advantage over existing straightforward decoders
when the encoded sources have different density/sparsity of the information
content. That nice property can be used for very efficient blockwise encoding
of the sets of data with a non-uniform distribution of the information. The
images are the most typical example of such sources. The important feature of
LGGA is its separation from the encoder. The decoder does not need any
additional side information from the encoder except for linear measurements and
the knowledge that those measurements created as a linear combination of
different sources.
