  The goal of this short note is to present a refined analysis of the modified
Basis Pursuit ($\ell_1$-minimization) approach to signal recovery in Compressed
Sensing with partially known support, as introduced by Vaswani and Lu. The
problem is to recover a signal $x \in \mathbb R^p$ using an observation vector
$y=Ax$, where $A \in \mathbb R^{n\times p}$ and in the highly underdetermined
setting $n\ll p$. Based on an initial and possibly erroneous guess $T$ of the
signal's support ${\rm supp}(x)$, the Modified Basis Pursuit method of Vaswani
and Lu consists of minimizing the $\ell_1$ norm of the estimate over the
indices indexed by $T^c$ only. We prove exact recovery essentially under a
Restricted Isometry Property assumption of order 2 times the cardinal of $T^c
\cap {\rm supp}(x)$, i.e. the number of missed components.
