Relation Between Compressed Sensing and Incoherence
The coherence can be used to bound the number of measurements $m$ required for successful reconstruction as seen for example in equation (6) in Candès &amp; Wakin 2008:
$$m ≥ C \cdot \mu^2(\Phi, \Psi)\cdot S  \cdot \log n$$
where $C$ is some positive constant, $S$ is the sparsity, and $n$ is the size of the sparse signal. 

One problem, however, is that this is not a very sharp bound. Donoho &amp; Tanner 2010, for example, describe some brief history of this in Section X of the linked paper. I have not followed the development on this aspect of compressed sensing for a long time, so I do not know if there are newer or better explanations. So, in my opinion, the best you can use the coherence for in practice is as a rule of thumb: the less coherence, the better.