The literature on compressive sensing (CS) frequently notes that CS relies on two principles: sparsity and incoherence. While I understand why the signal of interest should be sparse in some domain since CS relies on minimizing the norm, incoherence is much fuzzier to me. While equations exist that quantify the incoherence of $\Phi$ and $\Psi$, I have been trying to wrap my head around exactly why the property is important.
For reference, I am interested in applying CS to image scanning where each row of $\Phi$ is full of zeroes except for a one at a prescribed pixel (would this be called a spike basis?). It makes sense to me that $\Phi$ should be incoherent, if such means that the sampling occurs in an irregular pattern that allows maximal information to be obtained. And I see why the signal should be sparse in the $\Psi$ basis. However, I do not see how the CS result depends on the incoherence relationship between $\Phi$ and $\Psi$. Why is this so?