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In this work we analyse Cherenkov radiation by relativistic muons, positive and negative, channeled in optically transparent diamond and silicon crystals in comparison with ordinary Cherenkov radiation. We have shown that the maxima in the spectral angular distributions for both types of radiation are revealed at the derivative extrema for the media refractive index, while, due to the difference in scattering of positively and negatively charged particles at crystal channeling, the number of Cherenkov photons emitted by channeled positive muons might be over the one for negative muons. We have demonstrated that Cherenkov radiation by quasi free projectiles is described as one limiting approximation of a general expression for Cherenkov radiation by channeled projectiles, which takes into account non-zero derivative of the refractive index. The last may result in essential increase of radiation intensity.