I doubt that compressed sensing will be very effective for this problem. Many of the guarantees for compressive signal recovery require incoherence between measurements- meaning each measurement is sufficiently different from the others. In this case, each measurement is highly correlated with the others.

The Rice single pixel camera (discussed on the wiki page) effectively multiplies the image by a random mask before it is sensed by the photodiode. This is how they control the incoherence property.