I've never used this in any practical way, but would any of the techniques from Compressed Sensing help at all? http://en.wikipedia.org/wiki/Compressed_sensing
I guess applying compressed sensing techniques would mean making assumptions about your image which allow you to fill in some of the blanks created by the incomplete angle and position coverage (i.e. the fact that the system is underdetermined). If you could somehow assume a certain redundancy, you could probably make guesses about what goes in the spots that didn't get well-covered.
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.