1. A Fourier ptychographic X-ray imaging system, comprising: an X-ray optical element configured to receive radiation issuing from a specimen being imaged during operation; an X-ray radiation detector configured to receive radiation transmitted by the X-ray optical element and configured to capture a plurality of variably-illuminated, low-resolution intensity X-ray images of the specimen during operation; and a processor having instructions for reconstructing a high-resolution X-ray image of the specimen by iteratively determining the high resolution X-ray image that is self-consistent with the variably-illuminated, low-resolution intensity X-ray images captured by the X-ray radiation detector, wherein reconstructing the high-resolution X-ray image includes recovering phase image data.