Mixed transfer function and transport of intensity approach for phase retrieval in the Fresnel region

We present a method for phase retrieval in propagation-based x-ray imaging, based on the contrast transfer and transport of intensity equation approaches. We show that the contrast transfer model does not coincide with the transport of intensity in the limit of small propagation distances, and we derive a new model that alleviates this problem. Using this model, we devise an algorithm to retrieve the phase from slowly varying samples that is valid beyond the limit of small distances. We show its utility by imaging in three dimensions a biological sample that causes both strong absorption and phase shift.