X-Ray cone-beam phase tomography formulas based on phase-attenuation duality.

We present a detailed derivation of the phase-retrieval formula based on the phase-attenuation duality that we recently proposed in previous brief communication. We have incorporated the effects of x-ray source coherence and detector resolution into the phase-retrieval formula as well. Since only a single image is needed for performing the phase retrieval by means of this new approach, we point out the great advantages of this new approach for implementation of phase tomography. We combine our phase-retrieval formula with the Feldkamp-Davis-Kresss (FDK) cone-beam reconstruction algorithm to provide a three-dimensional phase tomography formula for soft tissue objects of relatively small sizes, such as small animals or human breast. For large objects we briefly show how to apply Katsevich's cone-beam reconstruction formula to the helical phase tomography as well.