Theoretical study on high order interior tomography.

In this paper, we study a new type of high order interior problems characterized by high order differential phase shift measurement. This problem is encountered in local x-ray phase-contrast tomography. Here we extend our previous theoretical framework from interior CT to interior differential phase-contrast tomography, and establish the solution uniqueness in this context. We employ the analytic continuation method and high order total variation minimization which we developed in our previous work for interior CT, and prove that an image in a region of interest (ROI) can be uniquely reconstructed from truncated high order differential projection data if the image is known a priori in a sub-region of the ROI or the image is piecewise polynomial in the ROI. Preliminary numerical experiments support the theoretical finding.