High-resolution, three-dimensional (3D) imaging of soft tissues requires the solution of two inverse problems: phase retrieval and the reconstruction of the 3D image from a tomographic stack of two-dimensional (2D) projections. The number of projections per stack should be small to accommodate fast tomography of rapid processes and to constrain X-ray radiation dose to optimal levels to either increase the duration of in vivo time-lapse series at a given goal for spatial resolution and/or the conservation of structure under X-ray irradiation. In pursuing the 3D reconstruction problem in the sense of compressive sampling theory, we propose to reduce the number of projections by applying an advanced algebraic technique subject to the minimisation of the total variation (TV) in the reconstructed slice. This problem is formulated in a Lagrangian multiplier fashion with the parameter value determined by appealing to a discrete L-curve in conjunction with a conjugate gradient method. The usefulness of this reconstruction modality is demonstrated for simulated and in vivo data, the latter acquired in parallel-beam imaging experiments using synchrotron radiation.

For coherent X-ray imaging of pure phase objects we study the reliability of linear relations in phase-retrieval algorithms based on a single intensity map after free-space propagation. For large phase changes and/or large propagation distances we propose two venues of working beyond linearity: Projection onto an effective, linear and local model in Fourier space and expansion of intensity contrast in powers of object-detector distance. We apply both algorithms successfully to simulated data.

Phase contrast in the object plane of a phase object is retrieved from intensity contrast at a single object-detector distance. Expanding intensity contrast and phase shift in the detector plane in powers of object-detector distance, phase retrieval is performed beyond the solution to the linearized transport-of-intensity equation. The expansion coefficients are determined by the entire paraxial wave equation. The Laplacian of the phase shift in the object plane thus is written as a local expression linear in the intensity contrast and nonlinear in the phase shift in the object plane. A perturbative approach to this expression is proposed and tested with simulated phantom data.