Phase retrieval from series of images obtained by defocus variation

We develop and compare three different methods of phase retrieval from series of image measurements obtained at different defocus values. The first approach is an approximate solution to the transport of intensity equation (TIE) based on Fourier transforms. The second is an exact solution of the TIE, using multigrid methods. Lastly an iterative approach, using the free space propagator between image planes, is discussed. The iterative scheme is robust in the presence of discontinuities in the phase, unlike those methods based on solution of the TIE. The performance of the different methods in the presence of noise is discussed. Application of these methods to a set of experimental images taken using X-ray imaging is investigated. A computer program which can reproduce the simulations and analyse experimental image data is briefly discussed.