Nonlinear, noniterative, single-distance phase retrieval and developmental biology

For coherent X-ray imaging, based on phase contrast through free-space Fresnel propagation, we discuss two noniterative, nonlinear approaches to the phase-retrieval problem from a single-distance intensity map of a pure-phase object. On one hand, a perturbative set-up is proposed where nonlinear corrections to the linearized transport-of-intensity situation are expanded in powers of the object-detector distance z and are evaluated in terms of the linear estimate. On the other hand, a nonperturbative projection algorithm, which is based on the (linear and local) contrast-transfer function (CTF), works with an effective phase in Fourier space. This effective phase obeys a modified CTF relation between intensity contrast at z > 0 and phase contrast at z= 0: Unphysical singularities of the local CTF model are cut off to yield 'quasiparticles' in analogy to the theory of the Fermi liquid. By identifying the positions of the zeros of the Fourier transformed intensity contrast as order parameters for the dynamical breaking of scaling symmetry we investigate the phase structure of the forward-propagation problem when interpreted as a statistical system. Results justify the quasiparticle approach for a wide range of intermediary phase variations. The latter algorithm is applied to data from biological samples recorded at the beamlines TopoTomo and ID19 at ANKA and ESRF, respectively.