  The phase problem can be considered as one of the cornerstones of quantum
mechanics intimately connected to the detection process and the uncertainty
relation. The latter impose fundamental limits on the manifold phase
reconstruction schemes invented to date in particular at small magnitudes of
the quantum wave. Here, we show that a rigorous solution of the Transport of
Intensity Reconstruction (TIE) scheme in terms of a linear elliptic partial
differential equation for the phase provides reconstructions even in the
presence of wave zeros if particular boundary conditions (BCs) are given. We
furthermore discuss how partial coherence hampers phase reconstruction and show
that a modified version of the TIE reconstructs the curl-free current density
at arbitrary (in-)coherence. This opens the way for a large variety of new
applications in fields as diverse as astrophysics, geophysics, photonics,
acoustics, and electron microscopy, where zeros in the respective wave field
are a ubiquitous feature.
