Local convergence of the error-reduction algorithm for real-valued objects

Fourier phasing is the problem of retrieving Fourier phase information from Fourier intensity data. The error-reduction (ER) algorithm consists of two projections on the subspaces generated by the Fourier magnitude constraint and the object-domain constraint. The random phase illumination (RPI) and the real-valued constraint on the object significantly reduce the complexity of the intersection of the two subspaces. In this paper, we study how to approximate the projection of the starting point onto the subspace generated by the Fourier magnitude constraint by its projection on the tangent plane and estimate the approximation error by orthogonal decompositions. Moreover, we prove that the local geometric convergence rate of the ER algorithm is less than one almost surely and can be characterized as the cosine of the angle between the two projection spaces. A theoretical estimate of the convergence rate is derived and validated by some numerical experiments.