  Coherent ptychographic imaging experiments often discard over 99.9 % of the
flux from a light source to define the coherence of an illumination. Even when
coherent flux is sufficient, the stability required during an exposure is
another important limiting factor. Partial coherence analysis can considerably
reduce these limitations. A partially coherent illumination can often be
written as the superposition of a single coherent illumination convolved with a
separable translational kernel. In this paper we propose the Gradient
Decomposition of the Probe (GDP), a model that exploits translational kernel
separability, coupling the variances of the kernel with the transverse
coherence. We describe an efficient first-order splitting algorithm GDP-ADMM to
solve the proposed nonlinear optimization problem. Numerical experiments
demonstrate the effectiveness of the proposed method with Gaussian and binary
kernel functions in fly-scan measurements. Remarkably, GDP-ADMM produces
satisfactory results even when the ratio between kernel width and beam size is
more than one, or when the distance between successive acquisitions is twice as
large as the beam width.
