Proximal Heterogeneous Block Implicit-Explicit Method and Application to Blind Ptychographic Diffraction Imaging

Abstract

We propose a general alternating minimization algorithm for nonconvex optimization problems with separable structure and nonconvex coupling between blocks of variables. To fix our ideas, we apply the methodology to the problem of blind ptychographic imaging. Compared to other schemes in the literature, our approach differs in two ways: (i) it is posed within a clear mathematical framework with practically verifiable assumptions, and (ii) under the given assumptions, it is provably convergent to critical points. A numerical comparison of our proposed algorithm with the current state-of-the-art on simulated and experimental data validates our approach and points toward directions for further improvement.

Quantitative Convergence Analysis of Iterated Expansive, Set-Valued Mappings

“…In particular, the cyclic projections algorithm can be expected to converge linearly on neighborhoods of stable fixed points regardless of whether or not the phase sets intersect. This improves, in several ways, the local linear convergence result obtained in Luke [49, theorem 5.1], which established local linear convergence of approximate alternating projections to local solutions with more general gauges for the case of two sets: first, the present theory handles more than two sets, which is relevant for wave front sensing (Luke [52], Hesse et al [35]); second, it does not require that the intersection of the constraint sets (which are expressed in terms of noisy, incomplete measurement data) be nonempty. This is in contrast to recent studies of the phase retrieval problem (of which there are too many to cite here), which require the assumption of feasibility, despite evidence, numerical and experimental, to the contrary.…”

“…Example 3.6 (Phase Retrieval). In the discrete version of the phase retrieval problem (Luke et al [52], Bauschke et al [13], Burke and Luke [23], Luke [47,49], Hesse et al [35], Luke [50]), the constraint sets are of the form…”

3D x-ray imaging of continuous objects beyond the depth of focus limit

“…12 of $$F(\text{x},\text{y},\text{z})=\sum _{j=1}^m{\Vert \left(\prod _{q=1\dots s}C\text{ diag }\left(s_{q,j}\text{y}\right)\right)\text{x}-{bold\text{z}}_j\Vert }_2^2$$ is a generalization of the one used for 2D ptychography [44] and represents the data mismatch based on the forward model in Eq. 11.…”

Phase retrieval for object and probe in the optical near-field

“…The operator pointing onto the set O cannot exactly be expressed as a projector [73]. For this reason, it shall be named Π O instead of P O .…”

“…The absolute value ofp j is far from zero: The entire field of view of the detector is illuminated by a partially coherent p j . Yet, if one would account for a probe covering only a finite section of the detector and for an object that has regions of zero transmission, it would be more accurate to describe equations 4.317 and 4.318 in a similar way as shown in [73] byô…”