Phase Retrieval with Sparse Phase Constraint

Abstract

In this paper, we propose and investigate the phase retrieval problem with the a priori constraint that the phase is sparse (SPR), which encompasses a number of practical applications, for instance, in characterizing phase-only objects such as microlenses, in phase-contrast microscopy, in optical path difference microscopy, and in Fourier ptychography, where the phase object occupies a small portion of the whole field. The considered problem is strictly more general than the sparse signal recovery problem, which assumes the sparsity of the signal because the sparsity of the signal trivially implies the sparsity of the phase, but the converse is not true. As a result, existing solution algorithms in the literature of sparse signal recovery cannot be applied to SPR and there is an appeal for developing new solution methods for it. In this paper, we propose a new regularization scheme which efficiently captures the sparsity constraint of SPR. The idea behind the proposed approach is to perform a metric projection of the current estimated signal onto the set of all the signals whose phase satisfies the sparsity constraint. The main challenge here is that the latter set is not convex and its associated projector in general does not admit a closed form. One novelty of our analysis is to establish an explicit form of that projector when restricted to those points which are relevant to the solutions of SPR. Note that this result is fundamentally different from the widely known calculation form for projections onto intensity constraint sets. Based on this new result, we propose an efficient solution method, named the sparsity regularization on phase (SROP) algorithm, for the SPR problem in the challenging setting where only one point-spread-function image is given, and we analyze its convergence. The algorithm is the combination of the Gerchberg--Saxton (GS) algorithm with the projection step described above. In view of the GS algorithm being equivalent to the alternating projection for an associated two-set feasibility, the SROP algorithm is shown to be the cyclic projection for an associated three-set feasibility, one of the sets being analyzed in this paper for the first time. Analyzing regularity properties of the involved sets, we obtain convergence results for the SROP algorithm based on our recent convergence theory for the cyclic projection method. Numerical results show clear effectiveness and efficiency of the proposed solution approach for the SPR problem.

Alternating Projections with Applications to Gerchberg-Saxton Error Reduction

“…In [33] sparsity of the phase in the frequency domain is considered with the prior A = {x ∈ C N : arg( x(ω)) = 0 for at most k frequencies ω}.…”

“…All we have to see is that (37) is just a way of encoding the spiral (33) in frequency coordinates, and bearing in mind that the fourth coordinate is fixed throughout. In other words, A = P(F (A)), and A = F (P (A)), and the same for B, B.…”

Phase retrieval based on the vectorial model of point spread function

“…This section demonstrates that VAM and VAM + are also efficient for phase retrieval with discontinuous phase, for example, phase retrieval with sparse phase constraint for applications in characterizing phase-only objects such as microlenses, phase-contrast microscopy, optical path difference microscopy and in Fourier ptychography, where the phase object occupies less than 10% of the whole field [40]. As mentioned in the introduction, it is not feasible to accurately approximate a discontinuous wavefront or its associated GPF with a weighted sum of Zernike modes because the continuity property is invariant with respect to linear combination.…”

Projection methods for high numerical aperture phase retrieval

“…We then study the geometry of the high-NA phase retrieval problem and the obtained results are subsequently used to establish convergence criteria of projection algorithms in the presence of noise. Making use of the vectorial PSF is, on the one hand, the key difference between this paper and the literature of phase retrieval mathematics which mostly deals with the scalar PSF, see, for example, [5,18,20,21,36,37,52,53,55,57]. The results of this paper, on the other hand, can be viewed as extensions of those concerning projection methods for low-NA phase retrieval.…”