Fast Phase Retrieval from Local Correlation Measurements

Abstract

We develop a fast phase retrieval method which can utilize a large class of local phaseless correlation-based measurements in order to recover a given signal x ∈ C d (up to an unknown global phase) in near-linear O d log 4 d -time. Accompanying theoretical analysis proves that the proposed algorithm is guaranteed to deterministically recover all signals x satisfying a natural flatness (i.e., non-sparsity) condition for a particular choice of deterministic correlation-based measurements. A randomized version of these same measurements is then shown to provide nonuniform probabilistic recovery guarantees for arbitrary signals x ∈ C d . Numerical experiments demonstrate the method's speed, accuracy, and robustness in practice -all code is made publicly available.In its simplest form, our proposed phase retrieval method employs a modified lifting scheme akin to the one utilized by the well-known PhaseLift algorithm. In particular, it interprets quadratic magnitude measurements of x as linear measurements of a restricted set of lifted variables, xixj, for |j − i| < δ d. This leads to a linear system involving a total of (2δ − 1)d unknown lifted variables, all of which can then be solved for using only O(δd) measurements. Once these lifted variables, xixj for |j − i| < δ d, have been recovered, a fast angular synchronization method can then be used to propagate the local phase difference information they provide across the entire vector in order to estimate the (relative) phases of every entry of x. In addition, the lifted variables corresponding to xjxj = |xj| 2 automatically provide magnitude estimates for each entry, xj, of x. The proposed phase retrieval method then approximates x by carefully combining these entry-wise phase and magnitude estimates.Finally, we conclude by developing an extension of the proposed method to the sparse phase retrieval problem; specifically, we demonstrate a sublinear-time compressive phase retrieval algorithm which is guaranteed to recover a given s-sparse vector x ∈ C d with high probability in just O(s log 5 s · log d)-time using only O(s log 4 s · log d) magnitude measurements. In doing so we demonstrate the existence of compressive phase retrieval algorithms with near-optimal linear-in-sparsity runtime complexities.

Lower Lipschitz bounds for phase retrieval from locally supported measurements

“…In [3], Balan et al showed that if α and β are injective on C d / ∼, then β is bi-Lipschitz with respect to d 1 , and α is bi-Lipschitz with respect to D 2 , where in both cases R N is equipped with the Euclidean norm. Motivated by applications such as (Fourier) ptychography [18,22] and related numerical methods [13,14], we will study frames which are constructed as the shifts of a family of locally supported measurement vectors. Specifically, we assume that {m 1 , m 2 , .…”

A direct solver for the phase retrieval problem in ptychographic imaging

“…Note that this already differs from the setup given in [14] where also circulant shifts are considered, i.e., L = N − 1 = 127. We compare the reconstruction quality using the exponential window (12) analyzed in [13,14] against the Gaussian window (13) with α = 0.99, which more closely approximates experimental setups. The results are shown in Figure 2.…”

“…The results are illustrated in Figure 3. For both projectors a Gaussian window (13) with α = 0.99 is used. As seen before, the reconstructed amplitude of both projectors is similar.…”

Measurement Matrix Design for Phase Retrieval Based on Mutual Information

“…Lemma 2 characterizes˜ MI via a set of fixed point equations 4 . Note that the matrix A (˜ MI ) is constructed such that˜ MI which solves (20) induces a covariance matrix of the MMSE estimate ofŨ from Y C , denoted (˜ MI ), whose eigenvalues satisfy (19).…”

“…Recovery algorithms as well as specialized deterministic measurement matrices were considered in several works. In particular, [17], [18] studied phase recovery from short-time Fourier transform measurements, [19] proposed a recovery algorithm and measurement matrix design based on sparse graph codes for sparse SOIs taking values on a finite set, [20] suggested an algorithm using correlation based measurements for flat SOIs, i.e., strictly non-sparse SOIs, and [21] studied recovery methods and the corresponding measurement matrix design for the noisy phase retrieval setup by representing the projections as complex polynomials.…”