Phase retrieval from coded diffraction patterns

Abstract

This paper considers the question of recovering the phase of an object from intensity-only measurements, a problem which naturally appears in X-ray crystallography and related disciplines. We study a physically realistic setup where one can modulate the signal of interest and then collect the intensity of its diffraction pattern, each modulation thereby producing a sort of coded diffraction pattern. We show that PhaseLift, a recent convex programming technique, recovers the phase information exactly from a number of random modulations, which is polylogarithmic in the number of unknowns. Numerical experiments with noiseless and noisy data complement our theoretical analysis and illustrate our approach.

Sparse superresolution phase retrieval from phase-coded noisy intensity patterns

“…14,25 Methodologically, these algorithms are developed for the Poissonian likelihood criterion, i.e., for Poissonian noisy observations. Simulation experiments confirm that these algorithm works precisely provided nearly noiseless observations.…”

“…It results in observations known as coded diffraction patterns (e.g., see Refs. [13][14][15] E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 5 ; 6 3 ; 4 4 4 y s ¼ jPfM s · u o gj 2 ; s¼ 1; : : : ; L;…”

On relaxed averaged alternating reflections (RAAR) algorithm for phase retrieval with structured illumination

“…And recently the local geometrical convergence to a solution for HIO is proved [24,25]. An important difference between the optimization approaches in [20][21][22] and the standard iterative projection methods is that their coded diffraction patterns are not oversampled. We emphasize that reducing the number of coded diffraction patterns is crucial for the diffract-before-destruct approach and oversampling is a small price to pay with current sensor technology [24].…”

“…A natural way is incorporating the structured illumination, i.e., to collect the diffraction patterns of the modulated object wpnqxpnq, where the waveforms or patterns wpnq are known. The phase retrieval from structured illuminations was formulated as a matrix completion problem, whose convex relaxation is a convex trace-norm minimization problem [20,21]. However, due to the lifting from vector to matrix, the approach is prohibitive for two-dimensional problem.…”

Fast compressive phase retrieval from Fourier measurements

“…None of the previously mentioned works focus on measurement sets that could model a Fourier optics system as in Fig. 1 (see, e.g., [13,14,15,16,17,18,19]). In fact, the first paper about compressive phase retrieval [20] was addressing the very problem that we are trying to solve in the present paper, i.e., the recovery problem of a k-sparse complex signal x ∈ C N from Fourier intensity measurements | x[ω]| 2 .…”