Phase retrieval from the magnitudes of affine linear measurements

Abstract

Abstract. In this paper, we consider the phase retrieval problem in which one aims to recover a signal from the magnitudes of affine measurements. Let {aj } m j=1 ⊂ H d and b = (b1, . . . , bm)⊤ ∈ H m , where H = R or C. We say {aj } m j=1 and b are affine phase retrievable for H d if any x ∈ H d can be recovered from the magnitudes of the affine measurements {| aj , x + bj|, 1 ≤ j ≤ m}. We develop general framework for affine phase retrieval and prove necessary and sufficient conditions for {aj } m j=1 and b to be affine phase retrievable. We establish results on minimal measurements and generic measurements for affine phase retrieval as well as on sparse affine phase retrieval. In particular, we also highlight some notable differences between affine phase retrieval and the standard phase retrieval in which one aims to recover a signal x from the magnitudes of its linear measurements. In standard phase retrieval, one can only recover x up to a unimodular constant, while affine phase retrieval removes this ambiguity. We prove that unlike standard phase retrieval, the affine phase retrievable measurements {aj } m j=1 and b do not form an open set in H m×d ×H m . Also in the complex setting, the standard phase retrieval requires 4d − O(log 2 d) measurements, while the affine phase retrieval only needs m = 3d measurements.

On stability of generalized phase retrieval and generalized affine phase retrieval

“…The first one can be viewed as a generalization of stability of phase retrieval in [3,4], or as a continuation of the work in [17]. The second one is an extension of the work in [9,10]. As all the results in this paper are obtained in real Hilbert space, the stability property in complex Hilbert space still needs to be addressed.…”

“…The standard affine phase retrieval introduced by Bing Gao et al in [9] can be used for recovering signals with prior knowledge. In this section, we consider generalized affine phase retrieval theoretically and give some basic mathematical properties at first, then we focus on its stability property.…”

“…Since d 2 2 (x,ỹ) = x +ỹ 2 x -ỹ 2 = ( x + y 2 + 4) xy 2 , there exist constants c 0 , c 1 such that (3.2) holds. Similarly, by Theorem 2.4, we haveM B,b (x) -M b,b (y) ỹ) = min{ x +ỹ 2 , x -ỹ 2 } = min{ x + y 2 + 4, xy 2 }, there exist constants C 0 , C 1 such that (3.3) holds.In contrast to Theorem 4.1 in[9], Theorem 3.4 leads to a slack constraint of the signal from a compact set to R d . Although affine phase retrieval is not bi-Lipschitz with respect to one metric, the mappings M B,b and √ M B,b is bounded by two metrics.…”

Phaseless Sampling and Reconstruction of Real-Valued Signals in Shift-Invariant Spaces

“…To study phaseless sampling and reconstruction of signals in V (Φ), we recall the concept of a phase retrievable frame [4,18,21,43].…”

On stability of generalized (affine) phase retrieval in the complex case

“…Generalized phase retrieval (GPR) is introduced by Yang Wang and Zhiqiang Xu [16], which unifies and enhances results from the standard phase retrieval, phase retrieval by projections, and low-rank matrix. Affine phase retrieval aims to recover signals from the magnitudes of affine measurements [8]. Necessary and sufficient conditions as well as minimal number of measurements are given in [10].…”