Phase retrieval from very few measurements

Abstract

In many applications, signals are measured according to a linear process, but the phases of these measurements are often unreliable or not available. To reconstruct the signal, one must perform a process known as phase retrieval. This paper focuses on completely determining signals with as few intensity measurements as possible, and on efficient phase retrieval algorithms from such measurements. For the case of complex M -dimensional signals, we construct a measurement ensemble of size 4M − 4 which yields injective intensity measurements; this is conjectured to be the smallest such ensemble. For the case of real signals, we devise a theory of "almost" injective intensity measurements, and we characterize such ensembles. Later, we show that phase retrieval from M + 1 almost injective intensity measurements is NP-hard, indicating that computationally efficient phase retrieval must come at the price of measurement redundancy.

Fast compressive phase retrieval from Fourier measurements

“…and where α and β are defined as in (5). Based on these masks, we can prove the following recovery result.…”

“…However, our approach gives immediately explicit constructions of measurement systems as well as corresponding recovery algorithms. In particular, there exist explicit constructions for matrices A which have property (i) [22], and there exist several known systems of vectors which have property (ii) [4,5].…”

“…It is proven in [2] that 4N − 4 measurements are sufficient, and in [3] that 4N − o(N ) measurements are necessary for perfect recovery up to a global phase factor. Minimal deterministic constructions yielding injectivity with 4N − 4 measurement vectors are provided in [4,5]. Also, explicit deterministic measurement ensembles ensuring injectivity for almost every signal in C N are proposed in [6,7].…”

Solving (most) of a set of quadratic equalities: composite optimization for robust phase retrieval

“…In this iteration, it is straightforward to cache the matrix (I n +B T B) −1 , or a Cholesky factorization of the matrix, so that we can repeatedly compute the multiplication (20) in time n 2 + nm. Following Parikh and Boyd [35], we define the primal and dual residuals…”

“…As stated, this is a combinatorial problem that is, in the worst case, NP-hard [20]. Yet it naturally arises in a number of real-world situations, including phase retrieval [21,22,24], in which one receives measurements of the form b i = | a i , x | 2 for known measurement vectors a i ∈ C n , while x ∈ C n is unknown.…”