Stable Phase Retrieval in Infinite Dimensions

Foundations of Computational Mathematics https://doi.org/10.1007/s10208-018-9399-7 Stable Phase Retrieval in Infinite Dimensions Rima Alaifari1 · Ingrid Daubechies2 · Philipp Grohs3 · Rujie Yin2 Received: 31 January 2017 / Revised: 15 May 2018 / Accepted: 15 May 2018 © The Author(s) 2018 Abstract The problem of phase retrieval is to determine a signal f ∈ H, with H a Hilbert space, from intensity measurements |F(ω)|, where F(ω) :=  f , ϕω  are measurements of f with respect to a measurement system (ϕω )ω∈ ⊂ H. Although phase retrieval is always stable in the finite-dimensional setting whenever it is possible (i.e. injectivity implies stability for the inverse problem), the situation is drastically different if H is infinite-dimensional: in that case phase retrieval is never uniformly stable (Alaifari and Grohs in SIAM J Math Anal 49(3):1895–1911, 2017; Cahill et al. in Trans Am Math Soc Ser B 3(3):63–76, 2016); moreover, the stability deteriorates severely in the dimension of the problem (Cahill et al. 2016). On the other hand, all empirically observed instabilities are of a certain type: they occur whenever the function |F| of intensity measurements is concentrated on disjoint sets D j ⊂ , i.e. when F = k j=1 F j where each F j is concentrated on D j (and k ≥ 2). Motivated by these considerations, we propose a new paradigm for stable phase retrieval by considering the problem of reconstructing F up to a phase factor that is not global, but that can be different for each of the subsets D j , i.e. recovering F up to the equivalence F∼ k  eiα j F j . j=1 We present concrete applications (for example in audio processing) where this new notion of stability is natural and meaningful and show that in this setting stable phase retrieval can actually be achieved, for instance, if the measurement system is a Gabor frame or a frame of Cauchy wavelets. Keywords Phase retrieval · Fourier optics · Stability · Entire functions Communicated by Thomas Strohmer. B Philipp Grohs Extended author information available on the last page of the article 123 Foundations of Computational Mathematics Mathematics Subject Classification 30Axx · 78Axx · 46Bxx · 42Axx 1 Introduction 1.1 Problem Formulation Suppose we are given a complex-valued function F :  → C on some (discrete or continuous) domain , and we can observe only its absolute values |F|. The problem of phase retrieval is to reconstruct F from these measurements, up to a global phase (meaning that the functions F and eiα F, α ∈ R, are not distinguished). Such problems are encountered in a wide variety of applications, ranging from Xray crystallography and microscopy to audio processing and deep learning algorithms [15,26,36,39]; accordingly, a large body of literature treating the mathematical and algorithmic solution of phase retrieval problems exists, with new approaches emerging in recent years [6,9,11,27,40]. In these applications, the domain of definition  is a finite set, for example  = {1, . . . , N }, and the function F arises from a finite number of linear measurements F(k) = x, ak  := d  xl (ak )l , ak ∈ Cd , k = 1, . . . , N l=1 ∈ Cd which one seeks to recover. Such problems arise as finite of some signal x approximations to various real-world problems; in diffraction imaging, for instance, the set-up can be interpreted as measuring the diffraction pattern of x, modulated with a number of different filters. Classically, the numerical solution of phase retrieval problems is treated via alternating projection algorithms that are simple to implement but lack a theoretical understanding [17,19]. More recent work [11] has introduced an algorithm named PhaseLift, based on a reformulation of the N-dimensional phase retrieval problem as a semidefinite optimization problem in an N 2 −dimensional space. As shown in [11], PhaseLift succeeds with high probability in recovering the signal x, up to a global phase, in a randomized setting (meaning that the vectors a1 , . . . , a N are drawn at random); moreover, PhaseLift is stable if the measurements |x, an | are corrupted by additive noise. More recently, it has been shown that gradient descent algorithms, together with a careful guess for their starting value, achieve the same theoretical guarantees while being vastly more efficient [12]. 1.2 Infinite-Dimensional Phase Retrieval The vector x ∈ Cd typically arises as a digital representation of a physical quantity. For instance, x could represent a finite-dimensional approximation of a continuous function describing an infinite-dimensional object. This naturally leads one to consider the more general infinite-dimensional phase retrieval problem, where one seeks to recover a signal f ∈ H, with H a (possibly infinite-dimensional) Hilbert space, from the phaseless measurements |F(ω)|, with 123 Foundations of Computational Mathematics F(ω) :=  f , ϕω , ω ∈ , (1.1) and where (ϕω )ω∈ ⊂ H is a (possibly infinite) parameterized family of measurement functions, typically normalized so that ϕω = 1 for all ω ∈ . We mention a few examples. • Consider the classical n-dimensional phase retrieval problem of reconstructing a function f from intensity measurements of its Fourier transform  f . For a compact f (ω), ω ∈ , subset D ⊂ Rn , let H = L 2 (D) and consider f ∈ H. Let F(ω) =  where  is either all of Rn or a suitable discrete subset of Rn (since f has compact support, there exist ϕω ∈ H such that F(ω) =  f , ϕω ). Applications of this setup include coherent diffraction imaging, X-ray crystallography and many more, in which one typically can measure only intensities, corresponding to | f , ϕω |2 . The classical phase retrieval problem is in general not uniquely solvable [1]; recent work [34] has established the uniqueness of the solution, if the intensities of the Fourier transforms of certain structured modulations of f are measured instead. • Related to the previous example, the work [38] studies the reconstruction of a bandlimited real-valued function f from unsigned samples (| f (ω)|)ω∈ with  a suitable (discrete) sampling set; more general settings are considered in [2,14]. Note that the real-valued case (where only the sign ±1 is missing from each measurement) is qualitatively simpler than the complex-valued case where each measurement lacks a phase factor eiα , α ∈ R. • In order to overcome the problem of nonuniqueness of the classical phase retrieval problem and to be able to apply techniques in diffraction imaging also to extended objects, one often records local illuminations of different overlapping parts of the object, which mathematically amounts to a windowed (or short-time) Fourier transform (STFT) F = Vg f , where for f ∈ L 2 (R)  Vg f (x, y) := R f (t)g(t − x)e−2π it y dt (1.2) is defined by the window g ∈ L 2 (R) and the parameters (x, y) may vary over a discrete or continuous subset of R2 . See [36] for an excellent survey on phase retrieval from STFT measurements. • An (...truncated)