Stable Phase Retrieval in Infinite Dimensions
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Abstract
Keywords
Phase retrieval Fourier optics Stability Entire functionsMathematics Subject Classification
30Axx 78Axx 46Bxx 42Axx1 Introduction
1.1 Problem Formulation
Suppose we are given a complexvalued function \(F:\Omega \rightarrow \mathbb {C}\) on some (discrete or continuous) domain \(\Omega \), 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 \(e^{\mathrm {i}\alpha }F\), \(\alpha \in \mathbb {R}\), 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].
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 Ndimensional 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 \(a_1,\dots , a_N\) are drawn at random); moreover, PhaseLift is stable if the measurements \(\langle x, a_n\rangle \) 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 InfiniteDimensional Phase Retrieval

Consider the classical ndimensional phase retrieval problem of reconstructing a function f from intensity measurements of its Fourier transform \(\widehat{f}\). For a compact subset \(D\subset \mathbb {R}^n\), let \(\mathcal {H}=L^2(D)\) and consider \(f \in \mathcal {H}\). Let \(F(\omega )= \widehat{f}(\omega )\), \(\omega \in \Omega \), where \(\Omega \) is either all of \(\mathbb {R}^n\) or a suitable discrete subset of \(\mathbb {R}^n\) (since f has compact support, there exist \(\varphi _\omega \in \mathcal {H}\) such that \(F(\omega )=\langle f,\varphi _\omega \rangle \)). Applications of this setup include coherent diffraction imaging, Xray crystallography and many more, in which one typically can measure only intensities, corresponding to \(\langle f,\varphi _\omega \rangle ^2\). 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 realvalued function f from unsigned samples \((f(\omega ))_{\omega \in \Omega }\) with \(\Omega \) a suitable (discrete) sampling set; more general settings are considered in [2, 14]. Note that the realvalued case (where only the sign \(\pm 1\) is missing from each measurement) is qualitatively simpler than the complexvalued case where each measurement lacks a phase factor \(e^{\mathrm {i}\alpha }\), \(\alpha \in \mathbb {R}\).
 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 shorttime) Fourier transform (STFT) \( F = V_g f, \) where for \(f\in L^2(\mathbb {R})\)is defined by the window \(g\in L^2(\mathbb {R})\) and the parameters (x, y) may vary over a discrete or continuous subset of \(\mathbb {R}^2\). See [36] for an excellent survey on phase retrieval from STFT measurements.$$\begin{aligned} V_g f(x,y) :=\int _{\mathbb {R}}f(t)\overline{g(tx)}e^{2\pi \mathrm {i}t y}dt \end{aligned}$$(1.2)

Another instance of phase retrieval from STFT measurements arises in audio processing applications involving phase vocoders. A phase vocoder [18] is a tool that allows to modify an audio signal f by transforming its STFT. Given f, a phase vocoder first calculates \(V_g f(x,y)\) and then modifies it to some H(x, y) before it transforms back to the time domain by taking the inverse (discrete) STFT of H. Typical modifications include time scaling and pitch shifting. In general, the modified H may not result in an STFT of any signal. This leads to the socalled phase coherence problem [30] in which one aims to make modifications such that the modified H is an approximate STFT. One possible approach is to modify the amplitude F(x, y) only in a first step to obtain H(x, y) and then to recover the phase of H(x, y) in a coherent way.

More recent work [39] seeks to reconstruct a signal \(f\in L^2(\mathbb {R})\) from the magnitudes \(F(x,2^j)\) of semidiscrete wavelet measurements, where \( F(x,2^j) = w_\psi f(x, 2^j), \) with \(j\!\in \!\mathbb {N}\), \(x\!\in \!\mathbb {R}\) and \(w_\psi f(x,y)\!:=\! \int _{\mathbb {R}}f(t){y^{1/2}}\overline{\psi (y(tx))}dt\);^{1} the collection of these magnitudes is sometimes called the scalogram. The corresponding phase retrieval problem arises in, e.g. the reconstruction of f from the output of its socalled scattering transform as defined in [31].
1.3 (In)stability of (In)finiteDimensional Phase Retrieval
For phase retrieval problems in spaces of finite (and fixed) dimensions, stability and uniqueness typically go hand in hand [8, 11]. The situation changes drastically when we consider infinitedimensional spaces. A central finding of [4, 10] is that all infinitedimensional phase retrieval problems are unstable and that the stability of finitedimensional phase retrieval problems deteriorates severely as the dimension grows.
Example 1.1
Figure 1 shows the plot of the functions \(f_n\) and \(g_n\) for \(n=5\), illustrating that the two functions have almost identical absolute value despite being significantly different from each other. (Note that the two functions in this example are large on two distant domains and small in between. For the realvalued setting it was shown in [4] that this is the generic form of instabilities; in [23] it is shown that multicomponent signals are likewise the generic example for instability in the Gabor transform case.) Consequently, stable phase retrieval is not possible for infinitedimensional problems, or even for their finegrained (and thus finite but highdimensional) approximations.
1.4 Three Observations and a New Paradigm
 1.One way to construct phase retrieval problems leading to instabilities is to consider functions \(F= \sum _{j=1}^k F_j\) with \(F_j\) concentrated on disjoint sets \(D_j\) that are far apart from each other. In the sequel, we will occasionally refer to functions of this form as multicomponent functions. Clearly, any function of the formfor any \(\alpha _1,\dots , \alpha _k\in \mathbb {R}\), will result in an instability: the absolute values of F, G will be very close, due to the fact that the \(F_j\)’s are concentrated on wellseparated disjoint sets, but \(Fe^{\mathrm {i}\gamma } G\) need not be small at all, even for the optimal choice of \(\gamma \).$$\begin{aligned} G:=\sum _{j=1}^k e^{\mathrm {i}\alpha _j}F_j \end{aligned}$$(1.5)
The functions constructed in Example 1.1 are of this form with \(k=2\). In fact, in the general realvalued case it can be shown that all instabilities arise in this way [4]. In the complex case, it is not known whether this is the case as well.
 2.
One can investigate how existing concrete phase retrieval algorithms deal with finitedimensional approximations to the multicomponent F introduced above, under item 1. Figure 2 gives a typical albeit simplistic example. Consider an analytic^{2} signal f, e.g. as in Fig. 2a, whose Gabor transform \(F = V_\varphi f\) (as in Definition 1.2 with \(\varphi = e^{\pi t^2}\)) has two disconnected components \(F_1,\, F_2\), s.t. \(F= F_1 + F_2\), see Fig. 2b. Given the Gabor transform measurements \(F = V_{\varphi } f\), a reconstruction \(f^{rec}\) is obtained using the phase retrieval algorithm in [39], and the corresponding code from http://www.di.ens.fr/~waldspurger/wavelets_phase_retrieval.html.^{3} The relative error \(\Vert ff^{rec}\Vert /\Vert f\Vert \) in time domain is 8.61\(\times 10^{1}\), whereas the relative error \(\Vert FF^{rec}\Vert / \Vert F\Vert \) in the Gabor transform measurements is 1.27\(\times 10^{5}\). The large difference in the time domain (the ratio of the relative errors exceeds \(5\times 10^{4}\); see also Fig. 2c) is due to a nonuniform but piecewise constant phase shift in the time–frequency domain. Let \(F_1^{rec},\, F_2^{rec}\) be the two components of \(F^{rec}\) corresponding to \(F_1,\,F_2\). As shown in Fig. 2d, \(F_1\) and \(F_1^{rec}\) differ by only a phase factor \(e^{\mathrm {i}\alpha _1}\); similarly, \(F_2\) and \(F_2^{rec}\) differ by \(e^{\mathrm {i}\alpha _2}\); however, \(\alpha _1\ne \alpha _2\). So although it is hopeless to expect that any numerical algorithm could stably distinguish such a multicomponent function from \(\sum _{j=1}^k e^{\mathrm {i}\alpha _j}F_j\), algorithmic reconstruction up to the equivalence \(\sum _{j=1}^k F_j\sim \sum _{j=1}^k e^{\mathrm {i}\alpha _j}F_j\) seems to work quite well.
 3.
Being able to reconstruct (if this is indeed feasible) multicomponent functions of the type \(\sum _{j=1}^k F_j\) up to the equivalence \(\sum _{j=1}^k F_j \sim \sum _{j=1}^k e^{\mathrm {i}\alpha _j} F_j\) is of interest only if this equivalence is itself meaningful.
Our third observation is that this is indeed the case for some applications. We list two examples here.
Our first example is concerned with coherent diffraction imaging. Measurements of Xray diffraction intensities by complicated objects allow reconstruction of the object under certain constraints on the object; see, e.g. [29] for a mathematical uniqueness result, or [32] for an algorithm effective for finegrained reconstruction on physical data sets that are supported in a finite volume, without the exact location of this support being known. In its most strippeddown form, the problem consists in reconstruction of a function f supported on a compact domain \(\Omega \) from measurements of the magnitude of its Fourier transform, \(\widehat{f}(\xi )\). For the plainvanilla scattering implementation, the physical object to be reconstructed is illuminated by a plane wave. If the object is more extended, illumination by more narrowly concentrated beams might be easier to achieve; one then acquires scattering intensity data for each of several different beam illuminations, which corresponds to replacing the Fourier transform by an STFT. The methodology which we just described is widely used, for example, in Fourier ptychography [25, 35, 42].
If the scene to be reconstructed consisted of several disjoint objects, separated by “empty” space (the example in Fig. 1 in [32] illustrates such an example), then reconstruction of the individual objects might be numerically and mathematically much easier if it were allowed to reconstruct each object up to a uniform phase (for complex f) or up to a uniform sign (for real f). The simulation illustrated in Fig. 2, for a onedimensional Gabor transform, suggests as much.
Our second example is concerned with audio processing. It is well known that human audio perception is insensitive to a “global phase change”. One way to show this is to start with a (realvalued) audio signal f(t), with Fourier transform \(\widehat{f}(\xi )\), and carry out the following operations: first, take its analytic representation \(f_a\) by disregarding its negative frequency components: \(\widehat{f_a}(\xi ):=\widehat{f}(\xi )\chi _{\xi >0}\); next multiply it by an arbitrary (but fixed) phase \(e^{\mathrm {i}\alpha }\), \(\widehat{f_a^\alpha } (\xi ):= e^{\mathrm {i}\alpha } \widehat{f_a}(\xi )\). Finally, we turn it back into the Fourier transform of a realvalued function \(f^{\alpha }\) by “symmetrizing”, i.e. by setting \(\,\widehat{f^{\alpha }}(\xi )\,= \,e^{\mathrm {i}\alpha }\widehat{f}(\xi )\chi _{\xi >0}\,+\, e^{\mathrm {i}\alpha }\overline{\widehat{f}(\xi )}\chi _{\xi <0}\,\) (note that \(\overline{\widehat{f}(\xi )} = \widehat{f}(\xi )\) because f is realvalued). Equivalently, \(f^{\alpha }\) can be expressed in terms of the original signal f as \(f^{\alpha }(t)=\,\cos \alpha \cdot f(t) \,+\, \sin \alpha \cdot (Hf)(t)\), where Hf is the Hilbert transform of f. Then, even though the plot of f is typically very different from that of \(f^\alpha \) (if \(\alpha \) differs significantly from a multiple of \(2\pi \)), the two sound the same to the human ear, making them equivalent for most practical applications. Consider now an audio signal f consisting of two “bursts” of sound, separated by a short stretch of silence, i.e. \(f(t)\,=\,f_1(t)\,+\,f_2(t)\,\), with \({{\mathrm{supp \,}}}f_1=[t_1,T_1]\) and \({{\mathrm{supp \,}}}f_2=[t_2,T_2]\) where \(t_2T_1 > \tau \) for some preassigned positive \(\tau \) (typically of the order of a few tenths of seconds). Figure 3a plots such an example, for the utterance “cup, luck”, retrieved from the database at http://www.antimoon.com/how/pronuncsoundsipa.htm, with “cup” corresponding to \(f_1\), “luck” to \(f_2\). Because both \(f_1\) and \(f_2\) are highly oscillatory (as is customary for audio signals), \(Hf_1\) and \(Hf_2\) both have fast decay and are negligibly small outside \({{\mathrm{supp \,}}}f_1=[t_1,T_1]\) and \({{\mathrm{supp \,}}}f_2=[t_2,T_2]\), respectively. For such signals f, one can pick two different phases \(\alpha _1\) and \(\alpha _2\) and construct \(f^{\alpha _1,\alpha _2}=f_1^{\alpha _1}+f_2^{\alpha _2}\); the resulting audio signals again sound exactly the same as the original f. On https://services.math.duke.edu/~rachel/research/PhaseRetrieval/acoustic_result/acoustic_result.html, one can download and/or listen to f and \(f^{\alpha _1,\alpha _2}\).
We further note that signals remain undistinguishable to the human ear under a more general class of transformations: even for signals \(f=\sum _{j=1}^J f_j\) with \(J>2\) components, in which the \(f_j\) correspond to components \(F_j\) that are separated in the time–frequency domain (but not necessarily in time, or in frequency) replacing each \(F_j\) by \(e^{\mathrm {i}\alpha _j} F_j\) results in a signal that sounds exactly like the original signal f (see Fig. 4 for an example of such a signal and its Gabor transform; on https://services.math.duke.edu/~rachel/research/PhaseRetrieval/acoustic_result/acoustic_result.html one can listen to this example and componentwise phaseshifted versions).
If one seeks to reconstruct f only within the equivalence class of audio signals that are indistinguishable from f by human perception, then it is thus natural to treat all the functions of type (1.5) as equivalent, for all choices of \(\alpha _j\).
The question of whether bounds of the form (1.6) can actually be established for examples of practical interest will be the main subject of this article.
1.5 Stability for Atoll Functions
To study this question mathematically, we first need to make it more precise. Figure 4 suggests that a realistic model for Gabor transform measurements on acoustic signals are functions \(\sum _{j=1}^kF_j\) where each \(F_j\) is “large” on a domain \(D_j\), which we shall interpret as a strictly positive lower bound on \(F_j\). In practice, we expect that \(F_j\) may still have zeroes within \(D_j\), which means that there could be “holes” in \(D_j\) (reasonably small neighbourhoods of these zeroes) on which \(F_j\) could not be bounded below away from zero. This motivates the following definition:
Definition 1.2
(Atoll domains) Let \(D\subset \mathbb {C}\) be a domain. A domain \(D_0\subset D\) is called a hole of D if \(D_0\) is simply connected and \(\overline{D_0}\subset D\). By definition, D is called a domain with disjoint holes \((D_0^i)_{i=1}^l\) if \(D_0^i\) is a hole of D for all \(i=1,\dots , l\) and the sets \(\overline{D_0^i}, i=1, \dots , l\) are pairwise disjoint. For a set D with disjoint holes \((D_0^i)_{i=1}^l\), we call \(D_+:=D\setminus (\bigcup _{i=1}^l \overline{D_0^i})\) an atoll domain. The holes \((\overline{D_0^i})_{i=1}^l\) are called lagoons of the atoll domain.
Definition 1.3
The functions we want to consider for phase retrieval (and for which we will show that phase retrieval is uniformly stable) will correspond to a linear combination of atoll functions, each supported on different atolls. Furthermore, as proposed in Sect. 1.4, the reconstruction will be allowed to assign different phases to components supported on different atolls.
The present paper establishes such results; as an appetizer, we mention the following stability result which applies to the reconstruction of a function \(f\in L^2(\mathbb {R})\) from measuring absolute values of its Gabor transform \(V_\varphi f\) as defined in (1.2), with window function \(\varphi (t):=e^{\pi t^2}\).
Definition 1.4
Theorem 1.5
Suppose that \(f=\sum _{j=1}^k f_j\in L^2(\mathbb {R})\) such that ( 1.8 ) holds true with each \(f_j\) \(\varepsilon _j\)concentrated in \(D_j\).
The theorem states that a function that is the sum of components, each of which has a Gabor transform of type (1.8), can be stably reconstructed from the absolute values of its Gabor transform, whenever its Gabor transform is concentrated on a number of atolls with lagoons that are not too large.
Note that as the lagoons get large, more precisely, if we let \(r_j\) grow while keeping the ratios \(r_j/s_j\) fixed, the stability of reconstruction degenerates at most exponentially in their area. This is completely in line with the results of [10], and in particular with the example mentioned in Sect. 1.3 for which the stability of the reconstruction degenerates at least exponentially in the size of its corresponding lagoon. Therefore, we believe that such a decay is not a proof artefact but a fundamental barrier to stable phase retrieval, related to the TFlocalization properties of the window \(\varphi \), see also Remark 3.10 in Sect. 3.4.
One can construct an example of phase retrieval from Gabor measurements in the spirit of Example 1.1 of realvalued measurements in 1D: in [3], some of the authors construct two functions \(f_a^+\), \(f_a^\), for which the (Gabor transform) measurements are close to each other in absolute value but such that \(\Vert f_a^+e^{\mathrm {i}\alpha } f_a^\Vert _{L^2(\mathbb {R})}\) is not small for any phase factor \(e^{\mathrm {i}\alpha }\), \(\alpha \in \mathbb {R}\). The functions \(f_a^+\), \(f_a^\) are constructed such that their Gabor transforms are concentrated on two separated discs \(B_{r_0}((a,0))\) and \(B_{r_0}((a,0))\), so that they can be viewed as atoll functions. Applying Theorem 1.5 to this example gives stability of the phase retrieval problem with a stability constant that is independent of a. In contrast, in the classical sense (i.e. when \(V_\varphi f_a^+\), \(V_\varphi f_a^\) are not treated as atoll functions), phase retrieval is unstable in this example with the stability constant deteriorating exponentially in \(a^2\). We note, however, that the stability constant from Theorem 1.5 is not independent of the size of atolls, i.e. of the radius \(r_0\). In fact, it grows exponentially in \(r_0^2\). Recent work [23] by one of the authors has developed improved results that overcome this growth of the stability constant in the size of the atolls by replacing the exponential dependence on \(r_0^2\) by a loworder polynomial dependence. While the aforementioned work provides a rather complete picture of the local continuity properties of the inverse map \(f\mapsto V_\varphi f\), it is not immediate what can be shown in the noisy case where f is to be reconstructed from noisy measurements \(V_\varphi f+\mathrm {noise}\), unless rather stringent assumption on the noise hold true, see [23, Corollary 2.10] and also [7] for some general results in the finitedimensional case.
Theorem 1.5 is a special case of our much more general Theorem 3.1, proved in Sect. 3.3. Theorem 3.1, however, applies to a much wider class of measurement scenarios. Another application, discussed in Sect. 3.4, concerns the phase retrieval problem from measuring absolute values of the Cauchy wavelet transform of a signal.
1.6 Proof Strategy

At the backbone of Theorem 1.5 lies the wellknown fact that the Gabor transform \(F(x,y):= V_\varphi f(x,y)\) is a holomorphic function, up to normalization. More precisely, there exists a function \(\eta \) such that the product \(\eta \cdot F\) is holomorphic, see Theorem 3.6. In fact, in Theorem 3.1, we establish a general stability result for atoll functions which are, up to normalization, holomorphic.

A key insight leading to this result is the observation that, for a holomorphic function F, the rate of change of F is dominated by the rate of change of F. This fact, which is Lemma 4.1, follows directly from the Cauchy–Riemann equations.

Lemma 4.1 then allows us to prove a stability result for atoll functions, restricted to the atoll \(D_+\) on which a lower bound on their absolute value holds true.

In order to also establish a stability bound on the lagoons \((D_0^i)_{i=1}^l\), we use a version of the maximum principle and a trace theorem for Sobolev functions to prove that the reconstruction error on the lagoons \((D_0^i)_{i=1}^l\) can be dominated by the approximation error on the atoll \(D_+\) which has been controlled in the previous step. These two steps are carried out in Sect. 4. The proof turns out to be involved and dependent on a number of preparatory results which are summarized in Sect. 2.
1.7 Outline
The article is structured as follows. Section 2 provides a package of all the preparatory tools that will be needed later. In particular, we describe analytic Poincaré inequalities and the relation of the analytic Poincaré constant to the classical Poincaré constant in Sect. 2.1. Sections 2.2 and 2.3 outline the results that are needed to control the reconstruction error on the lagoons \((D_0^i)_{i=1}^l\). Stable point evaluations and the simultaneous control of two different constants that will appear in the main result of this paper are treated in Sect. 2.4.
Section 3 features our main result (Theorem 3.1) and gives its illustration for two concrete examples: the case of the domain \(D=D_+\) being a disc (Sect. 3.1) and the case of \(D_+\) being an annulus (Sect. 3.2). In the remainder of this section, the cases of magnitude measurements of the Gabor transform (Sect. 3.3) and of the Cauchy wavelet transform (Sect. 3.4) are studied and the stability constants are quantified. We give the proof of the main theorem (Theorem 3.1) in Sect. 4.
2 Preparatory Results
In the course of our work, we will use several auxiliary results that are summarized in this section. For an overview of the main results of this paper, the reader may want to visit Sect. 3 directly. We consider a pathconnected domain \(D\subset \mathbb {C}\) which is sufficiently nice (e.g. Lipschitz domain) and let \(\mathcal {O}(D)\) denote the space of holomorphic functions from D to \(\mathbb {C}\).
We will always write \(z=x+\mathrm {i}y\in \mathbb {C}\) and \(F(z)=u(x,y)+\mathrm {i}v(x,y)\). We denote \(F'(z)=u_x(x,y)+\mathrm {i}v_x(x,y)\) and \(\nabla F(z) = (\nabla u(x,y), \nabla v(x,y))\in \mathbb {R}^{2\times 2}\).
2.1 Analytic Poincaré Inequalities
Lemma 2.1
Proof
Essentially, Lemma 2.1 states that whenever \(z_0\) lies in a central location of D (i.e. not too close to \(\partial D\)), the constant \(C^a_\mathrm{poinc}(p,D,z_0)\) can be controlled by the classical Poincaré constant \(C_\mathrm{poinc}(p,D)\) which is well studied. For instance, the following result is known [33].
Theorem 2.2
For nonconvex domains, the determination of the optimal Poincaré constant is more difficult. For the annulus \(B_{r,s}(z)\), the following result is known.
Theorem 2.3
Proof
For more general domains which arise as a diffeomorphic image of a convex domain or an annulus, one can obtain estimates on the Poincaré constant by studying the Jacobian of the diffeomorphism, but in the present paper we are content with knowing the Poincaré constant on convex domains and on annuli.
2.2 Sobolev Trace Inequalities
The Sobolev trace inequality [16] provides an upper bound for this norm, which will be important for our purposes:
Theorem 2.4
The next result provides concrete estimates of the trace constant for discs and annuli. It says that the trace constant behaves nicely for annuli that are not too thin.
Theorem 2.5
Proof
2.3 Boundary Values of Holomorphic Functions
Another key fact we shall use is that the \(L^p\)norm of a holomorphic function on a simply connected domain is dominated by its \(L^p\)norm on the boundary.
Theorem 2.6
Proof
For discs \(B_r(z)\), a simple scaling argument leads to the following result.
Theorem 2.7
For all \(r>0\), \(z\in \mathbb {C}\) and \(D=B_r(z)\), we have \(C_\mathrm{bound}(p,D)\le r^{1/p}\).
For more general simply connected domains, the constant \(C_\mathrm{bound}(p,D)\) depends on upper and lower bounds on the Jacobian of the Riemann mapping from D to \(B_1(0)\).
2.4 Stable Point Evaluations
Given a function \(G\in L^p(D)\), the proof of our main result will require us to pick a point \(z\in D\) with a small sampling constant which is defined as follows.
Definition 2.8
To control the constant \(C(z_0,p,D_+,(D_0^i)_{i=1}^l)\) in our main result Theorem 3.1, it is necessary to control \(C_\mathrm{samp}(p,D_+,z_0,F_2F_1)\) and \(C^a_\mathrm{poinc}(p,D_+,z_0)\) simultaneously.
The purpose of this subsection is to show that this can indeed be achieved for general domains D and functions \(G\in L^p(D)\).
We start with the following lemma which shows that there exist “many” points with a given sampling constant.
Lemma 2.9
Proof
For “nice” domains, the quantity \(s_t(D)\) can be controlled easily. We mention the following result; the proof is an elementary calculus computation.
Lemma 2.10
Control of \(s_t\) lets us gain control over both the sampling constant and the analytic Poincaré constant. As an immediate consequence of Lemma 2.9, we have the following result.
Lemma 2.11
Proof
Lemma 2.1 now immediately implies the claimed bound for \(C^a_\mathrm{poinc}(p,D,z_0)\) (Fig. 7).
On the other hand, by the definition of C(t) and the fact that \(z_0\in D_{C(t)}(G)\), we get the desired bound on the sampling constant which proves the statement. \(\square \)
In order to make use of Lemma 2.11 to estimate the constants \(C_\mathrm{samp}(p,D,z_0,G)\) and \(C^a_\mathrm{poinc}(p,D,z_0)\), we need to control only the quantity \(s_t(D)\). For “nice” domains D, we expect that \(s_t(D)\) behaves like the diameter \(\mathrm {diam}(D)\) and also that \(\mathrm {diam}(D)^2\) behaves like D; hence, the quotient \(\frac{D}{\pi s_t(D)^2}\) would be uniformly bounded which implies that, for a suitable choice of \(z_0\in D\), the constant \(C^a_\mathrm{poinc}(p,D,z_0)\) is comparable to the classical Poincaré constant \(C_\mathrm{poinc}(p,D)\), while \(C_\mathrm{samp}(p,D,z_0,G)\) is bounded by a fixed constant. These considerations will give us full control of all underlying constants for sufficiently nice domains, needed in the estimates in the next section.
3 Stability of Phase Reconstruction from Holomorphic Measurements
The purpose of this section is to formulate the following fundamental result and discuss some of its implications.
Theorem 3.1
Suppose that \(F_1\) belongs to a class of atoll functions as in Definition 1.3, i.e. \(F_1\in \mathcal {H}(D,(D_0^i)_{i=1}^l,\delta ,\Delta )\). Assume further that \(F_2\in C^1(D)\) such that there exists a continuous function \(\eta : D \rightarrow \mathbb {C}\) for which both functions \(\eta \cdot F_1,\ \eta \cdot F_2\in \mathcal {O}(D)\). Suppose that \(1\le p\le \infty \).
Remark 3.2
By Lemma 2.11, the two constants \(C_\mathrm{samp}\) and \(C^a_\mathrm{poinc}(D_+)\) depending on \(z_0\) can be controlled simultaneously. To achieve the best possible \(C(z_0,p,D_+(D_0^i)_{i=1}^l)\), \(z_0\) should be picked s.t. \({{\mathrm{dist\,}}}(z_0,\partial D_+)\) is large and \(\Vert F_1(z_0)  F_2(z_0) \Vert \) is small.
Remark 3.3
In Theorem 3.1, we assume that there exists a normalization function \(\eta \), s.t. \(\eta \cdot F_1, \eta \cdot F_2\in \mathcal {O}(D)\). In Sects. 3.3 and 3.4, we show for F in the image domain of the Gabor or Cauchy wavelet transform, respectively, the existence of explicit functions \(\eta \) such that \(\eta \cdot F\) is holomorphic on the entire parameter domain and the results of [5] show that these are essentially the only functions which generate, up to normalization, holomorphic wavelet or STFT measurements. For more general measurements, such global \(\eta \) may not exist and for \(F\in \mathcal {H}(D, D_0,\delta ,\Delta )\), there might be accumulated zeros in \(D_0\). In this case, if the accumulated zero set \(D_O :=\overline{\{z; \ F_1(z)F_2(z) = 0\}^\circ }\subset D_0\) is simply connected with smooth boundary, then the bound (3.2) in Theorem 3.1 still holds with the domain of the \(L^p\)norm on the righthand side changing from \(D_+\) to D.^{4}
Before we provide the lengthy proof of Theorem 3.1 in Sect. 4, we pause and provide some special examples which might be illuminating. To give two simple examples, in Sect. 3.1 we shall see how to gain explicit estimates for the quantity \(C(z_0,p,D_+,D_0)\) for \(D=D_+\) a disc (i.e. \(D_0=\emptyset \)) and in Sect. 3.2 for \(D_+\) an annulus.
These examples should make clear that similar results also hold for more general domains.
3.1 Example I: A Disc
In this subsection, we shall treat the case \(D=D_+=B_r(z)\) and \(D_0=\emptyset \). The class \(\mathcal {H}(D_+,D_0,\delta ,\Delta )\) now consists of functions which are bounded from below by \(\delta \) and which (together with their gradient) are bounded from above by \(\Delta \) on all of \(B_r(z)\). We have the following result.
Theorem 3.4
Suppose that \(F_1\in \mathcal {H}(B_r(z),\emptyset ,\delta ,\Delta )\) for some \(r>0\) and \(z \in \mathbb {C}\). We further assume that \(F_2\in C^1(B_r(z))\) such that there exists a continuous function \(\eta : B_r(z) \rightarrow \mathbb {C}\) for which both functions \(\eta \cdot F_1,\ \eta \cdot F_2\in \mathcal {O}(B_r(z))\).
Proof
Now, it remains to employ Theorem 2.2 to get a suitable estimate on the quantity (3.3) for \(p=2\) which, together with Theorem 3.1, yields the desired result. \(\square \)
More general results can be obtained for domains D which are diffeomorphic to \(B_r(z)\) in an obvious way. The resulting bounds will depend on upper and lower bounds of the Jacobian of the mapping which maps D to \(B_r(z)\).
A similar result can also be established for general convex domains D where r in the theorem above may be replaced by \(\text{ diam }(D)\) and the constant c may depend on the geometry of D.
We omit the details.
3.2 Example II: An Annulus
To make the general result of Theorem 3.1 more accessible and to give an idea of the quantitative nature of the stability constant \(C(z_0,p,D_+,D_0)\), we treat here the case of an annulus \(D_+=B_{r,s}(z)\) and \(D_0=B_{r}(z)\) for \(s>r>0\) and some \(z \in \mathbb {C}\). It is interesting to observe the dependence of the stability constant on the size of the “lagoon” \(D_0\) on which the phaseless measurements are allowed to be arbitrarily small. We have the following result.
Theorem 3.5
Suppose that \(F_1\in \mathcal {H}(B_{s}(z),B_r(z),\delta ,\Delta )\) for \(s>r>0\). Furthermore, let \(F_2\in C^1(B_{s}(z))\) be such that there exists a continuous function \(\eta : B_s(z) \rightarrow \mathbb {C}\) for which both functions \(\eta \cdot F_1,\ \eta \cdot F_2\in \mathcal {O}(B_{s}(z))\).
Proof
Theorem 3.5 shows that stability can still be retained, even if the function \(F_1\) is allowed to be small on a large set. Again, more general results can be derived for domains which are diffeomorphic to an annulus.
3.3 Phase Retrieval from Gabor Measurements
Theorem 3.6
Now, consider the problem of stably reconstructing a function from the absolute values of its Gabor transform. By Theorem 3.6, we are in a position to apply Theorem 3.1 directly.
Theorem 3.7
Proof
The proof follows directly from Theorem 3.5 together with observing that \({{\mathrm{var}}}(\eta _{z_j},B_{r_j}(z_j))\le c\cdot e^{r_j^2 \pi /2}\) for a uniform constant \(c>0\). \(\square \)
We are now ready to conclude the proof of Theorem 1.5, as announced in Sect. 1.5.
Proof of Theorem 1.5
3.4 Phase Retrieval from Cauchy Wavelet Measurements
Theorem 3.8
Proof
Using Theorem 3.1, the statement of Theorem 3.8 immediately implies the following result related to the stability of phase retrieval from Cauchy wavelet measurements.
Theorem 3.9
Proof
We have that \({{\mathrm{var}}}(\eta ,B_{r_j}(z_j))\le c\cdot 1r_j/y_j^{s1/2}\) for a uniform constant \(c>0\), so that the statement is a direct consequence of Theorem 3.5. \(\square \)
Remark 3.10
It is interesting to observe how the stability bounds in Theorem 3.7 and 3.9 deteriorate as the size of the lagoons grows, that is, as the parameter \(r_j\) grows. In the case of Gabor measurements, this growth is of order \(e^{r_j^2 \pi /2}\), while in the case of Cauchy wavelets with s vanishing moments, the growth is of order \((\frac{1}{1r_j/y_j})^{s+1/2}\), becoming worse as the number of vanishing moments increases.
Interpreting these quantities in geometric terms, we note that the area of a lagoon in the parameter space of the Gabor transform is of order \(r_j^2 \pi \), that is, the stability decays exponentially in the area of the lagoon.
This behaviour is most likely related to the fact that Gabor systems are much more well localized in the time–frequency plane than Cauchy wavelets and that the localization properties of Cauchy wavelets increase as the number s of vanishing moments increases.
It is known that strong localization properties of the measurement system are an obstruction to stable phase retrieval [8] and in the light of this the stability behaviour of Theorems 3.7 and 3.9 is not really surprising.
4 Proof of Theorem 3.1
This section is devoted to prove Theorem 3.1 which is the main result of this paper. The proof follows several steps and relies on the following key lemma, see also [13] for related results.
Lemma 4.1
Proof
Having Lemma 4.1 at hand, we may now proceed to the proof of Theorem 3.1, which we restate here for convenience of the reader.
Theorem 3.1
Suppose that \(F_1\) belongs to a class of atoll functions as in Definition 1.3, i.e. \(F_1\in \mathcal {H}(D,(D_0^i)_{i=1}^l,\delta ,\Delta )\). Assume further that \(F_2\in C^1(D)\) such that there exists a continuous function \(\eta : D \rightarrow \mathbb {C}\) for which both functions \(\eta \cdot F_1,\ \eta \cdot F_2\in \mathcal {O}(D)\). Suppose that \(1\le p\le \infty \).
Proof of Theorem 3.1
Footnotes
 1.
Note that our \(w_\psi f(x,y)\) corresponds to \(W_\psi f(x,1/y)\) in the notation of [39].
 2.
i.e. \(\widehat{f}(\omega ) = 0, \forall \omega < 0\),
 3.
The original algorithm works on magnitude measurements of wavelet transforms such as Morlet wavelets and Cauchy wavelets. Here we apply it to dyadic Gabor wavelet, where the phenomenon of phase difference between the initial and reconstructed signal persists.
 4.
Notes
Acknowledgements
Open access funding provided by University of Vienna.
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