One-dimensional Phase Retrieval Research Articles

No existence of a linear algorithm for the one-dimensional Fourier phase retrieval

Fourier phase retrieval, which aims to reconstruct a signal from its Fourier magnitude, is of fundamental importance in fields of engineering and science. In this paper, we provide a theoretical understanding of algorithms for the one-dimensional Fourier phase retrieval problem. Specifically, we demonstrate that if an algorithm exists which can reconstruct an arbitrary signal x∈CN in Poly(N)log⁡(1/ϵ) time to reach ϵ-precision from its magnitude of discrete Fourier transform and its initial value x(0), then P=NP. This partially elucidates the phenomenon that, despite the fact that almost all signals are uniquely determined by their Fourier magnitude and the absolute value of their initial value |x(0)|, no algorithm with theoretical guarantees has been proposed in the last few decades. Our proofs employ the result in computational complexity theory that the Product Partition problem is NP-complete in the strong sense.

Coordinate-transformation iteration algorithm for phase modulation of arbitrary axial light distribution.

Axial light distribution modulation is widely applied in optical tweezers, hard-brittle material cutting, multilayer laser direct writing, etc. To generate arbitrary axial light distribution, the coordinate-transformation iteration (CTI) algorithm is presented. The CTI algorithm unifies equations in low and high numerical aperture (NA) scenarios, using the same iterative algorithm to produce phase computer-generated holograms. In a low NA scenario, twin-foci, flattop, and sin2 distributions have been achieved. In high NA scenarios, multirings, multifoci, and needle-like distributions have been realized in simulation with specific polarized incident beams. Since the CTI algorithm is inherently an efficient one-dimensional phase retrieval algorithm that is not limited by NA, this method has the potential to become a well-received solution for axial light distribution modulation.

One-dimensional phase retrieval: regularization, box relaxation and uniqueness

Recovering a signal from its Fourier magnitude is referred to as phase retrieval, which occurs in different fields of engineering and applied physics. This paper gives a new characterization of the phase retrieval problem. Particularly useful is the analysis revealing that the common gradient-based regularization does not restrict the set of solutions to a smaller set. Specifically focusing on binary signals, we show that a box relaxation is equivalent to the binary constraint for Fourier-types of phase retrieval. We further prove that binary signals can be recovered uniquely up to trivial ambiguities under certain conditions. Finally, we use the characterization theorem to develop an efficient denoising algorithm.

The Geometry of Ambiguity in One-Dimensional Phase Retrieval

We consider the geometry associated to the ambiguities of the one-dimensional Fourier phase retrieval problem for vectors in $\mathbb{C}^{N+1}$. Our first result states that the space of signals has a finite covering (which we call the root covering) where any two signals in the covering space with the same Fourier intensity function differ by a trivial covering ambiguity. Next we use the root covering to study how the nontrivial ambiguities of a signal vary as the signal varies. This is done by describing the incidence variety of pairs of signals with the same Fourier intensity function modulo global phase. As an application, we give a criterion for a real subvariety of the space of signals to admit generic phase retrieval. The extension of this result to multivectors played an important role in the author's work with Bendory and Eldar on blind phaseless short-time Fourier transform recovery.

Ambiguities in one-dimensional phase retrieval from magnitudes of a linear canonical transform

Phase retrieval problems occur in a wide range of applications in physics and engineering. Usually, these problems consist in the recovery of an unknown signal from the magnitudes of its Fourier transform. In some applications, however, the given intensity arises from a different transformation such as the Fresnel or fractional Fourier transform. More generally, we here consider the phase retrieval of an unknown signal from the magnitudes of an arbitrary linear canonical transform. Using the close relation between the Fourier and the linear canonical transform, we investigate the arising ambiguities of these phase retrieval problems and transfer the well-known characterizations of the solution sets from the classical Fourier phase retrieval problem to the new setting.

Enforcing uniqueness in one-dimensional phase retrieval by additional signal information in time domain

Considering the ambiguousness of the discrete-time phase retrieval problem to recover a signal from its Fourier intensities, one can ask the question: what additional information about the unknown signal do we need to select the correct solution within the large solution set? Based on a characterization of the occurring ambiguities, we investigate different a priori conditions in order to reduce the number of ambiguities or even to receive a unique solution. Particularly, if we have access to additional magnitudes of the unknown signal in the time domain, we can show that almost all signals with finite support can be uniquely recovered. Moreover, we prove that an analogous result can be obtained by exploiting additional phase information.

Non-negativity constraints in the one-dimensional discrete-time phase retrieval problem

Phase retrieval problems occur in a width range of applications in physics and engineering such as crystallography, astronomy, and laser optics. Common to all of them is the recovery of an unknown signal from the intensity of its Fourier transform. Because of the well-known ambiguousness of these problems, the determination of the original signal is generally challenging. Although there are many approaches in the literature to incorporate the assumption of non-negativity of the solution into numerical algorithms, theoretical considerations about the solvability with this constraint occur rarely. In this paper, we consider the one-dimensional discrete-time setting and investigate whether the usually applied a priori non-negativity can overcame the ambiguousness of the phase retrieval problem or not. We show that the assumed non-negativity of the solution is usually not a sufficient a priori condition to ensure uniqueness in one-dimensional phase retrieval. More precisely, using an appropriate characterization of the occurring ambiguities, we show that neither the uniqueness nor the ambiguousness are rare exceptions.

One-Dimensional Phase Retrieval with Additional Interference Intensity Measurements

The one-dimensional phase retrieval problem consists in the recovery of a complex-valued signal from its Fourier intensity. Due to the well-known ambiguousness of this problem, the determination of the original signal within the extensive solution set is challenging and can only be done under suitable a priori assumption or additional information about the unknown signal. Depending on the application, one has sometimes access to further interference measurements between the unknown signal and a reference signal. Beginning with the reconstruction in the discrete-time setting, we show that each signal can be uniquely recovered from its Fourier intensity and two further interference measurements between the unknown signal and a modulation of the signal itself. Afterwards, we consider the continuous-time problem, where we obtain an equivalent result. Moreover, the unique recovery of a continuous-time signal can also be ensured by using interference measurements with a known or an unknown reference which is unrelated to the unknown signal.

On the existence of the solution for one-dimensional discrete phase retrieval problem

We consider the discrete form of the one-dimensional phase retrieval (1-D DPhR) problem from the point of view of input magnitude data. The direct method can provide a solution to the 1-D DPhR problem if certain conditions are satisfied by the input magnitude data, namely the corresponding trigonometric polynomial must be nonnegative. To test positivity of a trigonometric polynomial a novel DFT-based criterion is proposed. We use this DFT criterion for different sets of input magnitude data to evaluate whether the direct method applied to the 1-D DPhR problem leads to a solution in all explored cases.

Ambiguities in One-Dimensional Discrete Phase Retrieval from Fourier Magnitudes

The present paper is a survey aiming at characterizing all solutions of the discrete phase retrieval problem. Restricting ourselves to discrete signals with finite support, this problem can be stated as follows. We want to recover a complex-valued discrete signal $$\mathbf{x} :\mathbb {Z}\rightarrow \mathbb {C}$$ with support $$\{ 0, \ldots , N-1 \}$$ from the modulus of its discrete-time Fourier transform $$\widehat{x}(\omega )$$ . We will give a full classification of all trivial and nontrivial ambiguities of the discrete phase retrieval problem. In our classification, trivial ambiguities are caused either by signal shifts in space, by multiplication with a rotation factor $$\mathrm {e}^{\mathrm {i}\alpha }$$ , $$\alpha \in [-\pi , \pi )$$ , or by conjugation and reflection of the signal. Furthermore, we show that all nontrivial ambiguities of the finite discrete phase retrieval problem can be characterized by signal convolutions. In the second part of the paper, we examine the usually employed a priori conditions regarding their ability to reduce the number of ambiguities of the phase retrieval problem or even to ensure uniqueness. For the corresponding proofs we can employ our findings on the ambiguity classification. The considerations on the structure of ambiguities also show clearly the ill-posedness of the phase retrieval problem even in cases where uniqueness is theoretically shown.

Uniqueness of a one-dimensional phase retrieval problem

A uniqueness theorem is proven for the problem of the reconstruction of a compactly supported complex valued function from the modulus of its Fourier transform in the one-dimensional case.

One-dimensional hard x-ray field retrieval using a moveable structure

We present a technique that allows measuring the field of an x-ray line focus using far-field intensity measurements only. One-dimensional phase retrieval with transverse translation diversity is used to recover a hard x-ray beam focused by a compound kinoform lens. The reconstruction is found to be in good agreement with independent knife-edge scan measurements taken at separated planes. The approach avoids the need for measuring the beam profile at focus and allows narrower beams to be measured than the traditional knife-edge scan.

Iterative phase retrieval from kinematic rocking curves in CBED patterns

In the two-beam limit, the intensity distribution in the dark field disc of a convergent beam electron diffraction (CBED) pattern represents a rocking curve mapped across the Bragg condition. For kinematic scattering from crystal planes which are bent or displaced within the illuminated column, the diffracted amplitude and phase is the Fourier transform of a phase function with an exponent proportional to the displacement normal to the planes. Recovery of the phase profile is formally equivalent to the one-dimensional phase retrieval problem for an object function with constant modulus within a compact support, given only the diffracted intensity distribution. In simulations using the error reduction algorithm, the computed solution always converged to the original phase profile. As a practical test, the asymmetric rocking curves diffracted from planes inclined to the surfaces in ion-thinned Si specimens were used as the diffraction constraint, with support width equal to the crystal thickness. The calculated displacement curve was S-shaped, interpreted as a dilation of 1% induced by Ar atoms implanted in the surface layers. The factors which limit the accuracy and spatial resolution for phase recovery along the beam direction are discussed.

Generalized noniterative maximum entropy procedure for phase retrieval problems in optical spectroscopy

An improved version of the recently proposed noniterative maximum entropy procedure for phase retrieval is presented. Unlike the old version, the present procedure can be applied to any practical case of one-dimensional phase retrieval problems arising in optical spectrum analysis. The applicability of the present method is shown by real and synthetic examples taken from reflectance spectroscopy of linear optics and coherent anti-Stokes Raman spectroscopy of nonlinear optics.