MichaelSFluhr1989

The STFT phase retrieval problem concerns the reconstruction of a function $f$ from the absolute value of its short-time Fourier transform (STFT) with respect to a window function $g$, i.e. the map $(x,\omega )\mapsto |V_{g}f(x,\omega )|=\left|\int _{\mathbb {R} }f(t){\overline {g(t-x)}}e^{-2\pi i\omega t}\,dt\right|$. The functions $f$ and $g$ are normally assumed to have finite energy which can phrased as $f,g\in L^{2}(\mathbb {R} )$ and $L^{2}(\mathbb {R} )$ denotes the Lebesgue space of square-integrable functions on the real line. It is the mathematical formulation of a variety of problems arising in imaging application such as ptychography. Abstractly, it can be regarded as a non-linear inverse problem and a special case of a phase retrieval problem (historically in this problem the STFT is replaced by the ordinary Fourier transform).

Problem formulation

The absolute value of the STFT is called the spectrogram. The spectrogram is defined for $(x,\omega )\in \mathbb {R} ^{2}$ and the Euclidean 2-space $\mathbb {R} ^{2}$ is known as the time-frequency plane in this context.

Problem. Let $A\subseteq \mathbb {R} ^{2}$ and let $g$ be a window function. Recover $f$ from the values $\{|V_{g}f(x,\omega )|:(x,\omega )\in A\}$.

Uniqueness

If $v,u\in L^{2}(\mathbb {R} )$ are two functions such that that $u=e^{i\phi }v$ with a real number $\phi \in \mathbb {R}$ then $u$ and $v$ are said to be equal up to a global phase factor. The global phase factor $e^{i\phi }$ is a value on the unit circle in the complex plane and has absolute value one. It follows that if $u=e^{i\phi }v$ then the spectrogram of $u$ and the spectrogram of $v$ agree at every point in the time-frequency plane, i.e. $|V_{g}u(x,\omega )|=|V_{g}v(x,\omega )|$. Therefore, a global phase factor of the form $e^{i\phi }$ constitutes an ambiguity of the STFT phase retrieval problem and a reconstruction of $f$ from $|V_{g}f(x,\omega )|$ is only possible up to this ambiguity. A natural question arising at this point is the uniqueness problem, i.e. under which assumptions equality of two spectrograms of $v,u$ implies equality up to a global phase of $v$ and $u$. The uniqueness problem depends on the set $A$ where the spectrogram is given, the window function $g$ and prior assumptions on the functions $u$ and $v$.

Problem. Let $S\subset L^{2}(\mathbb {R} )$ be a class of functions, $g$ a window function and $A$ a subset of the time-frequency plane. Is it true that $u,v\in S$ agree up to a global phase, provided that $|V_{g}u(x,\omega )|=|V_{g}v(x,\omega )|$ for every $(x,\omega )\in \mathbb {R} ^{2}$?

Several uniqueness results are provided in the following list:

• Let $g$ be a window function and $Ag$ the ambiguity function of $g$, $Ag(x,\omega) = ....$. If the zero set of $Ag$ has a dense complement in $\R^2$ then every $f \in L^2(\mathbb R)$ is determined up to a global phase from $|V_gf(A)|$ with $A=\mathbb{R}^2$.
• If $g(t)=e^{-\pi t^2}$ is a Gaussian then every $f \in L^2(\R)$ is determined up to a global phase from $|V_gf(A)|$ whenever $A$ contains an open set
• If $g(t)=e^{-\pi t^2}$ is a Gaussian then every $f \in L^2(\R)$ is determined up to a global phase from $|V_gf(A)|$ whenever $A$ contains two lines passing through the origin so that the angle between the two lines is not a multiple of $\pi \mathbb Q$.

Discrete sampling sets

In order to discretize the STFT phase retrieval problem, one is interested in the question whether a discrete sampling set $A$ gives uniqueness of the STFT phase retrieval problem. The following negative result shows that if $A$ is a lattice then uniqueness is never achieved if the class of functions $S$ is the entire space $L^2(\mathbb R)$.

Theorem (Grohs & Liehr, 2021). If $A$ is a lattice and $g$ is an arbitrary window function then one can always find two functions $u$ and $v$ which do not agree up to a global phase but $|V_gu(A)| = |V_gv(A)|$. In particular, the uniqueness problem is never satisfied in this setting.

However, $A$ can be chosen to be a lattice if the signal class $S$ is restricted to a proper subspace of $L^2(\R)$.

• If $S=L^4[-c/2,c/2], A=\frac{1}{2c}\Z \times \Z$ and $g$ is a Gaussian then every $f \in S$ is determined up to a global phase by $|V_gf(A)|$ (Grohs & Liehr, 2020).
• If $S$ consists of all real-valued band-limited functions with bandwidth $c$ then every $f \in S$ is determined up to a global phase by $|V_gf(A)|$ if $A=\frac{1}{2c}\Z \times \{ 0 \}$ and $g$ is a Gaussian (Alaifari & Wellershoff, 2020)