Stable Gabor phase retrieval in Gaussian shift-invariant spaces via biorthogonality

TL;DR

This work addresses the challenge of phaseless signal reconstruction from Gabor spectrogram magnitudes for signals in Gaussian shift-invariant spaces. It derives an explicit, stable inversion on vertical lines in the time-frequency plane using a biorthogonal (tensor-product) expansion and proves stability under a local non-vanishing condition on the signal. A provably stable finite-sample algorithm is introduced, with rigorous error and sample-size bounds that enable convergence to the true signal on compact intervals and extend to Paley-Wiener spaces. The results are significant for applications in time-frequency analysis and ptychography, providing both a theoretical reconstruction formula and a practical, robust numerical scheme.

Abstract

We study the phase reconstruction of signals $f$ belonging to complex Gaussian shift-invariant spaces $V^\infty(\varphi)$ from spectrogram measurements $|\mathcal{G} f(X)|$ where $\mathcal{G}$ is the Gabor transform and $X \subseteq \mathbb{R}^2$. An explicit reconstruction formula will demonstrate that such signals can be recovered from measurements located on parallel lines in the time-frequency plane by means of a Riesz basis expansion. Moreover, connectedness assumptions on $|f|$ result in stability estimates in the situation where one aims to reconstruct $f$ on compact intervals. Driven by a recent observation that signals in Gaussian shift-invariant spaces are determined by lattice measurements [Grohs, P., Liehr, L., Injectivity of Gabor phase retrieval from lattice measurements, Appl. Comput. Harmon. Anal. 62 (2023), pp. 173-193] we prove a sampling result on the stable approximation from finitely many spectrogram samples. The resulting algorithm provides a provably stable and convergent approximation technique. In addition, it constitutes a method of approximating signals in function spaces beyond $V^\infty(\varphi)$, such as Paley-Wiener spaces.

Figures (9)

• Figure 1: Visualization of condition (P): the points $p_1, \dots, p_9$ satisfy the properties $|f(p_j)| \geq \gamma$ for all $j=1, \dots,9$ and $p_{j+1}-p_j \leq r$ for all $j=1, \dots,8$.
• Figure 2: Reconstruction of a complex-valued function $f \in V_\beta^\infty(\varphi)$ ($\beta = 1, \varphi(t)=e^{-\pi t^2}$) on the interval $[-16,16]$ using the numerical approximation routine $\mathcal{R}$ with spectrogram samples located at $X= \tfrac{1}{2} \{ -40, \dots, 40 \} \times \tfrac{1}{12}\{ -60, \dots, 60 \}$. To each spectrogram sample, Gaussian noise with mean zero and standard deviation 0.001 is added, i.e. the measurement matrix $\mathfrak{S}$ takes the form $\mathfrak{S} = |\mathcal{G}f(X)|^2+\mathcal{N}(0,\sigma^2), \sigma=0.001$.
• Figure 3: Dual generators of Gaussians $\varphi^\sigma(t)$ associated with the shift-invariant space $V_\beta^2(\varphi)$ with step-sizes $\beta = 0.8, 0.6, 0.2$ and $\sigma=\beta$.
• Figure 4: The contour plot depicts the spectrogram of a function $f \in V_\beta^\infty(\varphi)$ and the white dots visualize the location of the samples $\mathfrak{S}$. The values $\frac{2}{\beta}$ and $\frac{1}{h}$ are the densities in time ($x$) resp. frequency ($\omega$) direction.
• Figure 5: Visualization of step 2 of Algorithm 1: plot of the absolute value $|f|$ of a function $f \in V_\beta^\infty(\varphi)$ on the interval $I=[-40,40]$. The black dots represent points $(p_j,F(p_j))$ with $p_{j+1}-p_j \leq r = 2$ and $F(p_j) \geq \gamma = 0.5$.

Theorems & Definitions (58)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1
• Definition 2.2
• Theorem 2.3
• proof
• Corollary 2.4
• proof