arXiv subjects

Lukas Liehr

Publications and source records attributed to Lukas Liehr.

25 records · Page 2Linked to original sources

arXiv2024

Arithmetic progressions and holomorphic phase retrieval

We study the determination of a holomorphic function from its absolute value. Given a parameter $θ\in \mathbb{R}$, we derive the following characterization of uniqueness in terms of rigidity of a set $Λ\subseteq \mathbb{R}$: if $\mathcal{F}$ is a vector space of entire functions containing all exponentials $e^{ξz}, \, ξ\in \mathbb{C} \setminus \{ 0 \}$, then every $F \in \mathcal{F}$ is uniquely determined up to a unimodular phase factor by $\{|F(z)| : z \in e^{iθ}(\mathbb{R} + iΛ)\}$ if and only if $Λ$ is not contained in an arithmetic progression $a\mathbb{Z}+b$. Leveraging this insight, we establish a series of consequences for Gabor phase retrieval and Pauli-type uniqueness problems. For instance, $\mathbb{Z} \times \tilde{\mathbb{Z}}$ is a uniqueness set for the Gabor phase retrieval problem in $L^2(\mathbb{R}_+)$, provided that $\tilde{\mathbb{Z}}$ is a suitable perturbation of the integers.

math.CV↗

arXiv2024

Phaseless sampling on square-root lattices

Due to its appearance in a remarkably wide field of applications, such as audio processing and coherent diffraction imaging, the short-time Fourier transform (STFT) phase retrieval problem has seen a great deal of attention in recent years. A central problem in STFT phase retrieval concerns the question for which window functions $g \in L^2(\mathbb{R}^d)$ and which sampling sets $Λ\subseteq \mathbb{R}^{2d}$ is every $f \in L^2(\mathbb{R}^d)$ uniquely determined (up to a global phase factor) by phaseless samples of the form $$ |V_gf(Λ)| = \left \{ |V_gf(λ)| : λ\in Λ\right \}, $$ where $V_gf$ denotes the short-time Fourier transform (STFT) of $f$ with respect to $g$. The investigation of this question constitutes a key step towards making the problem computationally tractable. However, it deviates from ordinary sampling tasks in a fundamental and subtle manner: recent results demonstrate that uniqueness is unachievable if $Λ$ is a lattice, i.e $Λ= A\mathbb{Z}^{2d}, A \in \mathrm{GL}(2d,\mathbb{R})$. Driven by this discretization barrier, the present article centers around the initiation of a novel sampling scheme which allows for unique recovery of any square-integrable function via phaseless STFT-sampling. Specifically, we show that square-root lattices, i.e., sets of the form $$ Λ= A \left ( \sqrt{\mathbb{Z}} \right )^{2d}, \ \sqrt{\mathbb{Z}} = \{ \pm \sqrt{n} : n \in \mathbb{N}_0 \}, $$ guarantee uniqueness of the STFT phase retrieval problem. The result holds for a large class of window functions, including Gaussians.

math.FA↗

arXiv2023

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

We study the phase reconstruction of signals $f$ belonging to complex Gaussian shift-invariant spaces $V^\infty(φ)$ 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 $V^\infty(φ)$, such as Paley-Wiener spaces.

math.FA↗

arXiv2023

Non-uniqueness theory in sampled STFT phase retrieval

The reconstruction of a function from its spectrogram (i.e., the absolute value of its short-time Fourier transform (STFT)) arises as a key problem in several important applications, including coherent diffraction imaging and audio processing. It is a classical result that for suitable windows any function can, in principle, be uniquely recovered up to a global phase factor from its spectrogram. However, for most practical applications only discrete samples - typically from a lattice - of the spectrogram are available. This raises the question of whether lattice samples of the spectrogram contain sufficient information for determining a function $f\in L^2(\mathbb{R}^d)$ up to a global phase factor. In the present paper, we answer this question in the negative by providing general non-identifiability results which lead to a non-uniqueness theory for the sampled STFT phase retrieval problem. Precisely, given any dimension $d$, any window function $g$ and any (symplectic or separable) lattice $\mathcal{L} \subseteq \mathbb{R}^d$, we construct pairs of functions $f,h\in L^2(\mathbb{R}^d)$ that do not agree up to a global phase factor, but whose spectrograms agree on $\mathcal{L}$. Our techniques are sufficiently flexible to produce counterexamples to unique recoverability under even more stringent assumptions; for example, if the window function is real-valued, the functions $f,h$ can even be chosen to satisfy $|f|=|h|$. Our results thus reveal the non-existence of a critical sampling density in the absence of phase information, a property which is in stark contrast to uniqueness results in time-frequency analysis.

math.FA↗

arXiv2022

Injectivity of Gabor phase retrieval from lattice measurements

We establish novel uniqueness results for the Gabor phase retrieval problem: if $\mathcal{G} : L^2(\mathbb{R}) \to L^2(\mathbb{R}^2)$ denotes the Gabor transform then every $f \in L^4[-\tfrac{c}{2},\tfrac{c}{2}]$ is determined up to a global phase by the values $|\mathcal{G}f(x,ω)|$ where $(x,ω)$ are points on the lattice $b^{-1}\mathbb{Z} \times (2c)^{-1}\mathbb{Z}$ and $b>0$ is an arbitrary positive constant. This for the first time shows that compactly-supported, complex-valued functions can be uniquely reconstructed from lattice samples of their spectrogram. Moreover, by making use of recent developments related to sampling in shift-invariant spaces by Gröchenig, Romero and Stöckler, we prove analogous uniqueness results for functions in shift-invariant spaces with Gaussian generator. Generalizations to nonuniform sampling are also presented. Finally, we compare our results to the situation where the considered signals are assumed to be real-valued.

math.FA↗

arXiv2022

On foundational discretization barriers in STFT phase retrieval

We prove that there exists no window function $g \in L^2(\mathbb{R})$ and no lattice $\mathcal{L} \subset \mathbb{R}^2$ such that every $f \in L^2(\mathbb{R})$ is determined up to a global phase by spectrogram samples $|V_gf(\mathcal{L})|$ where $V_gf$ denotes the short-time Fourier transform of $f$ with respect to $g$. Consequently, the forward operator $f \mapsto |V_gf(\mathcal{L})|$ mapping a square-integrable function to its spectrogram samples on a lattice is never injective on the quotient space $L^2(\mathbb{R}) / {\sim}$ with $f \sim h$ identifying two functions which agree up to a multiplicative constant of modulus one. We will further elaborate this result and point out that under mild conditions on the lattice $\mathcal{L}$, functions which produce identical spectrogram samples but do not agree up to a unimodular constant can be chosen to be real-valued. The derived results highlight that in the discretization of the STFT phase retrieval problem from lattice measurements, a prior restriction of the underlying signal space to a proper subspace of $L^2(\mathbb{R})$ is inevitable.

math.FA↗

arXiv2019

On the Mathematical Validity of the Higuchi Method

In this paper, we discuss the Higuchi algorithm which serves as a widely used estimator for the box-counting dimension of the graph of a bounded function $f : [0,1] \to \R$. We formulate the method in a mathematically precise way and show that it yields the correct dimension for a class of non-fractal functions. Furthermore, it will be shown that the algorithm follows a geometrical approach and therefore gives a reasonable estimate of the fractal dimension of a fractal function. We conclude the paper by discussing the robustness of the method and show that it can be highly unstable under perturbations.

math.MG↗