# Injectivity of Gabor phase retrieval from lattice measurements

Philipp Grohs and Lukas Liehr Faculty of Mathematics, Universität Wien, Oskar-
Morgenstern-Platz 1, 1090 Wien, Austria<EMAIL_ADDRESS>and Johann
Radon Institute of Applied and Computational Mathematics, Austrian Academy of
Sciences, Altenbergstrasse 69, 4040 Linz, Austria,
<EMAIL_ADDRESS>of Mathematics, Universität Wien, Oskar-
Morgenstern-Platz 1, 1090 Wien, Austria<EMAIL_ADDRESS>
###### Abstract

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 $f\in L^{4}[-c/2,c/2]$ is determined up to a global
phase by the values $|\mathcal{G}f(x,\omega)|$ where $(x,\omega)$ are points
on the lattice ${\mathbb{Z}}\times(2c)^{-1}{\mathbb{Z}}$. This for the first
time shows that compactly supported 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
general shift-invariant spaces with Gaussian generator. Generalizations to
nonuniform lattices are also presented. The results are based on a combination
of certain Müntz-type theorems as well as uniqueness theorems in shift-
invariant spaces.

Keywords. phase retrieval, spectrogram sampling, Gabor transform, shift-
invariant spaces, lattice measurements

AMS subject classifications. 94A12, 94A20, 42A65

##  1 Introduction

###  1.1 Gabor Phase Retrieval

The phase retrieval problem concerns the reconstruction of a given signal
$f\in\mathcal{C}$ in a signal class $\mathcal{C}$ contained in a given Banach
space $\mathcal{B}$ from phaseless linear measurements. These measurements are
of the form $\\{|\Phi_{\lambda}(f)|:\lambda\in X\\}$ with
$(\Phi_{\lambda})_{\lambda\in X}\subset\mathcal{B}^{\prime}$ constituting a
family of linear functionals on $\mathcal{B}$. Problems of this kind arise in
a large number of applications, most notably in coherent diffraction imaging
and audio processing where the measurements $(\Phi_{\lambda}(f))_{\lambda\in
X}\subset{\mathbb{C}}$ typically arise as (samples of) the Fourier- or Gabor
transform of a signal $f$, see for example the recent surveys [7, 8] and the
references therein. The question of when any given $f\in\mathcal{C}$ is
uniquely defined from its phaseless linear measurements _up to a global phase_
, meaning that $|\Phi_{\lambda}(f)|=|\Phi_{\lambda}(h)|$ for all $\lambda\in
X$ and some $f,h\in\mathcal{C}$ implies that there exists a global phase
factor $\tau\in{\mathbb{T}}\coloneqq\\{z\in{\mathbb{C}}:|z|=1\\}$ with $f=\tau
h$, in general constitutes a difficult mathematical problem, especially when
$\mathcal{B}$ is infinite dimensional and $X$ is discrete as it would be the
case in most practical applications. In the present paper we study the phase
retrieval problem arising from measurements of the Gabor transform: Let
$\varphi$ be the Gaussian windows function $\varphi(t)=e^{-\pi t^{2}}$ and
denote by $\mathcal{G}:{L^{2}({\mathbb{R}})}\to L^{2}({\mathbb{R}}^{2})$ the
Gabor transform defined via

$\mathcal{G}f(x,\omega)=\int_{\mathbb{R}}f(t)\varphi(t-x)e^{-2\pi
it\omega}\,dt.$

For a subset $X\subseteq{\mathbb{R}}^{2}$ of the time-frequency plane we are
interested in reconstructing $f\in\mathcal{C}\subseteq L^{2}({\mathbb{R}})$
from $\\{|\mathcal{G}f(x,\omega)|:(x,\omega)\in X\\}$, the corresponding
spectrogram measurements of the Gabor transform given on $X$. Such problems
arise in Ptychography [6] and audio processing [21].

######  Definition 1.1.

Let $\mathcal{C}\subseteq{L^{2}({\mathbb{R}})}$ and
$X\subseteq{\mathbb{R}}^{2}$. We say that the pair $(\mathcal{C},X)$ satisfies
condition (U) if every $f\in\mathcal{C}$ is determined up to a global phase by
the values $\\{|\mathcal{G}f(z)|:z\in X\\}$. In formulas,

$\forall f,h\in\mathcal{C}:(\,|\mathcal{G}f(z)|=|\mathcal{G}h(z)|\ \forall
z\in X\implies\exists\tau\in{\mathbb{T}}:f=\tau h\,).$

###  1.2 Previous Work

We provide a list of pairs $(\mathcal{C},X)$ which are known to satisfy
$(\textbf{U})$:

  1. 1.

It is well-known that every $f\in{L^{2}({\mathbb{R}})}$ is determined up to a
global phase by the measurements
$\\{|\mathcal{G}f(x,\omega)|:(x,\omega)\in{\mathbb{R}}^{2}\\}$ and therefore
$({L^{2}({\mathbb{R}})},{\mathbb{R}}^{2})$ satisfies condition (U). This is a
direct consequence of the ambiguity-relation

${\mathcal{F}}(|\mathcal{G}f|^{2})(x,\xi)=A\hat{f}(x,\xi)\,\overline{A\varphi(x,\xi)},$

and the fact that $A\varphi$ does not vanish on ${\mathbb{R}}^{2}$ [9, Theorem
A.2]. In practice, the measurements $|\mathcal{G}f(x,\omega)|$ are never given
on whole ${\mathbb{R}}^{2}$ and one seeks to proof similar uniqueness results
where $X$ is ”small” in the sense that $X$ is, for instance, discrete or a set
of measure zero.

  2. 2.

Assume that
$e^{i\alpha_{1}}{\mathbb{R}},e^{i\alpha_{2}}{\mathbb{R}}\subset{\mathbb{C}}$
are two lines which satisfy the condition
$\alpha_{1}-\alpha_{2}\notin\pi{\mathbb{Q}}$. A result due to Jaming shows
that every entire function of finite order is determined up to a global phase
from its modulus on $e^{i\alpha_{1}}{\mathbb{R}}\cup
e^{i\alpha_{2}}{\mathbb{R}}$ [16, Theorem 3.3]. The Gabor transform is, except
for a non-zero weighting factor and a reflection, an entire function. This
follows from the relation

$\mathcal{G}f(x,-\omega)=e^{\pi ix\omega}Bf(z)e^{-\pi|z|^{2}/2}$

where $z=x+i\omega$ and $Bf$ is the Bargmann transform of $f$ [11, Proposition
3.4.1]. The Bargmann transform of a function $f\in{L^{2}({\mathbb{R}})}$ is an
entire function of order less or equal than two [13, Theorem 2]. As a direct
consequence, Jaming’s theorem yields the following choice of $\mathcal{C}$ and
$X$.

######  Theorem 1.2.

Let $\alpha_{1},\alpha_{2}\in[0,2\pi)$ with
$\alpha_{1}-\alpha_{2}\notin\pi{\mathbb{Q}}$. Set
$\mathcal{C}={L^{2}({\mathbb{R}})}$ and $X=e^{i\alpha_{1}}{\mathbb{R}}\cup
e^{i\alpha_{2}}{\mathbb{R}}$. Then $(\mathcal{C},X)$ satisfies condition (U).

Jaming’s result is based on Hadmard’s factorization theorem and its proof
reveals that the two lines
$e^{i\alpha_{1}}{\mathbb{R}},e^{i\alpha_{2}}{\mathbb{R}}$ can be replaced by
sets $A\subseteq e^{i\alpha_{1}}{\mathbb{R}},B\subseteq
e^{i\alpha_{2}}{\mathbb{R}}$ where $A,B$ are sets of uniqueness of entire
functions of finite order (i.e. $A$ and $B$ contain at least a limit point)
[16, Remark 3.4]. In this case, $({L^{2}({\mathbb{R}})},A\cup B)$ satisfies
condition (U) as well. It follows readily from Theorem 1.2 that
$({L^{2}({\mathbb{R}})},\mathcal{O})$ satisfies condition (U) whenever
$\mathcal{O}$ is an open subset of ${\mathbb{R}}^{2}$. For if
$\mathcal{O}\subseteq{\mathbb{R}}^{2}\simeq{\mathbb{C}}$ is open then we can
find two lines
$e^{i\alpha_{1}}{\mathbb{R}},e^{i\alpha_{2}}{\mathbb{R}}\subset{\mathbb{C}}$
satisfying $\alpha_{1}-\alpha_{2}\notin\pi{\mathbb{Q}}$ such that
$A=e^{i\alpha_{1}}{\mathbb{R}}\cap\mathcal{O}$ and
$B=e^{i\alpha_{2}}{\mathbb{R}}\cap\mathcal{O}$ contain a limit point.

  3. 3.

Going from sets which contain a limit point to fully discrete sets, a result
due to Alaifari and Wellershoff shows that if
$\mathcal{C}=PW_{b}^{2}({\mathbb{R}},{\mathbb{R}})$ is the class of real-
valued Paley-Wiener functions,

$PW_{b}^{2}({\mathbb{R}},{\mathbb{R}})=\\{f\in
L^{2}({\mathbb{R}},{\mathbb{R}}):{\mathrm{supp}}(\hat{f})\subseteq[-b/2,b/2]\\},$
(1)

then the choice $X=(2b)^{-1}{\mathbb{Z}}\times\\{0\\}$ implies that
$(\mathcal{C},X)$ satisfies condition (U) [1, Theorem 2.5].

To summarize, 1 and 2 above consider non-discrete sampling sets. In practice
however the spectrogram can only be sampled on a discrete set $X$. The only
result applying to this case is 3 which is however only valid for a very
restricted class of (real-valued and band-limited) signals, whereas
applications in diffraction imaging would rather require results for more
general function classes. In particular, it is unknown which classes of
complex-valued functions (if any) can be uniquely recovered from discrete
samples of their spectrogram.

###  1.3 Contributions

In the present paper we significantly improve on previous work by adding two
more realistic examples of pairs $(\mathcal{C},X)$ which satisfy condition
$(\textbf{U})$:

Our first main result, Theorem 3.1, shows that condition $(\mathbf{U})$ holds
for arbitrary $c>0$ with the signal class $\mathcal{C}=L^{4}[-c/2,c/2]$ of
compactly supported _complex-valued_ functions and the discrete sampling set
$X={\mathbb{Z}}\times(2c)^{-1}{\mathbb{Z}}$. The proof uses Shannon’s sampling
theorem to sample in frequency space as well as Zalik’s theorem (Theorem 2.6)
on the completeness of Gaussian translates to obtain the desired sampling rate
in $x$.

Our second main result, Theorem 3.6, shows that condition $(\mathbf{U})$ holds
for arbitrary $\beta\in{\mathbb{R}}_{>0}\setminus{\mathbb{Q}}_{>0}$ and
$\varepsilon>0$ with the signal class

$\mathcal{C}=V_{\beta}^{p}(\varphi)\coloneqq\\{f\in
L^{p}({\mathbb{R}}):f=\sum_{k\in{\mathbb{Z}}}c_{k}\varphi(\cdot-\beta k),\
c\in\ell^{p}({\mathbb{Z}})\\},$

a Gaussian shift-invariant space and the discrete sampling set

$X=\frac{\beta}{2+\varepsilon}{\mathbb{Z}}\times{\mathbb{Z}}.$

The proof makes use of recent uniqueness results [12] for shift-invariant
spaces as well as Weyl’s equidistribution theorem. Observe that in this result
the elements of the signal class $\mathcal{C}$ are neither compactly supported
nor band-limited and/or real-valued which thus represents a new level of
generality compared to previous results. In addition, a uniqueness result
which holds for arbitrary $\beta>0$ will be presented.

Finally, we show that our results can be generalized to non-uniform sampling
as well as phase space rotations (i.e., fractional Fourier transforms) of the
considered signal classes.

###  1.4 Open Questions

We would like to mention two important questions that are left open:

Stability.  Additionally to the uniqueness property $(\textbf{U})$ practical
applications require _stable reconstruction_ in the sense that a small
perturbation in the measurements results in a comparable reconstruction error.
In the context of Gabor phase retrieval this problem is now quite well
understood _in the continuous case $X={\mathbb{R}}^{2}$_ [9, 10] but very
little is known in the discrete case (the very recent work [5] being a notable
exception).

Sharpness. In our results the density of the sampling set $X$ either depends
on the support size $c$ (for $\mathcal{C}=L^{4}[-c/2,c/2]$) or the ”regularity
parameter” $\beta$ (for $\mathcal{C}=V_{\beta}^{p}(\varphi)$). Due to the
translation and modulation equivariance of the Gabor transform one can
speculate that similar sampling results might hold for sampling sets $X$ which
are independent (or only mildly dependent) on $c$, resp. $\beta$. In this
sense, it would be interesting to study to which extent our results are sharp.

##  2 Basic Results

In this section we collect some basic results that will be used later on in
Section 3 where our main results will be proven. In Section 2.1 we provide
some well known uniqueness results in Paley-Wiener and shift-invariant spaces,
Section 2.2 presents some density results for translates of functions, and
Section 2.3 collects some basic properties of the fractional Fourier
transform.

###  2.1 Uniqueness sets

Our uniqueness results make use of uniqueness theorems for Paley-Wiener spaces
and certain shift-invariant spaces with Gaussian generator. In general, the
term uniqueness set can be defined as follows.

######  Definition 2.1.

Let $Q\subseteq C({\mathbb{R}})$ be a class of real- or complex-valued,
continuous functions. A set $\Lambda=(\lambda_{n})_{n}\subset{\mathbb{R}}$ is
called uniqueness set for $Q$ if for every $f\in Q$ the implication

$f(\lambda_{n})=0\ \forall n\in{\mathbb{N}}\implies f\equiv 0$

holds true.

Sufficient conditions on a set $\Lambda$ to be a uniqueness set for Paley-
Wiener spaces and shift-invariant spaces are given by means of their so-called
effective Beurling-Malliavin density and lower Beurling-density, respectively.
In the following we give a brief summary of the relevant results which will be
used to prove new uniqueness results for the Gabor phase retrieval problem.

####  2.1.1 Paley-Wiener spaces

For $b>0$ we denote by $PW_{b}^{2}({\mathbb{R}})$ the Paley-Wiener space,
defined via

$PW_{b}^{2}({\mathbb{R}})=\\{f\in
L^{2}({\mathbb{R}})\,|\,{\mathrm{supp}}(\hat{f})\subseteq[-b/2,b/2]\\}.$

The classical sampling theorem by Shannon, Whittaker and Kotel’nikov implies
that every $f\in PW_{b}^{2}({\mathbb{R}})$ is determined by its samples at
$b^{-1}{\mathbb{Z}}$ [14, Theorem 6.13]. In particular, $f$ can be recovered
by the cardinal sine series

$f(t)=\sum_{n\in{\mathbb{Z}}}f\left(\frac{n}{b}\right)\mathrm{sinc}(bt-n)$

and $\Lambda=b^{-1}{\mathbb{Z}}$ is a uniqueness set for
$PW_{b}^{2}({\mathbb{R}})$. Clearly, if $f\in PW_{b}^{2}({\mathbb{R}})$ then
it follows from the representation

$f(x)=\int_{-b/2}^{b/2}\hat{f}(\omega)e^{2\pi i\omega x}\,d\omega$

that a set of real numbers $\Lambda=(\lambda_{n})_{n}$ is a uniqueness set for
$PW_{b}^{2}({\mathbb{R}})$ if and only if the complex exponentials

$\mathcal{E}_{\Lambda}=\\{e^{2\pi i\lambda_{n}x}\\}_{n}$

are complete in $L^{2}[-b/2,b/2]$. Determining which assumptions on $\Lambda$
imply completeness of $\mathcal{E}_{\Lambda}$ is commonly known as the
completeness problem of complex exponentials. Equivalently, one could ask to
establish a formula for the radius of completeness of $\Lambda$ which is
defined to be

$R(\Lambda)=\sup\\{a\geq 0:\mathcal{E}_{\Lambda}\text{ is complete in
}L^{2}[0,a]\\}.$

A solution to this problem is a consequence of a deep theory of Beurling and
Malliavin [3, 4]. To state the relevant result, we start by defining the so-
called effective Beurling Malliavin density. Several equivalent definitions of
this density are known. We follow the approach given in Chapter 5 of the book
by Poltoratski [20].

For an interval $I=[a,b]\subset{\mathbb{R}}$ denote by $|I|$ its length and by
$\mathrm{dist}(0,I)$ its distance to the origin. A sequence $(I_{n})_{n}$ of
disjoint intervals on ${\mathbb{R}}$ is called short if

$\sum_{n}\frac{|I_{n}|^{2}}{1+\mathrm{dist}^{2}(0,I_{n})}<\infty$

and long otherwise. For a sequence
$\Lambda=(\lambda_{n})_{n}\subset{\mathbb{R}}$ of real numbers the _effective
Beurling Malliavin density (effective BM density)_ is defined as

$D^{*}(\Lambda)\coloneqq\sup\\{d\geq 0:\exists\text{ long }(I_{n})_{n}\text{
such that }\\#(\Lambda\cap I_{n})\geq d|I_{n}|\ \forall n\\},$

where $\\#(\Lambda\cap I_{n})$ denotes the number of elements in $\Lambda\cap
I_{n}$. The Beurling-Malliavin theorem reveals that $D^{*}(\Lambda)$ coincides
with $R(\Lambda)$ [20, Theorem 26].

######  Theorem 2.2 (Beurling-Malliavin).

If $\Lambda$ is a discrete real sequence then $R(\Lambda)=D^{*}(\Lambda).$
Consequently, if $a<D^{*}(\Lambda)<b$ for some $0<a<b$ then
$\mathcal{E}_{\Lambda}$ is complete in $L^{2}[-a/2,a/2]$ and incomplete in
$L^{2}[-b/2,b/2]$.

####  2.1.2 Shift-invariant spaces

Recent developments in sampling theory due to Gröchenig, Romero and Stöckler
show that assumptions on a suitable density of a sequence can be used to
obtain uniqueness results for certain shift-invariant spaces [12]. More
precisely, they consider shift-invariant spaces generated by a totally
positive function of Gaussian type. We start by defining shift-invariant
spaces. Let $1\leq p\leq\infty$ and let $\beta>0$ be a step size. The shift-
invariant space $V_{\beta}^{p}(\varphi)$, generated by the Gaussian
$\varphi(t)=e^{-\pi t^{2}}$, is defined to be

$V_{\beta}^{p}(\varphi)\coloneqq\\{f\in
L^{p}({\mathbb{R}}):f=\sum_{k\in{\mathbb{Z}}}c_{k}\varphi(\cdot-\beta k),\
c\in\ell^{p}({\mathbb{Z}})\\}.$ (2)

Note that functions in $V_{\beta}^{p}(\varphi)$ are continuous and if $1\leq
p\leq q\leq\infty$ then the inclusion

$V_{\beta}^{1}(\varphi)\subset V_{\beta}^{p}(\varphi)\subset
V_{\beta}^{q}(\varphi)$

holds true [2, Corollary 2.5]. Clearly, shift-invariant spaces can be defined
in a more general fashion by replacing the Gaussian generator $\varphi$ by
another generator $g$. In particular, the choice $g(t)=\mathrm{sinc}(bt)$ for
some $b>0$ implies that $V_{1/b}^{2}(g)=PW_{b}^{2}({\mathbb{R}})$ and we are
in the situation of Section 2.1.1. For a general introduction to shift-
invariant spaces we refer to the survey [22].

Compared to the previous subsection, to obtain uniqueness results for shift-
invariant spaces, the effective lower Beurling-Malliavin density will be
replaced by the _lower Beurling density_. If
$\Lambda=(\lambda_{n})_{n}\subset{\mathbb{R}}$ then the lower Beurling density
of $\Lambda$ is defined to be

$D^{-}(\Lambda)\coloneqq\liminf_{r\to\infty}\left(\inf_{a\in{\mathbb{R}}}\frac{\\#(\Lambda\cap[a,a+r))}{r}\right).$

In case where the generator is the Gaussian $\varphi$, we have the following
uniqueness theorem [12, Theorem 4.4].

######  Theorem 2.3.

If $\Lambda\subset{\mathbb{R}}$ has lower Beurling density
$D^{-}(\Lambda)>\beta^{-1}$ then $\Lambda$ is a uniqueness set for
$V_{\beta}^{\infty}(\varphi)$.

###  2.2 Müntz-Szász type approximations

We wish to combine the above-stated uniqueness theorems with certain results
concerning the density of translates. These results are based on the classical
Müntz-Szász theorem which we recall briefly [25, p. 78]. While the original
form of this theorem is stated in $C[0,1]$, the following result due to
Luxemburg and Korevaar provides a generalization to $L^{p}[a,b]$ whenever
$a>0$ [17, Theorem 6.1].

######  Theorem 2.4 (Müntz-Szász).

Let $[a,b]$ be a compact interval such that $a>0$ and let
$(\lambda_{n})_{n}\subset{\mathbb{R}}$ be a sequence of distinct real numbers.
Let $Q$ be one of the spaces $C[a,b]$ or $L^{p}[a,b]$ with $1\leq p<\infty$.
Then the sequence of powers $\\{t^{\lambda_{1}},t^{\lambda_{2}},\dots\\}$ is
complete in $Q$ if and only if

$\sum_{n}{}^{{}^{\prime}}\frac{1}{\lambda_{n}}=\infty,$

where the symbol $\sum^{\prime}$ indicates that sum runs over all terms with
nonvanishing denominator.

The Müntz-Szász theorem can be used to investigate under which assumptions a
function space can be spanned by discrete translates of a single function
$\psi$. For $\tau,\nu\in{\mathbb{R}}$ and $\psi\in{L^{2}({\mathbb{R}})}$ we
define the translation operator $T_{\tau}$ and the modulation operator
$M_{\nu}$ via $T_{\tau}\psi(t)=\psi(t-\tau)$ and $M_{\nu}\psi(t)=e^{2\pi i\nu
t}\psi(t)$, respectively.

######  Definition 2.5.

Let $I\subseteq{\mathbb{R}}$ be an interval and let
$\Lambda\subseteq{\mathbb{R}}$ be a set of shifts. We say that a function
$\psi\in{L^{2}({\mathbb{R}})}$ is a $\Lambda$-generator for $L^{2}(I)$ if

${\mathrm{span}\,}\\{T_{\lambda}\psi\,|\,\lambda\in\Lambda\\}$

is dense in $L^{2}(I)$.

For instance, Wiener’s Tauberian theorem [24, Theorem 1] implies that every
$\psi$ which has an a.e. non-vanishing Fourier transform is an
${\mathbb{R}}$-generator for ${L^{2}({\mathbb{R}})}$. It is natural to ask if
$\Lambda$ can be chosen to be discrete. It then depends on $\Lambda$ and the
topology of $I$ whether or not there exists a function that generates
$L^{2}(I)$. For example, there exists no ${\mathbb{Z}}$-generator for
${L^{2}({\mathbb{R}})}$ but an arbitrary perturbation of ${\mathbb{Z}}$ of the
form $\Lambda=\\{n+a_{n}\,|\,a_{n}\neq 0,a_{n}\to 0\\}$ admits a
$\Lambda$-generator for ${L^{2}({\mathbb{R}})}$. This is Olevskii’s theorem
[18]. If $\psi$ is a Gaussian then the following result due to Zalik provides
a Müntz-type condition on $\Lambda$ such that $\psi$ is a $\Lambda$-generator
for $L^{2}(I)$ whenever $I$ is a compact interval [26, Theorem 4]. We state
the result in a slightly modified way.

######  Theorem 2.6 (Zalik).

Let $I\subset{\mathbb{R}}$ be a compact interval and let
$\psi(t)=ae^{-b^{2}(t-c)^{2}}$ be a Gaussian, $a,b>0,c\in{\mathbb{R}}$.
Further, let $\Lambda=(\lambda_{n})_{n}\subset{\mathbb{R}}$ be a sequence of
distinct real numbers. Then $\psi$ is a $\Lambda$-generator for $L^{2}(I)$ if
and only if the series $\sum^{\prime}|\lambda_{n}|^{-1}$ diverges.

###  2.3 Fractional Fourier transform

Let $(h_{n})_{n}\subset{L^{2}({\mathbb{R}})}$ be the Hermite basis functions
for ${L^{2}({\mathbb{R}})}$,

$h_{n}(t)=\frac{2^{1/4}}{\sqrt{n!}}\left(-\frac{1}{\sqrt{2\pi}}\right)^{n}e^{\pi
t^{2}}\left(\frac{d}{dt}\right)^{n}e^{-2\pi t^{2}}.$

For $\theta\in{\mathbb{R}}$, the operator
${\mathcal{F}}_{\theta}:{L^{2}({\mathbb{R}})}\to{L^{2}({\mathbb{R}})}$,
defined by

${\mathcal{F}}_{\theta}f=\sum_{n}e^{-i\theta n}\langle f,h_{n}\rangle h_{n},$

is called the fractional Fourier transform of order $\theta$. If we denote by
$u_{\theta}:{\mathbb{R}}\to{\mathbb{C}}$ the chirp modulation function
$u_{\theta}(t)=e^{-i\pi|t|^{2}\cot\theta}$ then the fractional Fourier
transform satisfies the relation

${\mathcal{F}}_{\theta}f(\omega)=c_{\theta}u_{\theta}(\omega){\mathcal{F}}(u_{\theta}f)(\tfrac{\omega}{\sin\theta})$

where $c_{\theta}=\sqrt{1-i\cot\theta}$ is a normalization factor and
${\mathcal{F}}:{L^{2}({\mathbb{R}})}\to{L^{2}({\mathbb{R}})}$ denotes the
ordinary Fourier transform, defined on
$L^{1}({\mathbb{R}})\cap{L^{2}({\mathbb{R}})}$ by

${\mathcal{F}}f(\omega)=\int_{\mathbb{R}}f(t)e^{-2\pi it\omega}\,dt$

and on ${L^{2}({\mathbb{R}})}$ via a standard extension argument. If we define
in addition the rotation matrix $R_{\theta}\in{\mathbb{R}}^{2\times 2}$ by

$R_{\theta}\coloneqq\left({\begin{array}[]{cc}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\\\ \end{array}}\right)$ (3)

then the fractional Fourier transform has the following properties [15, 19] :

  1. 1.

${\mathcal{F}}_{\theta}\varphi=\varphi$ for every $\theta\in{\mathbb{R}}$
whenever $\varphi$ is the Gaussian $\varphi(t)=e^{-\pi t^{2}}$

  2. 2.

${\mathcal{F}}_{\theta}{\mathcal{F}}_{\eta}={\mathcal{F}}_{\theta+\eta}$ for
every $\theta,\eta\in{\mathbb{R}}$

  3. 3.

${\mathcal{F}}_{\frac{\pi}{2}}={\mathcal{F}}$,
${\mathcal{F}}_{-\frac{\pi}{2}}={\mathcal{F}}^{-1}$ and
${\mathcal{F}}_{0}=\mathrm{Id}$

  4. 4.

$A(f,g)(R_{\theta}z)=A({\mathcal{F}}_{\theta}f,{\mathcal{F}}_{\theta}g)(z)$
for every $z\in{\mathbb{R}}^{2}$, $\theta\in{\mathbb{R}}$ and every
$f,g\in{L^{2}({\mathbb{R}})}$ where $A(f,g)$ denotes the cross-ambiguity
function of $f$ and $g$,

$A(f,g)=\int_{\mathbb{R}}f(t+\tfrac{x}{2})\overline{g(t-\tfrac{x}{2})}e^{-2\pi
it\omega}\,dt.$

Property 2 and 3 above show that
$\\{{\mathcal{F}}_{\theta}\\}_{\theta\in{\mathbb{R}}}$ defines a subgroup of
the group of bounded invertible linear operators on ${L^{2}({\mathbb{R}})}$.
By property 4, the action of the fractional Fourier transform on functions $f$
and $g$ constitutes a rotation of the time-frequency plane.

##  3 Main Results

In this section we prove the main results of this paper. In Section 3.1 we
derive uniqueness results for compactly supported functions and in Section 3.2
we establish analogous results for functions in shift-invariant spaces with
Gaussian generator.

###  3.1 Compactly supported functions

For a function $h:{\mathbb{R}}\to{\mathbb{C}}$ and a value
$\tau\in{\mathbb{R}}$ we introduce the shorthand notation

$h_{\tau}\coloneqq(T_{\tau}h)\overline{h},$ (4)

where $T_{\tau}$ is the translation operator, $T_{\tau}h=h(\cdot-\tau).$

######  Theorem 3.1.

Let $c>0$ and $\mathcal{C}=L^{4}[-c/2,c/2]$. Then the choice
$X={\mathbb{Z}}\times(2c)^{-1}{\mathbb{Z}}$ implies that $(\mathcal{C},X)$
satisfies condition (U). In other words, every $f\in L^{4}[-c/2,c/2]$ is
determined up to a global phase by the measurements

$\\{|\mathcal{G}f(x,\omega)|\,:\,(x,\omega)\in{\mathbb{Z}}\times(2c)^{-1}{\mathbb{Z}}\\}.$

###### Proof.

We denote by $\varphi(t)=e^{-\pi t^{2}}$ the Gaussian window function.
Further, we define $p_{x}(t)\coloneqq f(t+x/2)\overline{\varphi(t-x/2)}$. From
the definition of the cross-ambiguity function it follows that

$A(f,\varphi)(x,\omega)\overline{A(f,\varphi)(x,\omega)}={\mathcal{F}}p_{x}(\omega)\overline{{\mathcal{F}}p_{x}(\omega)}={\mathcal{F}^{-1}}({\mathcal{F}}^{2}p_{x}*\overline{p_{x}})(\omega).$

The convolution ${\mathcal{F}}^{2}p_{x}*\overline{p_{x}}$ can be written as

$({\mathcal{F}}^{2}p_{x}*\overline{p_{x}})(\omega)=\int_{\mathbb{R}}T_{\omega}f(k)\,\overline{f(k)}\,T_{x}[\overline{T_{\omega}\varphi(k)}\,\varphi(k)]\,dk=\langle
f_{\omega},T_{x}\varphi_{\omega}\rangle,$

where we used the change of variables $k\mapsto k-\frac{x}{2}$ in the first
equality as well as the notation as defined in 4. From the fact that
$\mathcal{G}f(x,\omega)=e^{-\pi ix\omega}A(f,\varphi)(x,\omega)$ we obtain the
identity

${\mathcal{F}}_{2}(|\mathcal{G}f|^{2})(x,\omega)=\langle
f_{\omega},T_{x}\varphi_{\omega}\rangle,$ (5)

where ${\mathcal{F}}_{2}$ denotes the Fourier transform with respect to the
second argument. Observe that $f_{\omega}=0$ a.e. whenever $|\omega|>c$.
Consequently, for every $x\in{\mathbb{R}}$ we have
$|\mathcal{G}f(x,\cdot)|^{2}\in PW_{2c}^{2}({\mathbb{R}})$. Now suppose that a
function $h\in L^{4}[-c/2,c/2]$ satisfies

$|\mathcal{G}f(n,k)|=|\mathcal{G}h(n,k)|$

for every $(n,k)\in{\mathbb{Z}}\times(2c)^{-1}{\mathbb{Z}}$. By Shannon’s
sampling theorem we know that $(2c)^{-1}{\mathbb{Z}}$ is a uniqueness set for
$PW_{2c}^{2}({\mathbb{R}})$. Consequently, the functions
$|\mathcal{G}f(n,\cdot)|^{2}$ and $|\mathcal{G}h(n,\cdot)|^{2}$ agree for ever
$n\in{\mathbb{Z}}$. Taking the Fourier transform with respect to the second
argument and consulting equation 5, we obtain the relation

$\langle f_{\omega}-h_{\omega},T_{n}\varphi_{\omega}\rangle=0$

which holds for every $\omega\in{\mathbb{R}}$ and every $n\in{\mathbb{Z}}$.
Observe, that for a fixed $\omega$, the function $\varphi_{\omega}$ is a
Gaussian of the form $\varphi_{\omega}(t)=ae^{-b^{2}(t-d)^{2}}$ for some
$a,b>0,d\in{\mathbb{R}}$. Clearly, the series
$\sum_{n\in{\mathbb{Z}}\setminus\\{0\\}}|n|^{-1}$ is divergent and it follows
from Zalik’s theorem that the system of translates
$\\{T_{n}\varphi_{\omega}:n\in{\mathbb{Z}}\\}$ is complete in
$L^{2}[-c/2,c/2]$. Note that $f_{\omega}$ and $h_{\omega}$ have support in
$[-c/2,c/2]$ and by Hölder’s inequality $f_{\omega}-h_{\omega}\in
L^{2}[-c/2,c/2]$. Therefore, $f_{\omega}=h_{\omega}$ for every
$\omega\in{\mathbb{R}}$. Choosing $\omega=0$ we have

$|f|^{2}=|h|^{2}$ (6)

and $h=0$ whenever $f=0$. If $f\neq 0$, select a point $p\in{\mathbb{R}}$ for
which $f(p)\neq 0$. Then $f(t)\overline{f(p)}=h(t)\overline{h(p)}$ for every
$t$. Hence, there exists a $\tau\in{\mathbb{C}}$ such that $f=\tau h$.
Equation 6 implies $\tau\in{\mathbb{T}}$. ∎

In the previous theorem the sampling set
$X={\mathbb{Z}}\times(2c)^{-1}{\mathbb{Z}}$ is a lattice in
${\mathbb{R}}^{2}$, i.e. $X=A{\mathbb{Z}}^{2}$ for some invertible
$A\in{\mathbb{R}}^{2\times 2}$. This lattice was chosen in such a way that
both Shannon’s sampling theorem and Zalik’s theorem were applicable. In view
of Theorem 2.2 and the assumptions of Zalik’s theorem, the lattice
${\mathbb{Z}}\times(2c)^{-1}{\mathbb{Z}}$ can be replaced by more general
sampling sets.

######  Theorem 3.2.

Suppose that $A=(a_{n})_{n}$ and $B=(b_{n})_{n}$ are sequences in
${\mathbb{R}}$ with the following properties:

  1. 1.

The elements of $(a_{n})_{n}$ are distinct and the series of reciprocals
$\sum^{\prime}|a_{n}|^{-1}$ diverges

  2. 2.

$B$ is a uniqueness set for $PW_{2c}^{2}({\mathbb{R}})$ for some $c>0$

Then every $f\in L^{4}[-c/2,c/2]$ is determined up to a global phase by the
values $|\mathcal{G}f(A\times B)|$. In particular, the pair
$(L^{4}[-c/2,c/2],A\times B)$ satisfies condition (U).

###### Proof.

Replacing ${\mathbb{Z}}\times(2c)^{-1}{\mathbb{Z}}$ with $A\times B$ in the
proof of Theorem 3.1 yields the result. ∎

Note that in Theorem 3.2 the assumption on $B$ to be a uniqueness set for
$PW_{2c}^{2}({\mathbb{R}})$ is satisfied if one requires that $B$ has
effective Beurling Malliavin density $D^{*}(B)>2c$. Observe further that the
previous theorem can be stated in an even more general way: The uniqueness set
$(b_{n})_{n}$ does not need to be a priori fixed. In fact, for every single
$a_{n}$ we can choose a different uniqueness set $(s_{k}^{n})_{k}$.

######  Theorem 3.3.

Let $c>0$ and let $A=(a_{n})_{n\in{\mathbb{N}}}$ be a sequence of distinct
real numbers with the property that the series of reciprocals
$\sum^{\prime}|a_{n}|^{-1}$ diverges. For every $n\in{\mathbb{N}}$ let
$(s_{k}^{n})_{k\in{\mathbb{N}}}$ be a uniqueness set for
$PW_{2c}^{2}({\mathbb{R}})$. Set

$X=\bigcup_{n\in{\mathbb{N}}}\\{(a_{n},s_{k}^{n}):k\in{\mathbb{N}}\\}$

and $\mathcal{C}=L^{4}[-c/2,c/2]$. Then $(\mathcal{C},X)$ satisfies condition
(U).

###### Proof.

The proof is essentially the same as the proof of Theorem 3.1. ∎

Consulting Section 2.3, new uniqueness results can be obtained via a phase-
space rotation. In particular, all of the previous theorems can be stated in
an analogous manner for band-limited functions by means of a rotation of the
lattice by 90 degrees.

######  Proposition 3.4.

If $\theta\in{\mathbb{R}}$ then every
$f\in\mathcal{C}_{\theta}\coloneqq{\mathcal{F}}_{-\theta}L^{4}[-c/2,c/2]$ is
determined up to a global phase by the measurements
$X_{\theta}\coloneqq\\{|\mathcal{G}f(n,k)|:(n,k)\in
R_{\theta}({\mathbb{Z}}\times(2c)^{-1}{\mathbb{Z}})\\},$ where $R_{\theta}$ is
the rotation matrix as defined in equation 3. In particular,
$(\mathcal{C}_{\theta},X_{\theta})$ satisfies condition (U). As a special
case, all band-limited functions in the space

$\mathcal{C}_{\frac{\pi}{2}}={\mathcal{F}^{-1}}L^{4}[-c/2,c/2]=\\{f\in
PW_{c}^{2}({\mathbb{R}},{\mathbb{C}}):\hat{f}\in L^{4}[-c/2,c/2]\\}$

are determined up to a global phase by the measurements

$\\{|\mathcal{G}f(n,k)|:(n,k)\in(2c)^{-1}{\mathbb{Z}}\times{\mathbb{Z}}\\}.$

###### Proof.

Suppose that $f,h\in\mathcal{C}_{\theta}$ satisfy

$|\mathcal{G}f(R_{\theta}(n,k))|=|\mathcal{G}h(R_{\theta}(n,k))|$

for every $(n,k)\in{\mathbb{Z}}\times(2c)^{-1}{\mathbb{Z}}$. By the properties
of the fractional Fourier transform (see section 2.3) we have

$|\mathcal{G}{\mathcal{F}}_{\theta}f(n,k)|=|\mathcal{G}{\mathcal{F}}_{\theta}h(n,k)|$

for every $(n,k)\in{\mathbb{Z}}\times(2c)^{-1}{\mathbb{Z}}$. By assumption,
${\mathcal{F}}_{\theta}f$ and ${\mathcal{F}}_{\theta}h$ are elements of
$L^{4}[-c/2,c/2]$ and the first part of the statement follows from Theorem
3.1. Since ${\mathcal{F}}_{-\frac{\pi}{2}}={\mathcal{F}^{-1}}$ and

$R_{\frac{\pi}{2}}=\left({\begin{array}[]{cc}0&-1\\\ 1&0\\\
\end{array}}\right)$

the second part is a special case of the first. ∎

###  3.2 Shift-invariant spaces with Gaussian generator

All of the theorems in the previous chapter used the fact that our signal
space $\mathcal{C}$ consists of functions which are compactly supported in the
time domain or the frequency domain (or more general: their fractional Fourier
transform has compact support). Uniqueness sets for the Paley-Wiener space
were then a crucial tool to obtain lattices which determine $f$ up to a global
phase from its corresponding spectrogram measurements. We wish to make use of
the uniqueness Theorem 2.3 which holds for shift-invariant spaces with
Gaussian generator to find sets $X$ consisting of separated points such that
$(V_{\beta}^{p}(\varphi),X)$ satisfies condition (U). The following Lemma will
be useful in our further considerations.

######  Lemma 3.5.

Let $\beta>0$ and $f=\sum_{k\in{\mathbb{Z}}}c_{k}\varphi(\cdot-\beta k)\in
V_{\beta}^{1}(\varphi)$. Then the modulus of the Gabor transform of $f$ can be
written as

$|\mathcal{G}f(x,\omega)|^{2}=\sum_{k}\sum_{j}c_{k}\overline{c_{j}}a(j,k,\beta)M_{\frac{\beta}{2}(j-k)}\varphi(\omega)T_{\frac{\beta}{2}(j+k)}\varphi(x),$

where $a(j,k,\beta)=\frac{1}{2}e^{-\frac{\pi\beta^{2}}{4}(k-j)^{2}}$ and $T,M$
is the translation- and modulation operator, respectively.

###### Proof.

If $f=\sum_{k\in{\mathbb{Z}}}c_{k}\varphi(\cdot-\beta k)$ then it follows from
the properties of the Gabor transform that

$\mathcal{G}f(x,\omega)=\sum_{k}c_{k}e^{-2\pi i\beta
k\omega}\mathcal{G}\varphi(x-\beta k,\omega).$

Note that the Gabor transform of the Gaussian $\varphi$ is given by
$\mathcal{G}\varphi(x,\omega)=\frac{1}{\sqrt{2}}e^{-\pi
ix\omega}e^{-\frac{\pi}{2}(x^{2}+\omega^{2})}$ and therefore

$\mathcal{G}f(x,\omega)=\sum_{k}\frac{c_{k}}{\sqrt{2}}e^{-2\pi i\beta
k\omega}e^{-\pi i(x-\beta k)\omega}e^{-\frac{\pi}{2}((x-\beta
k)^{2}+\omega^{2})}.$

Consequently, we obtain the identity

$|\mathcal{G}f(x,\omega)|^{2}=\sum_{k}\sum_{j}\frac{c_{k}\overline{c_{j}}}{2}e^{\pi
i\beta(j-k)\omega}e^{-\frac{\pi}{2}((x-\beta k)^{2}+(x-\beta
j)^{2}+2\omega^{2})}.$

But

$e^{-\frac{\pi}{2}((x-\beta k)^{2}+(x-\beta
j)^{2}+2\omega^{2})}=e^{-\frac{\pi\beta^{2}}{4}(k-j)^{2}}\varphi(\omega)\varphi(x-\tfrac{\beta}{2}(j+k))$

which yields the statement. ∎

The previous representation of $|\mathcal{G}f(x,\omega)|^{2}$ will be used to
provide a condition on the step size $\beta$ such that every function in
$V_{\beta}^{1}(\varphi)$ is determined up to a global phase from lattice
measurements. This is the content of

######  Theorem 3.6.

Let $\beta\in{\mathbb{R}}_{>0}\setminus{\mathbb{Q}}_{>0}$ and let
$\Lambda=(\lambda_{n})_{n}$ be a sequence in ${\mathbb{R}}$ with lower
Beurling density $D^{-}(\Lambda)>\frac{2}{\beta}$. Then every $f\in
V_{\beta}^{1}(\varphi)$ is determined up to a global phase by the measurements

$\\{|\mathcal{G}f(x,\omega)|:(x,\omega)\in\Lambda\times{\mathbb{Z}}\\}.$

In particular, for every $\varepsilon>0$ and every
$\beta\in{\mathbb{R}}_{>0}\setminus{\mathbb{Q}}_{>0}$ the pair
$(V_{\beta}^{1}(\varphi),\frac{\beta}{2+\varepsilon}{\mathbb{Z}}\times{\mathbb{Z}})$
satisfies condition (U).

###### Proof.

Suppose that $f=\sum_{k\in{\mathbb{Z}}}c_{k}\varphi(\cdot-\beta k)\in
V_{\beta}^{1}(\varphi)$ with $(c_{k})_{k}\in\ell^{1}({\mathbb{Z}}).$ Let
$a(j,k,\beta)$ be defined as in Lemma 3.5. Define a sequence
$(A_{n})_{n}\subset{\mathbb{R}}$ via

$A_{n}=\sum_{j,k\in{\mathbb{Z}},\,j+k=n}c_{k}\overline{c_{j}}a(j,k,\beta)M_{\frac{\beta}{2}(j-k)}\varphi(\omega)$

where $\omega\in{\mathbb{R}}$ is fixed. Moreover, for any fixed
$x\in{\mathbb{R}}$ define the sequence $(B_{n})_{n}\subset{\mathbb{R}}$ via

$B_{n}=\sum_{j,k\in{\mathbb{Z}},\,j-k=n}c_{k}\overline{c_{j}}a(j,k,\beta)T_{\frac{\beta}{2}(j+k)}\varphi(x).$

It follows from Lemma 3.5 that

$|\mathcal{G}f(x,\omega)|^{2}=\sum_{n\in{\mathbb{Z}}}A_{n}T_{\frac{\beta}{2}n}\varphi(x)=\sum_{n\in{\mathbb{Z}}}B_{n}M_{\frac{\beta}{2}n}\varphi(\omega).$

Since $|a(j,k,\beta)|\leq 1$ for every $j,k\in{\mathbb{Z}}$ and every
$\beta>0$ we have
$\left\|A\right\|_{{\ell^{1}({\mathbb{Z}})}}\leq\varphi(\omega)\left\|c\right\|_{{\ell^{1}({\mathbb{Z}})}}^{2}$
and
$\left\|B\right\|_{{\ell^{1}({\mathbb{Z}})}}\leq\left\|c\right\|_{{\ell^{1}({\mathbb{Z}})}}^{2}$.
We conclude that for every $\omega\in{\mathbb{R}}$, the map
$x\mapsto|\mathcal{G}f(x,\omega)|^{2}$ is an element of the shift-invariant
space $V_{\frac{\beta}{2}}^{1}(\varphi)$. In addition, for every
$x\in{\mathbb{R}}$ the map

$\omega\mapsto\frac{|\mathcal{G}f(x,\omega)|^{2}}{\varphi(\omega)}$

is continuous and $\frac{2}{\beta}$-periodic. Now suppose that $h\in
V_{\beta}^{1}(\varphi)$ satisfies

$|\mathcal{G}f(\lambda,n)|=|\mathcal{G}h(\lambda,n)|$

for every $(\lambda,n)\in\Lambda\times{\mathbb{Z}}$. According to the previous
arguments, we have $|\mathcal{G}f(\cdot,n)|,|\mathcal{G}h(\cdot,n)|\in
V_{\frac{\beta}{2}}^{1}(\varphi)$. By Theorem 2.3, $\Lambda$ is a uniqueness
set for $V_{\frac{\beta}{2}}^{1}(\varphi)$ and therefore
$|\mathcal{G}f(x,n)|=|\mathcal{G}h(x,n)|$ for every $x\in{\mathbb{R}}$ and
every $n\in{\mathbb{Z}}$. Fix $x\in{\mathbb{R}}$. It follows that

$\frac{|\mathcal{G}f(x,\omega)|^{2}}{\varphi(\omega)}=\frac{|\mathcal{G}h(x,\omega)|^{2}}{\varphi(\omega)}$
(7)

whenever $\omega\in{\mathbb{Z}}$. By periodicity, the left- and right-hand
side of equation 7 agree for every $\omega\in D$ where

$D\coloneqq\left\\{n\ \mathrm{mod}\
\frac{2}{\beta}:n\in{\mathbb{Z}}\right\\}=\frac{2}{\beta}\left\\{\frac{\beta}{2}n\
\mathrm{mod}\ 1:n\in{\mathbb{Z}}\right\\}.$

Since $\beta$ is irrational, it follows from Weyl’s criterion that the
sequence $(\frac{\beta}{2}n\ \mathrm{mod}\ 1)_{n}$ is uniformly distributed in
$[0,1]$ (see [23, Satz 2]). Hence both sides of equation 7 agree on a dense
subset of $[0,\frac{2}{\beta}]$. By periodicity and continuity, we conclude
that equation 7 holds for every $\omega\in{\mathbb{R}}$. Since $x$ was
arbitrary it follows that the modulus of the Gabor transform of $f$ agrees
with the modulus of the Gabor transform of $h$ on whole ${\mathbb{R}}^{2}$.
Consequently, there exists a $\tau\in{\mathbb{T}}$ such that $f=\tau h.$ ∎

In the previous theorem, the step size $\beta$ was chosen in such a way that
the sequence $(n\ \mathrm{mod}\ \frac{2}{\beta})_{n}$ is dense in
$[0,\frac{2}{\beta}]$. Clearly, we can drop the assumption on $\beta$ to be
irrational and instead work with a sampling set $\Lambda\times B$ where
$B=(b_{n})_{n}$ is such that $(b_{n}\ \mathrm{mod}\ \frac{2}{\beta})_{n}$ is
dense in $[0,\frac{2}{\beta}]$.

######  Theorem 3.7.

Let $\beta>0$ and suppose that $\Lambda=(\lambda_{n})_{n}$ and $B=(b_{n})_{n}$
are sequences in ${\mathbb{R}}$ with the following properties:

  1. 1.

The sequence $\Lambda$ has lower Beurling density
$D^{-}(\Lambda)>\frac{2}{\beta}$.

  2. 2.

The sequence $(b_{n}\ \mathrm{mod}\ \frac{2}{\beta})_{n}$ is dense in the
interval $[0,\frac{2}{\beta}]$.

Then every $f\in V_{\beta}^{1}(\varphi)$ is determined up to a global phase by
the measurements $|\mathcal{G}f(\Lambda\times B)|$ and the pair
$(V_{\beta}^{1}(\varphi),\Lambda\times B)$ satisfies condition (U). In
particular, the sequence $B$ can be chosen to be separated. In this case,
$\Lambda\times B$ constitutes a separated, discrete subset of
${\mathbb{R}}^{2}$.

Similar to the case of compactly supported functions, a rotation of the time-
frequency plane yields uniqueness results for functions which have the
property of being the fractional Fourier transform of a function in the shift-
invariant space $V_{\beta}^{1}(\varphi)$. More precisely we have

######  Proposition 3.8.

Let $\theta\in{\mathbb{R}}$,
$\beta\in{\mathbb{R}}_{>0}\setminus{\mathbb{Q}}_{>0}$ and suppose that
$\Lambda$ is a uniqueness set for $V_{\frac{2}{\beta}}^{1}(\varphi)$. Then
every $f\in{\mathcal{F}}_{-\theta}V_{\beta}^{1}(\varphi)$ is determined up to
a global phase by the measurements

$\\{|\mathcal{G}f(x,\omega)|:(x,\omega)\in
R_{\theta}(\Lambda\times{\mathbb{Z}})\\},$

where $R_{\theta}$ is the rotation matrix as defined in equation 3.

In particular, if $f$ is the product of a periodic function and the Gaussian
$\varphi$ then a rotation by 90 degrees implies the following result.

######  Corollary 3.9.

Assume that $f\in{L^{2}({\mathbb{R}})}$ factors as $f=\varphi u$ where $u\in
C^{1}({\mathbb{R}})$ is $\beta$-periodic. Let $\Lambda$ be a sequence in
${\mathbb{R}}$ with lower Beurling density $D^{-}(\Lambda)>2\beta$. If
$\beta\in{\mathbb{R}}_{>0}\setminus{\mathbb{Q}}_{>0}$ then $f$ is determined
up to a global phase by the measurements

$\\{|\mathcal{G}f(x,\omega)|:(x,\omega)\in{\mathbb{Z}}\times\Lambda\\}.$

###### Proof.

By assumptions, we can write $u$ as a Fourier series
$u(t)=\sum_{k}c_{k}e^{2\pi i\frac{1}{\beta}k}$, where the sequence
$(c_{k})_{k}$ is in $\ell^{1}({\mathbb{Z}})$. Consequently, we have

${\mathcal{F}}(\varphi u)=\sum_{k}c_{k}T_{\frac{1}{\beta}k}\varphi\in
V_{\frac{1}{\beta}}^{1}(\varphi).$

The statement follows from Theorem 3.6, since
$|\mathcal{G}f(x,\omega)|=|\mathcal{G}\hat{f}(\omega,-x)|$. ∎

##  4 Concluding remarks

We conclude the paper with some remarks about our results from Section 3 and
compare them with the previous work mentioned in Section 1.2.

  1. 1.

Let $s\in\\{-1,1\\}$ and $a\in{\mathbb{Z}}$. In both Theorem 3.1 and Theorem
3.6 the lattice ${\mathbb{Z}}\times(2c)^{-1}{\mathbb{Z}}$ and
$\frac{\beta}{2+\varepsilon}{\mathbb{Z}}\times{\mathbb{Z}}$ can be replaced by
$(s{\mathbb{N}}+a)\times(2c)^{-1}{\mathbb{Z}}$ and
$\frac{\beta}{2+\varepsilon}{\mathbb{Z}}\times(s{\mathbb{N}}+a)$,
respectively. Consequently, compactly supported functions and functions in
shift-invariant spaces with Gaussian generator are determined by spectrogram
samples lying in a half-plane.

  2. 2.

In the proof of Theorem 3.6 we started by sampling the spectrogram
$|\mathcal{G}f(x,\omega)|$ with respect to the time variable $x$. The desired
result then followed via an application of Weyl’s equidistribution theorem. On
the other hand, after applying the sampling Theorem 2.3 we could use the
identity $\mathcal{G}f(x,\omega)=e^{-2\pi
ix\omega}\mathcal{G}\hat{f}(\omega,-x)$ and then apply the Fourier transform
with respect to the second argument as we did in Theorem 3.1, the case of
compactly supported signals. This approach would yield the identity

${\mathcal{F}}|\mathcal{G}\hat{f}(\omega,\cdot)|^{2}(x)=\langle(\hat{f})_{x},T_{\omega}\varphi_{x}\rangle.$

Since $f$ is an element of $V_{\beta}^{1}(\varphi)$, the function
$(\hat{f})_{x}$ is certainly not compactly supported and Zalik’s theorem,
Theorem 2.6, is not applicable. At this point, one could use the following
generalization of Zalik’s theorem [26, Theorem 2].

######  Theorem 4.1.

Let $\psi(t)=ae^{-b^{2}(t-c)^{2}}$ be a Gaussian where $a,b>0$ and
$c\in{\mathbb{R}}$. Let $\Omega=(\omega_{n})_{n}$ be a sequence of distinct
real numbers and denote by $S(\varepsilon)$ the series

$\sum_{n}{}^{{}^{\prime}}\frac{1}{|\omega_{n}|^{2+\varepsilon}}.$

The divergence of $S(\varepsilon)$ for some some $\varepsilon>0$ suffices for
the sequence $(T_{\omega_{n}}\psi)_{n}$ to be complete in
${L^{2}({\mathbb{R}})}$, whereas the divergence of $S(0)$ constitutes a
necessary condition for completeness of $(T_{\omega_{n}}\psi)_{n}$ in
${L^{2}({\mathbb{R}})}$.

However, an application of this result is not meaningful for our purposes. For
if $\Omega$ is separated then $S(\varepsilon)$ certainly converges for every
$\varepsilon>0$. Therefore, this approach would yield sampling sets
$\Lambda\times\Omega$ which do not consist of separated points.

  3. 3.

At the beginning of the present paper it was shown that if
$e^{i\alpha_{1}}{\mathbb{R}},e^{i\alpha_{2}}{\mathbb{R}}\subset{\mathbb{C}}$
are two lines which satisfy the condition
$\alpha_{1}-\alpha_{2}\notin\pi{\mathbb{Q}}$ then the pair
$({L^{2}({\mathbb{R}})},e^{i\alpha_{1}}{\mathbb{R}}\cup
e^{i\alpha_{2}}{\mathbb{R}})$ satisfies condition (U). This was a consequence
of Jaming’s theorem [16]. A crucial step in the proof of Jaming’s theorem lies
in the observation that if $f$ is a non-zero entire function then for
arbitrary $z\in{\mathbb{C}}\setminus\\{0\\}$ the orbit
$D=\\{ze^{2i(\alpha_{1}-\alpha_{2})n}:n\in{\mathbb{Z}}\\}$ cannot be contained
in the zero set of $f$. For if $\alpha_{1}-\alpha_{2}\notin\pi{\mathbb{Q}}$
then by Weyl’s criterion the set $D$ contains a limit point. In Theorem 3.6,
Weyl’s criterion was used to obtain uniqueness results from fully discrete
measurements.

  4. 4.

Since the Fourier transform of the Gaussian $\varphi$ does not vanish, it is a
consequence of Wiener’s Tauberian theorem and the continuity of the
translation $\tau\mapsto T_{\tau}$ that the set

$\bigcup_{\beta\in{\mathbb{R}}_{>0}\setminus{\mathbb{Q}}_{>0}}V_{\beta}^{1}(\varphi)$

is dense in $L^{2}({\mathbb{R}})$. Theorem 3.6 implies the qualitative result
that the class of functions which are determined up to a global phase from
$|\mathcal{G}f(X_{f})|$, where $X_{f}$ is a lattice in ${\mathbb{R}}^{2}$
depending on $f$, constitutes a dense subset of $L^{2}({\mathbb{R}})$.

Acknowledgements. The authors would like to thank Karlheinz Gröchenig for
helpful remarks concerning Section 2.

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# Topological Analysis of Mouse Brain Vasculature via 3D Light-sheet
Microscopy Images

###### Abstract

Vascular networks play a crucial role in understanding brain functionalities.
Brain integrity and function, neuronal activity and plasticity, which are
crucial for learning, are actively modulated by their local environments,
specifically vascular networks. With recent developments in high-resolution 3D
light-sheet microscopy imaging together with tissue processing techniques, it
becomes feasible to obtain and examine large-scale brain vasculature in mice.
To establish a structural foundation for functional study, however, we need
advanced image analysis and structural modeling methods.

Existing works use geometric features such as thickness, tortuosity, etc.
However, geometric features cannot fully capture structural characteristics
such as the richness of branches, connectivity, etc. In this paper, we study
the morphology of brain vasculature through a topological lens. We extract
topological features111Our code is provided at
https://github.com/TopoXLab/VesselAnalysis based on the theory of topological
data analysis. Comparing of these robust and multi-scale topological
structural features across different brain anatomical structures and between
normal and obese populations sheds light on their promising future in studying
neurological diseases.

Index Terms—  Vasculature, Topological Data Analysis, Persistent Homology

##  1 Introduction

Vascular networks play a crucial role in understanding brain functionalities.
Neuronal activity and plasticity, which are crucial for learning, are actively
modulated by their local environments, including glial, lymphatic and vascular
networks. Despite only comprising 2% of the body weight, brain consumes 20% of
total body energy, which is supplied through the vascular system [1, 2, 3].
Local hemodynamics (the dynamics of blood flow) in the brain is tightly
correlated with the surrounding neuronal activity, continuously providing
nutrients and bloodborne factors and eliminates waste metabolites to regulate
neuronal activity [4, 5, 6, 7]. The coupling between neuronal activity and
blood flow, termed neurovascular coupling, is linked to various physiological
and pathological brain conditions. Chronic neurovascular coupling dysfunction
has been implicated as a key etiology underlying vascular diseases (e.g.,
stroke and vascular dementia), neurodegenerative disease (e.g., Alzheimer’s
disease and Parkinson’s disease), sleep disorders and psychiatric disorders
(e.g., autism). Many of these pathological conditions are associated with
learning deficits. Thus, it is of great importance to understand the function
of brain vasculature in learning.

However, current understanding of hemodynamics in neuroscience is far from
sufficient. Existing studies use Functional Magnetic Resonance Imaging (fMRI),
but lack the fundamental information on detailed brain vasculature structures
and cannot account for the highly variable microvascular structures across
individual subjects. We need novel statistical and machine learning structural
models of brain vasculature to provide important anatomical substrates for
functional and mechanistic studies in both physiological and pathological
conditions. With recent developments in high-resolution 3D light-sheet
microscopy imaging together with tissue processing techniques, it becomes
feasible to obtain and examine large-scale brain vasculature in mice [8]. To
establish a structural foundation for functional study, however, we need
advanced image analysis and structural modeling methods.

Existing works [9, 10, 11, 12] studied mouse brain vasculature using geometric
features such as thickness, tortuosity, etc. However, these geometric features
cannot fully capture structural characteristics such as the richness of
branches, connectivity, etc. These structural cues can be crucial for
understanding the physiology, hemodynamics, and eventually brain
functionality.

In this paper, we propose for the first time a suite of topological features
for the analysis of mouse brain vasculatures. Our features are based on the
theory of topological data analysis [13, 14]. In particular, using the theory
of persistent homology [15], we extract topological structural information in
a robust and multi-scale manner. A thorough analysis of these features on
mouse brain vasculature extracted from light-sheet images shows the promise of
these topological features in revealing insights into biological and clinical
outcomes.

##  2 Dataset Description

We tested our topological analysis framework on a public
dataset222https://zenodo.org/records/6025935 of 3D lightsheet fluorescence
microscopy (LSFM) images of blood vessels from various brains regions from
wild-type and obese mice and the corresponding vessel segmentation [9]. Here,
we briefly summarize the image acquisition and segmentation method.

First, blood vessels in nine to twelve-week-old male C57BL/6 wild-type mice
(group: healthy), or leptin-deficient ob/ob mice (group: obese) of the same
age, by perfusing with FITC-albumin-enriched hydrogel. A specific point
mutation in the leptin gene of these mice results in the onset of
hyperlipidemia, along with related disorders including hyperglycemia,
hyperinsulinemia, and infertility. Subsequently, images at various
rostrocaudal levels in the brain, such as the striatum, cortex, midbrain, and
hippocampus, were acquired with lightsheet fluorescence microscope,
UltraMicroscope Blaze™ (LaVision BioTec (Miltenyi), Bielefeld, Germany) with
bidirectional light sheet illumination. Finally, the LSFM images were
segmented using the image segmentation workflow developed in [9] and then
manually inspected by human experts for quality control. Our experiments in
this paper were directly applied on the released segmentation without any
changes. In total, the number of images are 12 healthy + 8 obese for cortex, 9
healthy + 5 obese for hippocampus, 6 healthy + 8 obese for midbrain, and 12
healthy + 10 obese for Striatum. An example is shown in Fig. 1.

Fig. 1: Vessel skeletons at four rostrocaudal levels of the same mouse,
observed from a consistent angle, are depicted. As demonstrated, the vessels
in the striatum exhibit more complex structures, aligning with the indications
from our topological features presented in Fig.3.

##  3 Topological Features

Our research employs an advanced topological analysis framework that utilizes
the Persistent Homology Transform (PHT) to identify and extract distinct
topological features from various shapes of vasculature. These extracted
features are then rigorously assessed using a permutation test to discern
their statistical significance and variability.

###  3.1 Persistent Homology

At the foundation of our framework lies the concept of persistent homology, a
pivotal tool in topological data analysis. This method focuses on
understanding and characterizing the persistence of specific topological
features, such as loops or holes, across different scales of observation. The
definition is built on homology classes from the classic algebraic topology,
which is instrumental in formalizing holes of various dimensions within a
topological space. For instance, 0-dimensional holes represent connected
components, 1-dimensional holes embody loops, and 2-dimensional holes are akin
to voids.

The essence of persistent homology is captured in the process of filtration.
This involves the sequential construction of a topological shape, where
elements are added based on a parameter, such as radius or threshold. As the
shape undergoes transformation, we track the persistence of features like
holes, enabling a deeper understanding of the topological dynamics of the
data.

For visualization and analysis, we rely on persistence diagrams, as shown in
Fig. 2. These diagrams are pivotal in illustrating the birth and demise times
of topological features, thereby shedding light on their persistence
throughout the data. In a Cartesian coordinate system, these diagrams plot
points above the line y=x, with each point’s distance from this line
indicating the feature’s lifespan—points nearer to the line signify ephemeral
features, whereas those further away indicate more enduring characteristics.

Fig. 2: In the case of an object comprising a solid square (marked by $) and a
void square (marked by #), we apply two distinct filtration functions. These
functions are based on the distance from varying directions. Each filtration
gives rise to a unique persistence diagram. Collectively, these diagrams from
all directions form what is known as a persistence homology transform.

###  3.2 Persistent Homology Transform

Our analysis is further enhanced by the implementation of Persistent Homology
Transform (PHT), as elucidated in [16]. PHT expands upon the principles of
persistent homology, offering a more nuanced analysis of shapes and forms
within datasets. This method is particularly advantageous in scenarios where
an in-depth understanding of a dataset’s geometric structure is essential.

PHT is rooted in persistent homology, focusing on the persistence of data
features like holes or loops across diverse scales, ascertained through
filtration. Distinctively, PHT employs multiple filtration functions,
diverging from traditional approaches that rely on a single filtration
perspective.

This approach in PHT involves applying persistent homology analysis from
multiple angles, culminating in a comprehensive depiction of the data. Such a
multifaceted representation allows PHT to encapsulate the complete topological
nature of a shape or dataset. The integration of varied filtration functions
enables PHT to adeptly highlight the complex geometric variations among
different shapes and forms, as depicted in Fig. 2.

Mathematically, PHT is expressed for an object $M$ as a set of persistence
diagrams, formulated as:

$PHT(M)=\\{PD_{v_{i}}(M)\\}_{i=1}^{k}.$ (1)

Here, each persistence diagram $PD_{v_{i}}$ is derived using a distinct
filtration function based on the distance from different directions $v_{i}$,
providing a multidimensional perspective on the topological features within
the data.

###  3.3 Betti Curves and Betti Curve Transform

To overcome the challenges posed by the multi-point nature of persistence
diagrams in machine learning applications, we employ Betti curves (BC). These
curves effectively convert the complex, multidimensional data from persistence
diagrams into a streamlined, one-dimensional vector format. The Betti curve
tracks the evolution of topological features (like connected components,
holes, and voids) across varying parameters, typically linked to the scale of
observation. By observing changes in Betti numbers—which quantify these
features—the curve offers a continuous and interpretable topological profile
of the data.

Integrating Betti curves with PHT, we compute a Betti curve for each
persistence diagram within the PHT framework. This results in a novel feature
representation, termed the Betti Curve Transform (BCT):

$BCT(M)=\\{BC_{v_{i}}(M)\\}_{i=1}^{k}.$ (2)

We utilize the BCT as a key feature in differentiating vasculatures across
diverse groups. This approach not only harnesses the topological richness of
the data but also adapts it into a format amenable for advanced machine
learning analysis, enhancing our ability to discern and categorize the
intricate patterns present in vascular structures.

##  4 Experiments and Discussion

We have applied the proposed methodology to distinguish between obese and
healthy mice by analyzing the characteristics of their brain vessels. Our
findings demonstrate that the topological features we introduced are effective
in detecting differences in the hippocampus and midbrain regions.

###  4.1 Hypothesis Test

We construct filtration functions from seven different orientations, computing
a one-dimensional Betti curve for each respective filtration. These functions
are formulated based on the distance from each direction, subsequently
normalized to a range of [0, 1]. To discern features between obese and healthy
mice, we employ a t-test on the mean feature values , as shown in Fig. 3. In
this figure, a higher value generally signifies an increased number of loops
within the vessel, as we are utilizing the 1-dimensional Betti number, which
quantifies the count of loops.

Fig. 3: The p-value of a t-test on the mean feature values among each group.
For Hippocampus and Midbrain, their p-values are 0.04 and 0.07, respectively.

Maximum mean discrepancy. We also employ a permutation test on the maximum
mean discrepancy (MMD) statistics. MMD is a statistical measure designed for
comparing the distributions of two datasets, proving particularly effective in
high-dimensional spaces where conventional statistical tests may be
inadequate. MMD functions by assessing the disparity between the mean
embeddings of two distributions within a feature space. This space is
typically defined through a kernel function, denoted as $K(\cdot,\cdot)$.
Consider two sample sets, $X$ and $Y$, comprising $m$ and $n$ samples
respectively. The MMD formula is expressed as follows:

$\displaystyle MMD^{2}(X,Y)$
$\displaystyle=\frac{1}{m^{2}}\sum_{i,j=1}^{m}K(x_{i},x_{j})+\frac{1}{n^{2}}\sum_{i,j=1}^{n}K(y_{i},y_{j})$
$\displaystyle\quad-\frac{2}{mn}\sum_{i=1}^{m}\sum_{j=1}^{n}K(x_{i},y_{j}),$

where $x_{i}$ and $y_{j}$ represent individual samples from $X$ and $Y$
respectively.

To determine the statistical significance of the MMD measure, a permutation
test is employed, typically involving $3\times 10^{3}$ random permutations.
This approach helps in computing the p-value for the MMD statistics, thereby
assessing the likelihood of the observed difference occurring by chance. The
results, as shown in Table 1, indicate that the p-value for the midbrain is
$0.049$, signifying a statistically significant difference in the vessel
structures within this region between obese and healthy mice.

Table 1: P-value for MMD statistics. | Cortex | Hippocampus | Striatum | Midbrain
---|---|---|---|---
p-value | 0.258 | 0.411 | 0.506 | 0.049

###  4.2 Compare with Geometric Features

Our topological features well complements existing geometric features. In a
prior study [9], the authors employed statistical analysis on the extracted
geometrical features. Within the same experimental conditions, they observed
significant differences (p-value $<0.05$) in the vasculature of the striatum
and cortex between obese and healthy mice, while no such distinction was noted
in the midbrain and hippocampus. In contrast, our analysis reveals that the
topological features exhibit significant differences (p-value $<0.05$) in the
vasculature of the hippocampus and midbrain. This comparison suggests a
complementary relationship between geometric and topological features.

##  5 Conclusion

Overall, the novel topological analysis using persistent homology presented
here provides unprecedented opportunities to gain valuable insights into
vascular systems. By labeling of the complete vasculature and comprehensively
scanning whole organ samples, we can further investigate the intricate details
of recovery processes after events, such as ischemia, in various organs,
including the brain and heart. The ability to specifically label certain
vascular components such as arteries and veins, in conjunction with the
staining of other brain cells such as neurons, astrocytes, pericytes and
microglia, provides a solid foundation for understanding network typology.
Analyzing complete brain scans also offers a unique opportunity to combine our
innovative topological analysis with specific topographical information within
distinct brain regions. This approach holds great promise for answering
important scientific questions, including elucidating the complexity of brain
development in health and disease, evaluating the efficacy of pharmaceutical
interventions, and deciphering age-related implications. We expect the
developed topological analysis suite paves the road to significantly improve
our understanding of vascular dynamics in different physiological and
pathological contexts.

##  6 Compliance with Ethical Standards

This research study was conducted retrospectively using microscopy images from
animal studies made available in open access in [9]. Ethical approval was not
required as confirmed by the license (Creative Commons Attribution 4.0
International) attached with the original open access data.

##  7 Acknowledgement

The research of J. Yao and C. Chen was partially supported by NSF 2144901 and
NIH R21CA258493. The research of J. Chen was partially supported by the
Federal Ministry of Education and Research (Bundesministerium für Bildung und
Forschung, BMBF) in Germany under the funding reference 161L0272 and the
Ministry of Culture and Science of the State of North Rhine-Westphalia
(Ministerium für Kultur und Wissenschaft des Landes Nordrhein-Westfalen, MKW
NRW).

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