# Off-axis Aberrations Improve the Resolution Limits of Incoherent Imaging

Kevin Liang1,2  1Physics Department, Adelphi University, Garden City, NY
11530, USA
2The Institute of Optics, University of Rochester, Rochester NY 14627, USA
<EMAIL_ADDRESS>
††journal: optcon††articletype: Research Article

The presence of off-axis tilt and Petzval curvature, two of the lowest-order
off-axis Seidel aberrations, is shown to improve the Fisher information of
two-point separation estimation in an incoherent imaging system compared to an
aberration-free system. Our results show that the practical localization
advantages of modal imaging techniques within the field of quantum-inspired
superresolution can be achieved with direct imaging measurement schemes alone.

##  1 Introduction

In well-corrected imaging systems, the propagation from the object to image
planes is often modeled with shift invariance [1]. This indicates that the
profile of the imaging system’s point spread function (PSF) is independent of
object impulse’s location. Recent works have recasted the associated
Rayleigh’s criterion, which states that the separation of two nearby point
sources cannot be precisely estimated, for such systems in terms of the
classical Fisher information (CFI) [2]. The CFI informs on the precision one
expects in extracting an unknown parameter with a given measurement scheme. In
this context, Rayleigh’s criterion states that the CFI for the separation of
two point sources, with direct intensity (DI) measurements, vanishes as the
separation between the point sources approaches zero.

A shift in the well-established understanding of the Rayleigh’s criterion was
provided by Tsang and Nair [3]. They showed that the quantum Fisher
information (QFI), an upperbound for the CFIs over all possible measurement
schemes, for the separation of two equally bright incoherent point sources
remains non-zero in the sub-Rayleigh regime. This result, along with the
introduction of modal imaging techniques like spatial mode demultiplexing
(SPADE), a measurement scheme that saturates the QFI, is responsible for the
wide net of recent research efforts that sought to extend the promise of this
so-called quantum-inspired superresolution [4, 5, 6, 7, 8, 9, 10]. Such
extensions include theoretical work in which strides have been made regarding
the generalization of the object’s spatial and coherence details [11, 12, 13,
14, 15, 7, 16]. Regarding experimental considerations, the advantages of
SPADE-type measurements have been tested and confirmed for a variety of
objects [17, 8].

In this work, calculations are provided for the CFI and QFI for the separation
between two equally bright incoherent point sources in imaging systems which
possess off-axis/field-dependent aberrations. The presence of these
aberrations causes imaging systems to be shift-variant; the analysis of such
systems have been thus far neglected in the context of quantum-inspired
superresolution. Remarkably, we show that the inclusion of simple low-order
off-axis Seidel aberrations can give rise to more precise estimation of source
separation when compared to an aberration-free system by improving on the CFI
and QFI. In particular, we provide the analysis for off-axis tilt (OAT) and
Petzval curvature. Both are shown to improve the CFI for both DI and modal
imaging measure schemes, with the former providing a global improvement and
the latter giving rise to improvements in the sub-Rayleigh regime. To derive
these results, a general framework for shift-variant imaging systems is
introduced in Section 2, from which a specialized treatment for two equally
bright incoherent point sources is considered. Our results, given in Section
3, provide context to the nascent field of quantum-inspired superresolution
and show that much improvement can be made through the intentional usage of
off-axis aberrations.

##  2 Theoretical Background

###  2.1 Shift variant imaging systems

In a shift-invariant imaging system, the optical field at the image plane may
be obtained from the optical field at the object plane via a convolution with
the system’s PSF, $\psi$ [1]. In the presence of off-axis aberrations,
however, the relation between object field, $U_{o}$, and image field, $U_{i}$,
is given by a more general integral transformation. For simplicity, we assume
one transverse spatial coordinate for which the aforementioned transformation
is given by

$\displaystyle U_{i}(x)=\int_{-\infty}^{\infty}U_{o}(\xi)\psi(x,\xi)\,{\rm
d}\xi,$ (1)

where $\xi$ and $x$ are the object and image plane spatial coordinates,
respectively. The departure from a shift-invariant imaging system is capture
by $\psi(x,\xi)$, which indicates that the system PSF may have a dependence on
both $\xi$ and $x$ that is not confined to their difference $x-\xi$. The form
for $\psi(x,\xi)$ may be obtained by considering the propagation of a quasi-
monochromatic point source with wavenumber $k=2\pi/\lambda$, modeled as a
Dirac delta impulse, located at $\xi$, through the imaging system. This
impulse is mapped onto a linear phase factor at the pupil plane. There, the
linear phase factor is modulated by a Gaussian pupil function of
characteristic width $\sigma_{p}$ and a phase aberration function, $\Delta W$.
The choice of a Gaussian pupil function, commonly made in related works, is
used for mathematical convenience [3, 12, 11]. An inverse Fourier
transformation is then performed on the pupil-plane field to give

$\displaystyle\psi(x,\xi)=U_{0}\int_{-\infty}^{\infty}\exp\left(-{\rm
i}k\frac{\xi
u}{f}\right)\exp\left(-\frac{u^{2}}{4\sigma_{p}^{2}}\right)\exp\left[-{\rm
i}k\Delta W(u,\xi)\right]\exp\left({\rm i}k\frac{ux}{f}\right)\,{\rm d}u,$ (2)

where $U_{0}$ is a constant with dimensions of optical field that ensures the
intensity-normalization of $\psi$. Furthermore, $u$ is the pupil-plane spatial
coordinate, and $f$ is the focal length of the lenses in a $4f$ imaging system
(although the results presented in this work are valid for arbitrary other
imaging configurations); the Fourier optical details regarding Eq. (2) is
described further in Supplement 1. A schematic of the imaging system is shown
in Fig. 1. It will be useful to define $\sigma=f\lambda/(4\pi\sigma_{p})$ as
the characteristic width of the diffraction-limited [$\Delta W(u,\xi)=0$] PSF.
Notice, as is true for the case of a shift-invariant imaging system, the
system PSF would explicitly be a function of the difference in coordinates
$x-\xi$ if $\Delta W$ were independent of the object location $\xi$. Such on-
axis/field-independent aberrations include well-known Zernike terms such as
defocus and spherical aberration. Our work focuses instead on examples of
$\Delta W$ which depend on $\xi$.

Figure 1: A schematic of a $4f$ imaging system with $\xi,u,$ and $x$ as the
spatial coordinates for the object, pupil, and image planes, respectively. An
illustration of two point sources separated by $s$ is shown along with their
image distributions in the presence of OAT and Petzval curvature of strengths
$T$ and $P$, respectively, which are defined in Eq. (4). The individual PSFs
have a half-width of $g$ (which depends on $P$), defined in Eq. (6), and their
separation is magnified by a factor of $1+2\pi T\sigma$. The pupil is modeled
as a Gaussian with half-width $\sigma_{p}$.

Before proceeding to the analysis of off-axis aberrations, we note that the
spatial profile of $\psi$ is affected by the strength of the aberration
$\Delta W$. For the purposes of quantifying this strength in later sections,
it is useful to use $2\sigma_{p}$ and $2\sigma$ as standard lengths for the
pupil coordinate $u$ and the object location $\xi$, respectively. The reason
for this choice is due to (1) the Gaussian pupil function strongly attenuating
any signal at the pupil plane located at $|u|>2\sigma_{p}$ and (2) the context
of quantum-inspired resolution is most relevant for objects located at
$|\xi|<2\sigma$; that is, for objects whose features are below twice the
diffraction-limited PSF. This regime is henceforth referred to as the sub-
Rayleigh regime.

###  2.2 Off-axis aberrations

From Eq. (1), it is clear that the PSF of a shift variant system is determined
by the aberration function $\Delta W(u,\xi)$. Although there is an infinitude
of options; we restrict our analysis to the case where

$\displaystyle\Delta
W(u,\xi)=W_{111}\left(\frac{\xi}{2\sigma}\right)\left(\frac{u}{2\sigma_{p}}\right)+W_{220}\left(\frac{\xi}{2\sigma}\right)^{2}\left(\frac{u}{2\sigma_{p}}\right)^{2},$
(3)

which is recognized as the superposition of the two low-order off-axis Seidel
aberrations commonly called OAT and Petzval curvature [18]. Notice that Eq.
(3) is written in terms of ratios for $\xi$ and $u$ so that $W_{111}$ and
$W_{220}$ can be interpreted as the maximal [when $\xi=2\sigma$ and
$u=2\sigma_{p}$, as discussed in the paragraph following Eq. (2)] optical path
difference caused by OAT and Petzval curvature, respectively. By further
defining $T$ and $P$ as

$\displaystyle
T\triangleq\frac{W_{111}}{\lambda\cdot(2\sigma)}\quad\text{and}\quad
P\triangleq\frac{W_{220}}{\lambda\cdot(2\sigma)^{2}},$ (4)

to be OAT and Petzval curvature strength parameters, respectively (they
measure rates, for $u=2\sigma_{p}$, at which the phase difference caused by
OAT and Petzval curvature, respectively, increases as a function of $\xi$),
one finds that the intensity-normalized shift-variant PSF is given through Eq.
(2) by

$\displaystyle\psi(x,\xi;P,T)=\frac{1}{[2\pi
g^{2}(\xi;P)]^{1/4}}\exp\left\\{-\frac{[x-\xi(1+2\pi
T\sigma)]^{2}}{4g^{2}(\xi,P)}+{\rm i}\Phi(x,\xi;P,T)\right\\},$ (5)

where

$\displaystyle g(\xi;P)\triangleq\sigma\sqrt{1+4\pi^{2}P^{2}\xi^{4}},$ (6)

is the characteristic width of $\psi$ and

$\displaystyle\Phi(x,\xi;P,T)\triangleq-\frac{1}{2}\left\\{\tan^{-1}(2\pi
P\xi^{2})+\frac{\pi P\xi^{2}[x-\xi(1+2\pi
T\sigma)]^{2}}{g^{2}(\xi;P)}\right\\},$ (7)

is the phase of $\psi$. Equation (5) shows why the choice of Eq. (3) was used:
OAT and Petzval curvature cause intuitive changes in the PSF’s mean location
and width, respectively. Therefore, much insight can be obtained from studying
Eq. (3) since, while other off-axis aberrations may induce higher order
effects on $\psi$, the effects of OAT and Petzval is expected to provide a
useful leading-order description.

OAT and Petzval curvature, among other Seidel aberrations, are common in
typical imaging systems and have well-known interpretations particularly in
the language of geometrical (ray) optics. For a system with OAT, the plane
wave emerging from the pupil associated with each object location $\xi$ is
proportionally tilted (posses an additional linear phase). This often arises
from unintentional tilts of the optics within the system and causes an
incorrect magnification in the image. This magnification factor, according to
Eq. (5) and Fig. 1, is $(1+2\pi T\sigma)$; in the case of an object scene with
two point sources, this magnification translates to an amplification of the
separation in the image field. Petzval curvature, on the other hand, describes
an imaging system in which the image is perfect when the intensity is measured
along a curved surface. If the intensity is instead measured, as it usually
is, along a flat surface, off-axis object points $\xi$ are mapped to a shift-
variant PSF whose width increases with $|\xi|$. This width, according to Eqs.
(5) and (6), this width is $g(\xi;P)$.

For the aberration-free case ($P=0$ and $T=0$), Eq. (5) reduces to the shift-
invariant Gaussian PSF with width $\sigma$. Furthermore, it should be noted
that the proceeding comparisons of Fisher information (FI) values between
systems where the PSF is given by $\psi(x,\xi;P,T)$ and the aberration-free
case $\psi(x,\xi;0,0)$ is fair, since $g(\xi;P)\geq\sigma$. In other words,
the presence of OAT and Petzval curvature does not reduce the diffraction-
limited spot size and therefore the improved FI is not attributed to the
reduction of $\sigma$.

###  2.3 Measurement schemes

Although this work will primarily be concerned with incoherent imaging, a
general framework is provided here for completeness and comprehension. To this
end: the density matrix, which represents the cross-spectral density of the
optical field at the image plane, for the case of two partially coherent point
sources with intensity ratio $A$, a known centroid taken to be at the origin
of the coordinate system, separated by a distance $s$ is given by

$\displaystyle\rho(s;P,T)=\frac{|\psi_{+}\rangle\langle\psi_{+}|+A|\psi_{-}\rangle\langle\psi_{-}|+\Gamma|\psi_{+}\rangle\langle\psi_{-}|+\Gamma^{*}|\psi_{-}\rangle\langle\psi_{+}|}{1+A+2\text{Re}[\Gamma
d(s;P,T)]},$ (8)

which is expressed over the non-orthogonal, intensity-normalized basis
$\\{|\psi_{+}\rangle,|\psi_{-}\rangle\\}$. These basis kets are defined over
position-space through Eq. (1) as

$\displaystyle|\psi_{\pm}\rangle$
$\displaystyle\triangleq\int_{-\infty}^{\infty}\psi_{\pm}(x;P,T)|x\rangle\,{\rm
d}x\triangleq\int_{-\infty}^{\infty}\psi\left(x,\pm\frac{s}{2};P,T\right)|x\rangle\,{\rm
d}x.$ (9)

Furthermore,

$\displaystyle
d(s;P,T)\triangleq\langle\psi_{-}|\psi_{+}\rangle=\int_{-\infty}^{\infty}\psi\left(x,\frac{s}{2};P,T\right)\psi^{*}\left(x,-\frac{s}{2};P,T\right)\,{\rm
d}x$ (10)

is the field-overlap integral between the two imaged PSFs and $\Gamma$ is a
complex-valued coherence parameter whose values are restricted by the
condition $|\Gamma|\leq A$. The normalization factor in Eq. (8) indicates that
the density matrix $\rho(s)$ uses the image-plane normalization scheme, in
which the unit-trace condition on $\rho$ is ensured by the total number of
received photons at the image plane. This choice is in contrast to the object-
plane normalization scheme; a detailed discussion of the two options is
provided elsewhere.

The act of performing a measurement on the received photons is captured
mathematically through computing the modulus-square coefficients over a
corresponding set of projections of $\rho$. These coefficients are taken to be
the probabilities of finding an image-plane photon in a certain mode (which
may be continuous or discrete). In this work, we focus on comparing the
measurement schemes of DI and SPADE. For DI, the set of projection modes is
the continuous position basis $\\{|x\rangle\\}$; the probability density of
finding a photon in the $x$ position of the image plane (conditioned on a
detection event) is given by

$\displaystyle p_{\text{I}}(x,s;P,T)$ $\displaystyle=\langle
x|\rho(s;P,T)|x\rangle$
$\displaystyle=\frac{|\psi_{+}(x;P,T)|^{2}+|\psi_{-}(x;P,T)|^{2}+2\text{Re}\left[\Gamma\psi_{+}(x;P,T)\psi_{-}(x;P,T)\right]}{1+A+2\text{Re}[\Gamma
d(s;P,T)]}.$ (11)

In Section 3, we focus on the case where $\Gamma=0$ and $A=1$ (equally bright
incoherent point sources). Figure 2 is provided for the visualization of
$p_{\text{I}}(x,s;P,T)$, which is a normalized version of the image-plane
intensity distribution, for various levels of OAT and Petzval curvature. The
aberration-free case shown in Fig. 2(a) shows the usual intensity pattern of
two Gaussian PSFs converging as the separation $s$ vanishes. The effects of
non-zero values of $T$ and $P$ are illustrated in Figs. 2(b) - (d): OAT causes
the intensity distributions from the two point sources to converge at a
different rate due to the magnification factor $(1+2\pi T\sigma)$ and Petzval
curvature gives the intensity distributions a $s$-dependent width. Regarding
OAT, it is clear from the comparison of Fig. 2(a) and (b) that non-zero values
of $T$ allows for the two point sources’ PSFs to be more easily distinguished
for smaller values of $s/\sigma$. This effect, which will be detailed later,
is responsible for larger values of CFI when using DI for imaging systems with
OAT. Petzval curvature, on the other hand, does not have a similarly clear
benefit in $s$-estimation whe compared to OAT. The resolution advantages
offered by non-zero values of $P$ instead comes from the sensitivity of the
width $p_{\text{I}}$ as $s$ approaches zero.

Figure 2: Density plots of $p_{\text{I}}(x,s;P,T)$ over $x/\sigma$ and
$s/\sigma$ for various values of $T\sigma$ and $P\sigma^{2}$ in (a) - (d). In
each, the solid red lines show the location of the two point sources. The
dashed blue lines in (b) and (d) show the centers of intensity distribution
from each point source for non-zero OAT. The dashed yellow lines in (c) and
(d) show the outer envelope of $p_{\text{I}}$ for non-zero Petzval curvature.

For SPADE, the projection is done over any set of discrete orthonormal modes.
A natural choice is the $\sigma$-matched Hermite-Gauss (HG) basis
$\\{\phi_{q}(x)\\}_{q=0}^{\infty}$, where the $q$-th mode is defined as

$\displaystyle\phi_{q}(x)=\frac{1}{(2\pi\sigma^{2})^{1/4}}\frac{1}{\sqrt{2^{q}q!}}H_{q}\left(\frac{x}{\sqrt{2}\sigma}\right)\exp\left(-\frac{x^{2}}{4\sigma^{2}}\right),$
(12)

and $H_{q}$ is the $q$-th physicist’s Hermite polynomial. The probability of
finding a photon (conditioned on a detection event) in the $q$-th mode is
therefore the projection

$\displaystyle p_{\text{II}}(q,s;P,T)$
$\displaystyle=\langle\phi_{q}|\rho(s;P,T)|\phi_{q}\rangle$
$\displaystyle=\frac{|\langle\phi_{q}|\psi_{+}\rangle|^{2}+|\langle\phi_{q}|\psi_{-}\rangle|^{2}+2\text{Re}\left(\Gamma\langle\phi_{q}|\psi_{+}\rangle\langle\psi_{-}|\phi_{q}\rangle\right)}{1+A+2\text{Re}[\Gamma
d(s;P,T)]}.$ (13)

The probability densities given by Eqs. (11) and (13) for DI and SPADE,
respectively, are needed in the calculation of the CFI of either measurement
scheme. For the case where only the separation $s$ is an unknown parameter,
the corresponding CFIs for DI and SPADE are given by

$\displaystyle F_{\text{I}}(s;P,T)$
$\displaystyle=\int_{-\infty}^{\infty}\frac{1}{p_{\text{I}}(x,s;P,T)}\left[\frac{\partial}{\partial
s}p_{\text{I}}(x,s;P,T)\right]^{2}\,{\rm d}x,$ (14)

and

$\displaystyle F_{\text{II}}(s;P,T)$
$\displaystyle=\sum_{q=0}^{\infty}\frac{1}{p_{\text{II}}(q,s;P,T)}\left[\frac{\partial}{\partial
s}p_{\text{II}}(q,s;P,T)\right]^{2},$ (15)

respectively. Although a full background on SPADE has been presented thus far,
it is sufficient to consider a simplified version of SPADE known as binary
SPADE (BSPADE) when one is interested in the sub-Rayleigh regime. In BSPADE, a
photon arriving from the object plane is measured either in the $q=0$ mode or
in the combined $q>0$ mode. Intuition for this simplification comes from the
fact that, as the object shrinks, most of the photons arriving at the image
plane will project into the lowest order modes and the higher order modes
contains progressively less information. In this case, the summation in Eq.
(15) simplifies and can can be explicitly expressed as

$\displaystyle
F_{\text{II}}(s;P,T)\approx\frac{1}{p_{\text{II}}(0,s;P,T)-p_{\text{II}}^{2}(0,s;P,T)}\left[\frac{\partial}{\partial
s}p_{\text{II}}(0,s;P,T)\right]^{2}.$ (16)

The quantities $F_{\text{I}}$ and $F_{\text{II}}$ are valuable in that, via
estimation theory, their reciprocals provide lower bounds for the variance in
the estimation of $s$ when the corresponding measurement scheme is used. In
other words, if $\check{s}_{\text{I}}$ and $\check{s}_{\text{II}}$ are
estimators for $s$ constructed from data acquired via DI or BSPADE
measurements, respectively, then

$\displaystyle\text{Var}(\hat{s}_{\text{I}})\geq
F_{\text{I}}^{-1}\quad\text{and}\quad\text{Var}(\hat{s}_{\text{II}})\geq
F_{\text{II}}^{-1}.$ (17)

The CFI for DI and BSPADE measurements are compared for various values of $T$
and $P$ in Section 3. There, it is shown that non-zero values of these
aberration strength parameters can lead to larger values of $F_{\text{I}}$ and
$F_{\text{II}}$.

###  2.4 Quantum FI

In addition to the CFI, which depends on the choice of measurement scheme, it
is informative to consider the QFI, $Q_{s}$, which provides an upperbound to
all possible CFI once the transformation between object and image planes is
stipulated. In other words,

$\displaystyle Q_{s}\geq F_{\text{I}}\quad\text{and}\quad Q_{s}\geq
F_{\text{II}}.$ (18)

As is standard in deriving the QFI, one first finds the symmetric logarithm
derivative (SLD), $L_{s}$, associated with the separation parameter $s$. The
SLD is defined implicitly through

$\displaystyle\frac{\partial\rho(s;P,T)}{\partial s}$
$\displaystyle=\frac{\rho(s;P,T)L_{s}+L_{s}\rho(s;P,T)}{2},$ (19)

where $\rho$ is the image-plane density matrix given by Eq. (8); one should
note that this prescription of $\rho$ corresponds to the image-plane
normalization scheme detailed in []. With $L_{s}$ in hand, the QFI for $s$ is
immediately obtained via

$\displaystyle Q_{s}(s;P,T)$
$\displaystyle=\text{Tr}\left[\frac{\partial\rho(s;P,T)}{\partial
s}L_{s}\right].$ (20)

Details pertaining to the derivation of the QFI for imaging systems with off-
axis aberrations are found in Supplement 1. It should be emphasized that such
calculations for the separation QFI for imaging systems in which the PSF is
shift-variant require care and themselves constitute a novel result from which
interesting discussions may arise. However, to keep the focus on well-known
measurement schemes like DI and BSPADE, the QFI will primarily serve as a
confirmation of CFI results through Eq. (18).

##  3 Results and Discussion

###  3.1 Results

Although the framework developed in Section 2 is general, we now specialize to
the case of two equally bright incoherent point sources. This choice
corresponds to the selection of $A=1$ and $\Gamma=0$ in Eqs. (8), (11), and
(13). We are interested in calculating the CFI for DI and BSPADE for various
values of $P$ and $T$ and comparing them to the case where the imaging system
is aberration-free (and therefore shift-invariant), i.e., for $P=0$ and $T=0$.

In the following, the CFI for DI, given by Eq. (14), is calculated
numerically. For BSPADE, it can be shown using Eqs. (5), (12), and (13) that

$\displaystyle
p_{\text{II}}(0,s;P,T)=\frac{1}{\sqrt{1+P^{2}\pi^{2}(s/2)^{4}}}\exp\left\\{-\frac{(s/2)^{2}(1+2\pi
T\sigma)^{2}}{4\sigma^{2}[1+P^{2}\pi^{2}(s/2)^{4}]}\right\\},$ (21)

which may be inserted into Eq. (16) to obtain $F_{\text{II}}$. Comparisons of
$F_{\text{I}}$ and $F_{\text{II}}$ for various values of $T$ and $P$ are
displayed in Fig. 3. When considering the $P=0$ case (the imaging system only
has OAT and no Petzval curvature), Fig. 3(a) shows that increasing values of
$T$ leads to greater $F_{\text{I}}$ for all values of $s/\sigma$: this global
improvement in $F_{\text{I}}$ for non-zero values of $T$ is one of the main
results of this analysis. For BSPADE, Fig. 3(b) shows that a similar
improvement in $F_{\text{II}}$ occurs when $T\neq 0$ in the sub-Rayleigh
regime. Additionally, notice that $F_{\text{II}}$ saturates the QFI at exactly
$s=0$ for all values of $T$, which indicates that BSPADE continues to be an
optimal measurement for $s$ even for shift-variant imagine systems. However,
we re-emphasize that the main purpose of this analysis to to show that
introducing off-axis aberrations in a system can vastly improve CFI even if
one only considers a DI measurement scheme. The introduction of OAT with
strength $T=0.4\sigma^{-1}$ leads to very high information (in comparison with
the QFI value of $0.25$ in the aberration-free case) well into the sub-
Rayleigh regime ($s<2\sigma$). That is, even though $F_{\text{I}}=0$ at
exactly $s=0$ always, one in principle can obtain any $F_{\text{I}}$ for non-
zero $s/\sigma$ by increasing the strength of OAT even if $s/\sigma$ is small.

Figure 3: CFI (solid curves) associated with the separation $s$ for (a,c) DI
and (b,d) BSPADE are shown for the case of (a,b) $P=0$, varying $T$ and (c,d)
$T=0$, varying $P$. The QFI (dashed curves), $Q_{s}$, is also plotted for the
corresponding values of $P$ and $T$. The black curves in each plot correspond
to the aberration-free (shift-invariant) imaging system.

Figures 3(c) and (d) consider the case where the imaging system contains
Petzval curvature, but no OAT. It can be seen, in a behavior similar to that
seen in Figs. 3(a) and (b) that larger values of $P$ lead to larger values of
$F_{\text{I}}$ and $F_{\text{II}}$ in the sub-Rayleigh regime, with the latter
saturating the QFI near $s=0$. Therefore, like OAT, Petzval curvature can
improve the performance of an imaging system regarding two-point separation
estimation. However, there are some notable differences between the behaviors
of the CFI when the system contains OAT and when it contains Petzval
curvature. First, increasing $P$ does not affect the value of the
$F_{\text{II}}$ at exactly $s=0$ ($F_{\text{I}}=0$ regardless of $P$ and $T$).
In other words, all the curves seen in Fig. 3(d) coincide in the limit of
vanishing separation. Second, there is an interval of $s$ within the sub-
Rayleigh regime in which DI outperforms BSPADE. For example, comparing the
$P=0.4\sigma^{-2}$ case for $F_{\text{I}}$ and $F_{\text{II}}$ in Figs. 3(c)
and (d), repsectively, it can be seen that the former reaches a peak of
roughly $1.5$ compared to the latter’s $0.75$. Finally, we note that Petzval
curvature may lead to the formation of local minima in $F_{\text{I}}$ and
$F_{\text{II}}$, which indicate that the corresponding measurement schemes
perform optimally neither at exactly $s=0$ nor at $s\rightarrow\infty$, but
rather at some intermediate value that, according to Figs. 3(c) and (d), occur
in the sub-Rayleigh regime.

For completeness, CFI and QFI curves for the case where both OAT and Petzval
curvature are present are shown in Fig. 4 for a fixed value of
$T=0.2\sigma^{-1}$ and varying values of $P$. As was observed from Fig. 3,
larger values of $T$ raise the maximal attainable information while various
values of $P$ lead to CFI curves that peak locally in the sub-Rayleigh regime
[although such peaks for the BSPADE CFIs in Fig. 4(b) are not apparent for
smaller values of $P$ since the nonzero value of $T$ raises the CFIs at
$s=0$].

It is prudent to point out the behavior of the QFI curves in Figs. 3 and 4 as
well. When only OAT is present, the QFI remains constant over $s$, with its
value given by

$\displaystyle Q_{s}(s;0,T)=F_{\text{II}}(0;0,T)=\frac{(1+2\pi
T\sigma)^{2}}{4\sigma^{2}},$ (22)

which indicates that $Q_{s}(s;0,T)$ grows quadratically as a function of $T$.
On the other hand, when Petzval curvature is present, the QFI increases from
the value given by Eq. (22) at $s=0$ to larger values as $s$ increases. This
peculiar behavior, which is had not been seen for incoherent quantum-inspired
superresolution studies of shift-invariant systems, is a result of the
presence of the second term in the phase, $\Phi$, of the PSF given by Eq. (7).
This quadratic (in $x$) phase contribution is due to the presence of Petzval
curvature in $\Delta W$, as stipulated in Eq. (3); Supplement 1 provides more
insight regarding the QFI’s behavior.

Figure 4: CFI (solid curves) associated with the separation $s$ for (a) DI and
(b) BSPADE are shown for the case of $T=0.2\sigma^{-1}$, varying $P$. The QFI
(dashed curves), $Q_{s}$, is also plotted for the corresponding values of $P$
and $T$.

Maximum likelihood estimation (MLE) simulations for the separation $s$ were
performed for DI to support the findings from Eq. (14) and Fig. 3(a) and (c).
For the $i$-th (out of $M$) iteration of the simulation, the number of photons
arriving at the image plane, $N_{i}$, was chosen using Poisson statistics
around an average photon number of $N$. That is, the probability mass function
for $N_{i}$ is given by

$\displaystyle p(N_{i})=\frac{N^{N_{i}}\exp(-N)}{N_{i}!}.$ (23)

Once $N_{i}$ is chosen, an equivalent number of photon positions in the image
plane is chosen using Eq. (11) as the probability density function for $N_{i}$
independent events. Given these $N_{i}$ photon locations, the unbiased MLE
estimator, denoted as $\hat{s}$, is used to find an estimate for the
separation $s$. The variance of $\hat{s}$, whose average value is $s$, is then
calculated over the ensemble of $M$ iterations. In order to compare the
simulation results to $F_{\text{I}}$, one must divide the reciprocal of the
aforementioned variance by $N$ to compute the per-photon CFI. This process is
repeated for different values of $s,T,$ and $P$ and the results of these
simulations are shown in Fig. 5 over the sub-Rayleigh regime $(s<2\sigma)$ for
$N=2000$ photons and $M=500$ iterations. As is done in Figs. 3(a) and (c), the
cases of $P=0$ and $T=0$ were analyzed, respectively. It is evident that there
is good agreement between the simulation results and $H_{\text{I}}$.

Figure 5: Comparisons of $F_{\text{I}}$ (solid curves) and
$\text{Var}^{-1}(\hat{s}-s)/N$ from MLE simulations (discrete points) for the
cases of (a) $P=0$, varying $T$ and (b) $T=0$, varying $P$. A mean photon
number of $N=2000$ was used for each iteration (each value of $s$) and the
variance was calculated over $M=500$ iterations.

###  3.2 Discussion

The primary result of this work is that it is possible to improve upon the
two-point resolution of an aberration-free imaging system by intentionally
introducing known off-axis aberrations. Particularly, this is true even for DI
measurement schemes as seen in Figs. 3 and 4. In fact, whether the improvement
is global (over all $s$, as it is when OAT is present) or only over the sub-
Rayleigh regime (when only Petzval curvature is present), the improvement can
exceed the improvements offer by modal imaging techniques like BSPADE for
aberration-free systems. In other words, if a sufficient amount of off-axis
aberration can be provided, DI imaging schemes can lead to large
$F_{\text{I}}$ even in the sub-Rayleigh regime. Since practical applications
of two-point imaging deal with non-zero values of $s/\sigma$, increasing $T$
can effectively allow DI measurement schemes to attain the same information
allowed by BSPADE in shift-invariant systems. However, this is not to say that
BSPADE (and other modal imaging schemes) are fruitless; it is clear from Figs.
3 and 4 that BSPADE benefits in a similar fashion to DI when off-axis
aberrations are introduced. Importantly, at least for OAT and Petzval
curvature, BSPADE still (as it did for the aberration-free case) yield a
$F_{\text{II}}$ that saturates the QFI in the sub-Rayleigh regime.

The findings in this work are somewhat counter-intuitive as many traditional
imaging systems start with substantial aberrations and a significant amount of
effort is often needed to minimize such errors towards the goal of an
aberration-free (diffraction-limited) system. However, at least in the context
of resolving two incoherent and equally bright point sources where the
separation is the only unknown parameter, it turns out that the presence of
off-axis aberrations can actually improve performance. It may be of interest
then, given the number of well-corrected imaging systems in existence, to
understand how one may intentionally introduce OAT and Petzval curvature back
into these imaging systems in order to take advantage of the larger DI or
BSPADE CFI. Of course, it is possible to purposely (re)introduce aberrations
by misaligning or adding/subtracting optical elements in the system. However,
we should mention that despite their simple geometrical optics
interpretations, OAT and Petzval curvature cannot be properly introduced by
simply tilting or curving the image plane, respectively. Although these
geometrical manipulations of the image surface create effective PSFs that are
SV, they do not have the necessary dependence on $\xi$, the object plane
coordinate, in order to bring about the results from a genuine $\Delta W$
given by Eq. (3).

##  4 Concluding Remarks

The analysis presented in this work inform on the CFI (for DI and BSPADE) and
QFI regarding the separation estimation between two equally bright incoherent
point sources when the imaging system includes off-axis aberrations. The
resulting shift-variant nature of the system’s field PSF gives rise to CFI and
QFI that are novel and, importantly, greater than their counterparts in the
aberration-free case. Specifically, OAT and Petzval curvature, which
constitute two of the lowest order off-axis Seidel aberrations, were shown to
provide improvements to the CFI for both DI and BSPADE measurement schemes. A
summary of their effects is provided as follows:

  * •

OAT [$\Delta W(\xi,u)$ is linear in both object ($\xi$) and pupil ($u$)
locations] induces a magnification on the image field. In other words, two
object points with separation $s$ are mapped to two PSFs with (larger)
separation $(1+2\pi T\sigma)s$. This magnified separation provides the
intuition for globally a larger CFI and QFI compared to the aberration-free
case.

  * •

Petzval curvature [$\Delta W(\xi,u)$ is quadratic in both object ($\xi$) and
pupil ($u$) locations] induces an image field where the width of the PSF
varies with object location. As two object points with separation $s$ approach
each other $(s\rightarrow 0)$, the width of the two PSFs approaches the
aberration-free case ($g\rightarrow\sigma$). This object-dependent width
increases the sensitivity of the image field, which in turn leads to a larger
CFI and QFI despite the fact that $g>\sigma$ for nonzero separation $s$.

QFI results were also developed and compared with the CFI. Once again, the
details of the QFI derivation are shown in Supplement 1.

To more fully appreciate the results of this work, it is valuable to
contextualize them with the majority of the research done so far regarding
quantum-inspired superresolution. The primary message in recent history is
that novel modal imaging schemes, like BSPADE, can provide an advantage in
resolution compared to traditional DI measurements. Such claims have been
extended, and supported through theory and experiments, to more complicated
object distributions (discrete or continuous). Furthermore, additional works
have sought to optimize modal imaging. Examples include the development of
practical adaptive imaging schemes that leverage advantages in both DI and
BSPADE as well as a theoretical analysis on the effect of photon statistics
[19, 9]. However, our present work demonstrates that the presence of off-axis
aberrations, whose study have been largely ignored in the context of quantum-
inspired superresolution, provides an improvement to both CFI and QFI that is
relatively intuitive and whose mathematical treatment is straightforward.

The findings here are reminiscent of computational imaging techniques in which
an imaging system is intentionally altered from the traditional aberration-
free DI scheme in order to benefit from the known adjustments. We show that
the introduction of OAT and Petzval curvature into imaging systems can improve
the CFI in two-point separation estimation and therefore provide a link
between the recent field of quantum-inspired superresolution with aspects of
computational imaging. However, although the derivation presented here and in
Supplement 1 for the CFI and QFI encompasses more than the case where the
object scene consists of two equally bright incoherent point sources,
realistic objects are much more complicated and require more care in their
treatment. In addition to this concern, which is relevant in all of quantum-
inspired superresolution analyses, realistic imaging systems have constraints
regarding the strength of off-axis aberrations like OAT and Petzval curvature
due to manufacturing and tolerancing limitations. Further studies are required
to determine the benefits of introducing off-axis imaging systems to resolve
general, realistic object scenes.

Acknowledgements. The author thanks S. A. Wadood, A. N. Vamivakas, and M. A.
Alonso for useful discussions.

Disclosures. The author declares no conflicts of interest.

See Supplement 1 for supporting content.

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# Cryogenic Multiplexing with Bottom-Up Nanowires

Dags Olsteins  Center For Quantum Devices, Niels Bohr Institute, University of
Copenhagen,
2100 Copenhagen, Denmark  Gunjan Nagda  Center For Quantum Devices, Niels Bohr
Institute, University of Copenhagen,
2100 Copenhagen, Denmark  Damon J. Carrad  Department of Energy Conversion and
Storage, Technical University of Denmark, 2800 Kgs.Lyngby, Denmark  Daria V.
Beznasiuk  Department of Energy Conversion and Storage, Technical University
of Denmark, 2800 Kgs.Lyngby, Denmark  Christian E. N. Petersen  Department of
Energy Conversion and Storage, Technical University of Denmark, 2800
Kgs.Lyngby, Denmark  Sara Martí-Sánchez  Catalan Institute of Nanoscience and
Nanotechnology (ICN2), CSIC and BIST, Campus UAB, Bellaterra, Barcelona,
Catalonia, Spain  Jordi Arbiol  Catalan Institute of Nanoscience and
Nanotechnology (ICN2), CSIC and BIST, Campus UAB, Bellaterra, Barcelona,
Catalonia, Spain  ICREA, Passeig de Lluís Companys 23, 08010 Barcelona,
Catalonia, Spain  Thomas Sand Jespersen  Center For Quantum Devices, Niels
Bohr Institute, University of Copenhagen,
2100 Copenhagen, Denmark  Department of Energy Conversion and Storage,
Technical University of Denmark, 2800 Kgs.Lyngby, Denmark<EMAIL_ADDRESS>
###### Abstract

Bottom-up grown nanomaterials play an integral role in the development of
quantum technologies. Among these, semiconductor nanowires (NWs) are widely
used in proof-of-principle experiments, however, difficulties in parallel
processing of conventionally-grown NWs makes scalability unfeasible. Here, we
harness selective area growth (SAG) to remove this road-block. We demonstrate
large scale integrated SAG NW circuits consisting of 512 channel
multiplexer/demultiplexer pairs, incorporating thousands of interconnected SAG
NWs operating under deep cryogenic conditions. Multiplexers enable a range of
new strategies in quantum device research and scaling by increase the device
count while limiting the number of connections between room-temperature
control electronics and the cryogenic samples. As an example of this potential
we perform a statistical characterization of large arrays of identical SAG
quantum dots thus establishing the feasibility of applying cross-bar gating
strategies for efficient scaling of future SAG quantum circuits.

Figure 1:  a Cross-sectional HAADF STEM micrograph of a SAG NW, showing the
InAs conducting channel atop the GaAs substrate and GaAs(Sb) MBE-grown buffer.
The NW exhibits an asymmetric triangular shape imposed by the (311) substrate
symmetry. The shape is not important for the present study which could equally
well have been based on NWs grown on, e.g., (111) or (100) substrates with
higher symmetry. b Combined schematic/3D atomic force microscope (AFM)
micrograph of a $2\,\mu\mathrm{m}$ section of a single NW. c AFM micrograph of
two SAG NWs. d Optical dark-field microscope image of a section of an InAs SAG
NW array. Each NW is $\sim 150\,\mathrm{nm}$ wide and $10\,\mu\mathrm{m}$ long
and individual NWs are spaced by $20\times 2\,\mu\mathrm{m}$. Inset:
Photograph of an as-grown sample. The large NW arrays are visible in green due
to diffraction of light.

Quantum electronics is rapidly maturing towards large scale integrated (LSI)
circuits incorporating a multitude of interacting quantum devices. There is
therefore an onus on potential quantum materials candidates to exhibit both
high-precision reproducibility and scalability potential. Semiconductor
nanowires (NWs) constitute an important platform for quantum electronics,
since the electronic confinement intrinsic to the structure simplifies
fabrication of complex devicesmourik:2012 ; hu:2007 ; hofstetter:2009 ;
fasth2005tunable , the flexibility of contact materials enables hybridization
with important quantum materials such as superconductors doh:2005 ; xiang:2006
; hofstetter:2009 ; mourik:2012 ; larsen:2015 ; krogstrup:2015 ; hays:2018 ,
and an increased capacity for strain relaxation over bulk materials enables
exploration of exotic heterostructureshu:2007 ; xiang:2006 ; krogstrup:2015 ;
SvenssonNL15 . Conventional NWs grown perpendicular to the substrate have been
highly successful for high-performance electronics, simple complementary
circuits, and in fundamental mesoscopic physicsfasth2005tunable ; hu:2007 ;
doh:2005 ; xiang:2006 ; hofstetter:2009 ; mourik:2012 ; larsen:2015 ;
krogstrup:2015 ; hays:2018 ; tomioka:2012 ; SvenssonNL15 . However, the
difficulty in fabricating individual vertical devices, and the insufficient
precision and yield of techniques for transferring NWs to the planar geometry
compatible with standard semiconductor processingfreer:2010a ; yao:2013 ;
li:2008  has thus far inhibited development of LSI NW circuits. A promising
alternative is the bottom-up growth of in-plane semiconductor NWs directly on
a suitable device substrate using selective area growthWang2019Jun ;
op2020plane ; Raya2020Jan ; Bollani2020Jan  (SAG). In SAG, the positions and
dimensions of NWs are controlled by lithographically defining openings in a
dielectric mask, enabling the controlled growth of large-scale networks and NW
arraysAseev2019Jan ; Krizek2018Sep ; op2020plane . While proof-of-principle
single NW devices, e.g., field effect transistors (FETs) beznasyuk:2022 ;
Friedl2018Apr , Hall crossesLee2019Aug , quantum
interferometersvaitiekenas2018selective ; Krizek2018Sep , hybrid
superconducting deviceshertel:2022 ; vaitiekenas2018selective , and quantum
dotstenkate:2022  have been reported, scalability towards integrated quantum
circuits – a central motivation behind the development of SAG and similar
bottom-up grown planar nanostructuresschmid:2015 ; gooth:2017  – has not been
addressed.

Here we make the first demonstration of LSI circuits based on SAG. Starting
from large arrays of thousands of SAG NWs we fabricate multiplexer (MUX)
circuits which operate at the deep cryogenic conditions relevant for quantum
electronics. Cryogenic multiplexers are key ingredients towards scaling of
quantum electronicspauka:2020 ; paquelet2020multiplexed ; smith:2014 ;
smith:2020  as the number of addressable devices scales exponentially – rather
than linearly – with the number of connecting control lines. This is crucial
for reducing heat load from wiring between the cryogenic sample and room-
temperature, and integrated MUX circuits allow highly dense packing of devices
utilizing chip-area conventionally required for bonding and routing. Our setup
allows us to address and measure 512 individual SAG quantum devices using only
37 control lines. Our architecture also includes de-multiplexers (d-MUX)
connected back-to-back with the corresponding MUX, enabling us to
unambiguously confirm the functionality of the circuit, to identify faulty
operation among the thousands of NWFETs, and self-correct against most failure
modes.

Introducing on-chip multiplexing to bottom-up grown nanostructures enables new
strategies in quantum electronics research, such as automated searches through
large ensembles of devices for rare or exotic phenomena, and systematic,
statistically significant exploration of the correlation between device
performance and e.g., materials properties or device geometry. To demonstrate
the potential of the latter we perform a statistical characterization of
device reproducibility within a large array of nominally identical SAG NW
quantum dots (QDs). QD arrays are promising candidates for implementing
quantum computation and simulationloss1998quantum ; tomioka2020rational ;
vandersypen2019semiconductor , and quantifying device-to-device
reproducibility – enabled by the MUX circuit – is crucial for the successful
development of crossbar gate architectures which constitute an important
strategy for limiting gate counts in realistic large scale
implementationsli2018crossbar ; Borsoi:2022 . We find that all QDs of the
array can be concurrently tuned to Coulomb Blockade using only three shared
crossbar gates further confirming the potential SAG as a scalable platform for
quantum devices,

Figure 2: a Conductance, $G$, as function of gate voltage VG for NWFETs based
on 1, 4, 8, 16, 32, and 64 SAG NWs in parallel. Inset shows an SEM micrograph
of a NWFET based on 4 SAG NWs, Ti/Au Ohmic contacts in gold, Ti/Au gate in
blue. The gate in dark gray is screened by the underlying ohmic contact as
illustrated by the cross-section schematic. b SEM micrograph of NWFETs
arranged in a MUX circuit. Gates are screened in alternating elements as
indicated by the blue/gray false coloring. c A schematic representation of the
circuit in b.

## Material and electrical properties

Our circuits are based on [0-11] oriented InAs SAG NWs grown using molecular
beam epitaxy (MBE) on GaAs (311)A substrates. See Methods and Supplementary
Section S1 for details. Figure 1a shows a cross-sectional high-angle annular
dark field scanning transmission electron microscope (HAADF STEM) micrograph
of a single NW, and Fig. 1b shows a combined schematic and atomic force
microscopy (AFM) micrograph of a NW section. The conducting InAs channel sits
atop an insulating GaAs substrate and GaAs(Sb) bufferKrizek2018Sep . The NWs
are terminated with 111A facets as a consequence of the (311)A substrate
symmetry, producing the asymmetric cross-section. Detailed structural analysis
is presented in Supplementary Section S2. Figures 1c and d illustrate the
capacity for scale-up inherent to SAG, through an AFM micrograph and dark-
field optical microscope micrograph of representative sections of a $512\times
16$ array of nominally identical $10~{}\mu$m-long NWs. The inset to Fig. 1d
shows a photograph of a cleaved $5\times 5\,\mathrm{mm}$ piece of the growth
wafer containing $\sim 18000$ SAG NWs, $9216$ of which were used for device
fabrication; the diffraction from the large arrays is visible.

Figure 3: a Optical microscope image of MUX/d-MUX circuit based on InAs SAG NW
arrays. Devices under test (DUT) are labeled $\alpha$ and labels
$\beta,\gamma,\delta$ indicate FET gates failing to deplete and related to
corresponding off-diagonal signals in panel c. For the device shown 500 of the
512 lines were connected at the DUT level. The number of DUT was doubled from
$2^{8}=256$ by using two source/drain pairs. b SEM micrograph of the DUT area
indicated in panel a, showing 8 devices connected to the MUX and d-MUX
channels. Every second channel is shorted for use as a reference to obtain the
series resistance of the adjacent channel. c Conductance matrix of all 65536
combinations of source and drain channels of the first 256 connections of the
circuit. The color of each pixel corresponds to the measured conductance of
the specific channel combination. The uninterrupted diagonal feature shows
that that all of the DUT are addressable and conducting. d Expanded view of
the conductance matrix within the blue square in c. e Schematic three-level
MUX/d-MUX. The colored crosses mark a FET failing to deplete and panel f shows
the corresponding signatures on the conductance matrix. When operated at the
diagonal, the circuit is immune to such errors except symmetric pairs such as
the red/purple. This case can, however, be identified in the conductance
matrix as regions with symmetric off-diagonal finite conductance, and
accounted for in subsequent measurements.  Figure 4: a SEM micrograph of an
array of NW QD devices. Contacts (gold) are spaced by 900 nm,
top/middle/bottom cross-bar gates (blue) are 300/200/300 nm wide and have a
150 nm spacing. The gates are shared between 50 devices of which 20 were
analysed in detail. Each device is paired with a shorted line for reference. b
Source-drain bias spectrum (differential conductance $dI/dV$ vs
$V_{\mathrm{sd}}$ and $V_{\mathrm{M}}$) of Dev1 showing Coulomb diamonds. When
sweeping $V_{\mathrm{M}}$ the barrier gates were compensated for a slight
capacitive cross coupling (Supplementary Section S10). $V_{\mathrm{T}}$ and
$V_{\mathrm{B}}$ starting values correspond to the red point in c. c
Conductance $G$ vs $V_{\mathrm{T}}$ and $V_{\mathrm{B}}$ for Dev1. d Map
showing $\overline{\Delta V_{\mathrm{M}}}$ of Dev1 for various
$V_{\mathrm{T}}$ and $V_{\mathrm{B}}$ combinations within the white square in
c. Crossed points have no identifiable Coulomb peaks. e Histograms of
$\overline{\Delta V_{\mathrm{M}}}$ for Dev1-Dev4 and Dev20 in the same
$V_{\mathrm{T}}$ and $V_{\mathrm{B}}$ range. f Distribution of
$\overline{\Delta V_{\mathrm{M}}}$ for Dev1-Dev20. The red dots indicate
$\overline{\overline{\Delta V_{\mathrm{M}}}}$ of each device and red dashed
line shows $\overline{\Delta V_{\mathrm{M}}}$ across all devices. g Histogram
showing overall distribution of $\overline{\Delta V_{\mathrm{M}}}$ for all
devices. Red dashed line indicate the average value and corresponds to the
line in f. h Number of devices with observable Coulomb peaks at each
combination of $V_{\mathrm{T}}$ and $V_{\mathrm{B}}$. White circles show
points where all 20 devices have $\overline{\Delta V_{\mathrm{M}}}$ within
$2\sigma$ of the common average.

The Fig. 2a inset shows a scanning electron microscope (SEM) micrograph of a
typical device along with a schematic cross section. The device includes $4$
SAG NWs connected in parallel by Ti/Au ohmic contacts and a $1\,\mu\mathrm{m}$
gated segment (see Methods for fabrication details). Two gates are seen: one
which acts on the exposed InAs NWs (blue), thereby controlling the
conductivity, and one which is screened by the metal contact, and thus has no
effect on the underlying NW (grey). This gate is, however, important for the
MUX operation as discussed below. Figure 2a shows the conductance, $G$, as a
function of gate voltage, $V_{\mathrm{G}}$, measured at a temperature of
$T=20$ mK for 6 different devices with varying numbers of NWs,
$M=1,4,16,32,64$ after subtraction of a constant series resistance
$R_{\mathrm{S}}$ (see Methods). The devices act as normally-on, $n$-type FETs
with identical threshold voltages and the dashed lines show a common fit to
the relation $G=KM(V_{\mathrm{G}}-V_{\mathrm{TH}})$ with fixed parameters
$K=0.12\,\mathrm{mS/V}$ and $V_{\mathrm{TH}}=-0.4\,\mathrm{V}$. Except for
$M=64$ where $G$ is somewhat lower than expected, $G$ is proportional to $M$
as expected for equally contributing NWs and the linear scaling with
$V_{\mathrm{G}}$ is typical for NW FETs. The deviation for $M=64$ may be due
to a high sensitivity to the estimate of $R_{\mathrm{S}}$ when the device
resistance is low (Methods). Importantly, Fig. 2a shows that SAG devices
manufactured in parallel exhibit consistent $G$ vs $V_{\mathrm{G}}$, with
reproducible, $M$-independent $V_{\mathrm{TH}}$, enabling the use of large-$M$
FETs as building blocks in LSI SAG circuits.

## SAG multiplexers

We utilize the $V_{\mathrm{TH}}$ reproducibility to operate the circuit shown
in Fig. 2b. SAG FETs are connected in a hierarchical MUX structure, with each
level consisting of devices fabricated on different rows of the SAG NW array
in Fig. 1d. Each gate spans the respective row, and the positions of the gated
segment alternate such that for each NW FET, one gate (blue) tunes the carrier
density of the NWs while the other is screened (grey). Input signals are thus
directed through the MUX as illustrated in the schematic in Fig. 2c. With this
design, each additional level doubles the number of outputs such that a $n$
level MUX has $2^{n}$ outputs and requires $2n$ gates for operation.

Figure 3a shows an optical micrographs of an 8-level MUX circuit connected
back-to-back to a corresponding 8-level d-MUX. The circuit has a footprint of
$\sim 0.6\times 1.1\,\mathrm{mm}$ and incorporates 8192 individual SAG NWs in
the form of 1996 interconnected FETs. Combining the 32 gate lines with two
separate source-drain pairs enables individual addressing of any of 512
devices under test (DUT) located in the gap between the MUX/d-MUX units. Where
the DUT themselves consist of FETs with a single common gate, 37 control lines
thereby enable experiments on 512 devices. In our case, the DUT in Fig. 3a
consist of SAG devices with different functionalities and properties. For
example, the SEM micrograph in Fig. 3b shows DUT devices #70 - 77. Odd-
numbered devices #71,73,75,77 are SAG NWFETs with a contact separation of 100
nm and a common top gate. The even-numbered channels consist of continuous
metal paths covering the NW. These allow confirmation of the MUX/d-MUX
function irrespective of the DUT performance and also provide for reference
measurements the MUX and d-MUX series resistance.

Before discussing DUT properties, we analyze the functionality of the
MUX/d-MUX circuit. Figure 3c shows the conductance of the circuit for each of
the 65536 combinations of the first 256 MUX and d-MUX channels. The
measurement was performed with positive voltage on the DUT gates, which were
therefore all conducting. Indeed, high conductance is observed along the main
diagonal, ($\alpha$), which corresponds to both MUX and d-MUX addressing the
same DUT channel. This confirms that none of the 1996 SAG NWFETs of the
MUX/d-MUX pair fail conduct which would lead to regions of no conductance
along the diagonal. In the case of negative $V_{\mathrm{G}}$ on the DUT level,
every second pixel of the diagonal has $G=0$ (Supplementary Section S5). In
the ideal case, the diagonal would be the only non-zero values of the
conductance matrix. However, finite off-diagonal conductivity also appears
following a repeating pattern every 4, 64, and 128 channels in Fig. 3c and d
($\beta,\gamma,\delta$). Since the FETs are conducting at
$V_{\mathrm{G}}=0\,\mathrm{V}$, finite current at these combinations of MUX
and d-MUX channels corresponds to rows of NW FETs failing to respond to the
gates. This was likely due to a break in gate-lines or a failing bond wire
which can occur for large, complex circuits. Figure 3e and f schematically
illustrates the correlation between the patterns of the matrix and FETs
failing to pinch off turn off at various positions in the circuit. For
example, a non-responsive gate, on the second d-MUX level from the DUT layer
(blue cross in Fig. 3e) would allow transport for (MUX,d-MUX) combinations
(2,0), (3,1), (6,4), and (7,5) as indicated by blue in Fig. 3f. Comparing to
the measurement in Fig. 3c and Fig. 3d the periodically repeating off-diagonal
pattern can be assigned to failures of one of the gates in the MUX levels
marked with the corresponding labels in Fig. 3a. The additional feature
appearing at d-MUX channel 192 ($*$) results from the combination of faulty
FETs at channels 64 and 128 (Supplementary Section S7 and S8 provides a
further analysis of the faults of the circuit).

Importantly, the MUX/d-MUX configuration allows for identifying and in most
cases self-corrects for malfunctioning elements of the SAG circuit; the double
redundancy makes the measurement tolerant towards non-symmetrical errors, as,
e.g., a non-functioning gate on the MUX side will be intrinsically corrected
for by the function of the corresponding d-MUX gate. While errors appearing
symmetrically in the MUX and d-MUX side of the circuit cannot be corrected
for, they can be identified in the conductance matrix and the corresponding
DUT can be excluded from experiments/analysis. This is schematically
illustrated in Fig. 3e and f: if the MUX/d-MUX FETs fail to pinch-off at the
symmetric red/purple positions, addressing levels 0,1,2,3 would also mix
signals from levels 4,5,6,7, respectively. Such a situation is readily
identified by the symmetric off-diagonal non vanishing elements in the
conductance matrix (purple and red in Fig. 3f). We note that FETs failing to
pinch-off would pass unnoticed in single-ended MUX layoutssmith:2020  where
DUT share a common ground. The opposite case, where MUX FETs fail to open
would result in periodic non-conductive elements in the diagonal of the
matrix. Other examples of MUX/d-MUX circuits are discussed in Supplementary
Section S6, showing that even with the amount of failures typical for
research-level devices, the self-correcting nature of MUX/d-MUX configuration
generally protects against a reduction in the available number of DUT.

As a final comment on MUX operation, we note that bandwidth is a key issue for
control electronics. In our experiments, bandwidth was limited by the
cryogenic setup, being optimized for low electron temperature, including $\sim
5\,\mathrm{kHz}$ low-pass filtering of each line. The MUX operation was
uninhibited up to these frequencies (Supplementary Section S9), and we expect
much higher bandwidth to be possible similar to other InAs NW electronics
operating at GHztomioka:2012 ; johansson:2014 ; egard:2010 .

## Multiplexing of Quantum dot arrays

Eliminating the device-count road block by the MUX/d-MUX circuit enables
fundamentally new experimental approaches in quantum electronics. For example,
adding statistical significance to the characterization and optimization of
device performance and material properties is crucial for efforts towards up-
scaling of quantum circuits. As an example, we demonstrate here the use of the
circuit for establishing the statistical reproducibility within large
ensembles of lithographically identical devices. An array of 50
lithographically identical SAG quantum dot devices were embedded in the DUT
layer as shown in Fig. 4a. Potentials ($V_{\mathrm{T}}$, $V_{\mathrm{M}}$,
$V_{\mathrm{B}}$) applied to three shared gates (top, middle, bottom)
simultaneously tune the electrostatics of all 50 devices. This cross-bar
approach is an important strategy for limiting the gate-count in up-scaling of
QD arrays, however, successful operation requires significant reproducible
between devicesli2018crossbar ; Borsoi:2022 . Here we benchmark the
consistency in the SAG NW QD array by comparing the statistical distributions
of QD parameters among devices labeled Dev1-Dev20. First, however, since InAs
SAG QDs have thus far not been demonstrated, we establish the characteristics
of a single device (Dev1).

Figure 4c shows $G$ vs. $V_{\mathrm{T}}$ and $V_{\mathrm{B}}$ for fixed
$V_{\mathrm{M}}=1\,\mathrm{V}$. Pinch-off is at $\sim-150\,\mathrm{mV}$ for
both $V_{\mathrm{T}}$ and $V_{\mathrm{B}}$, and horizontal/vertical structures
are attributed to resonances below each gate modulating the
transmissiondoh:2005 . With both gates near pinch-off, electrons are ideally
confined to a NW segment below the middle gate, thus defining a QD. Indeed,
fixing $V_{\mathrm{T}}$ and $V_{\mathrm{B}}$ at the position of the red dot in
Fig. 4c, diamond shaped regions of low conductance associated with Coulomb
blockade (CB) are observed in the map of the differential conductance,
$dI/dV_{\mathrm{SD}}$, vs. source bias $V_{\mathrm{SD}}$ and $V_{\mathrm{M}}$,
as shown in Fig. 4b. The $V_{\mathrm{SD}}$-height of the diamonds provides an
estimate of the QD addition energy, being the sum of the electrostatic
charging energy $E_{\mathrm{C}}$ and the single-particle level spacing $\Delta
E$. As discussed in Supplementary Section S10 the QDs have $\Delta E\ll
E_{\mathrm{C}}$ and from Fig. 4b we estimate $E_{\mathrm{C}}\sim
210\pm\,15\mu\mathrm{eV}$. The capacitance between the QD and the middle
plunger gate is estimated as $C_{\mathrm{G}}=e/\overline{\Delta
V_{\mathrm{M}}}=0.4\,\mathrm{fF}$ where $\overline{\Delta
V_{\mathrm{M}}}=0.46\,\mathrm{mV}$ is the average value CB peaks spacings in
the range of Fig. 4b. This $C_{\mathrm{G}}$ value agrees with the result
($0.37\mathrm{fF}$) of simple capacitance estimate based on the gate layout
(Supplementary Section S10) and thus support that in this particular gate-
configuration, the QD confinement is defined by gates as intended.

To investigate the sensitivity to the tuning, $G(V_{\mathrm{M}})$ was measured
at $11\times 11$ equally spaced $(V_{\mathrm{T}},V_{\mathrm{B}})$ points,
spanning the white square in Fig. 4c. CB peaks were identified at 108 of the
121 gate-tunings and Fig. 4d shows the corresponding map of $\overline{\Delta
V_{\mathrm{M}}}$. No systematic trend is observed and the distribution of
$\overline{\Delta V_{\mathrm{M}}}$ shown in Fig. 4e (top), is symmetric with a
mean $\overline{\overline{\Delta V_{\mathrm{M}}}}=464\,\mathrm{\mu V}$ and
standard deviation $\sigma=\pm 27\,\mathrm{\mu V}$, again consistent with a QD
defined between the top and bottom gates.

The choice of range for the cross-bar gate-tunings in Fig. 4b,d was based on
the gate characterization specific for Dev1 (Fig. 4c). We now use the MUX
circuit to gather statics for the different devices in order to probe the
consistency across the array while keeping these same tuning parameters.
Values of $\overline{\Delta V_{\mathrm{M}}}$ were extracted from
$G(V_{\mathrm{M}})$ traces measured for all devices and all gate-points. All
devices were operational and exhibited coulomb blockade and from the resulting
2420 gate-traces, 17924 CB peaks were identified and fitted. Examples of
measured data and peak analysis are presented in Supplementary Section S12 and
S13. The distributions of $\overline{\Delta V_{\mathrm{M}}}$ for all devices
are included in Supplementary Section S14 and examples for Dev2-5 are shown in
Fig. 4e. A comparison of the distributions and their mean values,
$\overline{\overline{\Delta V_{\mathrm{M}}}}$, among the devices of the array,
is shown in Fig. 4f. Except for Dev7, the distribution means fall within one
$\sigma$ of the overall common mean. The spread between devices could be
affected by structural variations between nanowires due to SAG processing or
to variation in post growth device processing. The spread within each devices
could be related to changes in the effective confinement potential with gate
tunings and may be different between devices due to random impurity in the
vicinity of the devices.

Finally, Fig. 4g shows the joint distribution of all $\overline{\Delta
V_{\mathrm{M}}}$ and Fig. 4h illustrates the number of devices displaying CB
for all measured combinations of $V_{\mathrm{T}}$, $V_{\mathrm{B}}$. In 27 out
of the 121 point in cross-bar gate space, all 20 devices simultaneously
exhibiting CB. The circles mark the 7 tunings where all devices fulfill the
stricter criterion of showing CB and having $\overline{\Delta V_{\mathrm{M}}}$
within $\pm 2\sigma$ of the joint mean peak spacing. Figure 4f-h constitute a
key result of the current study, establishing both a level of device to device
reproducibility supporting the potential of SAG for as a scalable platform for
quantum electronics. Further, the statistical bench-marking of the QD devices
explicitly demonstrates a key example of the new possibilities enabled by the
integration of MUX/d-MUX circuits.

## Conclusions and Outlook

In conclusion, we successfully fabricated and operated cryogenic
multiplexers/de-muliplexer circuits based on InAs NWs grown bottom-up by
selective area growth. The circuit removes the limitations on device count in
conventional cryogenic electronics thus enabling new experimental strategies
such as searches through large ensembles of devices for rare or exotic
phenomena, establishing the correlation between device performance and
materials properties or device geometry, and allows establishing the
statistical reproducibility among devices - a prerequisite for further scaling
quantum of circuits. This capacity was demonstrated by statistically
characterizing an ensemble of SAG quantum dots. In general, the methods
developed here enable optimization of quantum materials and devices based on
automated acquisition of statistically significant datasets rather than proof-
of-principle examples. This direction will be empowered by the ongoing
developments of advanced data evaluationlesage:2015  and machine
learningzwolak:2020 ; moon:2020 ; nguyen:2021  for unsupervised and optimized
acquisition and tuning of large ensembles of quantum devices with many tuning
parameterslennon:2019 ; darulova:2020 ; chatterjee:2022 . The circuit may be
expanded further by replacing the single lines in the current design by a
multi-channel busward:2013  to enable, e.g., integration of charge sensors,
multi-terminal devices, complex gate-architectures, and/or the operating the
MUX as a multi-channel DACpuddy:2015 .

## Methods

For SAG fabrication and synthesis, a $10\,\mathrm{nm}$ SiO2 mask layer was
first deposited on epi-ready GaAs (311)A substrates by plasma enhanced
chemical vapour deposition. $0.15\times 10\,\mu\mathrm{m}$ rectangular
openings were defined in the oxide along the [0-11] direction by e-beam
lithography (EBL) and dry etching. The openings were arranged in $512\times
16$ arrays with a pitch of $2\,\mu\mathrm{m}$ and $20\,\mu\mathrm{m}$ along
the [0-11] and [-233], respectively (Fig. 1d). GaAs(Sb)/InAs double layer NWs
were selectively grown in the openings where the GaAs(Sb) buffer was
introduced to improve the crystal surface for the subsequent InAs transport
channelKrizek2018Sep . Synthesis details and structural analysis are provided
in Supplementary Sections S1 and S2. For device fabrication, Ti/Au ohmic
contacts to the SAG NWs were defined on the growth substrate by standard EBL,
metal evaporation, and liftoff. Subsequently, a $15\,\mathrm{nm}$ HfO2 gate
dielectric was deposited by atomic layer deposition and top-gates were defined
by electron beam lithography, metal evaporation, and liftoff. The QD devices
in Fig. 4 have a contact separation of 900 nm, top/middle/bottom cross-bar
gates are 300/200/300 nm wide and have a 150 nm spacing. Electrical
measurements were carried out in a dilution refrigerator with a base
temperature of $20\,\mathrm{mK}$. The conductance, $G=I/V_{\mathrm{SD}}$,
where $I$ is the drain current generated by source voltage $V_{\mathrm{SD}}$,
was measured as a function of the gate potential, $V_{\mathrm{G}}$, using
standard lock-in techniques. The series resistance $R_{\mathrm{S}}$ for data
presented in Fig. 2 was estimated by fitting the $G(V_{\mathrm{G}})$ traces
with the standard expressiongul2015towards
$G=\left(R_{\mathrm{S}}+\frac{L^{2}}{\mu
C(V_{\mathrm{G}}-V_{\mathrm{TH}})}\right)^{-1}$ where $L$ is the gate length,
$C$ is gate capacitance simulated as described in Supplementary Section S3 and
S4, and $\mu_{\mathrm{FE}}$ is the electron mobility. When sweeping
$V_{\mathrm{M}}$ in the QD measurements, the barrier gates were compensated
for a slight capacitive cross coupling (Supplementary Section S10).

### Data availability

Full data sets for all figures, STEM micrographs, transport data, electronic
logbooks and other data that support the findings of this study are available
online at

    https://sid.erda.dk/

### Acknowledgements

This research was supported by research grants from Villum Fonden (00013157)
and the European Research Council (866158). ICN2 acknowledges funding from
Generalitat de Catalunya 2021SGR00457. The authors thank support from “ERDF A
way of making Europe”, by the “European Union”. ICN2 is supported by the
Severo Ochoa program from Spanish MCIN / AEI (Grant No.: CEX2021-001214-S) and
is funded by the CERCA Programme / Generalitat de Catalunya. Authors
acknowledge the use of instrumentation as well as the technical advice
provided by the National Facility ELECMI ICTS, node ”Laboratorio de
Microscopías Avanzadas” at University of Zaragoza. We acknowledge support from
CSIC Interdisciplinary Thematic Platform (PTI+) on Quantum Technologies (PTI-
QTEP+).

### Competing interests

The authors declare no competing financial interests

### Supporting Information

Supporting information contains extended information on sample preparation and
MBE growth, NW structure investigation via GPA, extended transport datasets
for Fig. 4b, details of the fitting procedures employed for Fig. 4c and d, as
well as a more in-depth description of the MUX/d-MUX circuit benchmarking,
measurement bandwidth, and switching speed.

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2021

[1,2]Antonio S Soares

[1]Wellcome/EPSRC Interventional and Surgical Sciences (WEISS) Centre,
University College London, 43-45 Foley Street, London, W1W 7JN, United Kingdom

2]Division of Surgery and Interventional Sciences, University College London

3]Department of Computer Science, University College London

4]Department of Medical Physics and Biomedical Engineering, University College
London

# Fluorescence angiography classification in colorectal surgery

A preliminary report

<EMAIL_ADDRESS>Sophia Bano<EMAIL_ADDRESS>Neil T Clancy
<EMAIL_ADDRESS>Laurence B Lovat<EMAIL_ADDRESS>Danail Stoyanov
<EMAIL_ADDRESS>Manish Chand<EMAIL_ADDRESS>*  [  [  [

###### Abstract

Background: Fluorescence angiography has shown very promising results in
reducing anastomotic leaks by allowing the surgeon to select optimally
perfused tissue. However, subjective interpretation of the fluorescent signal
still hinders broad application of the technique, as significant variation
between different surgeons exists. Our aim is to develop an artificial
intelligence algorithm to classify colonic tissue as ’perfused’ or ’not
perfused’ based on intraoperative fluorescence angiography data.

Methods: A classification model with a Resnet architecture was trained on a
dataset of fluorescence angiography videos of colorectal resections at a
tertiary referral centre. Frames corresponding to fluorescent and non-
fluorescent segments of colon were used to train a classification algorithm.
Validation using frames from patients not used in the training set was
performed, including both data collected using the same equipment and data
collected using a different camera. Performance metrics were calculated, and
saliency maps used to further analyse the output. A decision boundary was
identified based on the tissue classification.

Results: A convolutional neural network was successfully trained on 1790
frames from 7 patients and validated in 24 frames from 14 patients. The
accuracy on the training set was 100%, on the validation set was 80%. Recall
and precision were respectively 100% and 100% on the training set and 68.8%
and 91.7% on the validation set.

Conclusion: Automated classification of intraoperative fluorescence
angiography with a high degree of accuracy is possible and allows automated
decision boundary identification. This will enable surgeons to standardise the
technique of fluorescence angiography. A web based app was made available to
deploy the algorithm.

###### keywords:

fluorescence angiography, computer vision, colorectal surgery

##  1 Abstract

##  2 Introduction

Anastomotic leak is an important source of morbidity, mortality and rising
healthcare costs Ashraf2013  in colorectal surgery. Surgeons use fluorescence
angiography to optimise the perfusion of the tissue used to create the
anastomosisKeller_ICG . The use of fluorescence angiography has been shown to
reduce anastomotic leaks in non-randomized studies Blanco-Colino2018 ;
Degett2016  and has been recommended by the European Association for
Endoscopic Surgery [submitted]. The true utility of the technique remains
unquantified.

The interpretation of the signal in fluorescence angiography is dependent on
the surgeon and perhaps represents the actual ‘learning curve’ for this
technique. These decisions have a high degree of variability shown both for
static images [Soares, AS et al., Interobserver agreement study, under review]
and dynamic video hardy2021inter  assessments. Automating the interpretation
of the signal would eliminate this variability and provide an alternative
means of quantification, thereby leveraging the usefulness of surgical data
science Maier-Hein2022  in the clinical setting.

Convolutional neural networks are especially suited for image analysis. The
input for these neural networks are the pixel values of a given image and the
outputs are the probabilities of the image belonging to a particular pre-
defined class.

This paper describes the development of an artificial intelligence algorithm
to automatically classify tissue based on perfusion using fluorescence
angiography during colorectal surgery. A decision boundary is estimated based
on the classification of different areas of the tissue of interest.

##  3 Methods

###  3.1 Dataset

Patients undergoing colorectal surgery that involved construction of an
anastomosis were included in the study. De-identified data was collected
between May and November 2019 as previously described by our group
soares2022multisensor . Additional data used for validation was collected
between January and August 2021. This was a convenience sample. In all cases,
the bowel transection was performed following extracorporealisation of the
bowel. Prior to transection, the bowel was placed on clean white swabs to act
as ‘background’. A continuous sequence was recorded from the point of ICG
injection. 10 mg of ICG was injected intravenously in all cases. The recording
lasted until the operating surgeon decided fluorescence intensity was adequate
to choose the point of proximal transection (maximal fluorescence intensity),
as per current standard practice in FA. The training data was collected using
the PINPOINT camera (Stryker, Michigan, USA) and the validation data was
collected with PINPOINT and Stryker 1688 (Stryker, Michigan, USA) and Arthrex
(Arthrex Inc., Florida, USA) cameras. All patients provided written informed
consent for video collection, and standard clinical practice was followed in
all cases. No missing data occurred. The TRIPOD statementTRIPOD  was followed.

###  3.2 Model development

In the literature of deep learning, convolutional neural networks (CNNs) have
shown remarkable performance for the classification task. Transfer learning
was used based on a model trained on ResNet34. The neural network was then
fine-tuned using the Fast AI library in Python howard2018fastai . Before
loading the data for training, it was augmented using a randomly selected crop
from the complete 1440 x 1080 pixel frame. This process occurred for 4 epochs.
A random selection of 20% of the training data was used for initial
validation. The threshold defined for classification of fluorescent was a
probability output of the algorithm above 80%. The final result is an
algorithm which takes the colorectal image as input and output a binary
decision, whether the given input image has fluorescence in it or not.

###  3.3 Model evaluation

The model was deployed through a web interface using the Gradio Library and
Huggingface Spaces
(https://huggingface.co/spaces/asampaiosoares/fluorescence_id_app). This
platform enabled the serial testing of additional frames that were not used
for training the algorithm. Saliency maps were created to analyse the
predictions from the validation set.

###  3.4 Decision boundary estimation

The estimation of the decision boundary was performed on the images considered
positive for fluorescence in the validation set. These images were divided
into JPEG segments of 100 pixel width along the colonic longitudinal axis
using imageJ Schneider2012  and inputted into the algorithm. Both the binary
classification results as well as the relative probabilities were recorded for
each case. The decision boundary was defined as the limit between the most
distal area that was classified as fluorescent and the areas considered not
fluorescent.

###  3.5 Model deployment

The algorithm was deployed through an online platform called Gradio, which
allows the upload of images to be classified using the algorithm developed.

##  4 Results

A dataset of patients undergoing colorectal resection was created with the
characteristics shown in Table 1. Representative frames corresponding to the
training set can be found in figure 1.

| Training set | Validation set |  |
---|---|---|---|---
Number of patients | 7 | 14 |  |
Number of frames | 1790 | 30 |  |
Fluorescence positive frames | 19.6% | 53.3% |  |
Table 1: Dataset characteristics

###  4.1 Training and validation

Figure 1: Representative frames of the training set

Training took 21 minutes and 11 seconds. The validation on data not used in
the training set was done manually using the Gradio app. The performance
metrics can be seen in Table 2.

| Recall / Sensitiviy | Precision / PPV | Accuracy | F1 score
---|---|---|---|---
Training | 100.0% | 100.0% | 100.0% | 100.0%
Validation overall | 68.8% | 91.7% | 80.0% | 78.6%
Validation - internal data | 60.0% | 100.0% | 80.0% | 75.0%
Validation - external data | 83.3% | 83.3% | 80.0% | 83.3%

Table 2: Performance metrics for the classification algorithm. PPV positive
predictive value

Representative frames of true positive, true negative, false positive and
false negative results can be seen in figure 2, including the respective
saliency maps. These maps show areas that contribute more to activation of the
neural network in a darker shade of red.

Figure 2: Representative frames of predictions performed by the algorithm

###  4.2 Decision boundary estimation

The decision boundary was estimated for all the positive cases used for
validation as shown in figure 3.

Figure 3: Representative frame for decision boundary analysis, where green
boxes denote segments of the image classified as fluorescent and grey boxes
denote segments classified as not fluorescent with the decision boundary
between the two, shown in yellow

##  5 Discussion

These results have show that a neural network trained on clinical data has
achieved an accuracy over 90%. There was a drop in performance using data
acquired with a different equipment for validation but still maintaining an
accuracy over 80%.

The analysis of the saliency maps shows that in the cases where both false
negative and positive results were obtained the pixels corresponding to the
colon were not responsible for the main activation of the algorithm. This
leads us to hypothesize that an image analysis pipeline segmenting the colon
from the background first and, on this output, running the classification
algorithm could lead to improved results. Based on previous work from our
group [Soares, AS et al., Interobserver agreement study, under review] and
others hardy2021inter  significant variation exists between different
surgeons’ assessments. Standardisation of fluorescence angiography is a
necessary step to enable accurate comparison of results.

Our algorithm provides an easily accessible tool based on state of the art
deep learning techniques that eliminates interobserver variability in
interpreting the fluorescent signal. Previous efforts to standardise
fluorescence angiography assessments have focused on quantitative measurements
requiring post hoc signal processing wada2017icg ; son2019quantitative . Our
algorithm has the ability to simplify post-hoc processing as it was trained on
data captured intraoperatively and is designed to output a binary output
(”fluorescent” vs ”non fluorescent”).

Some limitations should be addressed. Although a high number of frames was
used for training, the variability of frames used for validation was high but
the overall number was small. This should be mitigated by including data from
more centres performing fluorescence angiography. Furthermore, the data
collected was of low quality to begin with, which limited the usefulness of
the algorithm. This should be interpreted with caution as even an experienced
surgeon would find some trouble analysing this data intraoperatively. A close
collaboration between engineers and clinicians has led to the development of
the algorithm presented in this work. Further work should aim to enrich the
dataset for further refinement of the algorithm. Clinical testing in a
randomized clinical trial will be necessary before large scale implementation
can occur.

##  6 Conclusion

An artificial intelligence algorithm to classify colonic perfusion based on
fluorescence angiography was developed using state of the art deep learning
techniques. This work paves the way for standardisation of fluorescence
angiography in colorectal surgery.

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