# Time-Resolved Focused Ion Beam Microscopy: Modeling, Estimation Methods, and
Analyses

Minxu Peng, John Murray-Bruce, and Vivek K Goyal M. Peng and V. K. Goyal are
with the Department of Electrical and Computer Engineering, Boston University,
Boston, MA 02215 USA (e-mail<EMAIL_ADDRESS>v.goyal@ieee.org).J. Murray-Bruce
is with the Department of Computer Science and Engineering, University of
South Florida, Tampa, FL 33620 USA (e-mail: murraybruce@usf.edu).This work was
supported in part by the US National Science Foundation under Grant No.
1422034 and Grant No. 1815896.

###### Abstract

In a focused ion beam (FIB) microscope, source particles interact with a small
volume of a sample to generate secondary electrons that are detected, pixel by
pixel, to produce a micrograph. Randomness of the number of incident particles
causes excess variation in the micrograph, beyond the variation in the
underlying particle–sample interaction. We recently demonstrated that joint
processing of multiple time-resolved measurements from a single pixel can
mitigate this effect of source shot noise in helium ion microscopy. This paper
is focused on establishing a rigorous framework for understanding the
potential for this approach. It introduces idealized continuous- and discrete-
time abstractions of FIB microscopy with direct electron detection and
estimation-theoretic limits of imaging performance under these measurement
models. Novel estimators for use with continuous-time measurements are
introduced and analyzed, and estimators for use with discrete-time
measurements are analyzed and shown to approach their continuous-time
counterparts as time resolution is increased. Simulated FIB microscopy results
are consistent with theoretical analyses and demonstrate that substantial
improvements over conventional FIB microscopy image formation are made
possible by time-resolved measurement.

###### Index Terms:

computational imaging, Fisher information, gallium ion microscopy, helium ion
microscopy, scanning electron microscopy, Poisson processes, shot noise,
statistical modeling and estimation.

##  I Introduction

The ability to image the structure of a sample at nanoscale resolution using
microscopes that scan samples with a focused beam of particles is critical in
material science and the life sciences. In a scanning electron microscope
(SEM) [1], a focused electron beam is raster scanned over the sample, causing
the sample to emit secondary electrons (SEs). An SEM is capable of providing
information regarding composition and distribution of sample components.
Resembling an SEM, a focused ion beam (FIB) microscope [2] instead uses a
focused beam of ions, such as gallium, helium, or xenon. As one member of the
FIB microscope family, a helium ion microscope (HIM) [3] offers many
advantages compared to an SEM, leading to widespread use in semiconductor and
biological imaging [4, 5, 6, 7]. The interaction volume with the sample is
much smaller for ions compared to that for electrons, resulting in higher
contrast [8]. Higher particle mass also leads to higher overall SE yield [9,
10]. Furthermore, to image insulating samples with an SEM, prior deposition of
a conductive coating is required to prevent the accumulation of electron
charges on the sample; an HIM uses an electron flood gun to prevent charge
accumulation, thus avoiding the masking of subtle features by a coating [11].

The number of incident ions determines the number of sample interactions that
can be measured. Hence, increasing this number—through increased beam current
or increased dwell time—will ideally create measurements that are more
informative about the mean number of SEs per incident ion, which is the sample
property of interest. However, the vastly greater mass of ions compared to
electrons (by a factor of $7.3\times 10^{3}$ for helium or $1.3\times 10^{5}$
for gallium) makes sputtering much more significant for FIB microscopy than
for SEM. Various studies have shown how the sputtering damage induced by
helium ions evolves with increasing numbers of incident ions [12, 13, 14], and
this damage is often determinative of the best possible image quality. Dose is
conventionally defined as the number of incident ions per unit area, and dose
limits to prevent significant damage have been measured for certain materials
and imaging configurations. For example, a safe imaging dose for suspended
graphene is as low as $10^{13}$ to $10^{14}$ per square centimeter [15]. It
follows, for example, that a $(10\,\mathrm{nm})^{2}$ pixel should be subjected
to only 10 to 100 ions. In our abstractions, we dispense with spatial
dimensions and hereafter express doses per pixel rather than per unit area;
expressed per pixel, finer resolution necessitates lower dose limits. While
randomized subsampling combined with regularized reconstruction can sometimes
yield high-quality images from reduced doses [16, 17, 18], this requires
piecewise smooth image structure. Here we restrict our attention to pixelwise
acquisition and estimation methods that do not rely on such assumptions.

We recently introduced the concept of _time-resolved_ (TR) measurements in FIB
microscopy to mean dividing any given pixel dwell time $t$ into $n$ dwell
times $t/n$ for some integer $n>1$ [19]. Without proofs, we gave theoretical
evidence that a set of TR measurements is fundamentally more informative than
a single measurement with the same total dose. Experiments with HIM data
demonstrated mean-squared error (MSE) improvement by about a factor of 4 at
doses of 1.0 and 2.5 incident ions per pixel. Importantly, this use of time
resolution is entirely for the purpose of making the measurements more
informative without increasing the total dose. It is not for imaging of
dynamic samples and hence not comparable to any previous use of time
resolution in microscopy.

The Zeiss ORION NanoFab HIM used in [19], like most commercial FIB
microscopes, uses indirect electron detection with a scintillator and
photomultiplier tube. Although not yet prevalent, direct electron detection
offers higher signal-to-noise ratio (SNR) as it avoids the statistical noises
brought by electron–photon conversion and additional readout noise [20]. For
example, Yamada et al. [21] demonstrated that the SNR of direct electron
detection is 2.5 times higher than that of indirect mode at low dose.
Furthermore, direct electron detection technology has also been applied to
imaging in transmission electron microscopy to improve resolution [22].

In this paper, we develop comprehensive theoretical results for FIB microscopy
with time-resolved direct SE detection. Indirect detection introduces many
sources of noise, including spatial nonuniformity in the scintillator
response, nonideal light transport from the scintillator to the
photomultiplier tube, and variations in pulses generated by the
photomultiplier tube. Though these effects were empirically modeled in [19],
including them here would make already lengthy expressions considerably more
complicated and more difficult to interpret. Restricting attention to direct
detection allows us to concentrate on the implications of TR sensing for
mitigation of source shot noise, separated from the effects of detection
noise. While some results presented here add rigor to statements in [19], we
more importantly introduce a new continuous-time abstraction for FIB
microscopy that yields to more elegant analyses while also representing the
ultimate limit of this technology. In addition to maximum likelihood (ML)
estimation, our study includes plug-in estimators for which we can complete
analytical performance analyses and that are easily generalized to settings in
which SE detection is indirect.

###  I-A Main Contributions

  * •

_A new continuous-time probabilistic model for FIB microscopy wherein the data
at any one pixel are related to a marked Poisson process._ Conventional and
continuous- and discrete-time time-resolved observation models are different
functions of the marked Poisson process.

  * •

_Fisher information analyses._ We show that, at any ion dose level,
continuous-time time-resolved measurements have Fisher information matching an
upper bound that conventional measurements meet only in a low-dose limit. For
conventional measurements, asymptotic expressions presented without proof in
[19] are proven here.

  * •

_Estimator analyses._ Biases and variances of quotient-mode estimators in both
the continuous- and discrete-time settings are derived. Convergence of the
discrete-time estimator’s performance to the performance of the continuous-
time estimator is proven.

###  I-B Outline

We introduce our mathematical abstraction for the operation of a FIB
microscope in Section II. This leads to four measurement models: an
unimplementable oracle model, conventional measurement, and continuous- and
discrete-time time-resolved measurement. The oracle and conventional cases are
analyzed within Section II. Section III develops the novel continuous-time
case in detail. Measurement distributions are derived, three estimators are
introduced and simulated, and the performance of a quotient-mode estimator is
rigorously analyzed. Section IV develops the discrete-time case first
introduced in [19] in detail. Three estimators are simulated, and the
performance of the discrete-time quotient-mode estimator is rigorously
analyzed. Convergences of Fisher information and quotient-mode estimator
performance to their continuous-time counterparts are shown. Section V
compares all the estimators in a simulated HIM experiment, demonstrating
substantial improvement of the time-resolved methods over the conventional
interpretation of the collected data. Section VI provides concluding comments
on how mean SE yield influences the advantages of TR methods, the time
resolution necessary to capitalize on these advantages, the roles of the
various estimators, and generalizations to indirect SE detection.

Table I summarizes the variables, symbols, and acronyms used in the
manuscript.

Table I: List of symbols and acronyms $\eta$ | mean secondary electron yield
---|---
$\widehat{\eta}_{\rm oracle}$ |  oracle estimator (8)
$\widehat{\eta}_{\rm baseline}$ |  baseline estimator (14)
$\widehat{\eta}_{\rm CTQM}$ |  continuous-time quotient mode estimator (28)
$\widehat{\eta}_{\rm CTLQM}$ |  continuous-time Lambert quotient mode estimator (29)
$\widehat{\eta}_{\rm CTML}$ |  continuous-time maximum likelihood estimator (30)
$\widehat{\eta}_{\rm DTQM}$ |  discrete-time quotient mode estimator (40)
$\widehat{\eta}_{\rm DTLQM}$ |  discrete-time Lambert quotient mode estimator (41)
$\widehat{\eta}_{\rm DTML}$ |  discrete-time maximum likelihood estimator (42)
$\lambda$ | ion dose per pixel
$\Lambda$ | dose per unit time
$\rho$ | $\mathrm{P}({X_{i}>0})=1-e^{-\eta}$
$i$ | ion index
$k$ | discrete time index
$n$ | number of subacquisitions
$p$ | $\mathrm{P}({Y_{k}>0})=1-\exp(-({{\lambda}/{n}})(1-e^{-\eta}))$
$t$ | dwell time
$\mathcal{I}_{Z}(\eta\,;\,\lambda)$ |  Fisher information about $\eta$ in $Z$ with $\lambda$ available
$L$ | number of subacquisitions with positive SEs
$\widetilde{L}$ |  zero-truncated version of $L$
$M$ | number of incident ions
$\widetilde{M}$ | number of incident ions yielding positive SEs
$T_{i}$ |  $i$th ion incidence time
$\widetilde{T}_{i}$ |  incidence time of $i$th ion to yield positive SEs
$X_{i}$ |  SEs detected due to $i$th incident ion
$\widetilde{X}_{i}$ |  SEs detected due to $i$th ion to yield positive SEs
$Y$ | total detected SEs
$Y_{k}$ |  detected SEs in $k$th subacquisition
$\widetilde{Y}_{k}$ | positive SE counts in a subacquisition
CRB | Cramér–Rao bound
CTTR |  continuous-time time-resolved (see observation (6))
DTTR |  discrete-time time-resolved (see observation (7))
FI | Fisher information
LQM | Lambert quotient mode
ML | maximum likelihood
QM | quotient mode
SE | secondary electron
TR | time-resolved

##  II Measurement Models and Basic Analyses

###  II-A Physical Abstractions

Throughout this paper, we model the incident ions at the sample to be imaged
as a Poisson process with known rate $\Lambda$ per unit time. Imaging proceeds
by raster scanning with known dwell time $t$ at each pixel. Hence, the number
of ions $M$ incident on a pixel is a Poisson random variable with known
parameter $\lambda=\Lambda t$. Since pixel area is not relevant in our
abstraction, we refer to $\lambda$ as the dose. The interaction of the $i$th
incident ion with the sample causes a number $X_{i}$ of SEs to be detected.
All of these SE counts are mutually independent Poisson random variables with
parameter $\eta$, independent of the incident-ion Poisson process, and
estimation of the mean SE yield $\eta$ is the objective of the imaging
experiment. Our analysis is for each pixel separately, so no pixel indexing is
necessary.

Note that a somewhat high 1 pA beam current corresponds to a rate of
$6.2\times 10^{6}$ ions per second, or a mean ion interarrival time of 160 ns.
The interaction between an incident ion and the sample and the subsequent
detection of SEs occurs within a few femtoseconds [23]. With the SE detections
happening so quickly, we abstract the SE detections caused by an incident ion
to be simultaneous with the ion incidence. Thus, the model can be described as
a marked Poisson process $\\{(T_{1},X_{1}),\,(T_{2},X_{2}),\,\ldots\\}$, where
$(T_{1},\,T_{2},\,\ldots)$ is the arrival time sequence of the ions. The ion
count $M$ is the largest $i$ such that $T_{i}\leq t$ (with $M=0$ when
$T_{1}>t$). One realization on an interval $[0,t]$ is illustrated in Fig.
1(a). Note that the arrival times (horizontal) are arbitrary positive real
numbers and the marks (vertical) are nonnegative integers.

(a) Underlying marked Poisson process.

(b) Process observed where mark is positive.

(c) Dwell time divided into $n=10$ subintervals.

(d) Discrete-time time-resolved measurement, $n=10$.

Figure 1: Illustration of the random processes generated in the abstraction of
FIB microscopy through one possible realization. (a) The underlying marked
Poisson process $\\{(T_{1},X_{1}),\,(T_{2},X_{2}),\,\ldots\\}$, with ion
incident at times $T_{1},\,T_{2},\,\ldots$ generating detected SE counts
$X_{1},\,X_{2},\,\ldots$. (b) The marked Poisson process
$\\{(\widetilde{T}_{1},\widetilde{X}_{1}),\,(\widetilde{T}_{2},\widetilde{X}_{2}),\,\ldots\\}$,
produced by discarding the ions for which no SEs are detected. (c)
Illustration of dividing dwell time of $t=20$ s into $n=10$ subintervals of
equal length. (d) The resulting discrete-time SE count process.

Since cases of $X_{i}=0$ produce no detected SEs, the corresponding ion
arrival time is not observable in practice. Thus consider also the thinned
process
$\\{(\widetilde{T}_{1},\widetilde{X}_{1}),\,(\widetilde{T}_{2},\widetilde{X}_{2}),\,\ldots\\}$,
where $\widetilde{T}_{i}$ is the arrival time of the $i$th ion that produces a
_positive_ number of detected SEs and $\widetilde{X}_{i}$ is the corresponding
number of detected SEs. Define $\widetilde{M}$ to be the largest $i$ such that
$\widetilde{T}_{i}\leq t$ (with $\widetilde{M}=0$ when $\widetilde{T}_{1}>t$).
Note that the thinned process is also a marked Poisson process because the
events of the form $\\{X_{i}=0\\}$, which determine whether an arrival in the
original Poisson process is retained, are independent of the arrival time
process. Fig. 1(b) illustrates the thinned process for the realization of the
underlying process in Fig. 1(a).

Now suppose the observation time interval $[0,t]$ is evenly divided into $n$
subintervals of length $t/n$. Counting the total number of SEs detected in
each subinterval produces a discrete-time, discrete-valued random process:

$Y_{k}=\sum_{\\{i\ :\ T_{i}\in[(k-1)t/n,\,kt/n)\\}}X_{i},\qquad
k=1,\,2,\,\ldots,\,n.$ (1)

We call an observation over a subinterval a _subacquisition_. Fig. 1(c)
illustrates the partition of $[0,t]$ into subintervals and Fig. 1(d)
illustrates the resulting discrete-time process for the realization of the
underlying process in Fig. 1(a). Because of the independence of a Poisson
process over disjoint intervals, $\\{Y_{k}\\}_{k=1}^{n}$ is an independent and
identically distributed (i.i.d.) process. We can also view
$\\{Y_{k}\\}_{k=1}^{n}$ as a marked Bernoulli process, where $\\{Y_{k}>0\\}$
indicates an arrival in discrete time slot $k$ and the “mark” is then the
(nonzero) value $Y_{k}$.

The abstraction described here applies similarly to SEM and FIB microscopy.
The main difference is the typical values of the mean SE yield $\eta$. In SEM,
neglecting topographical effects (which tend to increase yield), for a sample
with atomic number up to 83, the SE yield at the maximizing electron energy is
typically 0.6 to 2 [24]. In FIB microscopy, SE yield is typically between 1
and 8 [25]. The advantages of TR measurements that are established in this
paper diminish at smaller $\eta$ values. Furthermore, in FIB microscopy,
sample damage is more of an impediment to improving image quality by
increasing dose [12, 13, 14]. Thus, we concentrate on FIB microscopy.

###  II-B Measurement Models

We consider four measurement models for the probabilistic experiment described
in Section II-A:

  * •

_Oracle_ : Observe

$\\{M,\,(T_{1},X_{1}),\,(T_{2},X_{2}),\,\ldots,\,(T_{M},X_{M})\\}.$ (2)

Though no current instrument provides this information, this measurement model
provides a useful benchmark.

  * •

_Conventional_ : Observe only

$Y=\sum_{i=i}^{M}X_{i}.$ (3)

This would be the standard operation of a FIB microscope that has direct
detection of SEs. Note that

$Y=\sum_{i=1}^{\widetilde{M}}\widetilde{X}_{i}$ (4)

and

$Y=\sum_{k=1}^{n}Y_{k}$ (5)

are equivalent to the definition of $Y$. As the sum of a
$\operatorname{Poisson}(\lambda)$ number of mutually independent
$\operatorname{Poisson}(\eta)$ random variables, (3) is the simplest of the
expressions.

  * •

_Continuous-time time-resolved (CTTR)_ : Observe

$\\{\widetilde{M},\,(\widetilde{T}_{1},\widetilde{X}_{1}),\,(\widetilde{T}_{2},\widetilde{X}_{2}),\,\ldots\,\,(\widetilde{T}_{\widetilde{M}},\widetilde{X}_{\widetilde{M}})\\}.$
(6)

This is an idealization of a FIB microscope with direct detection of SEs with
perfect temporal precision.

  * •

_Discrete-time time-resolved (DTTR)_ : Observe

$\\{Y_{1},\,Y_{2},\,\ldots,\,Y_{n}\\}.$ (7)

This is a model for the use of a FIB microscope to collect a set of low-dose
subacquisitions.

Having established these abstractions, we can reiterate that our principal
goal is to demonstrate substantial improvements from time-resolved
measurements. In our previous work [19], we introduced the concept of DTTR
measurement along with estimators to apply with these measurements, and we
showed empirical improvements over the trivial estimator that is routinely
applied with conventional measurements. The analysis of estimators in that
work is limited, and certain theoretical assertions are made without proofs.
The CTTR model introduced here is easier to analyze and represents a bound for
what can be done with DTTR measurements. We also provide new analyses of
estimators for the DTTR model. Through these results and Monte Carlo
simulations, the convergence of DTTR estimators to CTTR estimators as
$n\rightarrow\infty$ can be understood precisely (see Sections IV-B and IV-D).

###  II-C Analyses for Oracle Measurement

We initially assume $M=m>0$. Then the oracle measurement (2) includes nonempty
sets of ion arrival times $\\{T_{1},\,T_{2},\,\ldots,\,T_{m}\\}$ and of SE
counts $\\{X_{1},\,X_{2},\,\ldots,\,X_{m}\\}$. The arrival times have beta
distributions, with no dependence on parameter of interest $\eta$. Thus,
arrival times are immaterial to estimation of $\eta$ (i.e., redundant with
knowing $M$). The SE counts are i.i.d. observations with the
$\operatorname{Poisson}(\eta)$ distribution, and it is elementary to show that
$Y=X_{1}+X_{2}+\cdots+X_{m}$ is a sufficient statistic and that $Y/m$ is an
efficient estimator of $\eta$ from the available data. In the case of $M=0$,
there is no basis for any estimate of $\eta$, so we must assign some
arbitrarily chosen number $\eta_{0}$.

From the arguments above, we define the oracle estimator

$\widehat{\eta}_{\rm
oracle}(M,X_{1},X_{2},\ldots,X_{M})=\left\\{\begin{array}[]{@{\,}rl}\eta_{0},&M=0;\\\
{{Y}/{M}},&M>0.\end{array}\right.$ (8)

Conditioned on $M=m>0$, the mean-squared error (MSE) of this estimator is
$\eta/m$ because $Y$ has mean $m\eta$ and variance $m\eta$. Using the Poisson
distribution for $M$ and the total expectation theorem,

$\mathrm{MSE}(\widehat{\eta}_{\rm
oracle})=e^{-\lambda}(\eta-\eta_{0})^{2}+\sum_{m=1}^{\infty}\frac{\eta}{m}\frac{\lambda^{m}}{m!}e^{-\lambda}.$
(9)

For large enough $\lambda$, the arbitrary guess of $\eta_{0}$ when $M=0$ has
little impact on the MSE. Approximating the second term using (60) from
Appendix A gives

$\mathrm{MSE}(\widehat{\eta}_{\rm
oracle})\approx\frac{\eta}{\lambda}\qquad\mbox{for large $\lambda$}.$ (10)

###  II-D Analyses for Conventional Measurement

It is straightforward to show that the conventional measurement $Y$ has
_Neyman Type A_ probability mass function (PMF)

$\mathrm{P}_{Y}(y\,;\,\eta,\lambda)=\frac{e^{-\lambda}\eta^{y}}{y!}\sum_{m=0}^{\infty}\frac{(\lambda
e^{-\eta})^{m}m^{y}}{m!},\quad y=0,\,1,\,\ldots,$ (11)

mean

$\mathrm{E}\\!\left[\,{Y}\,\right]=\lambda\eta,$ (12)

and variance

$\mathrm{var}\\!\left({Y}\right)=\lambda\eta+\lambda\eta^{2}.$ (13)

Starting from (3), (11) follows from the law of total probability, (12) from
iterated expectation with conditioning on $M$, and (13) from the law of total
variance with conditioning on $M$ [19]. Reaching the same conclusions starting
from (4) or (5) involves more complicated computations.

From (12), the baseline estimator

$\widehat{\eta}_{\rm baseline}(Y)=\frac{Y}{\lambda}$ (14)

is unbiased, and from (13) its MSE is

$\mathrm{MSE}(\widehat{\eta}_{\rm baseline})=\frac{\eta(1+\eta)}{\lambda}.$
(15)

If $\lambda$ ions were deterministically incident upon the sample, $Y$ would
be a $\operatorname{Poisson}(\lambda\eta)$ random variable, with variance
$\lambda\eta$, and the MSE of the baseline estimator would be $\eta/\lambda$.
The excess variance in (13), consistent with experimental observations [26],
and excess MSE in (15) are due to the random variation in the number of
incident ions or the _source shot noise_. Our line of work mitigates this
noise.

Since estimation under a Neyman Type A observation model is not well known,
the potential efficiency of the baseline estimator is not evident. To this
end, it is natural to evaluate the Fisher information (FI) about $\eta$ in $Y$
with $\lambda$ as a known parameter, which we denote by
$\mathcal{I}_{Y}(\eta\,;\,\lambda)$.

The FI is defined as

$\mathcal{I}_{Y}(\eta\,;\,\lambda)=\mathrm{E}\\!\left[\,{\left(\frac{\partial\log\mathrm{P}_{Y}(y\,;\,\eta,\lambda)}{\partial\eta}\right)^{2}\,;\,\eta}\,\right],$
(16)

where a known non-random parameter in the expectation is emphasized by putting
it after a semicolon. From (11),

$\displaystyle\log$ $\displaystyle\mathrm{P}_{Y}(y\,;\,\eta,\lambda)$
$\displaystyle=-\lambda+y\log\eta-\log
y!+\log\Bigg{(}\sum_{m=0}^{\infty}\frac{(\lambda
e^{-\eta})^{m}m^{y}}{m!}\Bigg{)}.$

Then taking the derivative with respect to $\eta$, we find that

$\displaystyle\frac{\partial\log\mathrm{P}_{Y}(y\,;\,\eta,\lambda)}{\partial\eta}$
$\displaystyle\\!=\\!$
$\displaystyle\frac{y}{\eta}-\frac{\sum_{m=0}^{\infty}\dfrac{m(\lambda
e^{-\eta})^{m}m^{y}}{m!}}{\sum_{m=0}^{\infty}\dfrac{(\lambda
e^{-\eta})^{m}m^{y}}{m!}}$
$\displaystyle\\!\stackrel{{\scriptstyle(a)}}{{=}}\\!$
$\displaystyle\frac{y}{\eta}-\frac{{\mathrm{P}_{Y}(y+1\,;\,\eta,\lambda)}\bigg{/}{\dfrac{e^{-\lambda}\eta^{y+1}}{(y+1)!}}}{{\mathrm{P}_{Y}(y\,;\,\eta,\lambda)}\bigg{/}{\dfrac{e^{-\lambda}\eta^{y}}{y!}}}$
$\displaystyle\\!=\\!$
$\displaystyle\frac{y}{\eta}-\frac{\mathrm{P}_{Y}(y+1\,;\,\eta,\lambda)}{\mathrm{P}_{Y}(y\,;\,\eta,\lambda)}\frac{y+1}{\eta},$

where (_a_) follows from (11). The FI is the second moment of the above
expression:

$\mathcal{I}_{Y}(\eta\,;\,\lambda)=\sum_{y=0}^{\infty}\left(\frac{y}{\eta}{-}\frac{\mathrm{P}_{Y}(y+1\,;\,\eta,\lambda)}{\mathrm{P}_{Y}(y\,;\,\eta,\lambda)}\frac{y+1}{\eta}\right)^{\\!2}\mathrm{P}_{Y}(y\,;\,\eta,\lambda).$
(17)

While (17) is not readily comprehensible, it can be used to numerically
evaluate $\mathcal{I}_{Y}(\eta\,;\,\lambda)$ and to derive certain useful
asymptotic approximations and limits. One can interpret the ratio
$\mathcal{I}_{Y}(\eta\,;\,\lambda)/\lambda$ as the information gain per
incident ion. As illustrated in Fig. 2, this _normalized Fisher information_
is a decreasing function of $\lambda$, with

$\lim_{\lambda\rightarrow
0}\frac{\mathcal{I}_{Y}(\eta\,;\,\lambda)}{\lambda}=\frac{1}{\eta}-e^{-\eta}$
(18)

and

$\lim_{\lambda\rightarrow\infty}\frac{\mathcal{I}_{Y}(\eta\,;\,\lambda)}{\lambda}=\frac{1}{\eta(1+\eta)}=\frac{1}{\eta}-\frac{1}{1+\eta}.$
(19)

Detailed derivations are provided in Appendix B.

Figure 2: Normalized Fisher information
$\mathcal{I}_{Y}(\eta\,;\,\lambda)/\lambda$ as a function of $\lambda$ for
$\eta=3$. The marked asymptotes are derived in Appendix B.

Using (19) to write

$\mathcal{I}_{Y}(\eta\,;\,\lambda)\approx\frac{\lambda}{\eta(1+\eta)}\qquad\mbox{for
large $\lambda$},$ (20)

we have a match to the reciprocal of the MSE in (15). Thus, the baseline
estimator achieves the Cramér–Rao bound (CRB) asympotically as
$\lambda\rightarrow\infty$, but not otherwise (since the normalized FI is a
decreasing function of $\lambda$).

One could seek improved estimators for low $\lambda$ or improved lower bounds
to demonstrate that substantially better estimators do not exist. We do not
pursue those goals here. In practice, improved estimators for low $\lambda$
may be of limited interest—even if they improve significantly upon the
baseline estimator. For example, with reference to Fig. 2, FI suggests that
one may be able to improve upon the baseline estimator by a factor of 3 at
dose $\lambda=10^{-2}$. However, even the lowest possible MSE would be quite
high at such a low dose.

The key observation from Fig. 2 and the limits in (18) and (19) is that low-
dose measurements are more informative per incident ion than high-dose
measurements. The remainder of the paper studies methods to realize
improvements related to this gap while operating at any dose level—not only
low dose. Furthermore, notice that the MSE in (15) has a simple inversely
proportional relationship with $\lambda$. Most performance bounds and
empirical performances in this paper share this simple $1/\lambda$ behavior,
so we place little emphasis on the performance as $\lambda$ is varied.
Instead, we concentrate on comparisons among different methods and the
performance dependence on $\eta$.

##  III Continuous-Time Time-Resolved Measurement

###  III-A Measurement Distributions

The CTTR measurement (6) contains the number of incident ions that result in
positive detected SEs $\widetilde{M}$, the arrival times of these ions
$\\{\widetilde{T}_{1},\,\widetilde{T}_{2},\,\ldots,\,\widetilde{T}_{\widetilde{M}}\\}$,
and the corresponding SE counts
$\\{\widetilde{X}_{1},\,\widetilde{X}_{2},\,\ldots,\,\widetilde{X}_{\widetilde{M}}\\}$.
As noted in Section II-A, the mutual independence of the arrival times in the
underlying process $\\{T_{1},\,T_{2},\,\ldots\\}$ and all events of the form
$\\{X_{i}=0\\}$ cause the Poisson process property to be preserved.

Since $\mathrm{P}({X_{i}=0})=e^{-\eta}$, the rate $\Lambda$ of the underlying
process is reduced to $\Lambda(1-e^{-\eta})$ for the thinned process. For the
thinned ion count over dwell time $t$, we have
$\widetilde{M}\sim\operatorname{Poisson}(\lambda(1-e^{-\eta}))$, or more
explicitly the PMF

$\mathrm{P}_{\widetilde{M}}(\widetilde{m}\,;\,\eta,\lambda)=\exp(-\lambda(1-e^{-\eta}))\frac{(\lambda(1-e^{-\eta}))^{\widetilde{m}}}{{\widetilde{m}}!},$
(21)

for $\widetilde{m}=0,\,1,\,\ldots$.

The distribution of the $\widetilde{X}_{i}$ variables is simply the zero-
truncation of the $\operatorname{Poisson}(\eta)$ distribution:

$\mathrm{P}_{\widetilde{X}_{i}}(j\,;\,\eta)=\frac{e^{-\eta}}{1-e^{-\eta}}\cdot\frac{\eta^{j}}{j!},\qquad
j=1,\,2,\,\ldots.$ (22)

While the interarrival times of the thinned process have a simple exponential
distribution, this is not relevant to our estimation tasks: Under the CTTR
measurement model, we have $\widetilde{M}$ available, and conditioned on
$\widetilde{M}$, the thinned arrival times have beta distributions with no
dependence on the parameter of interest $\eta$.

###  III-B Fisher Information

We would like to evaluate the FI about $\eta$ in the CTTR measurement (6) with
$\lambda$ as a known parameter:

$\displaystyle\mathcal{I}_{\rm CTTR}(\eta\,;\,\lambda)$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\mathcal{I}_{(\widetilde{T}_{1},\widetilde{X}_{1}),\ldots,(\widetilde{T}_{\widetilde{M}},\widetilde{X}_{\widetilde{M}})|\widetilde{M}}(\eta\,;\,\lambda)+\mathcal{I}_{\widetilde{M}}(\eta\,;\,\lambda)$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\mathcal{I}_{\widetilde{X}_{1},\ldots,\widetilde{X}_{\widetilde{M}}|\widetilde{M}}(\eta\,;\,\lambda)+\mathcal{I}_{\widetilde{M}}(\eta\,;\,\lambda)$
$\displaystyle\stackrel{{\scriptstyle(c)}}{{=}}\mathrm{E}\\!\left[\,{\widetilde{M}}\,\right]\mathcal{I}_{\widetilde{X}_{i}}(\eta\,;\,\lambda)+\mathcal{I}_{\widetilde{M}}(\eta\,;\,\lambda)$
$\displaystyle\stackrel{{\scriptstyle(d)}}{{=}}\lambda(1-e^{-\eta})\mathcal{I}_{\widetilde{X}_{i}}(\eta\,;\,\lambda)+\mathcal{I}_{\widetilde{M}}(\eta\,;\,\lambda),$
(23)

where (_a_) follows from the chain rule for FI [27]; (_b_) from the
conditional distribution of each $\widetilde{T}_{i}$ given $\widetilde{M}$
having no dependence on $\eta$; (_c_) from additivity of FI and the
independence of
$\\{\widetilde{M},\,\widetilde{X}_{1},\,\widetilde{X}_{2},\,\ldots,\,\widetilde{X}_{\widetilde{M}}\\}$;
and (_d_) from substitution of the mean of $\widetilde{M}$. Thus, we need to
evaluate $\mathcal{I}_{\widetilde{X}_{i}}(\eta\,;\,\lambda)$ and
$\mathcal{I}_{\widetilde{M}}(\eta\,;\,\lambda)$. The former, which represents
the FI for estimating $\eta$ from $\widetilde{X}_{i}$ is

$\displaystyle\mathcal{I}_{\widetilde{X}_{i}}(\eta\,;\,\lambda)$
$\displaystyle=\mathrm{E}\\!\left[\,{\left(\frac{\partial\log\mathrm{P}_{\widetilde{X}_{i}}(\widetilde{X}_{i}\,;\,\eta,\lambda)}{\partial\eta}\right)^{\\!2}\,;\,\eta}\,\right]$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\sum_{j=1}^{\infty}\left(\frac{j}{\eta}-\frac{1}{1-e^{-\eta}}\right)^{\\!2}\frac{e^{-\eta}}{1-e^{-\eta}}\frac{\eta^{j}}{j!}$
$\displaystyle=\frac{1}{1-e^{-\eta}}\frac{\eta+1}{\eta}-\frac{1}{(1-e^{-\eta})^{2}},$
(24)

where (_a_) uses the PMF in (22). Similarly,
$\mathcal{I}_{\widetilde{M}}(\eta\,;\,\lambda)$, which represents the FI for
estimating $\eta$ from $\widetilde{M}$, is

$\displaystyle\mathcal{I}_{\widetilde{M}}(\eta\,;\,\lambda)=\mathrm{E}\\!\left[\,{\left(\frac{\partial\log
P_{\widetilde{M}}(\widetilde{M}\,;\,\eta,\lambda)}{\partial\eta}\right)^{\\!2}\,;\,\eta}\,\right]$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\sum_{j=0}^{\infty}\left(\frac{e^{-\eta}}{1-e^{-\eta}}j-\lambda
e^{-\eta}\right)^{\\!2}\frac{e^{-\lambda(1-e^{-\eta})}[\lambda(1-e^{-\eta})]^{j}}{j!}$
$\displaystyle=\frac{\lambda e^{-\eta}}{e^{\eta}-1},$ (25)

where (_a_) uses the PMF in (21). Finally, substituting (24) and (25) into
(III-B) gives

$\displaystyle\mathcal{I}_{\rm CTTR}(\eta\,;\,\lambda)$
$\displaystyle=\lambda\left(\frac{1}{\eta}-{e^{-\eta}}\right).$ (26)

Notice that the FI for CTTR measurement matches the low-dose asymptote given
in (18). It is exact and holds for all values of $\lambda$. The greater FI for
CTTR measurement than for conventional measurement is suggestive of being able
to improve upon the baseline estimator (14). In the following sections, we
define new estimators and demonstrate their improvements.

###  III-C Estimators

In this section, we introduce estimators applicable to CTTR measurement (6).
From (4), it is clear that the total number of detected SEs $Y$ is available.
As we have explained in the derivation of $\mathcal{I}_{\rm
CTTR}(\eta\,;\,\lambda)$ in Section III-B, with $\widetilde{M}$ available,
there is no additional information about $\eta$ in the thinned ion incidence
times
$\\{\widetilde{T}_{1},\,\widetilde{T}_{2},\,\ldots,\,\widetilde{T}_{\widetilde{M}}\\}$.
Furthermore, with $Y$ and $\widetilde{M}$ available, there is no additional
information about $\eta$ in the positive SE counts
$\\{\widetilde{X}_{1},\,\widetilde{X}_{2},\,\ldots,\,\widetilde{X}_{\widetilde{M}}\\}$.
To see this, consider any observation vector
$(\widetilde{X}_{1},\,\widetilde{X}_{2},\,\ldots,\,\widetilde{X}_{\widetilde{M}})=(j_{1},\,j_{2},\,\ldots,\,j_{m})$
conditioned on $\widetilde{M}=\widetilde{m}$. Using independence and (22), the
likelihood of the observation (conditioned on $\widetilde{M}=\widetilde{m}$)
is

$\prod_{i=1}^{\widetilde{m}}\mathrm{P}_{\widetilde{X}_{i}}(j_{i}\,;\,\eta)=\left(\frac{e^{-\eta}}{1-e^{-\eta}}\right)^{\\!\widetilde{m}}\frac{\eta^{j_{1}+j_{2}+\cdots+j_{\widetilde{m}}}}{j_{1}!\,j_{2}!\,\cdots\,j_{\widetilde{m}}!}.$
(27)

As a function of $\eta$, the dependence on SE counts is only through their
sum. Hence, all the estimators in the section depend only on $Y$ and
$\widetilde{M}$.

####  III-C1 Continuous-Time Quotient Mode Estimator

Recall that the oracle estimator (8) divides the total SE count $Y$ by the
number of incident ions $M$. Using $\widetilde{M}$ as a proxy for the number
of incident ions yields the _continuous-time quotient mode_ (CTQM) estimator

$\widehat{\eta}_{\rm
CTQM}(\widetilde{M},Y)=\left\\{\begin{array}[]{@{\,}rl}0,&\widetilde{M}=0;\\\
{{Y}/{\widetilde{M}}},&\widetilde{M}>0.\end{array}\right.$ (28)

Note that the 0 estimate for $\widetilde{M}=0$ is not arbitrary, it is the ML
estimate of $\eta$ for this case. The name “quotient mode” is to acknowledge a
similar concept in a presentation by John Notte of Zeiss [28] and in a patent
application [29].

####  III-C2 Continuous-Time Lambert Quotient Mode Estimator

The CTTR measurement observes a thinned version of the underlying ion
incidence process, so $\widetilde{M}\leq M$. We can do better than to use
$\widetilde{M}$ as a proxy for $M$. Since

$\mathrm{E}\\!\left[\,{\widetilde{M}\,|\,M}\,\right]=(1-e^{-\eta})M,$

$(1-e^{-\eta})^{-1}\widetilde{M}$ would be an unbiased proxy for $M$.
Unfortunately, this has dependence on $\eta$, which is not known. In the
spirit of the oracle and CTQM estimators, we may seek an estimate
$\widehat{\eta}$ that satisfies

$\widehat{\eta}=\frac{Y}{(1-e^{-\widehat{\eta}})^{-1}\widetilde{M}}.$

The solution of this equation gives the _continuous-time Lambert quotient
mode_ (CTLQM) estimator:

$\widehat{\eta}_{\rm CTLQM}=W(-\widehat{\eta}_{\rm
CTQM}e^{-\widehat{\eta}_{\rm CTQM}})+\widehat{\eta}_{\rm CTQM},$ (29)

where $W(\cdot)$ is the Lambert W function [30].

####  III-C3 Continuous-Time Maximum Likelihood Estimator

Rather than use a heuristic approximation for the number of incident ions $M$,
one could instead use the statistically principled ML estimation approach as
follows. The ML estimate is the value of $\eta$ that maximizes the joint
likelihood of the full CTTR observation.

We have already seen that we can drop the times
$\\{\widetilde{T}_{1},\,\widetilde{T}_{2},\,\ldots\,\widetilde{T}_{\widetilde{M}}\\}$,
and the conditional likelihood of
$\\{\widetilde{X}_{1},\,\widetilde{X}_{2},\,\ldots\,\widetilde{X}_{\widetilde{M}}\\}$
given $\widetilde{M}$ was given in (27). Thus, we must maximize the product of
(21) and (27) over $\eta$. For observation
$(\widetilde{M},\,\widetilde{X}_{1},\,\widetilde{X}_{2},\,\ldots,\,\widetilde{X}_{\widetilde{M}})=(\widetilde{m},\,j_{1},\,j_{2},\,\ldots,\,j_{\widetilde{m}})$,
by dropping factors that do not depend on $\eta$, we obtain _continuous-time
ML_ (CTML) estimator

$\widehat{\eta}_{\rm
CTML}=\operatorname*{arg\,max}_{\eta}e^{-\lambda(1-e^{-\eta})}e^{-{\widetilde{m}}\eta}\eta^{y},$

where $y=j_{1}+j_{2}+\cdots+j_{\widetilde{m}}$. The unique maximizer satisfies

$\displaystyle\widehat{\eta}_{\rm CTML}=\frac{Y}{\widetilde{M}+\lambda
e^{-\widehat{\eta}_{\rm CTML}}},$ (30)

which can be solved by using an appropriate root-finding algorithm.

###  III-D Analyzing the Continuous-Time Quotient Mode Estimator

By computing the MSE of $\widehat{\eta}_{\rm CTQM}$, we can evaluate the
efficacy of the quotient mode estimator. We begin by noting that

$\displaystyle\mathrm{MSE}(\widehat{\eta}_{\rm
CTQM})=\operatorname{bias}(\widehat{\eta}_{\rm
CTQM})^{2}+\mathrm{var}\\!\left({\widehat{\eta}_{\rm CTQM}}\right).$ (31)

As detailed in Appendix C, the bias is given by

$\displaystyle\operatorname{bias}(\widehat{\eta}_{\rm CTQM})$
$\displaystyle=\mathrm{E}\\!\left[\,{\widehat{\eta}_{\rm CTQM}}\,\right]-\eta$
$\displaystyle=\left(\frac{e^{-\eta}-e^{-\lambda(1-e^{-\eta})}}{1-e^{-\eta}}\right)\eta.$
(32)

For fixed $\eta$, this is a nonzero bias even as dose
$\lambda\rightarrow\infty$, consistent with the motivation for defining the
CTLQM estimator to improve upon the CTQM estimator. If $\eta\rightarrow\infty$
as well, the bias vanishes, which is consistent with the convergence in
distribution of $\widetilde{M}$ to $M$.

As also detailed in Appendix C, the variance is given by

$\displaystyle\mathrm{var}\\!\left({\widehat{\eta}_{\rm CTQM}}\right)$
$\displaystyle=\frac{\eta^{2}}{\rho^{2}}e^{-\lambda\rho}(1-e^{-\lambda\rho})$
$\displaystyle\quad+\frac{\eta(\rho-\eta
e^{-\eta})}{\rho^{2}}\sum_{j=1}^{\infty}\frac{1}{j}\frac{e^{-\lambda\rho}(\lambda\rho)^{j}}{j!},$
(33)

where we introduce

$\rho=\mathrm{P}({X_{i}>0})={1-e^{-\eta}}$

as a shorthand to make certain expressions more compact. Expression (33) can
be combined with (60) from Appendix A to show that the variance vanishes as
$\lambda\rightarrow\infty$, decaying asymptotically as $\sim 1/\lambda$.
However, because of nonzero bias, the MSE is not inversely proportion to
$\lambda$, and the MSE of $\widehat{\eta}_{\rm CTQM}$ relative to other
estimates depends on $\lambda$. Substituting (III-D) and (33) into (31) gives
an expression for the MSE of the CTQM estimator:

$\displaystyle\mathrm{MSE}(\widehat{\eta}_{\rm CTQM})=$ $\displaystyle\
\frac{\eta^{2}}{\rho^{2}}\left(e^{-\eta}-e^{-\lambda\rho}\right)^{\\!2}+\frac{\eta^{2}}{\rho^{2}}e^{-\lambda\rho}(1-e^{-\lambda\rho})$
$\displaystyle+\frac{\eta(\rho-\eta
e^{-\eta})}{\rho^{2}}\sum_{j=1}^{\infty}\frac{1}{j}\frac{e^{-\lambda\rho}(\lambda\rho)^{j}}{j!}.$
(34)

Substituting the upper bound (61) from Appendix A for the series in (34) gives

$\displaystyle\mathrm{MSE}(\widehat{\eta}_{\rm CTQM})<$ $\displaystyle\
\frac{\eta^{2}}{\rho^{2}}\left(e^{-\eta}-e^{-\lambda\rho}\right)^{\\!2}+\frac{\eta^{2}}{\rho^{2}}e^{-\lambda\rho}(1-e^{-\lambda\rho})$
$\displaystyle+0.518\frac{\eta(\rho-\eta e^{-\eta})}{\rho^{2}}.$ (35)

(a) MSE across $\eta$.

(b) Bias across $\eta$.

(c) Variance across $\eta$.

Figure 3: Comparison of continuous-time time-resolved measurement estimators
as a function of $\eta$. Conventional, oracle, continuous-time quotient mode,
continuous-time Lambert quotient mode, and continuous-time maximum likelihood
estimators are simulated for dose rate $\Lambda=1/1600$ ions per ns and dwell
time $t=32\,000$ ns, hence total dose $\lambda=20$ ions per pixel. (a) MSE.
(b) Bias. (Conventional estimator omitted because its bias is zero.) (c)
Variance.

###  III-E Numerical Comparisons of Estimators

To demonstrate the benefits afforded by CTTR measurement, in Fig. 3 we compare
the conventional, oracle, CTQM, CTLQM and CTML estimators across ground truth
$\eta\in[0,10]$. The MSE values in Fig. 3(a) are computed from $150\,000$
independent Monte Carlo trials, using a dose rate $\Lambda=1/1600$ ions per ns
and dwell time $t=32\,000$ ns, for a total dose $\lambda=20$ ions per pixel.
The curve for $\widehat{\eta}_{\rm baseline}$ matches the theoretical MSE
expression (15). Although unimplementable, the curve for $\widehat{\eta}_{\rm
oracle}$ also matches the theoretical MSE in (9).111We did not need to choose
a value for $\eta_{0}$ because the event $\\{M=0\\}$, which has probability
$e^{-20}\approx 2\cdot 10^{-9}$, did not occur in any of the trials. We can
observe that $\widehat{\eta}_{\rm CTQM}$ has a large MSE for small $\eta$,
caused largely by $\widetilde{M}$ severely underestimating $M$ in these cases.
Predictably, $\widehat{\eta}_{\rm CTLQM}$ reduces MSE tremendously for small
values of $\eta$ because the role of $\widetilde{M}$ is modified the most in
these cases. For low $\eta$ (about 0 to 2.5), $\widehat{\eta}_{\rm CTML}$
achieves lowest MSE amongst all implementable estimators; for moderate $\eta$
(about 2.5 to 5.5), $\widehat{\eta}_{\rm CTQM}$ is slightly better than the
others; and for large $\eta$ (about 5.5 and above), $\widehat{\eta}_{\rm
CTQM}$, $\widehat{\eta}_{\rm CTLQM}$ and $\widehat{\eta}_{\rm CTML}$ all give
nearly identical performance. It is noteworthy that CTQM converges with the
oracle at $\eta$ above about 3.0, though the oracle is unimplementable in
practice.

The MSE trends and comparisons can be better understood through the biases in
Fig. 3(b) and variances in Fig. 3(c). Curves for $\widehat{\eta}_{\rm CTQM}$
coincide with the bias and variance expressions derived in (III-D) and (33).
As previously noted, the large bias of $\widehat{\eta}_{\rm CTQM}$ for small
values of $\eta$ is caused by the number of incident ions $M$ being severely
underestimated by $\widetilde{M}$. The bias of $\widehat{\eta}_{\rm CTQM}$ can
be corrected by the use of $\widehat{\eta}_{\rm CTLQM}$. The dashed cyan-
colored curve in Fig. 3(c) is the CRB for any unbiased estimator, which is the
reciprocal of (26). The variances of the implementable and approximately
unbiased $\widehat{\eta}_{\rm CTLQM}$ and $\widehat{\eta}_{\rm CTML}$
estimators approximately coincide with the CRB.

##  IV Discrete-Time Time-Resolved Measurement

###  IV-A Measurement Distributions

The DTTR measurement (7) is a length-$n$ vector of SE counts collected over
subacquisition dwell times of $t/n$. Thus, some modeling and analysis for DTTR
measurement follows from scaling of $\lambda$ in expressions from Section
II-D. Since $\\{Y_{1},\,Y_{2},\,\ldots,\,Y_{n}\\}$ are independent, their
joint PMF is simply

$\mathrm{P}_{Y_{1},Y_{2},\ldots,Y_{n}}(y_{1},y_{2},\ldots,y_{n}\,;\,\eta,\lambda)=\prod_{k=1}^{n}\mathrm{P}_{Y}(y_{k}\,;\,\eta,\lambda/n),$
(36)

written in terms of the PMF in (11).

###  IV-B Fisher Information

To evaluate the FI about $\eta$ in the DTTR measurement (7) with $\lambda$ as
a known parameter is also quite simple. Because FI is additive over
independent observations,

$\displaystyle\mathcal{I}_{\rm DTTR}(\eta\,;\,\lambda,n)$
$\displaystyle=n\,\mathcal{I}_{Y}(\eta\,;\,\lambda/n),$ (37)

expressed in terms of the FI in (17). While this FI inherits the complexity
and lack of interpretability of (17), the distinction is that the relevant
dose parameter in $\mathcal{I}_{Y}$ has been reduced from $\lambda$ to
$\lambda/n$, so it is more reasonable to approximate with the low-dose
asymptote (18). Specifically, we can write

$\displaystyle\mathcal{I}_{\rm DTTR}(\eta\,;\,\lambda,n)$
$\displaystyle=\lambda\,\frac{\mathcal{I}_{Y}(\eta\,;\,\lambda/n)}{\lambda/n}$
$\displaystyle\approx\lambda\left(\frac{1}{\eta}-e^{-\eta}\right),$ (38)

where the approximation holds for large enough $n$ because of (18). Note that
(38) has the same expression as (26), so as $n\rightarrow\infty$, the FI of
DTTR measurement converges from below to the FI of CTTR measurement.

###  IV-C Estimators

In this section, we present estimators applicable to DTTR measurement (7) that
were first introduced in [19]. The subsequent analyses are new.

####  IV-C1 Discrete-Time Quotient Mode Estimator

Similar to the principle behind the CTQM estimator (28), the number of
subacquisitions with positive SEs,

$L=\sum_{k=1}^{n}\mathbbm{1}_{\\{Y_{k}>0\\}},$ (39)

can be a proxy for the number of incident ions $M$. The _discrete-time
quotient mode_ (DTQM) estimator is then defined as

$\widehat{\eta}_{\rm DTQM}(L,Y)=\left\\{\begin{array}[]{@{\,}rl}0,&L=0;\\\
{{Y}/{L}},&L>0.\end{array}\right.$ (40)

####  IV-C2 Discrete-Time Lambert Quotient Mode Estimator

At low $\eta$, $M$ is severely underestimated by $L$. Correspondingly, the
_discrete-time Lambert quotient mode_ (DTLQM) estimator is obtained by
incorporating the correction factor $(1-e^{-\eta})^{-1}$,

$\widehat{\eta}=\frac{Y}{(1-e^{-\widehat{\eta}})^{-1}L}.$

The solution of this equation gives the estimator:

$\widehat{\eta}_{\rm DTLQM}=W(-\widehat{\eta}_{\rm
DTQM}e^{-\widehat{\eta}_{\rm DTQM}})+\widehat{\eta}_{\rm DTQM}.$ (41)

####  IV-C3 Discrete-Time Maximum Likelihood Estimator

The _discrete-time ML_ (DTML) estimate is the value of $\eta$ that maximizes
the joint likelihood in (36):

$\displaystyle\widehat{\eta}_{\rm
DTML}=\operatorname*{arg\,max}_{\eta}\,\prod_{k=1}^{n}\mathrm{P}_{Y}(y_{k}\,;\,\eta,\lambda/n),$
(42)

where $\mathrm{P}_{Y}(\cdot\,;\,\cdot,\cdot)$ is given by (11). Unlike in the
CTTR case, we have no fixed-point form for the estimator. Instead, it can be
computed by direct numerical optimization. Since the decision variable is
scalar, even a simple grid search is not impractical.

###  IV-D Analyzing the Discrete-Time Quotient Mode Estimator

Like in Section III-D, we analyze the MSE of $\widehat{\eta}_{\rm DTQM}$ by
finding expressions for its bias and variance. In both calculations, we make
use of a zero-truncated modification of the Neyman Type A distribution at ion
incidence parameter $\lambda/n$. Let $p=\mathrm{P}({Y_{k}>0})$. Then using
(11), we find

$p=1-\mathrm{P}_{Y}(0\,;\,\eta,\lambda/n)=1-\exp\\!\left(-\frac{\lambda}{n}(1-e^{-\eta})\right)\\!.$
(43)

If $\widetilde{Y}_{k}$ is the zero-truncated version of $Y_{k}$, then its PMF
is

$\mathrm{P}_{\widetilde{Y}_{k}}(y_{k})=\frac{1}{p}\frac{e^{-{{\lambda}/{n}}}\eta^{y_{k}}}{y_{k}!}\sum_{m=0}^{\infty}\frac{(\frac{\lambda}{n}e^{-\eta})^{m}m^{y_{k}}}{m!},\quad
y_{k}=1,2,\ldots,$ (44)

its mean is

$\mathrm{E}[\,{\widetilde{Y}_{k}}\,]=\frac{1}{p}\frac{\lambda\eta}{n},$ (45)

and its variance is

$\mathrm{var}({\widetilde{Y}_{k}})=\frac{1}{p}\left(\frac{\lambda}{n}\eta+\frac{\lambda}{n}\eta^{2}+\left(\frac{\lambda}{n}\eta\right)^{2}\right)-\frac{1}{p^{2}}\left(\frac{\lambda}{n}\eta\right)^{2}.$
(46)

Furthermore, as a sum of independent indicator random variables, $L$ is a
binomial random variable with $n$ trials and success probability $p$ for each
trial.

For $\ell>0$,

$\displaystyle\mathrm{E}\\!\left[\,{\widehat{\eta}_{\rm
DTQM}\,|\,L=\ell}\,\right]$
$\displaystyle=\mathrm{E}\\!\left[\,{\frac{Y}{L}\,|\,L=\ell}\,\right]=\frac{1}{\ell}\mathrm{E}\\!\left[\,{Y\,|\,L=\ell}\,\right]$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\frac{1}{\ell}\mathrm{E}\\!\left[\,{\sum_{j=1}^{\ell}\widetilde{Y}_{j}}\,\right]=\mathrm{E}\\!\left[\,{\widetilde{Y}_{k}}\,\right],$
(47)

where (_a_) follows from using $\widetilde{Y}_{j}$ to denote the $j$th
positive subacquistion $Y_{k}$. Trivially,

$\mathrm{E}\\!\left[\,{\widehat{\eta}_{\rm DTQM}\,|\,L=0}\,\right]=0.$ (48)

Using $\mathrm{P}({L>0})=1-(1-p)^{n}$ and the total expectation theorem to
combine (47) and (48) gives

$\mathrm{E}\\!\left[\,{\widehat{\eta}_{\rm
DTQM}}\,\right]=\mathrm{E}[\,{\widetilde{Y}_{k}}\,](1-(1-p)^{n})).$

The bias of $\widehat{\eta}_{\rm DTQM}$ is thus given by

$\displaystyle\operatorname{bias}(\widehat{\eta}_{\rm DTQM})$
$\displaystyle=\mathrm{E}\\!\left[\,{\widehat{\eta}_{\rm DTQM}}\,\right]-\eta$
$\displaystyle=\frac{\lambda\eta}{np}\left[1-(1-p)^{n}\right]-\eta,$ (49)

where (45) has been substituted.

Similar arguments, detailed in Appendix D, yield

$\displaystyle\mathrm{var}\\!\left({\widehat{\eta}_{\rm DTQM}}\right)$
$\displaystyle=\left(\mathrm{E}[\,{\widetilde{Y}_{k}}\,]\right)^{2}\left[1-(1-p)^{n}\right](1-p)^{n}$
$\displaystyle\quad+\mathrm{var}({\widetilde{Y}_{k}})\sum_{\ell=1}^{n}\frac{1}{\ell}\binom{n}{\ell}p^{\ell}(1-p)^{n-\ell},$
(50)

where $\mathrm{E}[\,{\widetilde{Y}_{k}}\,]$ is given in (45) and
$\mathrm{var}({\widetilde{Y}_{k}})$ is given in (46). Since the behavior of
the series in (IV-D) for large $n$ is not evident, we also derive in Appendix
D the lower bound

$\displaystyle\mathrm{var}\\!\left({\widehat{\eta}_{\rm DTQM}}\right)$
$\displaystyle\geq\left(\mathrm{E}[\,{\widetilde{Y}_{k}}\,]\right)^{2}\left[1-(1-p)^{n}\right](1-p)^{n}$
$\displaystyle\quad+\mathrm{var}({\widetilde{Y}_{k}})\frac{[1-(1-p)^{n}]^{2}}{np}.$
(51)

The expressions in (IV-D) and (IV-D) can be added to the square of the bias
from (IV-D) to obtain the MSE of $\widehat{\eta}_{\rm DTQM}$ and a lower bound
for this MSE.

(a) MSE across $n$

(b) Bias across $n$

(c) Variance across $n$

Figure 4: MSE, bias, and variance of $\widehat{\eta}_{\rm DTQM}$ as functions
of the number of subacquisitions $n$ and those of $\widehat{\eta}_{\rm CTQM}$
(dashed lines) for $\lambda=10$ and $\eta=5$. The Cramér–Rao lower bound for
time-resolved measurements (yellow, see (37)) is plotted as well.

#### High $n$ Limits

One of the themes of this paper is that the CTTR measurement model is easier
to analyze than the DTTR measurement model, yet it gives expressions relevant
to understanding the practical DTTR measurement setting. We would like to
examine properties of $\widehat{\eta}_{\rm DTQM}$ at the high $n$ limit, in
part to demonstrate that we obtain matches to $\widehat{\eta}_{\rm CTQM}$
behavior.

For the limit of the bias, we will use two facts proven in Appendix E:

$\displaystyle\lim_{n\to\infty}np$ $\displaystyle=\lambda(1-e^{-\eta}),$ (52)
$\displaystyle\lim_{n\to\infty}(1-p)^{n}$
$\displaystyle=e^{-\lambda(1-e^{-\eta})}.$ (53)

By substituting (52) and (53) into (IV-D),

$\displaystyle\lim_{n\to\infty}\operatorname{bias}(\widehat{\eta}_{\rm DTQM})$
$\displaystyle=\frac{\lambda\eta}{\lambda(1-e^{-\eta})}(1-e^{-\lambda(1-e^{-\eta})})-\eta$
$\displaystyle=\left(\frac{e^{-\eta}-e^{-\lambda(1-e^{-\eta})}}{1-e^{-\eta}}\right)\eta,$
(54)

an exact match to $\operatorname{bias}(\widehat{\eta}_{\rm CTQM})$ in (III-D).

(a) MSE across $\eta$

(b) Bias across $\eta$

(c) Variance across $\eta$

Figure 5: Comparison of discrete-time time-resolved estimators as a function
of $\eta$. Conventional, oracle, discrete-time quotient mode, discrete-time
Lambert quotient mode, and discrete-time maximum likelihood estimators are
simulated for total dose $\lambda=20$ split over $n=200$ subacquisitions. (a)
MSE. (b) Bias. (Conventional estimator excluded because its bias is zero.) (c)
Variance.

For the limit of the variance, we will use two additional facts proven in
Appendix E:

$\displaystyle\lim_{n\to\infty}\mathrm{E}[\,{\widetilde{Y}_{k}}\,]$
$\displaystyle=\frac{\eta}{1-e^{-\eta}},$ (55)
$\displaystyle\lim_{n\to\infty}\mathrm{var}({\widetilde{Y}_{k}})$
$\displaystyle=\frac{\eta-(\eta+\eta^{2})e^{-\eta}}{(1-e^{-\eta})^{2}}.$ (56)

Along with substitution of limits, we recognize that the summand in (IV-D)
includes a binomial probability for which there is a Poisson limit because
$np$ approaches a positive constant [31]:

$\binom{n}{\ell}p^{\ell}(1-p)^{n-\ell}\rightarrow\frac{(\lambda(1-e^{-\eta}))^{\ell}}{\ell!}e^{-\lambda(1-e^{-\eta})}.$
(57)

It is now tedious but straightforward to substitute (52), (53), (55), (56),
and (57) into (IV-D) to obtain

$\displaystyle\lim_{n\to\infty}\mathrm{var}\\!\left({\widehat{\eta}_{\rm
DTQM}}\right)$ $\displaystyle=\mathrm{var}\\!\left({\widehat{\eta}_{\rm
CTQM}}\right),$ (58)

where $\mathrm{var}\\!\left({\widehat{\eta}_{\rm CTQM}}\right)$ is given in
(33).

###  IV-E Numerical Comparisons of Estimators

Fig. 4 shows the MSE, bias, and variance of $\widehat{\eta}_{\rm DTQM}$ as
functions of the number of subacquisitions $n$ when $\lambda=10$ and $\eta=5$.
Fig. 4(b) shows the bias approaching the asymptote given by (IV-D). Fig. 4(a)
and Fig. 4(c) show the MSE and variance and their lower bounds based on
(IV-D). When $n$ is sufficiently large, these are small and close to the
Cramér–Rao bound.

Fig. 5 compares the conventional, oracle, DTQM, DTLQM, and DTML estimators
across ground truth $\eta\in[0,10]$. MSE, bias, and variance are computed by
Monte Carlo simulation using total dose $\lambda=20$ split over $n=200$
subacquisitions. The conventional curves match those in the CT setting in Fig.
3. The DTQM estimator has a large bias for smaller $\eta$, which is absent
from the DTLQM estimator. Unlike in CT, where the bias for all the studied
estimators vanish at moderate and high $\eta$, both the DTQM and DTLQM
estimator have substantial bias that is approximately linear in $\eta$ for
moderate and high $\eta$. The lack of bias of the DTML estimator explains its
uniform superiority over the DTQM and DTLQM estimators; this contrasts with
the CT setting in which CTQM, CTLQM, and CTML estimators have nearly equal MSE
at moderate and high $\eta$.

|  | Absolute errors of time-resolved estimators
---|---|---
Ground truth |  | DT $n=50$ | DT $n=100$ | DT $n=200$ | DT $n=500$ | CT
|  QM |  |  |  |  |
Absolute error of |  |  |  |  |  |
conventional estimator |  |  |  |  |  |
|  LQM |  |  |  |  |
Absolute error of |  |  |  |  |  |
oracle estimator |  |  |  |  |  |
|  ML |  |  |  |  |

$\mathrm{MSE}=1.111$

$\mathrm{MSE}=0.566$

$\mathrm{MSE}=0.392$

$\mathrm{MSE}=0.313$

$\mathrm{MSE}=0.272$

$\mathrm{MSE}=0.599$

$\mathrm{MSE}=0.877$

$\mathrm{MSE}=0.398$

$\mathrm{MSE}=0.259$

$\mathrm{MSE}=0.206$

$\mathrm{MSE}=0.181$

$\mathrm{MSE}=0.135$

$\mathrm{MSE}=0.246$

$\mathrm{MSE}=0.211$

$\mathrm{MSE}=0.191$

$\mathrm{MSE}=0.178$

$\mathrm{MSE}=0.166$

Figure 6: Simulated FIB microscopy experiment with time-resolved estimators
under discrete- and continuous-time settings. Aside from the ground truth in
the upper-left corner, with $\eta\in[1,8]$, all images are of the absolute
value of the error. All results are for total dose $\lambda=20$ mean incident
ions per pixel. The columns of time-resolved estimators are for increasing
numbers of subacquisitions $n$, culminating in the limiting continuous-time
case. Quotient mode, Lambert quotient mode, maximum likelihood estimators are
compared, with the conventional estimator (14) and oracle estimator (8)
provided for context. None of these estimators include spatial regularization.

##  V Simulated Microscopy Results

Figures 3 and 5 show that time-resolved estimators improve upon the
conventional processing of abstracted FIB microscope data, and Fig. 4 shows
that the performances of DTTR estimators improve with increasing numbers of
subacquisitions, converging to the performances of corresponding CTTR
estimators. We conclude with visual results to demonstrate these properties in
simulated FIB microscopy experiments.

We use the _Hairstyle_ 222A SEM image of the upper part of the style and
stigma from an Arabidopsis flower,
https://www.flickr.com/photos/fei_company/9316514268/in/set-72157634429801580/
image from ThermoFisher Scientific (upper-left of Fig 6) as the ground truth
image, scaled to have SE yield $\eta\in[1,\,8]$. All experiments use total
dose $\lambda=20$. Fig. 6 displays absolute error images for quotient mode,
Lambert quotient mode and maximum likelihood estimators. CTTR measurements are
simulated and, by division of the dwell time, also interpreted as DTTR
measurements for $n\in\\{50,\,100,\,200,\,500\\}$ subacquisitions. Fig. 7
complements Fig. 6 by plotting MSEs as functions of $n$ for these and
additional values of $n$.

In Fig. 6, each estimator shows improving performance as $n$ increases, with
$n=500$ coming close to the CT performance. With $n$ increased to 2000 in Fig.
7, convergence to CT performance is more clearly indicated. For each value of
$n$, the ranking of estimators has ML best, LQM second, and QM worst. For
$n\geq 100$, all the TR methods perform better than the conventional
estimator. One way to summarize is to see that with a factor of 4.4 separating
the MSEs of the conventional and oracle estimators, the CTML estimator
achieves a 3.6 times lower MSE than the conventional estimator.

Figure 7: Mean-squared errors as functions of number of subacquisitions $n$ in
simulated FIB microscopy of the image in Fig. 6 using conventional, oracle,
quotient mode, Lambert quotient mode, and maximum-likelihood estimators. Total
dose is $\lambda=20$, and mean secondary yield $\eta$ of the ground truth
image is scaled to $[1,8]$.

##  VI Conclusion

In this work, we establish an abstract framework for TR measurement in FIB
microscopy with direct electron detection. Through estimation-theoretic
analyses, analyses of estimators, and Monte Carlo FIB imaging simulations, we
show the extent to which source shot noise can be mitigated by TR measurement
methods. The most easily interpreted conclusion comes from the Fisher
information of continuous-time TR measurements $\lambda(1/\eta-e^{-\eta})$ in
(26); when $\eta$ is not too small, this is only slightly smaller than the
$\lambda/\eta$, which is equal to the FI that would be obtained with a
deterministic incident particle beam. The dependence on mean SE yield $\eta$
shows that TR methods have greater potential in FIB microscopy than in SEM.

Continuous-time measurement is not implementable in any foreseeable
technology. Instead, it is intended as a greatly simpler way to understand the
limits of performance under very fine time resolution than to consider
$n\rightarrow\infty$ limits for discrete-time results. Through performance
comparisons such as those in Figs. 4, 6, and 7, one can predict the time
resolution that is necessary to approach the CT limit within a desired margin.
Necessary time resolution can also be understood through the use of Fig. 2 to
choose a sufficiently small value for $\lambda/n$. For example, for the
illustrated value of $\eta=3$, the normalized Fisher information plot suggests
that when the time resolution is fine enough for $\lambda/n<0.1$, at least 83%
of the improvement created by time-resolved measurement will be attained.

We study three types of unconventional estimators for both continuous- and
discrete-time TR measurements. QM estimators are the simplest to implement and
are similar to estimators proposed by Zeiss but not made commercially
available. LQM estimators greatly reduce a source of bias at low $\eta$ and
merely require a table lookup to be applied to QM estimates. ML estimators
require root-finding or minimization of a non-convex function. The
relationships among the estimators are nontrivial: though generally best, the
ML estimator does not outperform the others uniformly over $\eta$; and though
far better at low $\eta$, the LQM estimator does not outperform the QM
estimator uniformly over $\eta$.

The improvements presented here seem to be rooted entirely in making the
number of incident ions estimable, and this is potentially applicable even
without direct SE detection. Indirect electron detection creates uncertainty
in the number of detected SEs, including uncertainty in whether any SEs were
detected and hence in whether an ion was incident. However, the experimental
results of [19] suggest that the degradation in mitigating source shot noise
can be small, since the improvements presented therein are similar to the
results presented here. The QM and LQM estimators may be extended to cases in
which the probabilistic model relating measurements to numbers of SEs is
complicated or uncertain. For an advantage from TR measurement, it may be
enough to have a mean instrument output that is monotonically increasing with
the number of detected SEs. Then a QM or LQM estimator can use $\widetilde{M}$
or $(1-e^{-\eta})^{-1}\widetilde{M}$ to normalize a sum of nonlinearly scaled
TR measurements to mitigate source shot noise. This is one of several lines of
inquiry suggested by the results presented here.

##  Appendix A High-Dose Performance of Oracle Estimator

Let

$g(x)=e^{-x}\sum_{m=1}^{\infty}\frac{1}{m}\frac{x^{m}}{m!}.$ (59)

This function appears in the performance of the oracle estimator (9) and the
variance (33) and MSE (34) of the CTQM estimator. We are interested in
approximating and bounding $g(x)$ to better understand those expressions.

For $x\ll 1$, the first term is dominant, so $g(x)\approx x$. The behavior at
large $x$ is less obvious. The series in (59) converges to
$\mathrm{Ei}(x)-\gamma-\log x$, where $\mathrm{Ei}(x)$ is the exponential
integral function and $\gamma$ is the Euler–Mascheroni constant [32].
Asympototically for $x\rightarrow\infty$, $\mathrm{Ei}(x)\sim{{e^{x}}/{x}}$,
meaning that the ratio of the expressions approaches 1. Thus,

$g(x)\sim\frac{1}{x}.$ (60)

The log-log plot of $g(x)$ in Fig. 8 shows the accuracies of the low- and
high-$x$ approximations. Being nonnegative, continuous, and vanishing as
$x\rightarrow\infty$, $g(x)$ has a finite upper bound:

$g(x)\leq 0.518,\qquad\mbox{for all $x\in[0,\infty)$.}$ (61)

Figure 8: Function $g(x)$ in (59) along with low- and high-$x$ asymptotes.

##  Appendix B Normalized Fisher Information Limits

###  B-A Low-Dose Limit

To evaluate $\lim_{\lambda\rightarrow
0}{{\mathcal{I}_{Y}(\eta\,;\,\lambda)}/{\lambda}}$, we first find
$\lambda\rightarrow 0$ limits of expressions that appear in (17), including
both the PMF in (11) and the probability ratio
${{\mathrm{P}_{Y}(y+1\,;\,\eta,\lambda)}/{\mathrm{P}_{Y}(y\,;\,\eta,\lambda)}}$.

For $y=0$,

$\displaystyle\mathrm{P}_{Y}(0\,;\,\eta,\lambda)$
$\displaystyle=\frac{e^{-\lambda}\eta^{0}}{0!}\sum_{m=0}^{\infty}\frac{(\lambda
e^{-\eta})^{m}m^{0}}{m!}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}e^{-\lambda}\sum_{m=0}^{\infty}\frac{(\lambda
e^{-\eta})^{m}}{m!}\stackrel{{\scriptstyle(b)}}{{=}}e^{-\lambda}\exp(\lambda
e^{-\eta}),$ (62)

where (_a_) follows from $m^{0}=1$; and (_b_) from identifying the series
expansion of the exponential function. Similarly, for $y=1$,

$\displaystyle\mathrm{P}_{Y}(1\,;\,\eta,\lambda)$
$\displaystyle=\frac{e^{-\lambda}\eta^{1}}{1!}\sum_{m=0}^{\infty}\frac{(\lambda
e^{-\eta})^{m}m^{1}}{m!}$ $\displaystyle=(e^{-\lambda}\eta)(\lambda
e^{-\eta})\exp(\lambda e^{-\eta}),$ (63)

and for $y=2$,

$\displaystyle\mathrm{P}_{Y}(2\,;\,\eta,\lambda)$
$\displaystyle=\frac{e^{-\lambda}\eta^{2}}{2!}\sum_{m=0}^{\infty}\frac{(\lambda
e^{-\eta})^{m}m^{2}}{m!}$
$\displaystyle=\frac{e^{-\lambda}\eta^{2}}{2}(\lambda e^{-\eta})(1+\lambda
e^{-\eta})\exp(\lambda e^{-\eta}).$ (64)

For general $y>0$,

$\displaystyle\mathrm{P}_{Y}(y\,;\,\eta,\lambda)$
$\displaystyle=\frac{e^{-\lambda}\eta^{y}}{y!}\sum_{m=0}^{\infty}\frac{(\lambda
e^{-\eta})^{m}m^{y}}{m!}$
$\displaystyle=\frac{e^{-\lambda}\eta^{y}}{y!}(\lambda
e^{-\eta})\,\mathrm{poly}_{y-1}(\lambda e^{-\eta})\exp(\lambda e^{-\eta}),$
(65)

where $\mathrm{poly}_{y}(\lambda e^{-\eta})$ is a degree-$y$ polynomial in
$\lambda e^{-\eta}$ with unit constant term. This allows us to conclude, for
any $y>0$,

$\lim_{\lambda\rightarrow
0}\frac{\mathrm{P}_{Y}(y\,;\,\eta,\lambda)}{\lambda}=\frac{\eta^{y}}{y!}e^{-\eta}.$
(66)

From (62) and (63), we obtain, for $y=0$,

$\frac{\mathrm{P}_{Y}(y+1\,;\,\eta,\lambda)}{\mathrm{P}_{Y}(y\,;\,\eta,\lambda)}=\frac{\mathrm{P}_{Y}(1\,;\,\eta,\lambda)}{\mathrm{P}_{Y}(0\,;\,\eta,\lambda)}=\eta\lambda
e^{-\eta}.$ (67)

From (63) and (64), we obtain, for $y=1$,

$\frac{\mathrm{P}_{Y}(y+1\,;\,\eta,\lambda)}{\mathrm{P}_{Y}(y\,;\,\eta,\lambda)}=\frac{\mathrm{P}_{Y}(2\,;\,\eta,\lambda)}{\mathrm{P}_{Y}(1\,;\,\eta,\lambda)}=\frac{1}{2}\eta(1+\lambda
e^{-\eta}).$ (68)

For general $y>0$, it follows from (66) that

$\lim_{\lambda\rightarrow
0}\frac{\mathrm{P}_{Y}(y+1\,;\,\eta,\lambda)}{\mathrm{P}_{Y}(y\,;\,\eta,\lambda)}=\frac{\eta}{y+1}.$
(69)

Now to evaluate $\lim_{\lambda\rightarrow
0}{{\mathcal{I}_{Y}(\eta\,;\,\lambda)}/{\lambda}}$, we can pass the limit
through to each term in (17). The first term is

$\displaystyle\lim_{\lambda\rightarrow
0}\left(\frac{0}{\eta}-\frac{\mathrm{P}_{Y}(1\,;\,\eta,\lambda)}{\mathrm{P}_{Y}(0\,;\,\eta,\lambda)}\frac{1}{\eta}\right)^{2}\frac{\mathrm{P}_{Y}(0\,;\,\eta,\lambda)}{\lambda}$
$\displaystyle\quad\stackrel{{\scriptstyle(a)}}{{=}}\lim_{\lambda\rightarrow
0}\left(\frac{0}{\eta}-\eta\lambda
e^{-\eta}\frac{1}{\eta}\right)^{2}\frac{e^{-\lambda}\exp(\lambda
e^{-\eta})}{\lambda}=0,$

where (_a_) follows from (62) and (67). By substituting (66) and (69) in (17),
the remaining terms give

$\displaystyle\lim_{\lambda\rightarrow
0}\frac{\mathcal{I}_{Y}(\eta\,;\,\lambda)}{\lambda}=\sum_{y=1}^{\infty}\left(\frac{y}{\eta}-\frac{\eta}{y+1}\frac{y+1}{\eta}\right)^{2}\frac{\eta^{y}}{y!}e^{-\eta}$
$\displaystyle\quad=\sum_{y=1}^{\infty}\left(\frac{y}{\eta}-1\right)^{2}\frac{\eta^{y}}{y!}e^{-\eta}=\left(\frac{e^{\eta}}{\eta}-1\right)e^{-\eta}=\frac{1}{\eta}-e^{-\eta}.$

This proves (18), as desired.

###  B-B High-Dose Limit

Let us first compute the Fisher information for the parameter $\eta$ when a
Gaussian random variable has mean $\eta$ and variance $f(\eta)$ for some
twice-differentiable function $f$. Let $S\sim\mathcal{N}(\eta,\,f(\eta))$.
Then the log-likelihood of $S$ is

$\log f_{S}(s\,;\,\eta)=-\frac{1}{2}\log(2\pi)-\frac{1}{2}\log
f(\eta)-\frac{(s-\eta)^{2}}{2f(\eta)}.$ (70)

The derivative of $\log f_{S}(s\,;\,\eta)$ with respect to $\eta$ is

$\displaystyle\frac{\partial\log f_{S}(s\,;\,\eta)}{\partial\eta}$
$\displaystyle=-\frac{f^{\prime}(\eta)}{2f(\eta)}-\frac{2(\eta-s)f(\eta)-(\eta-s)^{2}f^{\prime}(\eta)}{2f(\eta)^{2}}.$

The second derivative is then

$\displaystyle\frac{\partial^{2}\log f_{S}(s\,;\,\eta)}{\partial\eta^{2}}$
$\displaystyle=-\frac{f^{\prime\prime}(\eta)f(\eta)-f^{\prime}(\eta)^{2}}{2f(\eta)^{2}}-\frac{1}{f(\eta)}$
$\displaystyle\quad+\frac{2f^{\prime}(\eta)}{f(\eta)^{2}}(\eta-s)$
$\displaystyle\quad-\frac{2[f^{\prime}(\eta)]^{2}-f^{\prime\prime}(\eta)f(\eta)}{2f(\eta)^{3}}(\eta-s)^{2}.$

The Fisher information for the estimation of $\eta$ is

$\displaystyle\mathcal{I}_{S}(\eta)$
$\displaystyle=\mathrm{E}\\!\left[\,{-\frac{\partial^{2}\log
f_{S}(S\,;\,\eta)}{\partial\eta^{2}}\,;\,\eta}\,\right]$
$\displaystyle=\frac{f^{\prime\prime}(\eta)f(\eta)-f^{\prime}(\eta)^{2}}{2f(\eta)^{2}}+\frac{1}{f(\eta)}-\frac{2f^{\prime}(\eta)}{f(\eta)^{2}}\mathrm{E}\\!\left[\,{\eta-S}\,\right]$
$\displaystyle\quad+\frac{2[f^{\prime}(\eta)]^{2}-f^{\prime\prime}(\eta)f(\eta)}{2f(\eta)^{3}}\mathrm{E}\\!\left[\,{(\eta-S)^{2}}\,\right]$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\frac{f^{\prime\prime}(\eta)f(\eta)-f^{\prime}(\eta)^{2}}{2f(\eta)^{2}}+\frac{1}{f(\eta)}-\frac{2f^{\prime}(\eta)}{f(\eta)^{2}}\cdot
0$
$\displaystyle\quad+\frac{2[f^{\prime}(\eta)]^{2}-f^{\prime\prime}(\eta)f(\eta)}{2f(\eta)^{3}}\cdot
f(n)$
$\displaystyle=\frac{1}{f(\eta)}+\frac{[f^{\prime}(\eta)]^{2}}{2f(\eta)^{2}},$
(71)

where (_a_) follows from substituting
$\mathrm{E}\\!\left[\,{\eta-S}\,\right]=0$ and
$\mathrm{E}[\,{(\eta-S)^{2}}\,]=\mathrm{var}\\!\left({S}\right)=f(\eta)$.
(Note that this simplifies to the familiar reciprocal of the variance when
$f(\eta)$ is a constant.)

At high dose, $Y/\lambda$ is well-approximated as a
$\mathcal{N}(\eta,\,\eta(\eta+1)/\lambda)$ random variable [33, Sect. IV].
Thus, define $f(\eta)={{\eta(\eta+1)}/{\lambda}}$ so that $Y/\lambda$ is
approximated well by $S$. Substituting
$f^{\prime}(\eta)={{(2\eta+1)}/{\lambda}}$ into (71) gives

$\displaystyle\mathcal{I}_{S}(\eta)$
$\displaystyle=\frac{\lambda}{\eta(\eta+1)}+\frac{(2\eta+1)^{2}}{2\eta^{2}(\eta+1)^{2}}.$
(72)

Since $Y\approx\lambda S$,

$\displaystyle\lim_{\lambda\rightarrow\infty}\frac{\mathcal{I}_{Y}(\eta\,;\,\lambda)}{\lambda}$
$\displaystyle=\lim_{\lambda\rightarrow\infty}\left[\frac{1}{\eta(\eta+1)}+\frac{(2\eta+1)^{2}}{2\lambda\eta^{2}(\eta+1)^{2}}\right]$
$\displaystyle=\frac{1}{\eta(\eta+1)},$

as desired.

##  Appendix C Derivation of Mean-Squared Error for Continuous-Time Quotient
Mode Estimator

###  C-A Bias of $\widehat{\eta}_{\rm CTQM}$

For $m>0$,

$\displaystyle\mathrm{E}\\!\left[\,{\widehat{\eta}_{\rm
CTQM}\,|\,\widetilde{M}=m}\,\right]=\mathrm{E}\\!\left[\,{\frac{Y}{\widetilde{M}}\,|\,\widetilde{M}=m}\,\right]$
$\displaystyle\quad=\frac{1}{m}\mathrm{E}\\!\left[\,{Y\,|\,\widetilde{M}=m}\,\right]\stackrel{{\scriptstyle(a)}}{{=}}\frac{1}{m}\mathrm{E}\\!\left[\,{\sum_{i=1}^{m}\widetilde{X}_{i}}\,\right]=\mathrm{E}\\!\left[\,{\widetilde{X}_{i}}\,\right]$
$\displaystyle\quad\stackrel{{\scriptstyle(b)}}{{=}}\frac{\eta}{1-e^{-\eta}},$
(73)

where (_a_) follows from using (4) as an expression for $Y$; and (_b_) from
the effect of zero-truncation on the $\operatorname{Poisson}(\eta)$
distribution. Trivially,

$\displaystyle\mathrm{E}\\!\left[\,{\widehat{\eta}_{\rm
CTQM}\,|\,\widetilde{M}=0}\,\right]=0.$ (74)

Using $\mathrm{P}({\widetilde{M}>0})=1-e^{-\lambda(1-e^{-\eta})}$ from (21)
and the total expectation theorem to combine (73) and (74) gives

$\mathrm{E}\\!\left[\,{\widehat{\eta}_{\rm
CTQM}}\,\right]=\left(1-e^{-\lambda(1-e^{-\eta})}\right)\frac{\eta}{1-e^{-\eta}}.$
(75)

Subtracting $\eta$ gives (III-D).

###  C-B Variance of $\widehat{\eta}_{\rm CTQM}$

From (73) and (74), $\mathrm{E}[\,{\widehat{\eta}_{\rm
CTQM}\,|\,\widetilde{M}}\,]$ is a two-valued random variable equal to
$\eta/(1-e^{-\eta})$ with probability $1-e^{\lambda(1-e^{-\eta})}$ and equal
to 0 otherwise. Thus,

$\displaystyle\mathrm{var}\\!\left({\mathrm{E}\\!\left[\,{\widehat{\eta}_{\rm
CTQM}\,|\,\widetilde{M}}\,\right]}\right)$ $\displaystyle\
=\frac{\eta^{2}}{(1-e^{-\eta})^{2}}e^{-\lambda(1-e^{-\eta})}\left(1-e^{-\lambda(1-e^{-\eta})}\right)$
(76)

by direct calculation.

Toward computing $\mathrm{E}[\,{\mathrm{var}({\widehat{\eta}_{\rm
CTQM}\,|\,\widetilde{M}})}\,]$, let us first examine
$\mathrm{var}({\widehat{\eta}_{\rm CTQM}\,|\,\widetilde{M}=m})$. For $m>0$,

$\displaystyle\mathrm{var}\\!\left({\widehat{\eta}_{\rm
CTQM}\,|\,\widetilde{M}=m}\right)=\mathrm{var}\\!\left({\frac{Y}{\widetilde{M}}\,|\,\widetilde{M}=m}\right)$
$\displaystyle\quad=\frac{1}{m^{2}}\mathrm{var}\\!\left({Y\,|\,\widetilde{M}=m}\right)\stackrel{{\scriptstyle(a)}}{{=}}\frac{1}{m^{2}}\mathrm{var}\\!\left({\sum_{i=1}^{m}\widetilde{X}_{i}}\right)$
$\displaystyle\quad=\frac{1}{m}\mathrm{var}\\!\left({\widetilde{X}_{i}}\right)\stackrel{{\scriptstyle(b)}}{{=}}\frac{1}{m}\left(\frac{\eta+\eta^{2}}{1-e^{-\eta}}-\frac{\eta^{2}}{(1-e^{-\eta})^{2}}\right),$
(77)

where (_a_) follows from using (4) as an expression for $Y$; and (_b_) from
the effect of zero-truncation on the $\operatorname{Poisson}(\eta)$
distribution. Trivially,

$\displaystyle\mathrm{var}\\!\left({\widehat{\eta}_{\rm
CTQM}\,|\,\widetilde{M}=0}\right)=0.$ (78)

Using the PMF of $\widetilde{M}$ from (21) and the total expectation theorem
to combine (77) and (78) gives

$\displaystyle\mathrm{E}\\!\left[\,{\mathrm{var}\\!\left({\widehat{\eta}_{\rm
CTQM}\,|\,\widetilde{M}}\right)}\,\right]$
$\displaystyle=\left(\frac{\eta+\eta^{2}}{1-e^{-\eta}}-\frac{\eta^{2}}{(1-e^{-\eta})^{2}}\right)\\!e^{-\lambda(1-e^{-\eta})}$
$\displaystyle\qquad\cdot\sum_{m=1}^{\infty}\frac{1}{m}\frac{[\lambda(1-e^{-\eta})]^{m}}{m!}.$
(79)

By summing (76) and (C-B) we obtain the variance of $\widehat{\eta}_{\rm
CTQM}$, which verifies (33).

##  Appendix D Derivation of Mean-Squared Error for Discrete-Time Quotient
Mode Estimator

###  D-A Variance of $\widehat{\eta}_{\rm DTQM}$

From (47) and (48), $\mathrm{E}[\,{\widehat{\eta}_{\rm DTQM}\,|\,L}\,]$ is a
two-valued random variable equal to $\mathrm{E}[\,{\widetilde{Y}_{k}}\,]$ with
probability $1-(1-p)^{n}$ and equal to 0 otherwise. Thus,

$\displaystyle\mathrm{var}\\!\left({\mathrm{E}\\!\left[\,{\widehat{\eta}_{\rm
DTQM}\,|\,L}\,\right]}\right)=\left(\mathrm{E}\\!\left[\,{\widetilde{Y}_{k}}\,\right]\right)^{2}[1-(1-p)^{n}](1-p)^{n},$
(80)

where $\mathrm{E}[\,{\widetilde{Y}_{k}}\,]$ is given in (45).

Toward computing $\mathrm{E}[\,{\mathrm{var}({\widehat{\eta}_{\rm
DTQM}\,|\,L})}\,]$, let us first examine $\mathrm{var}({\widehat{\eta}_{\rm
DTQM}\,|\,L=\ell})$. For $\ell>0$,

$\displaystyle\mathrm{var}\\!\left({\widehat{\eta}_{\rm
DTQM}\,|\,L=\ell}\right)=\mathrm{var}\\!\left({\frac{Y}{L}\,|\,L=\ell}\right)=\frac{1}{\ell^{2}}\mathrm{var}\\!\left({Y\,|\,L=\ell}\right)$
$\displaystyle\quad=\frac{1}{\ell^{2}}\mathrm{var}\\!\left({\sum_{j=1}^{\ell}\widetilde{Y}_{j}}\right)=\frac{1}{\ell}\mathrm{var}\\!\left({\widetilde{Y}_{k}}\right),$
(81)

where $\mathrm{var}({\widetilde{Y}_{k}})$ is given in (46). Trivially,

$\displaystyle\mathrm{var}\\!\left({\widehat{\eta}_{\rm
DTQM}\,|\,L=0}\right)=0.$ (82)

Using the binomial PMF of $L$ and the total expectation theorem to combine
(81) and (82) gives

$\displaystyle\mathrm{E}\\!\left[\,{\mathrm{var}\\!\left({\widehat{\eta}_{\rm
DTQM}\,|\,L}\right)}\,\right]$
$\displaystyle=\mathrm{var}\\!\left({\widetilde{Y}_{k}}\right)\sum_{\ell=1}^{n}\frac{1}{\ell}\binom{n}{\ell}p^{\ell}(1-p)^{n-\ell}.$
(83)

By summing (80) and (83) we obtain the variance of $\widehat{\eta}_{\rm
DTQM}$, which verifies (IV-D).

###  D-B Variance Lower Bound

Let $\widetilde{L}$ be the zero-truncated version of binomial random variable
$L$, with PMF

$\mathrm{P}_{\widetilde{L}}(\ell)=\frac{1}{1-(1-p)^{n}}\binom{n}{\ell}p^{\ell}(1-p)^{n-\ell},\quad\ell=1,2,\ldots,n.$

Then the series in (IV-D) equals
$(1-(1-p)^{n})\,\mathrm{E}[\,{1/\widetilde{L}}\,]$. Using Jensen’s inequality,
we can bound $\mathrm{E}[\,{1/\widetilde{L}}\,]$ using
$\mathrm{E}[\,{\widetilde{L}}\,]$, which has a simple closed-form expression:

$\mathrm{E}\\!\left[\,{\widetilde{L}}\,\right]=\frac{1}{1-(1-p)^{n}}\mathrm{E}\\!\left[\,{L}\,\right]=\frac{np}{1-(1-p)^{n}}.$
(84)

Specifically,

$\displaystyle\sum_{\ell=1}^{n}\frac{1}{\ell}\binom{n}{\ell}p^{\ell}(1-p)^{n-\ell}=(1-(1-p)^{n})\,\mathrm{E}\\!\left[\,{\frac{1}{\widetilde{L}}}\,\right]$
$\displaystyle\quad\stackrel{{\scriptstyle(a)}}{{\geq}}(1-(1-p)^{n})\,\frac{1}{\mathrm{E}\\!\left[\,{\widetilde{L}}\,\right]}\stackrel{{\scriptstyle(b)}}{{=}}\frac{(1-(1-p)^{n})^{2}}{np},$
(85)

where (_a_) follows from Jensen’s inequality; and (_b_) from (84).
Substituting (85) in (IV-D) gives (IV-D).

##  Appendix E Derivation of High $n$ Limits

###  E-A Proof of (52)

$\displaystyle\lim_{n\to\infty}np$
$\displaystyle=\lim_{n\to\infty}\frac{1-e^{-{({{\lambda}/{n}})}(1-e^{-\eta})}}{1/n}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\lim_{n\to\infty}\frac{-e^{-{({{\lambda}/{n}})}(1-e^{-\eta})}(1-e^{-\eta})\lambda/n^{2}}{-1/n^{2}}$
$\displaystyle=\lim_{n\to\infty}e^{-{({{\lambda}/{n}})}(1-e^{-\eta})}(1-e^{-\eta})\lambda$
$\displaystyle=\lambda(1-e^{-\eta}),$

where (_a_) follows from L’Hôpital’s rule.

###  E-B Proof of (53)

$\displaystyle\lim_{n\to\infty}(1-p)^{n}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\lim_{n\to\infty}\left(\exp\\!\left(-\frac{\lambda}{n}(1-e^{-\eta})\right)\right)^{n}$
$\displaystyle=\exp(-\lambda(1-e^{-\eta}))$

where (_a_) follows from substitution of (43).

###  E-C Proof of (55)

$\displaystyle\lim_{n\to\infty}\mathrm{E}[\,{\widetilde{Y}_{k}}\,]$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\lim_{n\to\infty}\frac{\lambda\eta}{np}$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\lim_{n\to\infty}\frac{\lambda\eta}{\lambda(1-e^{-\eta})}=\frac{\eta}{1-e^{-\eta}},$

where (_a_) follows from substitution of (45); and (_b_) from substitution of
(52).

###  E-D Proof of (56)

$\displaystyle\lim_{n\to\infty}\mathrm{var}({\widetilde{Y}_{k}})$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\lim_{n\to\infty}\frac{1}{p}\\!\left(\frac{\lambda}{n}\eta+\frac{\lambda}{n}\eta^{2}+\left(\frac{\lambda}{n}\eta\right)^{\\!2}\right)-\frac{1}{p^{2}}\\!\left(\frac{\lambda}{n}\eta\right)^{\\!2}$
$\displaystyle=\lim_{n\to\infty}\frac{\lambda\eta}{np}+\frac{\lambda\eta^{2}}{np}+\frac{\lambda^{2}\eta^{2}}{n^{2}p}-\frac{\lambda^{2}\eta^{2}}{n^{2}p^{2}}$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\lim_{n\to\infty}\frac{\lambda\eta+\lambda\eta^{2}}{\lambda(1-e^{-\eta})}-\frac{\lambda^{2}\eta^{2}}{\lambda^{2}(1-e^{-\eta})^{2}}$
$\displaystyle=\frac{\eta-(\eta+\eta^{2})e^{-\eta}}{(1-e^{-\eta})^{2}},$

where (_a_) follows from substitution of (46); and (_b_) from substitution of
(52) and noting that the third term vanishes.

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