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.
We prove Clifford theoretic results on the representations of finite groups which only hold in characteristic $2$. Let $G$ be a finite group, let $N$ be a normal subgroup of $G$ and let $varphi$ be an irreducible $2$-Brauer character of $N$ which i
s self-dual. We prove that there is a unique self-dual irreducible Brauer character $theta$ of $G$ such that $varphi$ occurs with odd multiplicity in the restriction of $theta$ to $N$. Moreover this multiplicity is $1$. Conversely if $theta$ is an irreducible $2$-Brauer character of $G$ which is self-dual but not of quadratic type, the restriction of $theta$ to $N$ is a sum of distinct self-dual irreducible Brauer character of $N$, none of which have quadratic type. Let $b$ be a real $2$-block of $N$. We show that there is a unique real $2$-block of $G$ covering $b$ which is weakly regular.
Passive non-line-of-sight imaging methods are often faster and stealthier than their active counterparts, requiring less complex and costly equipment. However, many of these methods exploit motion of an occluder or the hidden scene, or require knowle
dge or calibration of complicated occluders. The edge of a wall is a known and ubiquitous occluding structure that may be used as an aperture to image the region hidden behind it. Light from around the corner is cast onto the floor forming a fan-like penumbra rather than a sharp shadow. Subtle variations in the penumbra contain a remarkable amount of information about the hidden scene. Previous work has leveraged the vertical nature of the edge to demonstrate 1D (in angle measured around the corner) reconstructions of moving and stationary hidden scenery from as little as a single photograph of the penumbra. In this work, we introduce a second reconstruction dimension: range measured from the edge. We derive a new forward model, accounting for radial falloff, and propose two inversion algorithms to form 2D reconstructions from a single photograph of the penumbra. Performances of both algorithms are demonstrated on experimental data corresponding to several different hidden scene configurations. A Cramer-Rao bound analysis further demonstrates the feasibility (and utility) of the 2D corner camera.
Focused ion beam (FIB) microscopy suffers from source shot noise - random variation in the number of incident ions in any fixed dwell time - along with random variation in the number of detected secondary electrons per incident ion. This multiplicity
of sources of randomness increases the variance of the measurements and thus worsens the trade-off between incident ion dose and image accuracy. Time-resolved sensing combined with maximum likelihood estimation from the resulting sets of measurements greatly reduces the effect of source shot noise. Through Fisher information analysis and Monte Carlo simulations, the reduction in mean-squared error or reduction in required dose is shown to be by a factor approximately equal to the secondary electron yield. Experiments with a helium ion microscope (HIM) are consistent with the analyses and suggest accuracy improvement for a fixed source dose, or reduced source dose for a desired imaging accuracy, by a factor of about 3.
Let $p$ be an odd prime and let $B$ be a $p$-block of a finite group which has cyclic defect groups. We show that all exceptional characters in $B$ have the same Frobenius-Schur indicators. Moreover the common indicator can be computed, using the can
onical character of $B$. We also investigate the Frobenius-Schur indicators of the non-exceptional characters in $B$. For a finite group which has cyclic Sylow $p$-subgroups, we show that the number of irreducible characters with Frobenius-Schur indicator $-1$ is greater than or equal to the number of conjugacy classes of weakly real $p$-elements in $G$.
Estimating the parameter of a Bernoulli process arises in many applications, including photon-efficient active imaging where each illumination period is regarded as a single Bernoulli trial. Motivated by acquisition efficiency when multiple Bernoulli
processes are of interest, we formulate the allocation of trials under a constraint on the mean as an optimal resource allocation problem. An oracle-aided trial allocation demonstrates that there can be a significant advantage from varying the allocation for different processes and inspires a simple trial allocation gain quantity. Motivated by realizing this gain without an oracle, we present a trellis-based framework for representing and optimizing stopping rules. Considering the convenient case of Beta priors, three implementable stopping rules with similar performances are explored, and the simplest of these is shown to asymptotically achieve the oracle-aided trial allocation. These approaches are further extended to estimating functions of a Bernoulli parameter. In simulations inspired by realistic active imaging scenarios, we demonstrate significant mean-squared error improvements: up to 4.36 dB for the estimation of p and up to 1.80 dB for the estimation of log p.
For each positive integer $n$, we construct a bijection between the odd partitions and the distinct partitions of $n$ which extends Bressouds bijection between the odd-and-distinct partitions of $n$ and the splitting partitions of $n$. We compare o
ur bijection with the classical bijections of Glaisher and Sylvester, and also with a recent bijection due to Chen, Gao, Ji and Li.
We determine the quadratic type of the 2-modular principal indecomposable modules of the double covers of alternating groups.
Let $P$ be a principal indecomposable module of a finite group $G$ in characteristic $2$ and let $varphi$ be the Brauer character of the corresponding simple $G$-module. We show that $P$ affords a non-degenerate $G$-invariant quadratic form if and on
ly if there are involutions $s,tin G$ such that $st$ has odd order and $varphi(st)/2$ is not an algebraic integer. We then show that the number of isomorphism classes of quadratic principal indecomposable $G$-modules is equal to the number of strongly real conjugacy classes of odd order elements of $G$.
For the Klein-Four Group $G$ and a perfect field $k$ of characteristic two we determine all indecomposable symplectic $kG$-modules, that is, $kG$-modules with a symplectic, $G$-invariant form which do not decompose into smaller such modules, and clas
sify them up to isometry. Also we determine all quadratic forms that have one of the above symplectic forms as their associated bilinear form and describe their isometry classes.
