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David El-Chai Ben-Ezra

Publications and source records attributed to David El-Chai Ben-Ezra.

9 recordsLinked to original sources
arXiv2026

Event Driven Clustering Algorithm

This paper introduces a novel asynchronous, event-driven algorithm for real-time detection of small event clusters in event camera data. Similar to hierarchical agglomerative clustering methods, the proposed algorithm detects clusters based on their spatio-temporal proximity. However, it explicitly leverages the asynchronous structure of event camera data and employs a simple yet efficient decision mechanism, achieving a linear time complexity of $Θ(N)$, where $N$ is the number of events. Furthermore, the runtime is independent of the sensor resolution, i.e., the number of pixels.

cs.CV↗
arXiv2026

On the Theory of Bias Tuning in Event Cameras

This paper lays the foundation of a theory for bias tuning in neuromorphic cameras, a novel sensing technology also known as "event cameras". We begin by formulating the high-level effect of the sensitivity biases on the camera's event rate in mathematical terms. We then show that, as a corollary of the Poincare-Miranda theorem, the commonly used tuning principles of rate budgeting and polarity balancing lead to a unique configuration of the sensitivity biases. As a corollary, we show how by adopting these principles, the multi-variable bias-tuning problem reduces to a two-parameter problem that can be resolved experimentally.

cs.CV↗
arXiv2024

Probabilistic Approach for Detection of High-Frequency Periodic Signals using an Event Camera

Being inspired by the biological eye, event camera is a novel asynchronous technology that pose a paradigm shift in acquisition of visual information. This paradigm enables event cameras to capture pixel-size fast motions much more naturally compared to classical cameras. In this paper we present a new asynchronous event-driven algorithm for detection of high-frequency pixel-size periodic signals using an event camera. Development of such new algorithms, to efficiently process the asynchronous information of event cameras, is essential and being a main challenge in the research community, in order to utilize its special properties and potential. It turns out that this algorithm, that was developed in order to satisfy the new paradigm, is related to an untreated theoretical problem in probability: let $0\leqτ_{1}\leqτ_{2}\leq\cdots\leqτ_{m}\leq1$, originated from an unknown distribution. Let also $ε,δ\in\mathbb{R}$, and $d\in\mathbb{N}$. What can be said about the probability $Φ(m,d)$ of having more than $d$ adjacent $τ_{i}$-s pairs that the distance between them is $δ$, up to an error $ε$ ? This problem, that reminds the area of order statistic, shows how the new visualization paradigm is also an opportunity to develop new areas and problems in mathematics.

cs.CV↗
arXiv2020

On Pro-$2$ Identities of $2\times2$ Linear Groups

Let $\hat{F}$ be a free pro-$p$ non-abelian group, and let $Δ$ be a commutative Noetherian complete local ring with a maximal ideal $I$ such that $\textrm{char}(Δ/I)=p>0$. In [Zu], Zubkov showed that when $p\neq2$, the pro-$p$ congruence subgroup $$GL_{2}^{1}(Δ)=\ker(GL_{2}(Δ)\overset{Δ\toΔ/I}{\longrightarrow}GL_{2}(Δ/I))$$ admits a pro-$p$ identity, i.e., there exists an element $1\neq w\in\hat{F}$ that vanishes under any continuous homomorphism $\hat{F}\to GL_{2}^{1}(Δ)$. In this paper we investigate the case $p=2$. The main result is that when $\textrm{char}(Δ)=2$, the pro-$2$ group $GL_{2}^{1}(Δ)$ admits a pro-$2$ identity. This result was obtained by the use of trace identities that originate in PI-theory.

math.GR↗
arXiv2020

The Congruence Subgroup Problem for finitely generated Nilpotent Groups

The congruence subgroup problem for a finitely generated group $Γ$ and $G\leq Aut(Γ)$ asks whether the map $\hat{G}\to Aut(\hatΓ)$ is injective, or more generally, what is its kernel $C\left(G,Γ\right)$? Here $\hat{X}$ denotes the profinite completion of $X$. In the case $G=Aut(Γ)$ we denote $C\left(Γ\right)=C\left(Aut(Γ),Γ\right)$. Let $Γ$ be a finitely generated group, $\barΓ=Γ/[Γ,Γ]$, and $Γ^{*}=\barΓ/tor(\barΓ)\cong\mathbb{Z}^{(d)}$. Denote $Aut^{*}(Γ)=\textrm{Im}(Aut(Γ)\to Aut(Γ^{*}))\leq GL_{d}(\mathbb{Z})$. In this paper we show that when $Γ$ is nilpotent, there is a canonical isomorphism $C\left(Γ\right)\simeq C(Aut^{*}(Γ),Γ^{*})$. In other words, $C\left(Γ\right)$ is completely determined by the solution to the classical congruence subgroup problem for the arithmetic group $Aut^{*}(Γ)$. In particular, in the case where $Γ=Ψ_{n,c}$ is a finitely generated free nilpotent group of class $c$ on $n$ elements, we get that $C(Ψ_{n,c})=C(\mathbb{Z}^{(n)})=\{e\}$ whenever $n\geq3$, and $C(Ψ_{2,c})=C(\mathbb{Z}^{(2)})=\hat{F}_ω$ = the free profinite group on countable number of generators.

math.GR↗
arXiv2019

The IA-congruence kernel of high rank free Metabelian groups

The congruence subgroup problem for a finitely generated group $Γ$ and $G\leq Aut(Γ)$ asks whether the map $\hat{G}\to Aut(\hatΓ)$ is injective, or more generally, what is its kernel $C\left(G,Γ\right)$? Here $\hat{X}$ denotes the profinite completion of $X$. In this paper we investigate $C\left(IA(Φ_{n}),Φ_{n}\right)$, where $Φ_{n}$ is a free metabelian group on $n\geq4$ generators, and $IA(Φ_{n})=\ker(Aut(Φ_{n})\to GL_{n}(\mathbb{Z}))$. We show that in this case $C(IA(Φ_{n}),Φ_{n})$ is abelian, but not trivial, and not even finitely generated. This behavior is very different from what happens for free metabelian group on $n=2,3$ generators, or for finitely generated nilpotent groups.

math.KT↗
arXiv2019

The Congruence Subgroup Problem for the Free Metabelian group on $n\geq4$ generators

The congruence subgroup problem for a finitely generated group $Γ$ asks whether the map $\hat{Aut\left(Γ\right)}\to Aut(\hatΓ)$ is injective, or more generally, what is its kernel $C\left(Γ\right)$? Here $\hat{X}$ denotes the profinite completion of $X$. It is well known that for finitely generated free abelian groups $C\left(\mathbb{Z}^{n}\right)=\left\{ 1\right\}$ for every $n\geq3$, but $C\left(\mathbb{Z}^{2}\right)=\hat{F}_ω$, where $\hat{F}_ω$ is the free profinite group on countably many generators. Considering $Φ_{n}$, the free metabelian group on $n$ generators, it was also proven that $C\left(Φ_{2}\right)=\hat{F}_ω$ and $C\left(Φ_{3}\right)\supseteq\hat{F}_ω$. In this paper we prove that $C\left(Φ_{n}\right)$ for $n\geq4$ is abelian. So, while the dichotomy in the abelian case is between $n=2$ and $n\geq3$, in the metabelian case it is between $n=2,3$ and $n\geq4$.

math.GR↗
arXiv2016

The Congruence Subgroup Problem for low rank Free and Free Metabelian groups

The congruence subgroup problem for a finitely generated group $Γ$ asks whether $\widehat{Aut\left(Γ\right)}\to Aut(\hatΓ)$ is injective, or more generally, what is its kernel $C\left(Γ\right)$? Here $\hat{X}$ denotes the profinite completion of $X$. In this paper we first give two new short proofs of two known results (for $Γ=F_{2}$ and $Φ_{2}$) and a new result for $Γ=Φ_{3}$: 1. $C\left(F_{2}\right)=\left\{ e\right\}$ when $F_{2}$ is the free group on two generators. 2. $C\left(Φ_{2}\right)=\hat{F}_ω$ when $Φ_{n}$ is the free metabelian group on $n$ generators, and $\hat{F}_ω$ is the free profinite group on $\aleph_{0}$ generators. 3. $C\left(Φ_{3}\right)$ contains $\hat{F}_ω$. Results 2. and 3. should be contrasted with an upcoming result of the first author showing that $C\left(Φ_{n}\right)$ is abelian for $n\geq4$.

math.GR↗