BZPEER

preprint2026arXiv

GraphMAR: Geometry-Aware Graph Learning Framework for Spatially Adaptive CT Metal Artifact Reduction

Computed tomography (CT) metal artifact reduction (MAR) aims to reduce the severe streaking artifacts induced by metallic implants and other high-density objects. Effective MAR generally requires both accurate artifact localization and artifact removal. Sinogram-domain methods can exploit explicit geometric cues, such as metal traces, to identify metal-corrupted measurements, while requiring raw projection data, which is often unavailable in clinical and practical scenarios. Image-domain methods are more flexible and widely applicable, yet they usually lack comparable geometric guidance, limiting their ability to localize artifacts and leading to suboptimal results. To address this limitation, we propose GraphMAR, a geometry-aware learning framework for explicit artifact identification and spatially adaptive MAR in the image domain. The key idea is to introduce graph-based geometric modeling as an image-domain analogue of sinogram metal traces. Specifically, we first construct a geometric graph from the metal mask and derive a geometric density graph that coarsely localizes artifact-prone regions according to inter-implant geometry. We then design GraphMoE, a graph-routed mixture-of-experts module that builds a polar-coordinate artifact graph in feature space and adaptively routes different experts to different spatial regions for MAR. By aligning the learned routing maps with the geometric density graph, GraphMAR provides explicit and interpretable artifact localization while enabling region-adaptive artifact reduction. Experiments on both simulated and real-world datasets demonstrate that GraphMAR achieves superior MAR performance compared with existing methods. To the best of our knowledge, this is the first work to introduce graph-based modeling for CT MAR and to enable explicit artifact identification in the image domain, improving both restoration quality and interpretability.

preprint2020arXiv

On clean, weakly clean, and feebly clean commutative group rings

A ring $R$ is said to be clean if each element of $R$ can be written as the sum of a unit and an idempotent. $R$ is said to be weakly clean if each element of $R$ is either a sum or a difference of a unit and an idempotent, and $R$ is said to be feebly clean if every element $r$ can be written as $r=u+e_1-e_2$ , where $u$ is a unit and $e_1,e_2$ are orthogonal idempotents. Clearly clean rings are weakly clean rings and both of them are feebly clean. In a recent article (J. Algebra Appl. 17 (2018), 1850111(5 pages)), McGoven characterized when the group ring $\mathbb Z_{(p)}[C_q]$ is weakly clean and feebly clean, where $p, q$ are distinct primes. In this paper, we consider a more general setting. Let $K$ be an algebraic number field, $\mathcal O_K$ its ring of integers, $\mathfrak p\subset \mathcal O$ a nonzero prime ideal, and $\mathcal O_{\mathfrak p}$ the localization of $\mathcal O$ at $\mathfrak p$ . We investigate when the group ring $\mathcal O_{\mathfrak p}[G]$ is weakly clean and feebly clean, where $G$ is a finite abelian group, and establish an explicit characterization for such a group ring to be weakly clean and feebly clean for the case when $K=\mathbb Q(ζ_n)$ is a cyclotomic field or $K=\mathbb Q(\sqrt{d})$ is a quadratic field.

preprint2015arXiv

Noetherian Rings Whose Annihilating-Ideal Graphs Have finite Genus

Let $R$ be a commutative ring and ${\Bbb{A}}(R)$ be the set of ideals with non-zero annihilators. The annihilating-ideal graph of $R$ is defined as the graph ${\Bbb{AG}}(R)$ with vertex set ${\Bbb{A}}(R)^*={\Bbb{A}}\setminus\{(0)\}$ such that two distinct vertices $I$ and $J$ are adjacent if and only if $IJ=(0)$ . We characterize commutative Noetherian rings $R$ whose annihilating-ideal graphs have finite genus $γ(\Bbb{AG}(R))$ . It is shown that if $R$ is a Noetherian ring such that $0<γ(\Bbb{AG}(R))<\infty$ , then $R$ has only finitely many ideals.

preprint2014arXiv

Minimal zero-sum sequence of length five over finite cyclic groups of prime power order

Let $G$ be a finite cyclic group. Every sequence $S$ of length $l$ over $G$ can be written in the form $S=(x_1g)\cdot\ldots\cdot(x_lg)$ where $g\in G$ and $x_1, \ldots, x_l\in[1, \ord(g)]$, and the index $\ind(S)$ of $S$ is defined to be the minimum of $(x_1+\cdots+x_l)/\ord(g)$ over all possible $g\in G$ such that $\langle g \rangle =G$ . Recently the second and the third authors determined the index of any minimal zero-sum sequence $S$ of length 5 over a cyclic group of a prime order where $S=g^2(x_2g)(x_3g)(x_4g)$ . In this paper, we determine the index of any minimal zero-sum sequence $S$ of length 5 over a cyclic group of a prime power order. It is shown that if $G=\langle g\rangle$ is a cyclic group of prime power order $n=p^μ$ with $p \geq 7$ and $μ\geq 2$ , and $S=(x_1g)(x_2g)(x_2g)(x_3g)(x_4g)$ with $x_1=x_2$ is a minimal zero-sum sequence with $\gcd(n,x_1,x_2,x_3,x_4,x_5)=1$ , then $\ind(S)=2$ if and only if $S=(mg)(mg)(m\frac{n-1}{2}g)(m\frac{n+3}{2}g)(m(n-3)g)$ where $m$ is a positive integer such that $\gcd(m,n)=1$ .

preprint2014arXiv

On the index of length four minimal zero-sum sequences

Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)\cdot\ldots\cdot(n_lg)$ where $g\in G$ and $n_1, \ldots, n_l\in[1, \ord(g)]$, and the index $\ind(S)$ of $S$ is defined to be the minimum of $(n_1+\cdots+n_l)/\ord(g)$ over all possible $g\in G$ such that $\langle g \rangle =G$ . A conjecture on the index of length four sequences says that every minimal zero-sum sequence of length 4 over a finite cyclic group $G$ with $\gcd(|G|, 6)=1$ has index 1. The conjecture was confirmed recently for the case when $|G|$ is a product of at most two prime powers. However, the general case is still open. In this paper, we make some progress towards solving the general case. Based on earlier work on this problem, we show that if $G=\langle g\rangle$ is a finite cyclic group of order $|G|=n$ such that $\gcd(n,6)=1$ and $S=(x_1g)(x_2g)(x_3g)(x_4g)$ is a minimal zero-sum sequence over $G$ such that $x_1,\cdots,x_4\in[1,n-1]$ with $\gcd(n,x_1,x_2,x_3,x_4)=1$ , and $\gcd(n,x_i)>1$ for some $i\in[1,4]$ , then $\ind(S)=1$. By using an innovative method developed in this paper, we are able to give a new (and much shorter) proof to the index conjecture for the case when $|G|$ is a product of two prime powers.

preprint2014arXiv

On zero divisors and prime elements of po-semirings

A semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. A po-semiring is a semiring equipped with a compatible bounded partial order. In this paper, properties of zero divisors and prime elements of a po-semiring are studied. In particular, it is proved that under some mild assumption the set $Z(A)$ of nonzero zero divisors of $A$ is $A\setminus \{0,1\}$ , each prime element of $A$ is a maximal element, and the zero divisor graph $\G(A)$ of $A$ is a finite graph if and only if $A$ is finite. For a po-semiring $A$ with $Z(A)=A\setminus \{0,1\}$ , it is proved that $A$ has finitely many maximal elements if ACC holds either for elements of $A$ or for principal annihilating ideals of $A$ . As applications of prime elements, it is shown that the structure of a po-semiring $A$ is completely determined by the structure of integral po-semirings if either $|Z(A)|=1$ or $|Z(A)|=2$ and $Z(A)^2\not=0$ . Applications to the ideal structure of commutative rings are considered.

preprint2013arXiv

An upper bound for Davenport constant of finite groups

Let $G$ be a finite (not necessarily abelian) group and let $p=p(G)$ be the smallest prime number dividing $|G|$ . We prove that $d(G)\leq \frac{|G|}{p}+9p^2-10p$ , where $d(G)$ denotes the small Davenport constant of $G$ which is defined as the maximal integer $\ell$ such that there is a sequence over $G$ of length $\ell$ contains no nonempty one-product subsequence.

preprint2013arXiv

Minimal zero-sum sequences of length five over finite cyclic groups

Let $G$ be a finite cyclic group. Every sequence $S$ of length $l$ over $G$ can be written in the form $S=(n_1g)\cdot\ldots\cdot(n_lg)$ where $g\in G$ and $n_1, \ldots, n_l\in[1, \ord(g)]$, and the index $\ind(S)$ of $S$ is defined to be the minimum of $(n_1+\cdots+n_l)/\ord(g)$ over all possible $g\in G$ such that $\langle g \rangle =G$ . In this paper, we determine the index of any minimal zero-sum sequence $S$ of length 5 when $G=\langle g\rangle$ is a cyclic group of a prime order and $S$ has the form $S=g^2(n_2g)(n_3g)(n_4g)$ . It is shown that if $G=\langle g\rangle$ is a cyclic group of prime order $p \geq 31$ , then every minimal zero-sum sequence $S$ of the above mentioned form has index 1 except in the case that $S=g^2(\frac{p-1}{2}g)(\frac{p+3}{2}g)((p-3)g)$ .

preprint2013arXiv

Minimal zero-sum sequences of length four over finite cyclic groups II

Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)\cdot\ldots\cdot(n_lg)$ where $g\in G$ and $n_1, \ldots, n_l\in[1, \ord(g)]$, and the index $\ind(S)$ of $S$ is defined to be the minimum of $(n_1+\cdots+n_l)/\ord(g)$ over all possible $g\in G$ such that $\langle g \rangle =G$ . An open problem on the index of length four sequences asks whether or not every minimal zero-sum sequence of length 4 over a finite cyclic group $G$ with $\gcd(|G|, 6)=1$ has index 1. In this paper, we show that if $G=\langle g\rangle$ is a cyclic group with order of a product of two prime powers and $\gcd(|G|, 6)=1$ , then every minimal zero-sum sequence $S$ of the form $S=(g)(n_2g)(n_3g)(n_4g)$ has index 1. In particular, our result confirms that the above problem has an affirmative answer when the order of $G$ is a product of two different prime numbers or a prime power, extending a recent result by the first author, Plyley, Yuan and Zeng.

preprint2011arXiv

On the Index of Sequences over Cyclic Groups

Let $G$ be a finite cyclic group of order $n \ge 2$ . Every sequence $S$ over $G$ can be written in the form $S=(n_1g)\cdot ... \cdot (n_lg)$ where $g\in G$ and $n_1,..., n_l \in [1,\ord(g)]$, and the index $\ind (S)$ of $S$ is defined as the minimum of $(n_1+ ... + n_l)/\ord (g)$ over all $g \in G$ with $\ord (g) = n$. In this paper we prove that a sequence $S$ over $G$ of length $|S| = n$ having an element with multiplicity at least $\frac{n}{2}$ has a subsequence $T$ with $\ind (T) = 1$, and if the group order $n$ is a prime, then the assumption on the multiplicity can be relaxed to $\frac{n-2}{10}$ . On the other hand, if $n=4k+2$ with $k \ge 5$ , we provide an example of a sequence $S$ having length $|S| > n$ and an element with multiplicity $\frac{n}{2}-1$ which has no subsequence $T$ with $\ind (T) = 1$. This disproves a conjecture given twenty years ago by Lemke and Kleitman.

Yuanlin Li

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10 publications

GraphMAR: Geometry-Aware Graph Learning Framework for Spatially Adaptive CT Metal Artifact Reduction

On clean, weakly clean, and feebly clean commutative group rings

Noetherian Rings Whose Annihilating-Ideal Graphs Have finite Genus

Minimal zero-sum sequence of length five over finite cyclic groups of prime power order

On the index of length four minimal zero-sum sequences

On zero divisors and prime elements of po-semirings

An upper bound for Davenport constant of finite groups

Minimal zero-sum sequences of length five over finite cyclic groups

Minimal zero-sum sequences of length four over finite cyclic groups II

On the Index of Sequences over Cyclic Groups

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