基于双树复小波与波原子的图像扩散滤波

图像的非线性扩散滤波来源于热方程的思想,其关键在于计算适当的扩散系数和控制扩散方向. 在已有的扩散模型中,由于扩散系数仅依赖于图像的梯度,因而这类模型容易受噪声的干扰;同时,图像的细节信息(如纹理)容易被误认为是噪声而被去除. 为克服这些不足,首先给出了一种采用双树复小波变换计算扩散系数的方法;然后设计了一种用于图像滤波的非线性扩散模型,最后提出了基于双树复小波变换和波原子阈值相结合的图像滤波算法. 仿真结果表明,所提出的算法在对含噪图像滤波的同时,能够较好地保持图像的边缘和纹理等细节信息. 关键词： 图像扩散滤波 非线性扩散 波原子 双树复小波变换     

Edge Partition of Graphs Embeddable in the Projective Plane and the Klein Bottle

In a previous paper by the author joint with Baogang XU published in Discrete Math in 2018, we show that every non-planar toroidal graph can be edge partitioned into a planar graph and an outerplanar graph. This edge partition then implies some results in thickness and outerthickness of toroidal graphs. In particular, if each planar graph has outerthickness at most $2$ (conjectured by Chartrand, Geller and Hedetniemi in 1971 and the confirmation of the conjecture was announced by Gon\c{c}alves in 2005), then the outerthickness of toroidal graphs is at most 3 which is the best possible due to $K_7$. In this paper we continue to study the edge partition for projective planar graphs and Klein bottle embeddable graphs. We show that (1) every non-planar but projective planar graph can be edge partitioned into a planar graph and a union of caterpillar trees; and (2) every non-planar Klein bottle embeddable graph can be edge partitioned into a planar graph and a subgraph of two vertex amalgamation of a caterpillar tree with a cycle with pendant edges. As consequences, the thinkness of projective planar graphs and Klein bottle embeddabe graphs are at most $2$, which are the best possible, and the outerthickness of these graphs are at most $3$.     

几乎无爪图的叶子数较少的生成树

A spanning tree with no more than 3 leaves is called a spanning 3-ended tree.In this paper, we prove that if G is a k-connected(k ≥ 2) almost claw-free graph of order n and σ_(k+3)(G) ≥ n + k + 2, then G contains a spanning 3-ended tree, where σk(G) =min{∑_(v∈S)deg(v) : S is an independent set of G with |S| = k}.     

具有匹配对网络模型的优美性

在当今网络研究中,人们需要将某些特殊的图分解为指定的结构.优美图可以被运用到图分解中.得到一些构造优美图的可算法化的方法,并构造较为复杂的优美图.     

Degree Conditions for Spanning Brooms

A broom is a tree obtained by subdividing one edge of the star an arbitrary number of times. In (E. Flandrin, T. Kaiser, R. Ku?el, H. Li and Z. Ryjá?ek, Neighborhood Unions and Extremal Spanning Trees, Discrete Math 308 (2008), 2343–2350) Flandrin et al. posed the problem of determining degree conditions that ensure a connected graph G contains a spanning tree that is a broom. In this article, we give one solution to this problem by demonstrating that if G is a connected graph of order with , then G contains a spanning broom. This result is best possible.     

Hop constrained Steiner trees with multiple root nodes

We consider a network design problem that generalizes the hop and diameter constrained Steiner tree problem as follows: Given an edge-weighted undirected graph with two disjoint subsets representing roots and terminals, find a minimum-weight subtree that spans all the roots and terminals so that the number of hops between each relevant node and an arbitrary root does not exceed a given hop limit H. The set of relevant nodes may be equal to the set of terminals, or to the union of terminals and root nodes. This article proposes integer linear programming models utilizing one layered graph for each root node. Different possibilities to relate solutions on each of the layered graphs as well as additional strengthening inequalities are then discussed. Furthermore, theoretical comparisons between these models and to previously proposed flow- and path-based formulations are given. To solve the problem to optimality, we implement branch-and-cut algorithms for the layered graph formulations. Our computational study shows their clear advantages over previously existing approaches.     

Spanning trees with variable degree bounds

In this paper, we introduce and study a generalization of the degree constrained minimum spanning tree problem where we may install one of several available transmission systems (each with a different cost value) in each edge. The degree of the endnodes of each edge depends on the system installed on the edge. We also discuss a particular case that arises in the design of wireless mesh networks (in this variant the degree of the endnodes of each edge depend on the transmission system installed on it as well as on the length of the edge). We propose three classes of models using different sets of variables and compare from a theoretical perspective as well as from a computational point of view, the models and the corresponding linear programming relaxations. The computational results show that some of the proposed models are able to solve to optimality instances with 100 nodes and different scenarios.     

Sizes of the largest clusters for supercritical percolation on random recursive trees

We consider Bernoulli bond‐percolation on a random recursive tree of size , with supercritical parameter for some fixed. We show that with high probability, the largest cluster has size close to whereas the next largest clusters have size of order only and are distributed according to some Poisson random measure. Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 29–44, 2014     

Asymptotics of trees with a prescribed degree sequence and applications

Let t be a rooted tree and n_bi(t) the number of nodes in t having i children. The degree sequence of t satisfies , where denotes the number of nodes in t. In this paper, we consider trees sampled uniformly among all plane trees having the same degree sequence ; we write for the corresponding distribution. Let be a list of degree sequences indexed by κ corresponding to trees with size . We show that under some simple and natural hypotheses on the trees sampled under converge to the Brownian continuum random tree after normalisation by . Some applications concerning Galton–Watson trees and coalescence processes are provided.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 290‐316, 2014     

Exact boundary controllability of nodal profile for quasilinear wave equations in a planar tree‐like network of strings

Based on the theory of semi‐global piecewise C² solutions to 1D quasilinear wave equations, the local exact boundary controllability of nodal profile for quasilinear wave equations in a planar tree‐like network of strings with general topology is obtained by a constructive method. The principles of providing nodal profiles and of choosing and transferring boundary controls are presented, respectively. Copyright © 2013 John Wiley & Sons, Ltd.