Caroline Silva

Frédéric Havet, Clément Rambaud, Caroline Silva

In an oriented graph $\vec{G}$, the {\it inversion} of a subset $X$ of vertices consists in reversing the orientation of all arcs with both endvertices in $X$. The {\it $(\leq p)$-inversion graph} of a labelled graph $G$, denoted by ${\mathcal{I}}^{\leq p}(G)$, is the graph whose vertices are the labelled orientations of $G$ in which two labelled orientations $\vec{G}_1$ and $\vec{G}_2$ of $G$ are adjacent if and only if there is a set $X$ with $|X|\leq p$ whose inversion transforms $\vec{G}_1$ into $\vec{G}_2$. In this paper, we study the {\it $(\leq p)$-inversion diameter} of a graph, denoted by $\mathrm{id}^{\leq p}(G)$, which is the diameter of its $(\leq p)$-inversion graph. We show that there exists a smallest number $Ψ_p$ with $\frac{1}{4}p - \frac{3}{2} \leq Ψ_p \leq \frac{1}{2}p^2$ such that $\mathrm{id}^{\leq p}(G) \leq \left\lceil\frac{|E(G)|}{\lfloor p/2\rfloor}\right \rceil + Ψ_p$ for all graph $G$. We then establish better upper bounds for several families of graphs and in particular trees and planar graphs. Let us denote by $\mathrm{id}^{\leq p}_{\cal F}(n)$ (resp. $\mathrm{id}^{\leq p}_{\cal P}(n)$) the maximum $(\leq p)$-inversion diameter of a tree (resp. planar graph) of order $n$. For trees, we show $\mathrm{id}^{\leq 3}_{\cal F}(n) = \left\lceil \frac{n-1}{2}\right\rceil$, $\mathrm{id}^{\leq 4}_{\cal F}(n)=\frac{3}{8}n + Θ(1)$, $\mathrm{id}^{\leq 5}_{\cal F}(n)= \frac{2}{7}n + Θ(1)$, and $\mathrm{id}^{\leq p}_{\cal F}(n) \leq \frac{n-1}{p- c\sqrt{p}} + 2$ with $c = \sqrt{2 + \sqrt{2}}$ for all $p\geq 6$. For planar graphs, we prove $\mathrm{id}^{\leq 3}_{\cal P}(n) \leq \frac{11n}{6} - \frac{8}{3}$, $\mathrm{id}^{\leq 4}_{\cal P}(n) \leq \frac{4n}{3} + \frac{10}{3}$, and $\mathrm{id}^{\leq p}_{\cal P}(n) \leq \left\lceil\frac{3n-6}{\lfloor p/2\rfloor}\right \rceil + 8\lfloor p/2\rfloor - 8$ for all $p\geq 6$.

Thierry Bouwmans, Caroline Silva, Cristina Marghes et al.

Background modeling has emerged as a popular foreground detection technique for various applications in video surveillance. Background modeling methods have become increasing efficient in robustly modeling the background and hence detecting moving objects in any visual scene. Although several background subtraction and foreground detection have been proposed recently, no traditional algorithm today still seem to be able to simultaneously address all the key challenges of illumination variation, dynamic camera motion, cluttered background and occlusion. This limitation can be attributed to the lack of systematic investigation concerning the role and importance of features within background modeling and foreground detection. With the availability of a rather large set of invariant features, the challenge is in determining the best combination of features that would improve accuracy and robustness in detection. The purpose of this study is to initiate a rigorous and comprehensive survey of features used within background modeling and foreground detection. Further, this paper presents a systematic experimental and statistical analysis of techniques that provide valuable insight on the trends in background modeling and use it to draw meaningful recommendations for practitioners. In this paper, a preliminary review of the key characteristics of features based on the types and sizes is provided in addition to investigating their intrinsic spectral, spatial and temporal properties. Furthermore, improvements using statistical and fuzzy tools are examined and techniques based on multiple features are benchmarked against reliability and selection criterion. Finally, a description of the different resources available such as datasets and codes is provided.