-\begin{figure}
+\begin{figure*}
\begin{center}
-\includegraphics[scale=0.3]{reseau.png}
+\includegraphics[scale=0.2]{SensorNetwork.png}
\begin{scriptsize}
-An example of a sensor network ofsize 10. All nodes are video sensor
-except the 5 and the 9 one which is the sink.
-\JFC{reprendre la figure, trouver un autre titre}
-\end{scriptsize}
-\caption{SN with 10 sensor}\label{fig:sn}.
+An example of a sensor network of size 10.
+All nodes are video sensors (depicted as small discs)
+except the 9 one which is the sink (depicted as a rectangle).
+Large lircles represent the maximum
+transmission range which is set to 20 in a square region which is
+$50 m \times 50 m$.
+\end{scriptsize}
+\caption{Illustration of a SN of size 10}\label{fig:sn}.
\end{center}
-\end{figure}
+\end{figure*}
Let us first recall the basics of the~\cite{HLG09} article.
The video sensor network is memorized as a connected non oriented
-oriented labelled graph.
+graph.
In this one,
the nodes, in a set $N$, are sensors, links, or the sink.
Furthermore, there is an edge from $i$ to $j$ if $i$ can
-send a message to $j$. The set of all edges is further denoted as
+send a message to $j$, \textit{i. e.}, the distance betwween
+$i$ and $j$ is less than a given maximum
+transmission range.
+All the possible edges are stored into a sequence
$L$.
Figure~\ref{fig:sn} gives an example of such a network.
where
$a_{il}$ is 1 if $l$ starts with $i$, is -1 if $l$ ends width $i$
and 0 otherwise.
-
+Moreover, the outgoing links(resp. the incoming links) are represented
+with the $A^+$ matrix (res. with the $A^-$ matrix), whose elements are defined:
+$a_{il}^+$ (resp. $a_{il}^-$) is 1 if the link $l$ is an outgoing link from $i$
+(resp an incoming link into $i$) and 0 otherwise.
Let $V \subset N $ be the set of the video sensors of $N$.
Let thus $R_h$, $R_h \geq 0$,
be the encoding rate of video sensor $h$, $h \in V$.
-Let $\eta_{hi}$ be the production rate of the node $i$,
-for the session initiated by $h$. More precisely, we have
+Let $\eta_{hi}$ be the rate inside the node $i$
+of the production that has beeninitiated by $h$. More precisely, we have
$ \eta_{hi}$ is equal to $ R_h$ if $i$ is $h$,
is equal to $-R_h$ if $i$ is the sink, and $0$ otherwise.
-We are then left to focus on the flows in this network.
+Let us focus on the flows in this network.
Let $x_{hl}$, $x_{hl}\geq 0$, be the flow inside the edge $l$ that
-issued from the $h$ session and
+issued from the node $h$ and
let $y_l = \sum_{h \in V}x_{hl} $ the sum of all the flows inside $l$.
Thus, what is produced inside the $i^{th}$ sensor for session $h$
is $ \eta_{hi} = \sum_{l \in L }a_{il}x_{hl} $.
The encoding power of the $i$ node is $P_{si}$, $P_{si} > 0$.
-
-The distortion is bounded $\sigma^2 e^{-\gamma . R_h.P_{sh}^{}2/3} \leq D_h$.
-
+The emmission distortion of the $i$ node is
+$\sigma^2 e^{-\gamma . R_i.P_{si}^{}2/3}$
+where $\sigma^2$ is the average input variance and
+$\gamma$ is the encoding efficiency coefficient.
+This distortion
+is bounded by a constant value $D_h$.
The initial energy of the $i$ node is $B_i$.
-
+The transmission consumed power of node $i$ is
+$P_{ti} = c_l^s.y_l$ where $c_l^s$ is the transmission energy
+consumption cost of link $l$, $l\in L$. This cost is defined
+as foolows: $c_l^s = \alpha +\beta.d_l^{n_p} $ wehre
+$d_l$ represents the distance of the link $l$,
+$\alpha$, $\beta$, and $n_p$ are constant.
+The reception consumed power of node $i$ is
+$P_{ri} = c^r \sum_{l \in L } a_{il}^-.y_l$
+where $c^r$ is a reception energy consumption cost.
The overall consumed power of the $i$ node is
$P_{si}+ P_{ti} + P_{ri}=
P_{si}+ \sum_{l \in L}a_{il}^{+}.c^s_l.y_l +
\label{eq:obj2}
\end{equation}
where $\delta$ is a regularisation factor.
-This indeed introduces quadratic fonctions on variables $x_{hl}$ and
+This indeed introduces quadratic functions on variables $x_{hl}$ and
$R_{h}$ and makes some of the functions strictly convex.
The authors then apply a classical dual based approach with Lagrange multiplier
\end{equation}
The proposed algorithm iteratively computes the following variables
-untill the variation of the dual function is less than a given threshold.
+until the variation of the dual function is less than a given threshold.
\begin{enumerate}
\item $ u_{hi}^{(k+1)} = u_{hi}^{(k)} - \theta^{(k)}. \left(
\eta_{hi}^{(k)} - \sum_{l \in L }a_{il}x_{hl}^{(k)} \right) $
\item
$\begin{array}{l}
\lambda_{i}^{(k+1)} = \max\left\{0, \lambda_{i}^{(k)} - \theta^{(k)}.\left(
- q^{(k)}.B_i \right. \left.\\
+ q^{(k)}.B_i \right. \right.\\
\qquad\qquad\qquad -\sum_{l \in L}a_{il}^{+}.c^s_l.\left( \sum_{h \in V}x_{hl}^{(k)} \right) \\
\qquad\qquad\qquad - \left.\left. \sum_{l \in L} a_{il}^{-}.c^r.\left( \sum_{h \in V}x_{hl}^{(k)} \right) - P_{si}^{(k)} \right) \right\}
\end{array}
=
\arg \min_{p > 0}
\left(
-v_h^{(k)}.\dfrac{\ln(\sigma^2/D_h)}{\gamma p ^{2/3}} + \lambda_h^{(k)}p
+v_h^{(k)}.\dfrac{\ln(\sigma^2/D_h)}{\gamma p^{2/3}} + \lambda_h^{(k)}p
\right)
$
-Let us first have a discussion on the stop criterion of the citted algorithm.
+Let us first have a discussion on the stop criterion of the cited algorithm.
We claim that even if the variation of the dual function is less than a given
-threeshold, this does not ensure that the lifetime has been maximized.
+threshold, this does not ensure that the lifetime has been maximized.
Minimizing a function on a multiple domain (as the dual function)
-may indeed easilly fall into a local trap because some of introduced
+may indeed easily fall into a local trap because some of introduced
variables may lead to uniformity of the output.
\begin{figure}
\label{fig:convergence:scatterplot}
\end{figure}
-Experimentations have indeed shown that even if the dual
+Experiments have indeed shown that even if the dual
function seems to be constant
(variations between two evaluations of this one is less than $10^{-5}$)
not all the $q_i$ have the same value.
The maximum amplitude rate of the sequence of $q_i$ --which is
$\frac{\max_{i \in N} q_i} {\min_{i \in N}q_i}-1$--
-is represented in $y$-coordonates
+is represented in $y$-coordinate
with respect to the
-value of the threeshold for dual function that is represented in
-$x$-coordonates.
+value of the threshold for dual function that is represented in
+$x$-coordinate.
This figure shows that a very small threshold is a necessary condition, but not
a sufficient criteria to observe convergence of $q_i$.
In the following, we consider the system are $\epsilon$-stable if both
-maximum amplitude rate and the dual function are less than a threeshold
+maximum amplitude rate and the dual function are less than a threshold
$\epsilon$.
v_h^{(k)}.\dfrac{\ln(\sigma^2/D_h)}{\gamma p ^{2/3}} + \lambda_h^{(k)}p
\right)
$.
-The function inside the $\arg \min$ is stricly convex if and only if
-$\lamda_h$ is not null. This asymptotic configuration may arrise due to
+The function inside the $\arg \min$ is strictly convex if and only if
+$\lambda_h$ is not null. This asymptotic configuration may arise due to
the definition of $\lambda_i$. Worth, in this case, the function is
-stricly decreasing and the minimal value is obtained when $p$ is the infinity.
+strictly decreasing and the minimal value is obtained when $p$ is the infinity.
To prevent this configuration, we replace the objective function given
in equation~(\ref{eq:obj2}) by
\begin{equation}
\sum_{i \in N }q_i^2 +
-\delta_x \sum_{h \in V, l \in L } .x_{hl}^2
-+ \delta_r\sum_{h \in V }\delta.R_{h}^2
-+ \delta_p\sum_{h \in V }\delta.P_{sh}^{\frac{8}{3}}.
+\delta_x \sum_{h \in V, l \in L } x_{hl}^2
++ \delta_r\sum_{h \in V }R_{h}^2
++ \delta_p\sum_{h \in V }P_{sh}^{\frac{8}{3}}.
\label{eq:obj2}
\end{equation}
In this equation we have first introduced new regularisation factors
(namely $\delta_x$, $\delta_r$, and $\delta_p$)
instead of the sole $\delta$.
This allows to further study the influence of each modification separately.
-Next, the introduction of the rationnal exponent is motivated by the goal of
-providing a stricly convex function.
+Next, the introduction of the rational exponent is motivated by the goal of
+providing a strictly convex function.
+
+Let us now verify that the induced function is convex.
+Let $f: \R^{+*} \rightarrow \R$ such that $
+f(p)= v_h.\dfrac{\ln(\sigma^2/D_h)}{\gamma p^{2/3}} + \lambda_h p
++ \delta_p p^{8/3}$. This function is differentiable and
+for any $x \in \R^{+*}$ and we have
+$$
+\begin{array}{rcl}
+f'(p) &=& -2/3.v_h.\dfrac{\ln(\sigma^2/D_h)}{\gamma p^{5/3}} + \lambda_h +
+8/3.\delta_p p^{5/3} \\
+&& \dfrac {8/3.\delta_p p^{10/3} + \lambda_h p^{5/3} -2/3.v_h\ln(\sigma^2/D_h) }{p^{5/3}}
+\end{array}
+$$
+which is positive if and only if the numerator is.
+Provided $p^{5/3}$ is replaced by $P$, we have a quadratic function which is strictly convex, for any value of $\lambda_h$.
\ No newline at end of file
