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\begin{document}


\begin{slide}
\begin{center}
{\Large\bf The strong chromatic index of $C_4$-free graphs}
\\[15mm] {\small \it Mohammad Mahdian}
\\[25mm] {\small M.Sec. Student \\ Department of Computer Science} \\[2mm]
$$f(\vec{u})=||\vec{r}(\vec{u})- \vec{p})||^2$$ $$S=\vec{r}(x, y)$$
{\small University of Toronto}
\end{center}
\end{slide}



\begin{slide}
A {\bf strong edge-colouring} of a graph is an 
edge-colouring such that every two 
edges \underline{\it of} \underline{\it distance at most two} 
receive different colours. 
\vspace{2mm}

Two edges are of distance at most two iff
\vspace{-10mm}
\begin{itemize}
\item they share an endpoint, or
\vspace{-10mm}
\item one of their endpoints are adjacent.
\end{itemize}

%{\small Example: $e_1$ and $e_2$ must receive different colours.}
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\end{slide}


\begin{note}
\begin{itemize}
\item Every colour class is an induced matching.
\end{itemize}
\end{note}


\begin{slide}
Quaternion-based algorithm for obtaining the least squares rotation and translation
\begin{itemize}
\item Each rotation in 3D is  denoted by {\em unit quaternion}, a four vector $\vec{q_R}= [q_0,q_1,q_2, q_3]^T$ where
$q_0\geq 0$ and $q_0^2+q_1^2+q_2^2+q_3^2= 1$.
\item Rotation matrix $R$ generated by a unit \\ rotation quaternion is:
{\small
$$\left[
{\begin{array}{ccc}
q_0^2+q_1^2-q_2^2-q_3^2 & 2(q_1q_2-q_0q_3) & 2(q_1q_3+q_0q_2) \\
2(q_1q_2+ q_0q_3) & q_0^2+q_2^2-q_1^2-q_3^2 & 2(q_2q_3+q_0q_2)\\
2(q_1q_3-q_0q_2) & 2(q_2q_3+q_0q_1) & q_0^2+q_3^2-q_1^2-q_2^2
\end{array}}
\right]
$$
}
\item Translation vector $\vec{q_T}= [q_4,q_5,q_6]^T$
\item Complete registration is denoted by $\vec{q}= [\vec{q_R}|\vec{q_T}]^T$
\end{itemize}
\end{slide}








\begin{slide}
\begin{itemize}
\item For aligning {\em data point set} $P= \{\vec{p_i}\}$ to {\em model point set}  $X= \{\vec{x_i}\}$, we wish to minimize the mean square objective function
$$f(\vec{q})= {1\over |P|}\sum_{i= 1}^{|P|}||\vec{x_i}-{\cal R}(\vec{p_i})-\vec{q_T}||^2$$ 

\item Using the concepts of {\em center of mass} and {\em cross-covariance matrix}, we can obtain procedure ${\cal Q}(P, X)$ [Horn 87], more precisely: 

\item {\bf Center of mass:}\\ $\vec{\mu_p}= {1\over |P|}\sum_{i= 1}^{|P|}\vec{p_i}$ and $\vec{\mu_x}= {1\over |X|}\sum_{i= 1}^{|X|}\vec{x_i}$

\item {\bf Cross-covariance Matrix:} $$\Sigma_{px}= {1\over |P|}\sum_{i= 1}^{|P|}[(\vec{p_i}-\vec{\mu_p})(\vec{x_i}-\vec{\mu_x})]$$ 

\item Let anti-symmetric matrix \\ $A_{ij}= (\Sigma_{px}- \Sigma_{px}^T)_{ij}$ and $\Delta=[A_{23}\ A_{31}\ A_{12}]^T$.

\item {\small  $$Q(\Sigma_{px})= \left[
{\begin{array}{cc}
tr(\Sigma_{px}) & \Delta^T\\
\Delta & \Sigma_{px}+\Sigma_{px}^T-tr(\Sigma_{px}){\bf I_3}
\end{array}}
\right] $$}

\item $ \vec{q_R}=[q_0\ q_1\ q_2\ q_3]^T$ is the unit vector corresponding to the maximum eigenvalue of $Q(\Sigma_{px})$ and $\vec{q_R}= \vec{\mu_x}- {\cal R}(\vec{q_R}\vec{\mu_p})$

\item ${\cal Q}(P, X)$ returns $\vec{q}$ and $d_{ms}$ (mean square point matching error).

\item Thus the running time of ${\cal Q}(P, X)$ is in $O(|P|)$. 
\end{itemize}
\end{slide}





\begin{slide}
\begin{itemize}
\item Overall iterative algorithm:
\begin{enumerate}
\item $Y= Correspond(P, S)$  \ \ \{$O(|P|\log(N_x))$ where $N_x$ is the number of points or triangles\}.
\item $(\vec{q}, d_{ms})= {\cal Q}(P, Y)$  \ \  \{$O(|P|)$\}
\item $P\rightarrow \vec{q}(P)$  \ \ \{$O(|P|)$\}
\item go to step 1 until $d_{ms}^k-d_{ms}^{k+ 1}< threshold$
\end{enumerate}
\item For $r$ iterations, we have running time $O(r|P|\cdot \log(N_x))$.
\item Usually $r\leq 10$.
\end{itemize}   
\end{slide}

\begin{slide}
\begin{itemize}
\item We can prove that {\em the ICP algorithm \\ always converges monotonically to  a {\bf local minimum} with respect to the mean-square distance objective function}.
\item Thus we consider some uniform initial states (rotation, translation) to go to the {\bf global minimum}.
\item We can perform the algorithm using \\ various randomely-selected subsets of points and selecting the best
\item We can also consider both the distance \\ between points and the distance between \\ colors  
\end{itemize}   
\end{slide}


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