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{\it The Final-exam of the Introduction to Pattern Recognition
             (15 Jan 2001)}\\
\begin{itemize}
\item[1.] Obtain the {\it roundness measure} 
          of the following three profiles.

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   \epsfile{file=final-exam-1.eps,width=12cm,height=4cm}
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\item[2.] Calculate the {\it central moment} 
          $\mu_1, \mu_2, \mu_3$, and  $\mu_4$ 
          for the three quantities of 1, 3 and 5.
\item[3.] Obtain the {\it Haar transform matrix} for $L=2^2=4$.
\item[4.] Answer the following questions for the image:

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 \begin{center}
   \epsfile{file=final-exam-2.eps,width=8cm,height=4cm}
 \end{center}
\end{figure}

   \begin{itemize}
   \item[1)] Obtain the {\it co-occurrence matrix} $A^0(1)$.
   \item[2)] Obtain the {\it co-occurrence matrix} $A^{45}(2)$.
   \item[3)] Obtain the {\it angular second moment} assuming,
             $A^0(1)=P(I_1, I_2)$.
   \item[4)] Obtain the {\it run length matrix} $Q^{0}$.
   \item[5)] Obtain the {\it short run emphasis}.
   \end{itemize}
\item[5.] Assume that $p(\mbox{\bf x} \vert \omega_1)$ and 
          $p(\mbox{\bf x} \vert \omega_2)$  follow the Gaussian
          distributions whose means and standard deviations are
          $(\mu_1, \mu_2) = (1, 3)$ and $(\sigma_1, \sigma_2)=(1, 1)$,
          respectively. Then obtain the dicision surface (the single
          point in this case) $x_0$.
\end{itemize}

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\noindent
{\it The Final-exam of the Introduction to Pattern Recognition
             (15 Jan 2001)}\\
\noindent
{\it No.1} \hspace{.2in} Name [ \hspace{2in} ] \hspace{.1in} 
Student-ID [ \hspace{1in} ]
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1. 1)\hspace{2.2in}2)\hspace{2.2in}3)
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2.
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3.
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4. 1)
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4. 2)

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\noindent
{\it The Final-exam of the Introduction to Pattern Recognition 
             (15 Jan 2001)}\\
\noindent
{\it No.2}\hspace{.2in} Name [ \hspace{2in} ] \hspace{.1in} 
Student-ID [ \hspace{1in} ]
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4. 3)
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4. 4)
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4. 5)
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5.



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