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%TCIDATA{Created=Saturday, November 26, 2005 21:34:54}
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\begin{document}


\begin{center}
Bilateral Filter
\end{center}

\bigskip

Use some weighted average%
\begin{equation*}
\widehat{x}_{k}=\frac{\dsum\limits_{n}w_{kn}g_{k-n}}{\dsum\limits_{n}w_{kn}} 
\end{equation*}

\bigskip

\begin{enumerate}
\item $w_{kn}^{s}=e^{-\frac{d_{k,k-n}^{2}}{2\sigma _{s}^{2}}}$ \qquad 
Gaussian, physical distance


\item $w_{kn}^{r}=e^{-\frac{(g_{k}-g_{k-n})^{2}}{2\sigma _{r}^{2}}}$ \qquad
Gaussian, gray value distance
\end{enumerate}

\bigskip

\begin{equation*}
w_{kn}=w_{kn}^{s}w_{kn}^{r} 
\end{equation*}

\bigskip

\noindent $\sigma _{s}\;,\sigma _{r}$ \ large implies uniform non-adaptive , degrades signal

\bigskip

\noindent $\sigma _{s}\;,\sigma _{r}$ \ small implies no smoothing


\newpage 

\begin{center}
Weighted Least Squares
\end{center}

\bigskip 

\noindent X=measured image

\noindent Y=desired image

\bigskip 

\begin{align*}
\varepsilon _{\text{WLS}} =\frac{1}{2}\left( X-Y\right) ^{T}\left(
X-Y\right) &+\underbrace{\frac{\lambda }{2}L(X)^{T}W(Y)L(X)} \\
&\text{\qquad \qquad penalty}
\end{align*}

\bigskip 

\noindent $L(X)$ can be first or second derivative

\bigskip 

Robust Estmation:%
\begin{align*}
\varepsilon _{\text{WLS}} = \frac{1}{2}\left( X-Y\right) ^{T}\left(
X-Y\right) &+\frac{\lambda }{2}\underbrace{\rho (X)} \\
&\text{nonlinear}
\end{align*}

\end{document}
