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\input{tcilatex}

\begin{document}


\begin{center}
Averages
\end{center}

\bigskip

$N^{2}$ is number of pixels in the window

\bigskip

\begin{enumerate}
\item Arithmetic Mean \quad =$\frac{1}{N^2} \dsum\limits_{r,c} d(r,c)$ \ \ \
\ \ \ \ low pass, blurs images \newline

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ reduces all
noise, best for Gaussian \& uniform

\item Geometric Mean: \quad = $\left[ \dprod\limits_{r,c}d(r,c)\right] ^{%
\frac{1}{N^{2}}}$ \ \ best with Gaussian \newline

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
sharper than arithmetic, doesn't work on pepper

\item Harmonic mean: \quad $=\frac{N^{2}}{\dsum\limits_{r,c}\frac{1}{d(r,c)}}
$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ good on salt - bad on pepper

\item $Y_{p}$ mean \qquad $=\left[ \dsum\limits_{r,c}\frac{d^{p}(r,c)}{N^{2}}%
\right] ^{\frac{1}{p}}$ \ for $p>0$ good on salt - bad on pepper

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ for $p<0$ bad on salt - good on pepper

\item Contra-harmonic mean $=\frac{\dsum\limits_{r,c}d^{R-1}(r,c)}{%
\dsum\limits_{r,c}d^{R}(r,c)}$ \ \ $R>0$ eliminates salt

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $R<0$ eliminates pepper

\item alpha - trim (T) $=\frac{1}{N^{2}-2T-1}\dsum%
\limits_{j=T+1}^{N^{2}-T}I_{j}$ \ \ between arithmetic and median

\item Nonlinear mean $=u^{-1}\left\{ \frac{\dsum a_{ij}u\left[ g_{ij}\right] 
}{\dsum a_{ij}}\right\} $

\begin{itemize}
\item Arithmetic mean $u(g)=g$ \newline

\item Harmonic mean $u(g)=\frac{1}{g}$ \newline

\item Geometric mean $u(g)=log(g)$
\end{itemize}

\newpage

\item Minimum Mean-Square Error (MMSE)%
\begin{equation*}
MMSE=d(r,c)-\frac{\sigma _{n}^{2}}{\sigma _{l}^{2}}\left(
d(r,c)-m_{l}(r,c)\right) 
\end{equation*}
\begin{itemize}
\item $\sigma _{n}^{2}$ = noise variance

\item $\sigma _{l}^{2}$ = local variance inside window

\item $m_{l}(r,c)$=mean inside window
\end{itemize}

\end{enumerate}

\newpage

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

\bigskip

Use some weighted average%
\[
\widehat{x}_{k}=\frac{\dsum\limits_{n}w_{kn}g_{k-n}}{\dsum\limits_{n}w_{kn}}
\]

\bigskip

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

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

\bigskip

\[
w_{kn}=w_{kn}^s w_{kn}^r
\]

\bigskip

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

\bigskip

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

\newpage

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

\bigskip

X=measured image

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{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ penalty}
\end{align*}

\bigskip

L(X) can be first or second derivative

\bigskip

\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}
