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\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{problem}[theorem]{Problem}
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\begin{document}


\bigskip 

\bigskip 

\begin{center}
{\Large One Dimensional Motion}

\bigskip 
\end{center}

\[
g(x,y)=\int_{0}^{T}f(x-x_{0}(t),y-y_{0}(t))dt
\]

Fourier transform%
\begin{eqnarray*}
G(m,n) &=&\int_{-\infty }^{\infty }\int_{-\infty }^{\infty }g(x,y)e^{-2\pi
i(mx+ny)}dxdy \\
&=&\int_{0}^{T}\left( \int_{-\infty }^{\infty }\int_{-\infty }^{\infty
}f(x-x_{0},y-y_{0})e^{-2\pi i(mx+ny)}dxdy\right) dt \\
&=&\int_{0}^{T}F(u,v)e^{-2\pi i(mx_{0}+ny_{0})}dt
\end{eqnarray*}

So the transfer function is%
\[
H(m,n)=\int_{0}^{T}e^{-2\pi i(mx_{0}(t)+ny_{0}(t))}dt
\]

Assume $x_{0}=\frac{\alpha t}{T}$ $y_{0}=0$. Then

\[
H(m,n)=\int_{0}^{T}e^{-2\pi im\frac{\alpha t}{T}}dt=Te^{-\pi im\alpha }\frac{%
\sin (\pi m\alpha )}{\pi m\alpha }
\]

Note that $H(m,n)=0$ whenever $m\alpha $ is an integer.

Using sum instead of integral we get

$i_{T}=$ total number of pixels recorded by the same cell of the camera

$N=$ total number of pixels in a row of the image%
\[
H(m,n)=\frac{1}{i_{T}}\frac{\sin (\frac{nm}{N}i_{T})}{\sin (\frac{nm}{N})}%
e^{-\pi i\frac{m}{N}\alpha }
\]

\newpage 

\begin{center}
{\Large Wiener Filter}

\bigskip 
\end{center}

\[
M(u,v)=\frac{H^{\ast }(u,v)}{|H(u,v)|^{2}+\frac{S_{nn}(u,v)}{S_{ff}(u,v)}}
\]

Note: $S_{ff}(u,v)$ is autocorrelation of the \textbf{original} picture 

\bigskip 

Assumption: Noise is white: 
\[
S_{nn}(u,v)=S_{nn}(0,0)=\int_{-\infty }^{\infty }\int_{-\infty }^{\infty
}R_{nn}(x,y)dxdy
\]

can be found when there is no image $f(x,y)=0$

\bigskip 

\begin{center}
Wierner-Khinchine Theorem
\end{center}

The spatial autocorrelation function of a random field f(x,y) is equal to
the spectral density $|F(u,v)|^{2}$

\bigskip 

example: blurred image

\begin{eqnarray*}
H(m,n) &=&\frac{1}{i_{T}}\frac{\sin (\frac{nm}{N}i_{T})}{\sin (\frac{nm}{N})}%
e^{-\pi i\frac{m}{N}\alpha } \\
|H(m,n)|^{2} &=&\frac{1}{i_{T}^{2}}\frac{\sin ^{2}(\frac{nm}{N}i_{T})}{\sin
^{2}(\frac{nm}{N})}
\end{eqnarray*}

\newpage 

\begin{center}
Constrained Matrix
\end{center}

\bigskip 

\begin{eqnarray*}
g &=&Hf+n \\
n &=&g-Hf
\end{eqnarray*}

Assume statistics of the noise are known, eg $n^{T}n=\epsilon $

Let $Lf$ be a functional of the picture e.g. discrete Laplacian. We wish to
find

\bigskip 

\begin{eqnarray*}
\left( Lf\right) ^{T}(LF) &=&\text{minimum} \\
\left( g-Hf\right) ^{T}\left( g-Hf\right)  &=&\epsilon 
\end{eqnarray*}

Use Lagrange multipliers

\begin{eqnarray*}
f &=&\left[ H^{\ast }H+\gamma L^{\ast }L\right] ^{-1}H^{\ast }g \\
&=&\frac{H^{\ast }g}{|H|^{2}+\gamma L^{\ast }L}
\end{eqnarray*}

Note instead of $L=Laplacian$ take $L^{\ast }L=-Laplacian$.

Looks similar to Wiener $L^{\ast }L=\frac{S_{nn}}{S_{ff}}$

\end{document}
