wiener.tex

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\title{Introduction to Wiener Filtering}

\author{Mosè Giordano}
\date{26 November 2014}
\institute[UniSalento and INFN Lecce]{Università del Salento and INFN Lecce}
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\begin{document}
\begin{frame}[plain]
  \maketitle
\end{frame}

\begin{frame}
  \frametitle{The purpose of Wiener filtering}
  Reduce degradation and noise in images, audio signals, etc.
  \begin{figure}
    \centering
    \(\vcenter{\hbox{\includegraphics[width=0.2\textwidth]{moon.jpg}}}\) ~
    \tikz{\draw[-latex,mm2skyblue,thick] (0,0) -- node[above] {Degradation}
      (1.7,0);} ~
    \(\vcenter{\hbox{\includegraphics[width=0.2\textwidth]{figures/degraded_moon.jpg}}}\)
    ~
    \tikz{\draw[-latex,mm2skyblue,thick] (0,0) -- node[above] {Filtering}
      (1.7,0);} ~
    \(\vcenter{\hbox{\includegraphics[width=0.2\textwidth]{figures/filtered_moon.jpg}}}\)
    \caption{Picture of the Moon taken by the Galileo spacecraft on 7 December 1992}
  \end{figure}
\end{frame}

\begin{frame}
  \frametitle{The theory}
  Suppose you have a \alert{signal} \(S(\bm{t})\) in the time domain (whatever
  ``time'' is: actual time, point in space, pixel of an image, etc), degraded by
  \alert{known blurring shift-invariant function} \(B(\bm{t})\) and
  \alert{additive noise} \(N(\bm{t})\)
  \begin{equation*}
    X(\bm{t}) = (B * S)(\bm{t}) + N(\bm{t})
  \end{equation*}
  The Fourier transform in the frequency domain of this \alert{degraded signal}
  \(X(\bm{t})\) is
  \begin{equation*}
    \hat{X}(\bm{f}) = \mathcal{F}(X)(\bm{f}) = \hat{B}(\bm{f})\hat{S}(\bm{f}) +
    \hat{N}(\bm{f})
  \end{equation*}
\end{frame}

\begin{frame}
  \frametitle{The theory (cont.)}
  We want to find an appropriate filter \(W(\bm{f})\) such that the function
  \(W(\bm{f})\hat{X}(\bm{f})\) is as much close as possible to the Fourier
  transform of the original signal \(\hat{S}(\bm{f}) = \mathcal{F}(S)(\bm{f})\),
  i.e. we want to minimize the quantity
  \begin{equation*}
    \Braket{W(\bm{f})\hat{X}(\bm{f}) - \hat{S}(\bm{f})} =
    \Braket{W(\bm{f})(\hat{B}(\bm{f})\hat{S}(\bm{f}) + \hat{N}(\bm{f})) -
      \hat{S}(\bm{f})}
  \end{equation*}
  This condition is fulfilled by
  \begin{equation*}
    W(\bm{f}) = \frac{\hat{B}^{*}(\bm{f})}{|\hat{B}(\bm{f})|^{2} +
      |\hat{N}(\bm{f})|^{2}/|\hat{S}(\bm{f})|^{2}}
  \end{equation*}
  This is the \alert{Wiener filter} function and
  \(\mathcal{F}^{-1}(W\hat{X})(\bm{t})\) is the \alert{filtered signal}.  This
  function gives more importance to frequencies with \alert{higher signal to
    noise ratio}.  In absence of blurring
  \begin{equation*}
    W(\bm{f}) = \frac{|\hat{S}(\bm{f})|^{2}}{|\hat{S}(\bm{f})|^{2} +
      |\hat{N}(\bm{f})|^{2}}
  \end{equation*}
\end{frame}

\begin{frame}
  \frametitle{The theory (cont.)}
  Theoretically, in order to calculate the Wiener filter function we need to
  know
  \begin{itemize}
  \item the original signal
  \item the blurring function
  \item the noise
  \end{itemize}
  or at least their power spectra.  Actually, power spectra need not to be known
  exactly (\alert{noise power spectrum can often be easily estimated},
  e.g. white noise has constant spectrum) because
  \begin{itemize}
  \item most signals of the same class have fairly \alert{similar power spectra}
  \item the Wiener filter is \alert{insensitive to small variations} in the
    original signal power spectrum
  \end{itemize}
  We can estimate the original signal power spectrum using a
  \alert{representative of the class of signals being filtered}
\end{frame}

\begin{frame}
  \frametitle{Wiener filter applied to a temporal signal}
  IDL/GDL code:
  \lstinputlisting[linerange={25-47}]{filtro_wiener.pro}
\end{frame}

\begin{frame}
  \frametitle{Wiener filter applied to a temporal signal (cont.)}
  \begin{figure}
    \centering
    \includegraphics[width=0.85\textwidth]{figures/signal}
  \end{figure}
\end{frame}

\begin{frame}
  \frametitle{Wiener filter applied to a temporal signal (cont.)}
  \begin{figure}
    \centering
    \includegraphics[width=0.85\textwidth]{figures/signal-noise}
  \end{figure}
\end{frame}

\begin{frame}
  \frametitle{Wiener filter applied to a temporal signal (cont.)}
  \begin{figure}
    \centering
    \includegraphics[width=0.85\textwidth]{figures/signal-spectra}
  \end{figure}
\end{frame}

\begin{frame}
  \frametitle{Wiener filter applied to a temporal signal (cont.)}
  \begin{figure}
    \centering
    \includegraphics[width=0.85\textwidth]{figures/filter}
  \end{figure}
\end{frame}

\begin{frame}
  \frametitle{Wiener filter applied to a temporal signal (cont.)}
  \begin{figure}
    \centering
    \includegraphics[width=0.85\textwidth]{figures/result}
  \end{figure}
\end{frame}

\begin{frame}
  \frametitle{Wiener filter applied to an image}
  IDL/GDL code:
  \lstinputlisting[linerange={64-88}]{filtro_wiener.pro}
\end{frame}

\begin{frame}
  \frametitle{Wiener filter applied to an image (cont.)}
  \begin{figure}
    \centering
    \vspace{-0.5em}
    \includegraphics[width=0.6\textwidth]{lena}
    \caption{The original image, file \texttt{lena.jpg}}
    \vspace{-1em}
  \end{figure}
\end{frame}

\begin{frame}
  \frametitle{Wiener filter applied to an image (cont.)}
  \begin{figure}
    \centering
    \vspace{-0.5em}
    \includegraphics[width=0.6\textwidth]{figures/degraded_lena}
    \caption{The image has been degraded with a large noise}
    \vspace{-1em}
  \end{figure}
\end{frame}

\begin{frame}
  \frametitle{Wiener filter applied to an image (cont.)}
  \begin{figure}
    \centering
    \vspace{-0.5em}
    \includegraphics[width=0.6\textwidth]{elaine}
    \caption{The picture used to filter the degraded image, file
      \texttt{elaine.jpg}}
    \vspace{-1em}
  \end{figure}
\end{frame}

\begin{frame}
  \frametitle{Wiener filter applied to an image (cont.)}
  We can use a different picture to filter the degraded image because they have
  similar power spectra
  \begin{figure}
    \centering
    \includegraphics[width=\textwidth]{figures/power-spectra}
    \caption{Left: power spectrum of \texttt{lena.jpg} file; right: power
      spectrum of \texttt{elaine.jpg} file}
  \end{figure}
\end{frame}

\begin{frame}
  \frametitle{Wiener filter applied to an image (cont.)}
  \begin{figure}
    \centering
    \vspace{-0.5em}
    \includegraphics[width=0.8\textwidth]{figures/img-filter}
    \caption{The Wiener filter function in the frequency domain}
    \vspace{-1em}
  \end{figure}
\end{frame}

\begin{frame}
  \frametitle{Wiener filter applied to an image (cont.)}
  \begin{figure}
    \centering
    \vspace{-0.5em}
    \includegraphics[width=0.6\textwidth]{figures/filtered_lena}
    \caption{The filtered image}
    \vspace{-1em}
  \end{figure}
\end{frame}

\begin{frame}
  \frametitle{\refname{} and further reading}
  \nocite{*}
  \printbibliography{}
\end{frame}
\end{document}

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