\section{FFT Image Denoising}
\label{sec:fft_imdenoise}
Convolution of an input with with a linear filter in the termporal or spatial
domain is equivalent to multiplication by the fourier transforms of the input
and the filter in the spectral domain. This provides a conceptually simple way
to think about filtering: transform your signal into the frequency domain,
dampen the frequencies you are not interested in by multiplying the frequency
spectrum by the desired weights, and then apply the inverse transform to the
modified spectrum, back into the original domain. In the example below, we
will simply set the weights of the frequencies we are uninterested in (the high
frequency noise) to zero rather than dampening them with a smoothly varying
function. Although this is not usually the best thing to do, since sharp edges
in one domain usually introduce artifacts in another (eg high frequency
``ringing''), it is easy to do and sometimes provides satisfactory results.
The image in the upper left panel of Figure~\ref{fig:fft_imdenoise} is a
grayscale photo of the moon landing. There is a banded pattern of high
frequency noise polluting the image. In the upper right panel we see the 2D
spatial frequency spectrum. The FFT output in \texttt{scipy} is packed with
the lower freqeuencies starting in the upper left, and proceeding to higher
frequencies as one moves to the center of the spectrum (this is the most
efficient way numerically to fill the output of the FFT algorithm). Because
the input signal is real, the output spectrum is complex and symmetrical: the
transformation values beyond the midpoint of the frequency spectrum (the
Nyquist frequency) correspond to the values for negative frequencies and are
simply the mirror image of the positive frequencies below the Nyquist (this is
true for the 1D, 2D and ND FFTs in \texttt{numpy}).
In this exercise we will compute the 2D spatial frequency spectra of the
luminance image, zero out the high frequency components, and inverse transform
back into the spatial domain. We can plot the input and output images with the
\texttt{pylab.imshow} function, but the images must first be scaled to be
within the 0..1 luminance range. For best results, it helps to
\textit{amplify} the image by some scale factor, and then \textit{clip} it to
set all values greater than one to one. This serves to enhance contrast among
the darker elements of the image, so it is not completely dominated by the
brighter segments
\lstinputlisting[label=code:fft_imdenoise,caption={IGNORED}]{problems/fft_imdenoise.py}
\begin{figure}
\begin{centering} \includegraphics[width=4in]{fig/fft_imdenoise} \par
\end{centering}
\caption{\label{fig:fft_imdenoise}High freqeuency noise filtering of a 2D
image in the Fourier domain. The upper panels show the original image
(left) and spectral power (right) and the lower panels show the same data
with the high frequency power set to zero. Although the input and output
images are grayscale, you can provide colormaps to \texttt{pylab.imshow} to
plot them in psudo-color}
\end{figure}
