\subsection{Fourier Analysis of SEM Images}

The two dimensional Discrete Fourier Transform is given by:
\begin{align}
	F(k_x, k_y) &= \displaystyle\sum_{x=0}^{N-1}\displaystyle\sum_{y=0}^{N-1} f(x, y) e^{\frac{-2 \pi i}{N}\left(k_x x + k_y y\right)} \label{dft}
\end{align}

Where $f(x, y)$ is a discrete data value (in this case the pixel intensity of the image) co-ordinates $(x, y)$, $N \times N$ are the dimensions of the image, and $F(k_x, k_y)$ gives the Fourier Coefficient. If the image represents a region with dimensions of $L \times L$, then the largest frequency components that can be contained in $F$ are $\frac{N}{L}$ \cite{}.


Equation \eqref{dft} actually gives the Fourier coefficients of the infinite periodic extension of $f(x, y)$. If $f(x, y)$ is not periodic, then applying \eqref{dft} introduces extra high frequency components due to sharp discontinuities at the boundaries. These components are revealed as sharp lines running through the centre of the DFT images. Because the Black Au image has a greater contrast in intensity levels and therefore larger discontinuities are the boundaries, these lines are sharper in the Black Au DFT images. 


Windowing techniques may be used to reduce these high frequency components, however since this comes at a cost of reduced resolution we have not employed windowing here. 
 
\pagebreak
\begin{figure}[H]
	\centering
	\includegraphics[scale=0.35]{figures/sem/Au_BLACK_82pix_200nm_fft_abs.png} 
	\caption{Amplitude density plot of DFT for Black Au SEM image}
	
\end{figure}
\begin{figure}[H]
	\centering
	\includegraphics[scale=0.35]{figures/sem/Au_BRIGHT_42pix_100nm_fft_abs.png}	
	\caption{Amplitude density plot of DFT for Au-Bright SEM image}
\end{figure}
\begin{figure}[H]
	\centering
	\includegraphics[scale=0.35]{figures/sem/Au_BLACK_82pix_200nm_fft_phase.png} 
	\caption{Phase density plot of DFT for Black Au SEM image}
	\captionof[figure]{Amplitude density plot of DFT for Black Au}
\end{figure}
\begin{figure}[H]
	\centering
	\includegraphics[scale=0.35]{figures/sem/Au_BRIGHT_42pix_100nm_fft_phase.png}	
	\caption{Phase density plot of DFT for Au-Bright SEM image}
\end{figure}




