More small fixes to the paper

parent 45d9a5ed
......@@ -8,7 +8,7 @@ Since we are concerned with reproducing the inference results of spatial domain
\begin{figure}
\centering
\includegraphics[width=0.5\linewidth]{figures/network.pdf}
\includegraphics[width=\linewidth]{figures/network.pdf}
\caption{Simple network architecture. $T$ indicates the batch size.}
\label{fig:na}
\end{figure}
......
......@@ -21,8 +21,31 @@ where $c$ and $c'$ index the input and output channels of the image respectively
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figures/asm_relu.png}
\caption{Example of ASM ReLu on an $8 \times 8$ block. Green pixels are negative, red pixels are positive, and blue pixels are zero. Left: original image. Middle: ReLu. Right: ReLu approximation using ASM with 6 spatial frequencies.}
\begin{subfigure}{0.20\textwidth}
\captionsetup{width=.8\linewidth}
\centering
\includegraphics[width=\textwidth]{figures/asm_relu_original.png}
\caption{Original image}
\end{subfigure}%
\begin{subfigure}{0.2\textwidth}
\captionsetup{width=.8\linewidth}
\centering
\includegraphics[width=\textwidth]{figures/asm_relu_true.png}
\caption{True ReLu}
\end{subfigure}\\
\begin{subfigure}{0.2\textwidth}
\captionsetup{width=.8\linewidth}
\centering
\includegraphics[width=\textwidth]{figures/asm_relu_apx.png}
\caption{ReLu using direct approximation}
\end{subfigure}%
\begin{subfigure}{0.2\textwidth}
\captionsetup{width=.8\linewidth}
\centering
\includegraphics[width=\textwidth]{figures/asm_relu_asm.png}
\caption{ReLu using ASM approximation}
\end{subfigure}
\caption{Example of ASM ReLu on an $8 \times 8$ block. Green pixels are negative, red pixels are positive, and blue pixels are zero. 6 spatial frequencies are used for both approximations. Note that the direct approximation fails to preserve positive pixel values.}
\label{fig:asm}
\end{figure}
......@@ -83,12 +106,12 @@ We call the function $\nnm(x)$ the nonnegative mask of $x$. This is our binary m
\begin{equation}
r(x) = \nnm(x)x
\end{equation}
This new function can be computed efficiently from fewer spatial frequencies with much higher accuracy since only the sign of the original function needs to be correct. Figure \ref{fig:asm} gives an example of this algorithm on a random block, and pseudocode is given in the supplementary material. To extend this method from the DCT domain to the JPEG transform domain, the rest of the missing JPEG tensor can simply be applied as follows:
This new function can be computed efficiently from fewer spatial frequencies with much higher accuracy since only the sign of the original function needs to be correct. Figure \ref{fig:asm} gives an example of this algorithm on a random block and compares it to computing ReLu on the approximation directly. Note that in the ASM image the pixel values of all positive pixels are preserved, the only errors are in the mask. In the direct approximation, however, none of the pixel values are preserved and it suffers from masking errors. The magnitude of the error is tested in Section \ref{sec:exprla} and pseudocode for the ASM algorithm is given in the supplementary material.
To extend this method from the DCT domain to the JPEG transform domain, the rest of the missing JPEG tensor can simply be applied as follows:
\begin{equation}
H^{k ij}_{k'} = Z^{k}_{\gamma} \widetilde{S}^{\gamma}_{\alpha\beta} D^{\alpha\beta ij}D^{ij}_{\alpha'\beta'} S^{\alpha'\beta'}_{\gamma'} Z^{\gamma'}_{k'}
\end{equation}
Since the operation is the same for each block, and there are no interactions between blocks, the blocking tensor $B$ can be skipped.
\subsection{Batch Normalization}
......
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