1 Image Filtering
To filter an image, we take a weighted average of pixels in a window around each pixel. We will use the simplest weights we can, uniform weights, to create a filter function ${\prosedeflabel{filters}{{f}}}$ that applies to values ${\prosedeflabel{filters}{{x}}}$ in $[-1,1]$:
$$\DeclareMathOperator*{\argmax}{arg\,max}
\DeclareMathOperator*{\argmin}{arg\,min}
\begin{align*}
\idlabel{ {"onclick":"event.stopPropagation(); onClickSymbol(this, 'f', 'filters', 'def', false, 'f')", "id":"filters-f", "sym":"f", "func":"filters", "localFunc":"f", "type":"def", "case":"equation"} }{ {\mathit{f}} }\left( \idlabel{ {"onclick":"event.stopPropagation(); onClickSymbol(this, 'x', 'filters', 'use', true, 'f')", "id":"filters-x", "sym":"x", "func":"filters", "localFunc":"f", "type":"use", "case":"equation"} }{ {\mathit{x}} } \right) & = \begin{cases} 1 & \text{if } \left|\idlabel{ {"onclick":"event.stopPropagation(); onClickSymbol(this, 'x', 'filters', 'use', true, 'f')", "id":"filters-x", "sym":"x", "func":"filters", "localFunc":"f", "type":"use", "case":"equation"} }{ {\mathit{x}} }\right| < 1 \\ 0 & \text{otherwise} \end{cases}\\\eqlabel{ {"onclick":"event.stopPropagation(); onClickEq(this, 'filters', ['x', 'f'], true, 'f', ['x'], 'Zih4KSA9IHsgMSBpZiB8eHwgPCAxCiAgICAgICAgIDAgb3RoZXJ3aXNlIHdoZXJlIHgg4oiIIOKEnQ==');"} }{}
\end{align*}
\tag{1}\label{1}$$
