1 Image Filtering
To filter an image, we take a weighted average of pixels in a window around each pixel. We will use a quadratic Gaussian approximation 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, '')", "id":"filters-f", "sym":"f", "func":"filters", "localFunc":"", "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 - 3{\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}} }}^{2} & \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| < ⅓ \\1.5{\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}} }}^{2} - 3\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| + 1.5 & \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'], 'Zih4KSA9IHsgMS0zeMKyICAgICAgICAgICAgICAgaWYgfHh8IDwg4oWTCiAgICAgICAgIDEuNXjCsiAtIDN8eHwgKyAxLjUgIGlmIHx4fCA8IDEKICAgICAgICAgMCBvdGhlcndpc2Ugd2hlcmUgeCDiiIgg4oSd');"} }{}
\end{align*}
\tag{1}\label{1}$$

