ガウシアンフィルタのフーリエ変換。 https://www.overleaf.com/read/wfqqyhmggjnd ネタ元：https://best-friends.chat/@mira/106182662941893419

2021-05-06_gaussian-filter.tex

\documentclass[xelatex,ja=standard,jafont=noto]{bxjsarticle}
\usepackage{fullpage,enumitem,amssymb,amsmath,xcolor,cancel,gensymb,hyperref,graphicx}

\title{ガウシアンフィルタのフーリエ変換}
\author{}
\date{2021-05-06}

\begin{document}

\maketitle

ガウシアンフィルタ
\[
h_g(x,y) = \frac{1}{2\pi\sigma^2}\exp{\left(-\frac{x^2+y^2}{2\sigma^2}\right)}
\]

\begin{align*}
  & \mathcal{F}[h_g(x,y)] \\
= & \frac{1}{2\pi\sigma^2}
    \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}
        \exp{\left(-\frac{x^2+y^2}{2\sigma^2}\right)}\exp{(-j2\pi(ux+vy))}\,dx\,dy \\
= & \frac{1}{2\pi\sigma^2}
    \int_{-\infty}^{\infty} \exp{\left(-\frac{x^2}{2\sigma^2}-j2\pi ux\right)}\,dx
    \int_{-\infty}^{\infty} \exp{\left(-\frac{y^2}{2\sigma^2}-j2\pi vy\right)}\,dy
\end{align*}

指数関数の中身を平方完成：
\begin{align*}
-\frac{x^2}{2\sigma^2}-j2\pi ux &= -\frac{1}{2\sigma^2} (x^2+j4\pi\sigma^2ux) \\
 &= -\frac{1}{2\sigma^2}\left( (x+j2\sigma^2\pi u)^2+(2\pi\sigma^2)^2u^2 \right)
\end{align*}

ガウス積分 $\displaystyle \int_{-\infty}^{\infty}\exp\left(-ax^2\right) = \sqrt{\frac{\pi}{a}}$ を使って
\begin{align*}
  & \int_{-\infty}^{\infty} \exp{\left(-\frac{x^2}{2\sigma^2}-j2\pi ux\right)}\,dx \\
= & \int_{-\infty}^{\infty} \exp{\left[-\frac{1}{2\sigma^2}\left( (x+j2\pi\sigma^2 u)^2+(2\pi\sigma^2)^2u^2 \right)\right]}\,dx \\
= & \exp{(-2\pi^2\sigma^2 u^2)}
    \int_{-\infty}^{\infty} \exp{\left[-\frac{1}{2\sigma^2} (x+j2\pi\sigma^2 u)^2 \right]}\,dx\\
= & \exp{(-2\pi^2\sigma^2 u^2)} \sqrt{2\pi\sigma^2}.
\end{align*}

したがって
\begin{align*}
\mathcal{F}[h_g(x,y)] = &  \frac{1}{2\pi\sigma^2}
    \int_{-\infty}^{\infty} \exp{\left(-\frac{x^2}{2\sigma^2}-j2\pi ux\right)}\,dx
    \int_{-\infty}^{\infty} \exp{\left(-\frac{y^2}{2\sigma^2}-j2\pi vy\right)}\,dy \\
= & \frac{1}{2\pi\sigma^2} \exp{(-2\pi^2\sigma^2 u^2)} \sqrt{2\pi\sigma^2} \exp{(-2\pi^2\sigma^2 v^2)} \sqrt{2\pi\sigma^2}\\
= & \exp{[-2\pi^2\sigma^2(u^2+v^2)]}.
\end{align*}

\end{document}