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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
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\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} |
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