In $[14]$: # MN$\_2\_$Fourier$\_$Transfors. ipnybinport numpy as np

Inport pylab

import matplotlib.pyplot as plt

from PIL import Image, ImageDraw

w$,$h $=65,65$

shape $=[(10,10),($w $-10,$ h $-10)]$

im$1=$ Image.new$("$RGB$",(($ w$,$ h$))$

ing $=$ InageDraw.Draw$($im$1)$

img.ellipse$($shape$,$ fill $=$"red", outline $=$"blue"$)$

pylab.imshow$($im$1)$

pylab.title$(\text{'}$Circle$\text{'})$

pylab.show

im$2=$ Image.new$("$RGB$",(\}|,$h$)\backslash$mathrm$\{))$

ing $=$ ImageDraw.Draw$($im$2)$

ing.rectangle$($shape$,$ fill $=$"yellow", outline $=$"purple"$)$

pylab.inshow$($im$2)$

pylab.title$(\text{'}$Box$\text{'})$

pylab.show

f $=$ plt$.$figure$()$

f$.$add$\_$subplot$(1,2,1)$

plt$.$ inshow$($np$.$rot$9$e$($im$1,2))$

f$,$add$\_$subplot$(1,2,2)$

plt$.$ inshow$($np$.$rot$9$e$($in$,2))$

plt$.$show$($block$=$True$)$

Task $2$:

Perform a Fourier analysis on the two above images. The program should do the following:

draws $2$ separate images

takes the $2$D discrete Fourier transform of one of the images $($HINT: using the numpy package, look at fft$2$ and fftshift$)$

applies a gaussian filter to the image

takes the inverse transform of the image $($HINT: using the numpy package, look at ifft$2)$

displays the blurred image

There are many ways you can attempt each of these steps, so I would recommend doing some research online.

The SciPy documentation on the scipy website is helpful if you're unsure where to start.

Gaussian Filter: A Gaussian Filter is used in reducing noise in the image, and also the details of the image. The Gaussian function based on the size of $$

$($standard deviation$)$ is defined as:

$G_{}=\frac{1}{2{}^2}{e}^{-\frac{({}^2+x^2)}{2{}^2}}$

where $x$ is the distance from the origin in the horizontal axis, $y$ is the distance from the origin in the vertical axis, and $$ is the standard deviation of the

Gaussian distribution. For the purposes of this lab, you can experiment which different values of $0.$ If you are unsure, try a value of $10.$ Your output should look

something like this for the cross image shown above $($from left to right, $2$D discrete Fourier transform, gaussian filter applied, and the final blurred image$)$: