The following code do the $$Discrete Cosine Transform$($DCT$)$ and Inverse DCT with $64$ coefficients.

$\%$matplotlib inline

import matplotlib.pyplot as plt

from matplotlib.gridspec import GridSpec

import numpy as np

from sklearn.datasets import load$\_$sample$\_$image

from scipy.fftpack import dctn$,$ idctn

from scipy.spatial.distance import pdist

# helper functions to extract patches from an image

def im$\_$to$\_$patch$($im$,$ patch$\_$size$)$:

return im$.$reshape$($im$.$shape$[0]//$ patch$\_$size$[0],$ patch$\_$size$[0],$ im$.$shape$[1]//$ patch$\_$size$[1],$ patch$\_$size$[1]).$swapaxes$(1,2).$reshape$(-1,$ patch$\_$size$[0],$ patch$\_$size$[1]).$copy$()$

def patch$\_$to$\_$im$($patches$,$ patch$\_$size, im$\_$size$)$:

return patches.reshape$($im$\_$size$[0]//$ patch$\_$size$[0],$ im$\_$size$[1]//$ patch$\_$size$[1],$ patch$\_$size$[0],$ patch$\_$size$[1]).$swapaxes$(1,2).$reshape$($im$\_$size$).$copy$()$

def im$\_$to$\_$vec$($im$,$ patch$\_$size$)$:

return im$.$reshape$($im$.$shape$[0]//$ patch$\_$size$[0],$ patch$\_$size$[0],$ im$.$shape$[1]//$ patch$\_$size$[1],$ patch$\_$size$[1]).$swapaxes$(1,2).$reshape$(-1,$ patch$\_$size$[0]*$ patch$\_$size$[1]).$copy$()$

def vec$\_$to$\_$im$($vec$,$ patch$\_$size, im$\_$size$)$:

return vec.reshape$($im$\_$size$[0]//$ patch$\_$size$[0],$ im$\_$size$[1]//$ patch$\_$size$[1],$ patch$\_$size$[0],$ patch$\_$size$[1]).$swapaxes$(1,2).$reshape$($im$\_$size$).$copy$()$

# accuracy metric $-$ MSE

def mse$($original$,$ noisy$)$:

return np$.$sum$(($original $-$ noisy$)**2)$

china $=$ load$\_$sample$\_$image$(\text{'}$china$.$jpg$\text{'}).$mean$($axis$=2)[$:$424,$ :$]/256$

china$\_$dct $=$ dctn$($im$\_$to$\_$patch$($china$,[8,8]),$ axes$=(1,2))$

x$,$ y $=$ np$.$mgrid$[$:$8,$ :$8]$

n$\_$coefs $=64$

china$\_$dct$64=$ np$.$where$($np$.$logical$\_$and$($x $$ np$.$sqrt$($n$\_$coefs$),$ y $$ np$.$sqrt$($n$\_$coefs$)),$ china$\_$dct$,0)$

china$\_$rc$64=$ patch$\_$to$\_$im$($idctn$($china$\_$dct$64,$ axes$=(1,2)),[8,8],[424,640])/256$

print$(\text{'}64$ Components $($MSE $=\{$:g$\})\text{'}.$format$($mse$($china$,$ china$\_$rc$64)))$

plt$.$imshow$($china$\_$rc$64,$ cmap$=$'plasma'$)$

plt$.$show$()$

The following code visualizes the basis vectors used to reconstruct each patch of the image:

from matplotlib import colors

def make$\_$dct$\_$basis$($coefs$,$ patch$\_$size$)$:

basis $=$ np$.$empty$($tuple$($coefs$)+$ tuple$($patch$\_$size$))$

x$,$ y $=$ np$.$mgrid$[$:patch$\_$size$[0],$ :patch$\_$size$[1]]$

for i in range$($coefs$[0])$:

for j in range$($coefs$[1])$:

basis$[$i$,$ j$]=$ np$.$cos$($np$.$pi $*($x $+\mathrm{.}5)*$ i $/$ coefs$[0])*$ np$.$cos$($np$.$pi $*($y $+\mathrm{.}5)*$ j $/$ coefs$[1])$

return basis

dct$\_$vecs $=$ make$\_$dct$\_$basis$([8,8],[64,64])$

fig, axes $=$ plt$.$subplots$(8,8,$ figsize$=(9,9))$

ims $=[]$

for i in range$(8)$:

for j in range$(8)$:

ims $+=[$axes$[$i$,$ j$].$imshow$($dct$\_$vecs$[$i$,$ j$],$ cmap$=$'binary'$)]$

axes$[$i$,$ j$].$set$\_$axis$\_$off$()$

vmin $=$ min$([$im$.$get$\_$array$().$min$()$ for im in ims$])$

vmax $=$ max$([$im$.$get$\_$array$().$max$()$ for im in ims$])$

norm $=$ colors.Normalize$($vmin$=$vmin, vmax$=$vmax$)$

for im in ims:

im$.$set$\_$norm$($norm$)$

fig.suptitle$(\text{'}$DCT Basis Patches'$)$

plt$.$show$()$

According the things mentioned above, which DCT basis vectors will likely be used to represent the selected regions $($ROIs$)$ within the image below? Provide the rationale and write a python code to test this hypothesis.

ROI $1$

ROI $3$

ROI $2$