CHW$6\mathrm{.}2.$ Discrete Cosine Transforms and Image Compression

The discrete cosine transform is a powerful technique for expressing data as a sum of cosine waves, and has many applications in audio and image compression. We will use the "chunking" technique discussed in the lab activity here to compress a whole image.

Given a $512512$ image, stored in the variable image, you will break the image into smaller $88$ chunks. Then for each chunk, you will convert to the DCT basis, drop the frequencies with relatively small contribution, then convert back to the standard basis and clip the resulting pixel values so that they lie between $0$ and $255.$ This image is given in gray scale, where each pixel varies from $0$ to $255.$ To gauge which frequencies have a small contribution, use a tolerance of $10\%$ of the largest absolute value in each particular chunk after converting it into the DCT basis.

The setup code provides the helper function create$\_$dct$\_$basis$(n,$ which creates an $nn$ matrix whose columns make up the $n-$dimensional DCT basis.

To construct your solution, write a code snippet that:

defines the function compress$\_$chunk$($chunk$)$ that takes as argument a "chunk" of the image, with dimension $88,$ and returns the compressed "chunk".

make sure your compressed chunk returned from compress$\_$chunk$()$ contains only

defines the function compress$\_$chunk$($chunk$)$ that takes as argument a "chunk" of the image, with dimension $88,$ and returns the compressed "chunk".

make sure your compressed chunk returned from compress$\_$chunk$()$ contains only values between $0$ and $255,$ as in the original image. You may find the function numpy.clip$()$ useful.

use your function to construct the variable compressed, which contains the compressed image.

The setup code gives the following variable:

$\backslash$table$[[$Name$,$Type,Description$],[$image$,512512$ numpy array,Test image$],[$create$\_$dct$\_$basis,function,Creates the n$-$dimensional DCT basis$]]$

Your code snippet should define the following variable:

$\backslash$table$[[$Name$,$Type,Description$],[$compress$\_$chunk,function,Function to compress an $88$ chunk$],[$compressed$,512512$ numpy array,Compressed image$]]$

user$\_$code.py

import numpy as $np$

import numpy.linalg as la

import matplotlib.pyplot as plt

plt$.$figure$()$

plt$.$imshow$($image$,$ cmap$=$'gray'$)$

compressed $=$ image. copy$()$

## Add your code here

plt$.$figure$()$

plt$.$imshow $($compressed$,$ cmap$=$'gray'$)$