Problem $2$: Image compression and SVD

Singular Value Decomposition $($SVD$)$ is a very useful tool in data analysis. In this exercise, you will

use SVD to compress image. Implement the following steps in Matlab.

$1.$ Load the clown image. You can use the following command load clown.mat. Look at what are

loaded in the workspace.

$2.$ Do an SVD decomposition on the X matrix.

$3.$ Find the rank k approximation of X$,$ for k $=5,15,25.$ Denote the rank k approximation of

X as Xk$.$

$4.$ Generate the image for Xk using the command image$($Xk$).$

If you are using Python, you can use the numpy and matplotlib packages. For example, you

can use X $=$ np$.$loadtxt$("$clownImage$.$txt$")$ to load the matrix of the clown image, and use

plt$.$gray$(),$ plt$.$imshow$($X$),$ and plt$.$show$()$ to display the image from the matrix X$.$ Besides, the

package numpy also provides a quick command to do the SVD decomposition of a matrix.

You should submit your code $($Matlab or python$)$ and the images.