#!/usr/bin/env python
# coding: utf-8

# ## <center>NYU CSCI-UA 9473 Introduction to ML </center>
#    ### <center> Unsupervised Learning (Part III)</center>
# 
# 

# #### Exercise 1. Dictionnary learning and image inpainting. 
# 
# We consider the popular Lena image below which was corrupted by the overlaying of some text 

# In[1]:


from PIL import Image

img = Image.open('lena.png').convert('LA')
img.save('greyscale.png')


# In[2]:


import cv2
import numpy as np
import matplotlib.pyplot as plt
image = cv2.imread('greyscale.png')  
texted_image =cv2.putText(img=np.copy(image), text="hello", org=(200,200),fontFace=3, fontScale=3, color=(0,0,255), thickness=5)
plt.imshow(texted_image)
plt.show()


# In[5]:


im = Image.fromarray(texted_image)
img = im.convert('LA')
plt.imshow(img)
plt.show()


# Using the grayscale image above (or if you want to your implementation to be faster, on a downsampled version of this image), decompose the image into a series of overlapping patches (e.g. of size 8x8), store those patches as a list of vectors. Center the list and scale it (divide by the std). Once we have a first dictionnary, we can try to compute the recovered image by first expressing each of the patches from the original image as
# 
# \begin{align*}
# \underset{\mathbf{s}}{\operatorname{argmin}} \left\|\mathbf{y} - \mathbf{D}\mathbf{s}\right\| + \lambda \|\mathbf{s}\|_1 
# \end{align*}
# 
# This minimization can be solved for example through [Matching Pursuit](https://en.wikipedia.org/wiki/Matching_pursuit)

# In[ ]:


# implement the minimization step here




# Once the decomposition of the image in the dictionnary has been found, we can the update this dictionnary as 
# \begin{align*}
# \underset{\mathbf{D}}{\operatorname{argmin}} \left\|\mathbf{Y} - \mathbf{D}\mathbf{S}\right\|
# \end{align*}
# which can be solved in closed form through the steps
# \begin{align*}
# \mathbf{D} = \mathbf{Y}\mathbf{S}^T\left(\mathbf{S}\mathbf{S}^T\right)^{-1}
# \end{align*}
# followed by the normalization of the atoms (colums of the dictionnary).

# In[ ]:


# add the dictionnary learning step below