In [4]:
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.image as mpimg
from sklearn.cluster import KMeans
Function to transform a color image to grayscale. Here, we use the method employed in PAL and NTSC and described in ITU-R Rec.601. According to this standard the luma is obtained as \begin{equation*} Y^\prime = 0.2989\cdot R + 0.587\cdot G + 0.114\cdot B \end{equation*}
In [5]:
def rgb2gray(rgb):
return np.dot(rgb[...,:3], [0.2989, 0.5870, 0.1140])
Load an image, needs to be an RGB image (24 bit color information), transform it to gray scale and display
In [6]:
image = rgb2gray(mpimg.imread('Mandrill.png'))
plt.figure(1,figsize=(10,10))
plt.imshow(image,cmap=plt.get_cmap('gray'), vmin=0, vmax=1)
plt.show()
This function performs simple scalar quantization without any optimization of the quantization levels
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def quantize_scalar(data,bits):
levels = np.power(2.0,bits)
pdata = np.minimum(np.floor(data*levels), (levels-1)*np.ones(data.shape))/(levels-1)
return pdata
In [27]:
image_3 = quantize_scalar(image,2)
plt.figure(1,figsize=(16,8))
plt.subplot(1,2,1)
plt.imshow(image, cmap='gray', vmin=0, vmax=1)
plt.title('Original image (monochrome)')
plt.axis('off')
plt.subplot(1,2,2)
plt.imshow(image_3,cmap='gray', vmin=0, vmax=1)
plt.title('Unoptimized scalar quantization with 2 bit/pixel')
plt.axis('off')
plt.show()
Construct a more general scalar/vector quantizer.
We start by grouping neighboring pixels of size blocksize x blocksize and assigning to each bit an equivalent of bits pixel This means that the codebook contains $2^{\texttt{bits}\times \texttt{blocksize}\times \texttt{blocksize}}$ entries. After rearranging the picture, we use the Linde-Buzo-Gray algorithm to find the codebook entries. Based on the codebook entries, the image is put together. As the Linde-Buzo-Gray ALgorithm A is conceptually identical to the k-means algorithm known from machine learning, we use the fast k-means implementation from the scikit-learn library. You can try to use your own implementation of the LBG algorithm to verify.
In [9]:
def vector_quantize(data, blocksize, bits):
codebook_size = np.int(pow(2,blocksize*blocksize*bits))
i_range = np.arange(0,data.shape[0],blocksize)
j_range = np.arange(0,data.shape[1],blocksize)
points = np.zeros((len(i_range)*len(j_range), blocksize*blocksize))
ii=0
for i in i_range:
for j in j_range:
idx = np.ix_(np.arange(0,blocksize)+i,np.arange(0,blocksize)+j)
pixelgroup = np.reshape(data[idx],(1,blocksize*blocksize))
points[ii,:] = pixelgroup
ii += 1
kmeans = KMeans(n_clusters=codebook_size, init='random', random_state=0, n_jobs=8, n_init=1, precompute_distances=True).fit(points)
reconstruct = np.zeros_like(data)
ii = 0
for i in np.arange(0,data.shape[0],blocksize):
for j in np.arange(0,data.shape[1],blocksize):
idx = np.ix_(np.arange(0,blocksize)+i,np.arange(0,blocksize)+j)
reconstruct[idx] = np.reshape(kmeans.cluster_centers_[kmeans.labels_[ii]], (blocksize, blocksize))
ii += 1
return reconstruct
In [24]:
reconstruct = vector_quantize(image, 1, 2)
plt.figure(1,figsize=(16,8))
plt.subplot(1,2,1)
plt.imshow(image, cmap='gray', vmin=0, vmax=1)
plt.title('Original image (monochrome)')
plt.axis('off')
plt.subplot(1,2,2)
plt.imshow(reconstruct,cmap='gray', vmin=0, vmax=1)
plt.title('Optimized scalar quantization with 2 bit/pixel')
plt.axis('off')
plt.show()
In [25]:
reconstruct = vector_quantize(image, 2, 1)
plt.figure(1,figsize=(16,8))
plt.subplot(1,2,1)
plt.imshow(image, cmap='gray', vmin=0, vmax=1)
plt.title('Original image (monochrome)')
plt.axis('off')
plt.subplot(1,2,2)
plt.imshow(reconstruct,cmap='gray', vmin=0, vmax=1)
plt.title('Optimized vector quantization (2x2 pixels) with 1 bit/pixel')
plt.axis('off')
plt.show()
In [26]:
reconstruct = vector_quantize(image, 2, 2)
plt.figure(1,figsize=(16,8))
plt.subplot(1,2,1)
plt.imshow(image, cmap='gray', vmin=0, vmax=1)
plt.title('Original image (monochrome)')
plt.axis('off')
plt.subplot(1,2,2)
plt.imshow(reconstruct,cmap='gray', vmin=0, vmax=1)
plt.title('Optimized vector quantization (2x2 pixels) with 2 bit/pixel')
plt.axis('off')
plt.show()
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