EE 123 Spring 2016 Discussion Section 03
Frank Ong (presented by Jon Tamir)

This is a great demo of the Discrete Cosine Transform (DCT) and is the essence of JPEG encoding. The demo was originally created by Frank Ong.

In [1]:
# Import functions and libraries
import numpy as np
import matplotlib.pyplot as plt
import scipy

from numpy import pi
from numpy import sin
from numpy import zeros
from numpy import r_
from scipy import signal
from scipy import misc # pip install Pillow
import matplotlib.pylab as pylab

%matplotlib inline
pylab.rcParams['figure.figsize'] = (20.0, 7.0)

Display image

In [2]:
# im = misc.imread("einstein.tif").astype(float)
# im = misc.imread("house.tif").astype(float)
im = misc.imread("zelda.tif").astype(float)
# im = misc.imread("barbara.png").astype(float)

f = plt.figure()
<matplotlib.image.AxesImage at 0x10e8be2d0>

Define 2D DCT and IDCT

In [3]:
def dct2(a):
    return scipy.fftpack.dct( scipy.fftpack.dct( a, axis=0, norm='ortho' ), axis=1, norm='ortho' )

def idct2(a):
    return scipy.fftpack.idct( scipy.fftpack.idct( a, axis=0 , norm='ortho'), axis=1 , norm='ortho')

Perform a blockwise DCT

In [4]:
imsize = im.shape
dct = np.zeros(imsize)

# Do 8x8 DCT on image (in-place)
for i in r_[:imsize[0]:8]:
    for j in r_[:imsize[1]:8]:
        dct[i:(i+8),j:(j+8)] = dct2( im[i:(i+8),j:(j+8)] )

Extract 8x8 block and look at its DCT coefficients

In [5]:
pos = 128

# Extract a block from image
plt.title( "An 8x8 Image block")

# Display the dct of that block
plt.imshow(dct[pos:pos+8,pos:pos+8],cmap='gray',vmax= np.max(dct)*0.01,vmin = 0, extent=[0,pi,pi,0])
plt.title( "An 8x8 DCT block")
<matplotlib.text.Text at 0x110a57dd0>

Display all DCT blocks

In [6]:
# Display entire DCT
plt.imshow(dct,cmap='gray',vmax = np.max(dct)*0.01,vmin = 0)
plt.title( "8x8 DCTs of the image")
<matplotlib.text.Text at 0x110a22e10>

Threshold DCT coefficients

In [7]:
# Threshold
thresh = 0.012
dct_thresh = dct * (abs(dct) > (thresh*np.max(dct)))

plt.imshow(dct_thresh,cmap='gray',vmax = np.max(dct)*0.01,vmin = 0)
plt.title( "Thresholded 8x8 DCTs of the image")

percent_nonzeros = np.sum( dct_thresh != 0.0 ) / (imsize[0]*imsize[1]*1.0)

print "Keeping only %f%% of the DCT coefficients" % (percent_nonzeros*100.0)
Keeping only 5.534744% of the DCT coefficients

Compare DCT compressed image with original

In [8]:
im_dct = np.zeros(imsize)

for i in r_[:imsize[0]:8]:
    for j in r_[:imsize[1]:8]:
        im_dct[i:(i+8),j:(j+8)] = idct2( dct_thresh[i:(i+8),j:(j+8)] )
plt.imshow( np.hstack( (im, im_dct) ) ,cmap='gray')
plt.title("Comparison between original and DCT compressed images" )
<matplotlib.text.Text at 0x1117db590>