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This numerical tour overviews the use of Fourier and wavelets for image approximation.
from __future__ import division
import numpy as np
import scipy as scp
import pylab as pyl
import matplotlib.pyplot as plt
from nt_toolbox.general import *
from nt_toolbox.signal import *
import warnings
warnings.filterwarnings('ignore')
%matplotlib inline
%load_ext autoreload
%autoreload 2
Note: to measure the error of an image $f$ with its approximation $f_M$, we use the SNR measure, defined as
$$ \text{SNR}(f,f_M) = -20\log_{10} \pa{ \frac{ \norm{f-f_M} }{ \norm{f} } }, $$which is a quantity expressed in decibels (dB). The higer the SNR, the better the quality.
First we load an image $ f \in \RR^N $ of $ N = N_0 \times N_0 $ pixels.
n0 = 512
f = rescale(load_image("nt_toolbox/data/hibiscus.bmp", n0))
Display the original image.
plt.figure(figsize = (5,5))
imageplot(f, 'Image f')
Display a zoom in the middle.
plt.figure(figsize = (5,5))
imageplot(f[n0//2 - 32:n0//2 + 32,n0//2 - 32:n0//2 + 32], 'Zoom')
An image is a 2D array, it can be modified as a matrix.
plt.figure(figsize = (8,8))
imageplot(-f, '-f', [1, 2, 1])
imageplot(f[::-1,], 'Flipped', [1, 2, 2])
Blurring is achieved by computing a convolution $f \star h$ with a kernel $h$.
Compute the low pass kernel.
k = 9; #size of the kernel
h = np.ones([k,k])
h = h/np.sum(h) #normalize
Compute the convolution $f \star h$.
from scipy import signal
fh = signal.convolve2d(f, h, boundary = "symm")
Display.
plt.figure(figsize = (5,5))
imageplot(fh, 'Blurred image')
The Fourier orthonormal basis is defined as $$ \psi_m(k) = \frac{1}{\sqrt{N}}e^{\frac{2i\pi}{N_0} \dotp{m}{k} } $$ where $0 \leq k_1,k_2 < N_0$ are position indexes, and $0 \leq m_1,m_2 < N_0$ are frequency indexes.
The Fourier transform $\hat f$ is the projection of the image on this Fourier basis
$$ \hat f(m) = \dotp{f}{\psi_m}. $$The Fourier transform is computed in $ O(N \log(N)) $ operation using the FFT algorithm (Fast Fourier Transform). Note the normalization by $\sqrt{N}=N_0$ to make the transform orthonormal.
F = pyl.fft2(f)/n0
We check this conservation of the energy.
from pylab import linalg
print("Energy of Image: %f" %linalg.norm(f))
print("Energy of Fourier: %f" %linalg.norm(F))
Energy of Image: 205.747421 Energy of Fourier: 205.747421
Compute the logarithm of the Fourier magnitude $ \log\left(\abs{\hat f(m)} + \epsilon\right) $, for some small $\epsilon$.
L = pyl.fftshift(np.log(abs(F) + 1e-1))
Display. Note that we use the function fftshift to put the 0 low frequency in the middle.
plt.figure(figsize = (5,5))
imageplot(L, 'Log(Fourier transform)')
An approximation is obtained by retaining a certain set of index $I_M$
$$ f_M = \sum_{ m \in I_M } \dotp{f}{\psi_m} \psi_m. $$Linear approximation is obtained by retaining a fixed set $I_M$ of $M = \abs{I_M}$ coefficients. The important point is that $I_M$ does not depend on the image $f$ to be approximated.
For the Fourier transform, a low pass linear approximation is obtained by keeping only the frequencies within a square.
$$ I_M = \enscond{m=(m_1,m_2)}{ -q/2 \leq m_1,m_2 < q/2 } $$where $ q = \sqrt{M} $.
This can be achieved by computing the Fourier transform, setting to zero the $N-M$ coefficients outside the square $I_M$ and then inverting the Fourier transform.
Number $M$ of kept coefficients.
M = n0**2//64
Exercise 1
Perform the linear Fourier approximation with $M$ coefficients. Store the result in the variable $f_M$.
run -i nt_solutions/introduction_4_fourier_wavelets/exo1
## Insert your code here.
Compare two 1D profile (lines of the image). This shows the strong ringing artifact of the linea approximation.
plt.figure(figsize=(7,6))
plt.subplot(2, 1, 1)
plt.plot(f[: , n0//2])
plt.xlim(0,n0)
plt.title('f')
plt.subplot(2, 1, 2)
plt.plot(fM[: , n0//2])
plt.xlim(0,n0)
plt.title('f_M')
plt.show()
Non-linear approximation is obtained by keeping the $M$ largest coefficients. This is equivalently computed using a thresholding of the coefficients $$ I_M = \enscond{m}{ \abs{\dotp{f}{\psi_m}}>T }. $$
Set a threshold $T>0$.
T = .2
Compute the Fourier transform.
F = pyl.fft2(f)/n0
Do the hard thresholding.
FT = np.multiply(F,(abs(F) > T))
Display. Note that we use the function fftshift to put the 0 low frequency in the middle.
L = pyl.fftshift(np.log(abs(FT) + 1e-1))
plt.figure(figsize = (5,5))
imageplot(L, 'thresholded Log(Fourier transform)')
Inverse Fourier transform to obtain $f_M$.
fM = np.real(pyl.ifft2(FT)*n0)
Display.
plt.figure(figsize = (5,5))
imageplot(clamp(fM), "Linear, Fourier, SNR = %.1f dB" %snr(f, fM))
Given a $T$, the number of coefficients is obtained by counting the non-thresholded coefficients $ \abs{I_M} $.
m = np.sum(FT != 0)
print('M/N = 1/%d' %(n0**2/m))
M/N = 1/53
Exercise 2
Compute the value of the threshold $T$ so that the number of coefficients is $M$. Display the corresponding approximation $f_M$.
run -i nt_solutions/introduction_4_fourier_wavelets/exo2
## Insert your code here.
A wavelet basis $ \Bb = \{ \psi_m \}_m $ is obtained over the continuous domain by translating and dilating three mother wavelet functions $ \{\psi^V,\psi^H,\psi^D\} $.
Each wavelet atom is defined as $$ \psi_m(x) = \psi_{j,n}^k(x) = \frac{1}{2^j}\psi^k\pa{ \frac{x-2^j n}{2^j} } $$
The scale (size of the support) is $2^j$ and the position is $2^j(n_1,n_2)$. The index is $ m=(k,j,n) $ for $\{ j \leq 0 \}$.
The wavelet transform computes all the inner products $ \{ \dotp{f}{\psi_{j,n}^k} \}_{k,j,n} $.
Set the minimum scale for the transform to be 0.
Jmin = 0
Perform the wavelet transform, $f_w$ stores all the wavelet coefficients.
from nt_toolbox.perform_wavelet_transf import *
fw = perform_wavelet_transf(f, Jmin, + 1)
Display the transformed coefficients.
plt.figure(figsize=(10,10))
plot_wavelet(fw)
plt.title('Wavelet coefficients')
plt.show()
Linear wavelet approximation with $M=2^{-j_0}$ coefficients is obtained by keeping only the coarse scale (large support) wavelets:
$$ I_M = \enscond{(k,j,n)}{ j \geq j_0 }. $$It corresponds to setting to zero all the coefficients excepted those that are on the upper left corner of $f_w$.
Exercise 3
Perform linear approximation with $M$ wavelet coefficients.
run -i nt_solutions/introduction_4_fourier_wavelets/exo3
## Insert your code here.
A non-linear approximation is obtained by keeping the $M$ largest wavelet coefficients.
As already said, this is equivalently computed by a non-linear hard thresholding.
Select a threshold.
T = .15
Perform hard thresholding.
fwT = np.multiply(fw,(abs(fw) > T))
Display the thresholded coefficients.
plt.figure(figsize=(15,15))
plt.subplot(1, 2, 1)
plot_wavelet(fw)
plt.title('Original coefficients')
plt.subplot(1, 2, 2)
plot_wavelet(fwT)
plt.title('Thresholded coefficients')
plt.show()
Perform reconstruction.
fM = perform_wavelet_transf(fwT, Jmin, -1)
Display approximation.
plt.figure(figsize=(5,5))
imageplot(clamp(fM), "Approximation, SNR, = %.1f dB" %snr(f, fM))
Exercise 4
Perform non-linear approximation with $M$ wavelet coefficients by chosing the correct value for $T$. Store the result in the variable $f_M$.
run -i nt_solutions/introduction_4_fourier_wavelets/exo4
## Insert your code here.
Compare two 1D profile (lines of the image). Note how the ringing artifacts are reduced compared to the Fourier approximation.
plt.figure(figsize=(7,6))
plt.subplot(2, 1, 1)
plt.plot(f[:,n0//2])
plt.title('f')
plt.subplot(2, 1, 2)
plt.plot(fM[:,n0//2])
plt.title('f_M')
plt.show()