# Image Approximation with Fourier and Wavelets¶

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This numerical tour overviews the use of Fourier and wavelets for image approximation.

In [1]:
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

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.

In [2]:
n0 = 512

Display the original image.

In [3]:
plt.figure(figsize = (5,5))
imageplot(f, 'Image f')

Display a zoom in the middle.

In [4]:
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.

In [5]:
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.

In [6]:
k = 9; #size of the kernel
h = np.ones([k,k])
h = h/np.sum(h) #normalize

Compute the convolution $f \star h$.

In [7]:
from scipy import signal
fh = signal.convolve2d(f, h, boundary = "symm")

Display.

In [8]:
plt.figure(figsize = (5,5))
imageplot(fh, 'Blurred image')

## Fourier Transform¶

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.

In [9]:
F = pyl.fft2(f)/n0

We check this conservation of the energy.

In [10]:
from pylab import linalg

print("Energy of Image:   %f" %linalg.norm(f))
print("Energy of Fourier: %f" %linalg.norm(F))
Energy of Image:   255.983054
Energy of Fourier: 255.983054

Compute the logarithm of the Fourier magnitude $\log\left(\abs{\hat f(m)} + \epsilon\right)$, for some small $\epsilon$.

In [11]:
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.

In [12]:
plt.figure(figsize = (5,5))
imageplot(L, 'Log(Fourier transform)')

## Linear Fourier Approximation¶

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.

In [13]:
M = n0**2//64

Exercise 1

Perform the linear Fourier approximation with $M$ coefficients. Store the result in the variable $f_M$.

In [14]:
run -i nt_solutions/introduction_4_fourier_wavelets/exo1
In [15]:

Compare two 1D profile (lines of the image). This shows the strong ringing artifact of the linea approximation.

In [16]:
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 Fourier Approximation¶

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 $T>0$.

In [17]:
T = .2

Compute the Fourier transform.

In [18]:
F = pyl.fft2(f)/n0

Do the hard thresholding.

In [19]:
FT = np.multiply(F,(abs(F) > T))

Display. Note that we use the function fftshift to put the 0 low frequency in the middle.

In [20]:
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 $f_M$.

In [21]:
fM = np.real(pyl.ifft2(FT)*n0)

Display.

In [22]:
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 $\abs{I_M}$.

In [23]:
m = np.sum(FT != 0)
print('M/N = 1/%d'  %(n0**2/m))
M/N = 1/31

Exercise 2

Compute the value of the threshold $T$ so that the number of coefficients is $M$. Display the corresponding approximation $f_M$.

In [24]:
run -i nt_solutions/introduction_4_fourier_wavelets/exo2
In [25]:

## Wavelet Transform¶

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.

In [26]:
Jmin = 0

Perform the wavelet transform, $f_w$ stores all the wavelet coefficients.

In [27]:
from nt_toolbox.perform_wavelet_transf import *

fw = perform_wavelet_transf(f, Jmin, + 1)

Display the transformed coefficients.

In [28]:
plt.figure(figsize=(10,10))

plot_wavelet(fw)
plt.title('Wavelet coefficients')

plt.show()