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This numerical tour overviews the use of Fourier and wavelets for image approximation.
library(pracma)
library(imager)
# Importing the libraries
for (f in list.files(path="nt_toolbox/toolbox_general/", pattern="*.R")) {
source(paste("nt_toolbox/toolbox_general/", f, sep=""))
}
for (f in list.files(path="nt_toolbox/toolbox_signal/", pattern="*.R")) {
source(paste("nt_toolbox/toolbox_signal/", f, sep=""))
}
Loading required package: plyr Loading required package: magrittr Attaching package: ‘magrittr’ The following objects are masked from ‘package:pracma’: and, mod, or Attaching package: ‘imager’ The following object is masked from ‘package:magrittr’: add The following object is masked from ‘package:plyr’: liply The following objects are masked from ‘package:stats’: convolve, spectrum The following object is masked from ‘package:graphics’: frame The following object is masked from ‘package:base’: save.image Attaching package: ‘tuneR’ The following objects are masked from ‘package:imager’: channel, play Attaching package: ‘akima’ The following object is masked from ‘package:imager’: interp
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.
n = 512
f = load_image("nt_toolbox/data/lena.png", n)
Display the original image.
options(repr.plot.width=4, repr.plot.height=4)
imageplot(f, 'Image f')
Display a zoom in the middle.
imageplot(f[c(((n/2) - 32):((n/2) + 32)), c(((n/2) - 32):((n/2) + 32))], 'Zoom')
An image is a 2D array, it can be modified as a matrix.
options(repr.plot.width=5, repr.plot.height=5)
imageplot(-f, '-f', c(1, 2, 1))
imageplot(f[dim(f)[1]:1,], 'Flipped', c(1, 2, 2))
Blurring is achieved by computing a convolution $f \star h$ with a kernel $h$.
Compute the low pass kernel.
k = 9
h = matrix(1, k, k)
h = h / sum(h) #normalize
Compute the convolution $f \star h$.
fh = convolution(f[,], h)
Display.
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 = fft(f[,]) / n
We check this conservation of the energy.
print(paste("Energy of Image: ", toString(norm(f[,]))))
print(paste("Energy of Fourier: ", toString(norm(Mod(F[,])))))
[1] "Energy of Image: 279.584762635359" [1] "Energy of Fourier: 279.584762635359"
Compute the logarithm of the Fourier magnitude $ \log\left(\abs{\hat f(m)} + \epsilon\right) $, for some small $\epsilon$.
L = fftshift(log(Mod(F) + 1e-1))
Display. Note that we use the function fftshift to put the 0 low frequency in the middle.
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 = n**2 / 64
Exercise 1
Perform the linear Fourier approximation with $M$ coefficients. Store the result in the variable $f_M$.
source("nt_solutions/introduction_4_fourier_wavelets/exo1.R")
# Insert your code here.
Compare two 1D profile (lines of the image). This shows the strong ringing artifact of the linea approximation.
# Insert your code here.
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 = 0.2
Compute the Fourier transform.
F = fft(f) / n
Do the hard thresholding.
FT = F * (abs(F) > T)
Display. Note that we use the function fftshift to put the 0 low frequency in the middle.
L = fftshift(log(abs(FT[,]) + 1e-1))
imageplot(L, 'thresholded Log(Fourier transform)')
Inverse Fourier transform to obtain $f_M$.
fM = Re(fft(FT, inverse=TRUE) * n)
# Normalize to 0-1
fM = (fM-min(fM))/(max(fM)-min(fM))
Display.
imageplot(fM, paste("Non - Linear, Fourier, SNR = ", round(snr(f, fM), 1), "dB"))
Given a $T$, the number of coefficients is obtained by counting the non-thresholded coefficients $ \abs{I_M} $.
m = sum(FT != 0)
print(paste("M/N = 1 /",as.integer(n**2/m)))
[1] "M/N = 1 / 30"
Exercise 2
Compute the value of the threshold $T$ so that the number of coefficients is $M$. Display the corresponding approximation $f_M$.
source("nt_solutions/introduction_4_fourier_wavelets/exo2.R")
# 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.
fw = perform_wavelet_transf(f, Jmin + 1, 1)
Display the transformed coefficients.
options(repr.plot.width=6, repr.plot.height=6)
plot_wavelet(fw)
title(main="Wavelet coefficients")
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.
source("nt_solutions/introduction_4_fourier_wavelets/exo3.R")
# 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 = fw * (abs(fw) > T)
Display the thresholded coefficients.
plot_wavelet(fw)
title(main='Original coefficients')
plot_wavelet(fwT)
title(main='Thresholded coefficients')
Perform reconstruction.
fM = perform_wavelet_transf(fwT, Jmin + 1, -1)
Display approximation.
# Normalize to 0-1
fM = (fM-min(fM))/(max(fM)-min(fM))
imageplot(fM, paste("Approximation, SNR = ", round(snr(f, fM), 1), "dB"))
Exercise 4
Perform non-linear approximation with $M$ wavelet coefficients by chosing the correct value for $T$. Store the result in the variable $f_M$.
source("nt_solutions/introduction_4_fourier_wavelets/exo4.R")
# Insert your code here.
Compare two 1D profile (lines of the image). Note how the ringing artifacts are reduced compared to the Fourier approximation.
# Insert your code here.