Image Approximation with Fourier and Wavelets

Important: Please read the installation page for details about how to install the toolboxes. $\newcommand{\dotp}[2]{\langle #1, #2 \rangle}$ $\newcommand{\enscond}[2]{\lbrace #1, #2 \rbrace}$ $\newcommand{\pd}[2]{ \frac{ \partial #1}{\partial #2} }$ $\newcommand{\umin}[1]{\underset{#1}{\min}\;}$ $\newcommand{\umax}[1]{\underset{#1}{\max}\;}$ $\newcommand{\umin}[1]{\underset{#1}{\min}\;}$ $\newcommand{\uargmin}[1]{\underset{#1}{argmin}\;}$ $\newcommand{\norm}[1]{\|#1\|}$ $\newcommand{\abs}[1]{\left|#1\right|}$ $\newcommand{\choice}[1]{ \left\{ \begin{array}{l} #1 \end{array} \right. }$ $\newcommand{\pa}[1]{\left(#1\right)}$ $\newcommand{\diag}[1]{{diag}\left( #1 \right)}$ $\newcommand{\qandq}{\quad\text{and}\quad}$ $\newcommand{\qwhereq}{\quad\text{where}\quad}$ $\newcommand{\qifq}{ \quad \text{if} \quad }$ $\newcommand{\qarrq}{ \quad \Longrightarrow \quad }$ $\newcommand{\ZZ}{\mathbb{Z}}$ $\newcommand{\CC}{\mathbb{C}}$ $\newcommand{\RR}{\mathbb{R}}$ $\newcommand{\EE}{\mathbb{E}}$ $\newcommand{\Zz}{\mathcal{Z}}$ $\newcommand{\Ww}{\mathcal{W}}$ $\newcommand{\Vv}{\mathcal{V}}$ $\newcommand{\Nn}{\mathcal{N}}$ $\newcommand{\NN}{\mathcal{N}}$ $\newcommand{\Hh}{\mathcal{H}}$ $\newcommand{\Bb}{\mathcal{B}}$ $\newcommand{\Ee}{\mathcal{E}}$ $\newcommand{\Cc}{\mathcal{C}}$ $\newcommand{\Gg}{\mathcal{G}}$ $\newcommand{\Ss}{\mathcal{S}}$ $\newcommand{\Pp}{\mathcal{P}}$ $\newcommand{\Ff}{\mathcal{F}}$ $\newcommand{\Xx}{\mathcal{X}}$ $\newcommand{\Mm}{\mathcal{M}}$ $\newcommand{\Ii}{\mathcal{I}}$ $\newcommand{\Dd}{\mathcal{D}}$ $\newcommand{\Ll}{\mathcal{L}}$ $\newcommand{\Tt}{\mathcal{T}}$ $\newcommand{\si}{\sigma}$ $\newcommand{\al}{\alpha}$ $\newcommand{\la}{\lambda}$ $\newcommand{\ga}{\gamma}$ $\newcommand{\Ga}{\Gamma}$ $\newcommand{\La}{\Lambda}$ $\newcommand{\si}{\sigma}$ $\newcommand{\Si}{\Sigma}$ $\newcommand{\be}{\beta}$ $\newcommand{\de}{\delta}$ $\newcommand{\De}{\Delta}$ $\newcommand{\phi}{\varphi}$ $\newcommand{\th}{\theta}$ $\newcommand{\om}{\omega}$ $\newcommand{\Om}{\Omega}$

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

In [2]:
using PyPlot
using NtToolBox
# using Autoreload
# arequire("NtToolBox")

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.

Image Loading and Displaying

First we load an image $ f \in \RR^N $ of $ N = N_0 \times N_0 $ pixels.

In [3]:
n0 = 512
f = rescale(load_image("NtToolBox/src/data/lena.png", n0));

Display the original image.

In [4]:
figure(figsize = (5,5))
imageplot(f, "Image f");

Display a zoom in the middle.

In [4]:
figure(figsize = (5,5))
imageplot(f[Int(n0/2 - 32) : Int(n0/2 + 32), Int(n0/2 - 32) : Int(n0/2 + 32)], "Zoom")
Out[4]:
PyObject <matplotlib.text.Text object at 0x000000001E7DA240>

An image is a 2D array, it can be modified as a matrix.

In [5]:
figure(figsize = (8,8))
imageplot(-f, "-f", [1, 2, 1])
imageplot(f[end:-1:1, :], "Flipped", [1, 2, 2])
Out[5]:
PyObject <matplotlib.text.Text object at 0x000000001EF47898>

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 = ones(k, k)
h = h/sum(h); #normalize
Out[6]:
9×9 Array{Float64,2}:
 0.0123457  0.0123457  0.0123457  …  0.0123457  0.0123457  0.0123457
 0.0123457  0.0123457  0.0123457     0.0123457  0.0123457  0.0123457
 0.0123457  0.0123457  0.0123457     0.0123457  0.0123457  0.0123457
 0.0123457  0.0123457  0.0123457     0.0123457  0.0123457  0.0123457
 0.0123457  0.0123457  0.0123457     0.0123457  0.0123457  0.0123457
 0.0123457  0.0123457  0.0123457  …  0.0123457  0.0123457  0.0123457
 0.0123457  0.0123457  0.0123457     0.0123457  0.0123457  0.0123457
 0.0123457  0.0123457  0.0123457     0.0123457  0.0123457  0.0123457
 0.0123457  0.0123457  0.0123457     0.0123457  0.0123457  0.0123457

Compute the convolution $f \star h$.

In [7]:
fh = conv2(Array{Float64, 2}(f), h);

Display.

In [8]:
figure(figsize = (5,5))
imageplot(fh, "Blurred image")
Out[8]:
PyObject <matplotlib.text.Text object at 0x000000001F218C50>

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 = plan_fft(f)
F = (F*f)/n0;

We check this conservation of the energy.

In [10]:
println(@sprintf("Energy of Image:   %f", norm(f)))
println(@sprintf("Energy of Fourier:   %f", norm(F)))
Energy of Image:   262.554108
Energy of Fourier:   262.554138

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

In [11]:
L = fftshift(log(abs(F) + 1e-1));

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

In [12]:
figure(figsize = (5,5))
imageplot(L, "Log(Fourier transform)")
Out[12]:
PyObject <matplotlib.text.Text object at 0x00000000244FA978>

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 = Base.div(n0^2, 64);

Exercise 1

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

In [14]:
include("NtSolutions/introduction_4_fourier_wavelets/exo1.jl")
Out[14]:
PyObject <matplotlib.text.Text object at 0x0000000024A79908>
In [15]:
## Insert your code here.

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

In [15]:
figure(figsize = (7, 6))

subplot(2, 1, 1)
plot(f[: , Base.div(n0, 2)])
xlim(0, n0)
title("f")

subplot(2, 1, 2)
plot(fM[: , Base.div(n0, 2)])
xlim(0, n0)
title("f_M")

show()