Linear Image Denoising¶

This numerical tour introduces basic image denoising methods.

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

In [2]:

using PyPlot
using NtToolBox
using Distributions

Noisy Image Formation¶

In these numerical tour, we simulate noisy acquisition by adding some white noise (each pixel is corrupted by adding an independant Gaussian variable).

This is useful to test in an oracle maner the performance of our methods.

Size $N = n \times n$ of the image.

In [3]:

n = 256
N = n^2;

We load a clean image $x_0 \in \RR^N$.

In [5]:

name = "NtToolBox/src/data/flowers.png"
x0 = load_image(name, n);

libpng warning: iCCP: known incorrect sRGB profile
libpng warning: iCCP: known incorrect sRGB profile
libpng warning: iCCP: known incorrect sRGB profile

Display the clean image.

In [7]:

imageplot(x0)

Variance of the noise.

In [8]:

sigma = .08

Out[8]:

0.08

We add some noise to it to obtain the noisy signal $y = x_0 + w$. Here $w$ is a realization of a Gaussian white noise of variance $\si^2$.

In [9]:

y = x0 .+ sigma.*rand(Normal(), 256, 256);

Display the noisy image.

In [10]:

imageplot(clamP(y))

Linear Image Denoising¶

We consider a noising estimator $x \in \RR^N$ of $x_0$ that only depends on the observation $y$. Mathematically speaking, it is thus a random vector that depends on the noise $w$.

A translation invariant linear denoising is necessarely a convolution with a kernel $h$ $$ x = x_0 \star h $$ where the periodic convolution between two 2-D arrays is defined as $$ (a \star b)_i = \sum_j a(j) b(i-j). $$

It can be computed over the Fourier domain as $$ \forall \om, \quad \hat x(\om) = \hat x_0(\om) \hat h(\om). $$

In [11]:

cconv = (a, b) -> real(plan_ifft((plan_fft(a)*a).*(plan_fft(b)*b))*((plan_fft(a)*a).*(plan_fft(b)*b)))

Out[11]:

(::#1) (generic function with 1 method)

We use here a Gaussian fitler $h$ parameterized by the bandwith $\mu$.

In [12]:

normalize = h -> h/sum(h)

X = [0:n/2; -n/2:-2]'
Y = [0:n/2; -n/2:-2]

h = mu -> normalize(exp(-(X.^2 .+ Y.^2)/(2*(mu)^2)))

Out[12]:

(::#5) (generic function with 1 method)

Display the filter $h$ and its Fourier transform.

In [13]:

mu = 10
subplot(1,2, 1)
imageplot(fftshift(h(mu)))
title("h")
subplot(1,2, 2)
imageplot(fftshift(real(plan_fft(h(mu))*h(mu))))
title(L"$\hat h$")

Out[13]:

PyObject <matplotlib.text.Text object at 0x334959f90>

In [14]:

imageplot(h(mu))

Shortcut for the convolution with $h$.

In [15]:

denoise = (x, mu) -> cconv(h(mu), x)

Out[15]:

(::#7) (generic function with 1 method)

Display a denoised signal.

In [16]:

imageplot(denoise(y, mu))

Exercise 1: Display a denoised signal for several values of $\mu$.

In [17]:

include("NtSolutions/denoisingsimp_2b_linear_image/exo1.jl")