# Introduction to Image Processing¶

This numerical tour explores some basic image processing tasks.

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]:
library(imager)
library(png)
library(pracma)

# 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=""))
}


Several functions are implemented to load and display images.

path to the images

In [2]:
name = 'nt_toolbox/data/lena.png'
n = 256


We can display it. It is possible to zoom on it, extract pixels, etc.

In [3]:
options(repr.plot.width=4, repr.plot.height=4)
imageplot(M[c(((n/2) - 25):((n/2) + 25)), c(((n/2) - 25):((n/2) + 25))], 'Zoom', c(1, 2, 2))


## Image Modification¶

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

In [4]:
options(repr.plot.width=5, repr.plot.height=5)
imageplot(-M, '-M', c(1,2,1))
imageplot(M[nrow(M):1,], 'Flipped', c(1,2,2))


Blurring is achieved by computing a convolution with a kernel.

Compute the low pass Gaussian kernel. Warning, the indexes needs to be modulo $n$ in order to use FFTs.

In [5]:
sigma = 5.
t = c(0:((n/2)), -(n/2):-2)
mesh = meshgrid(t)
X = mesh$X Y = mesh$Y
h = exp( -(X**2 + Y**2)/(2.0 * sigma**2) )
h = h/sum(h)
imageplot(fftshift(h))


Compute the periodic convolution ussing FFTs

In [6]:
Mh = Re(fft(fft(M[,]) * fft(h), inverse=TRUE) / n)


Display

In [7]:
imageplot(M, 'Image', c(1, 2, 1))
imageplot(Mh, 'Blurred', c(1, 2, 2))


Several differential and convolution operators are implemented.

In [8]:
G = grad(M[,])
imageplot(G[,,1], 'd/ dx', c(1, 2, 1))
imageplot(G[,,2], 'd/ dy', c(1, 2, 2))


## Fourier Transform¶

The 2D Fourier transform can be used to perform low pass approximation and interpolation (by zero padding).

Compute and display the Fourier transform (display over a log scale). The function fftshift is useful to put the 0 low frequency in the middle. After fftshift, the zero frequency is located at position $(n/2+1,n/2+1)$.

In [9]:
Mf = fft(M[,])
Lf = fftshift(log(abs(Mf) + 1e-1))
imageplot(M, 'Image', c(1, 2, 1))
imageplot(Lf, 'Fourier transform', c(1, 2, 2))


Exercise 1: To avoid boundary artifacts and estimate really the frequency content of the image (and not of the artifacts!), one needs to multiply M by a smooth windowing function h and compute fft2(M*h). Use a sine windowing function. Can you interpret the resulting filter ?

In [10]:
source("nt_solutions/introduction_3_image/exo1.R")

In [11]:
# Insert code here.


Exercise 2: Perform low pass filtering by removing the high frequencies of the spectrum. What do you oberve ?

In [12]:
source("nt_solutions/introduction_3_image/exo1.R")

In [13]:
# Insert code here.