Introduction to Image Processing

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 explores some basic image processing tasks.

In [2]:

Image Loading and Displaying

Several functions are implemented to load and display images.

First we load an image.

path to the images

In [3]:
name = 'lena';
n = 256;
M = load_image(name, []);
M = rescale(crop(M,n));

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

In [4]:
imageplot(M, 'Original', 1,2,1);
imageplot(crop(M,50), 'Zoom', 1,2,2);

Image Modification

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

In [5]:
imageplot(-M, '-M', 1,2,1);
imageplot(M(n:-1:1,:), 'Flipped', 1,2,2);

Blurring is achieved by computing a convolution with a kernel.

compute the low pass kernel

In [6]:
k = 9;
h = ones(k,k);
h = h/sum(h(:));

compute the convolution

In [7]:
Mh = perform_convolution(M,h);


In [8]:
imageplot(M, 'Image', 1,2,1);
imageplot(Mh, 'Blurred', 1,2,2);

Several differential and convolution operators are implemented.

In [9]:
G = grad(M);
imageplot(G(:,:,1), 'd/dx', 1,2,1);
imageplot(G(:,:,2), 'd/dy', 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 [10]:
Mf = fft2(M);
Lf = fftshift(log( abs(Mf)+1e-1 ));
imageplot(M, 'Image', 1,2,1);
imageplot(Lf, 'Fourier transform', 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 ? ompute kernel h ompute FFT isplay

In [11]:
In [12]:
%% Insert your code here.

Exercise 2

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

In [13]:
In [14]:
%% Insert your code here.

It is possible to do image interpolating by adding high frequencies

In [15]:
p = 64;
n = p*4;
M = load_image('boat', 2*p); M = crop(M,p);
Mf = fftshift(fft2(M));
MF = zeros(n,n);
sel = n/2-p/2+1:n/2+p/2;
sel = sel;
MF(sel, sel) = Mf;
MF = fftshift(MF);
Mpad = real(ifft2(MF));
imageplot( crop(M), 'Image', 1,2,1);
imageplot( crop(Mpad), 'Interpolated', 1,2,2);

A better way to do interpolation is to use cubic-splines. It avoid ringing artifact because the spline kernel has a smaller support with less oscillations.

In [16]:
Mspline = image_resize(M,n,n);
imageplot( crop(Mpad), 'Fourier (sinc)', 1,2,1);
imageplot( crop(Mspline), 'Spline', 1,2,2);