Texture Synthesis Using 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 explores texture synthesis using wavelets.

Image synthesis is obtained by drawing an image at random that satisfies some modeling constraint, that are usually learned from a given exemplar texture.

In [2]:

addpath('toolbox_signal')
addpath('toolbox_general')
addpath('solutions/graphics_2_synthesis_wavelets')

Multi-scale Texture Synthesis¶

The decay of wavelet coefficients caraterize pointwise singularities in images and texture. Histogram equalization enable the synthesis of texture with singularities. This corresponds to the texture synthesis algorithm of Heeger and Bergen.

Load a texture.

In [3]:

n = 512;
name = 'texture';
M = load_image(name, n);
M = rescale( sum(M,3) );

For Scilab users: you should increase the size of the memory. Warning: execute this line only once.

In [4]:

extend_stack_size(4);

First we compute the wavelet coefficients of the texture. We use a translation invariant transform.

In [5]:

options.ti = 1;
Jmin = 4;
MW = perform_wavelet_transf(M(:,:,1), Jmin, +1, options);

We initialize the synthesis by a random noise with the same gray values.

In [6]:

M1 = perform_hist_eq(randn(n,n), M);

Display.

In [7]:

clf;
imageplot(M, 'Exemplar', 1,2,1);
imageplot(M1, 'Initial noise', 1,2,2);

We also compute the wavelet transform of the noise.

In [8]:

MW1 = perform_wavelet_transf(M1, Jmin, +1, options);

A random texture is obtained by histogram equalization of each wavelet scale.

In [9]:

for i=1:size(MW,3)
    MW1(:,:,i) = perform_hist_eq(MW1(:,:,i), MW(:,:,i));    
end

We retrieve the texture by inverse wavelet transform.

In [10]:

M1 = perform_wavelet_transf(MW1, Jmin, -1, options);

Display.

In [11]:

clf;
imageplot(M, 'Exemplar', 1,2,1);
imageplot(M1, 'Initial synthesis', 1,2,2);

Exercise 1

Iterate these two steps (spatial and wavelet histogram matching) until convergence to a stable step. pacial matching

In [12]:

exo1()