1-D Haar 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 1-D multiresolution analysis with Haar transform. It was introduced in 1910 by Haar Haar1910 and is arguably the first example of wavelet basis.

In [2]:

Forward Haar Transform

The Haar transform is the simplest orthogonal wavelet transform. It is computed by iterating difference and averaging between odd and even samples of the signal.

Size $N$ of the signal.

In [3]:
N = 512;

First we load a 1-D signal $f \in \RR^N$.

In [4]:
name = 'piece-regular';
f = rescale( load_signal(name, N) );

The Haar transform operates over $J = \log_2(N)-1$ scales. It computes a series of coarse scale and fine scale coefficients $a_j, d_j \in \RR^{N_j}$ where $N_j=2^j$.

In [5]:
J = log2(N)-1;

The forward Haar transform computed $ \Hh(f) = (d_j)_{j=0}^J \cup \{a_0\} $. Note that the set of coarse scale coefficients $(a_j)_{j>0}$ are not stored.

This transform is orthogonal, meaning $ \Hh \circ \Hh^* = \text{Id} $, and that there is the following conservation of energy $$ \sum_i \abs{f_i}^2 = \norm{f}^2 = \norm{\Hh f}^2 = \sum_j \norm{d_j}^2 + \abs{a_0}^2. $$

One initilizes the algorithm with $a_{J+1}=f$. The set of coefficients $d_{j},a_j$ are computed from $a_{j+1}$ as $$ \forall k=0,\ldots,2^j-1, \quad a_j[k] = \frac{a_{j+1}[2k] + a_{j+1}[2k+1]}{\sqrt{2}}$ \qandq d_j[k] = \frac{a_{j+1}[2k] - a_{j+1}[2k+1]}{\sqrt{2}}. $$

Shortcut to compute $a_j, d_j$ from $a_{j+1}$.

In [6]:
haar = @(a)[   a(1:2:length(a)) + a(2:2:length(a));
                    a(1:2:length(a)) - a(2:2:length(a)) ]/sqrt(2);

Display the result of the one step of the transform.

In [7]:
plot(f); axis('tight'); title('Signal');
plot(haar(f)); axis('tight'); title('Transformed');

The output of the forward transform is stored in the variable |fw|. At a given iteration indexed by a scale |j|, it will store in |fw(1:2^(j+1))| the variable $a_{j+1}$, and in the remaining entries the variable $d_{j+1},d_{j+2},\ldots,d_J$.

Initializes the algorithm at scale $j=J$.

In [8]:
j = J;

Initialize |fw| to the full signal.

In [9]:
fw = f;

At iteration indexed by $j$, select the sub-part of the signal containing $a_{j+1}$, and apply it the Haar operator.

In [10]:
fw(1:2^(j+1)) = haar(fw(1:2^(j+1)));

Display the signal and its coarse coefficients.

In [11]:
s = 400; t = 40; 
plot(f,'.-'); axis([s-t s+t 0 1]); title('Signal (zoom)');
plot(fw(1:2^j),'.-'); axis([(s-t)/2 (s+t)/2 min(fw(1:2^j)) max(fw(1:2^j))]); title('Averages (zoom)');

Exercise 1

Implement a full wavelet Haar transform that extract iteratively wavelet coefficients, by repeating these steps. Take care of choosing the correct number of steps.

In [12]:
In [13]:
%% Insert your code here.

Check that the transform is orthogonal, which means that the energy of the coefficient is the same as the energy of the signal.

In [14]:
fprintf('Should be 0: %.3f.\n', (norm(f)-norm(fw))/norm(f));
Should be 0: 0.000.

We display the whole set of coefficients |fw|, with red vertical separator between the scales. Can you recognize where are the low frequencies and the high frequencies?

In [15]:
axis([1 N -2 2]);

Backward Haar Transform

The backward transform computes a signal $f_1 = \Hh^*(h)$ from a set of coefficients $ h = (d_j)_{j=0}^J \cup \{a_0\} $

If $h = \Hh(f)$ are the coefficients of a signal $f$, one recovers $ f_1 = f $.

The inverse transform starts from $j=0$, and computes $a_{j+1}$ from the knowledge of $a_j,d_j$ as $$ \forall k = 0,\ldots,2^j-1, \quad a_{j+1}[2k] = \frac{ a_j[k] + d_j[k] }{\sqrt{2}}$ a_{j+1}[2k+1] = \frac{ a_j[k] - d_j[k] }{\sqrt{2}}$ $$

One step of inverse transform

In [16]:
ihaar = @(a,d)assign( zeros(2*length(a),1), ...
        [1:2:2*length(a), 2:2:2*length(a)], [a+d; a-d]/sqrt(2) );

Initialize the signal to recover $f_1$ as the transformed coefficient, and select the smallest possible scale $j=0$.

In [17]:
f1 = fw;
j = 0;

Apply one step of inverse transform.

In [18]:
f1(1:2^(j+1)) = ihaar(f1(1:2^j), f1(2^j+1:2^(j+1)));

Exercise 2

Write the inverse wavelet transform that computes |f1| from the coefficients |fw|.

In [19]:
In [20]:
%% Insert your code here.

Check that we have correctly recovered the signal.

In [21]:
fprintf('Should be 0: %.3f.\n', (norm(f-f1))/norm(f));
Should be 0: 0.000.

Haar Linear Approximation

Linear approximation is obtained by setting to zeros the Haar coefficient coefficients above a certain scale $j$.

Cut-off scale.

In [22]:
j = J-1;

Set to zero fine scale coefficients.

In [23]:
fw1 = fw; fw1(2^j+1:end) = 0;

Computing the signal $f_1$ corresponding to these coefficients |fw1| computes the orthogonal projection on the space of signal that are constant on disjoint intervals of length $N/2^j$.

Exercise 3

Display the reconstructed signal obtained from |fw1|, for a decreasing cut-off scale $j$.

In [24]:
In [25]:
%% Insert your code here.

Haar Non-linear Approximation

Non-linear approximation is obtained by thresholding low amplitude wavelet coefficients. It is defined as $$ f_T = \Hh^* \circ S_T \circ \Hh(f) $$ where $S_T$ is the hard tresholding operator $$ S_T(x)_i = \choice{ x_i \qifq \abs{x_i}>T, \\ 0 \quad \text{otherwise}. }. $$

In [26]:
S = @(x,T) x .* (abs(x)>T);

Set the threshold value.

In [27]:
T = .5;

Threshold the coefficients.

In [28]:
fwT = S(fw,T);

Display the coefficients before and after thresholding.

In [29]:
plot_wavelet(fw); axis('tight'); title('Original coefficients');
plot_wavelet(fwT); axis('tight'); title('Thresholded coefficients');

Exercise 4

Find the threshold $T$ so that the number of remaining coefficients in |fwT| is a fixed number $m$. Use this threshold to compute |fwT| and then display the corresponding approximation $f_1$ of $f$. Try for an increasing number $m$ of coeffiients. ompute the threshold T

In [30]:
In [31]:
%% Insert your code here.

The Shape of a Wavelet

A wavelet coefficient corresponds to an inner product of $f$ with a wavelet Haar atom $\psi_{j,k}$ $$ d_j[k] = \dotp{f}{\psi_{j,k}} $$

The wavelet $\psi_{j,k}$ can be computed by applying the inverse wavelet transform to |fw| where |fw[m]=1| and |fw[s]=0| for $s \neq m$ where $m = 2^j+k$.

Exercise 5

Compute wavelets at several positions and scales.

In [32]:
In [33]:
%% Insert your code here.