Expanding Functions into the Haar Wavelet Series

Overview

Let $f(x)$ be a square-integrable function for $x \in [a,b]$. The function can be expanded into Haar wavelets series

$$f(x) = \sum\limits_{i = 1}^{2M} {{c_i}{h_i}\left( x \right)} $$.

Where $c_i$ denotes the Haar wavelet coefficients.

Let ${x_l} = a + \frac{{\left( {b - a} \right)\left( {l - 0.5} \right)}}{{2M}}$ be the collocation points. Therefore, we would have;

$$f({x_l}) = \sum\limits_{i = 1}^{2M} {{c_i}{h_i}\left( {{x_l}} \right)} $$

$$ \Rightarrow f = cH$$

$$ \Rightarrow c = f{H^{ - 1}} $$

Haar Wavelets

Consider the interval $x \in [a,b]$, where a and b are the given constant.

Define $M = 2^J$, where J is the maximal level of the resolution.

Define $\Delta x = \frac{b-a}{2M}$

The parameter $j = 0,1, \ldots ,J$ and $k = 0,1, \ldots ,m - 1$(where $m=2^j$) is introduced.

The index $i$ is calculated from the formula $i=m+k+1$. Where $i=2$ is the minimal and $i=2(2^J)$ is the maximal.

The i-th $(i>1)$ Haar wavelet is defined as

$$ h_i(x) = \begin{cases} 1, & \text{for } x \in [\alpha,\beta ), \\ -1, & \text{for } x \in [\beta,\gamma ), \\ 0, & \text{otherwise } \end{cases} $$

Where

• $\alpha = a + 2k\mu \Delta x$
• $\beta = a + \left( {2k + 1} \right)\mu \Delta x$
• $\gamma = a + 2\left( {k + 1} \right)\mu \Delta x$
• $\mu = M/m$

and

$$ h_1(x) = \begin{cases} 1, & \text{for } x \in [a,b], \\ 0, & \text{otherwise } \end{cases} $$

Prerequisites

MATLAB R2022 or later

Project Structure

1. Haar.m: Haar function - This function is used to generate the wavelet function for any a<=x<=b
2. ExpandHaarSeries.m: This function reconstructs any function $f$ using Haar wavelets as a basis. Larger $J$ gives a better approximation.
3. HaarMain.m: A MATLAB script that shows how to implement the two functions.

Results

Result 1: Eight First Haar Wavelets

Result 2: Expanding Functions into wavelets series

Let $f(x) = \sqrt{x}$

Plot of Reconstrction	Plot of Coefficients, $J=4$

Let

$$ f(x) = \begin{cases} \frac{|x|}{x}, & \text{for } x \ne 0, \\ 0, & x = 0 \end{cases} $$

Plot of Reconstrction	Plot of Coefficients, $J=9$

References

Lepik, Ü., & Hein, H. (2014). Haar wavelets. In Haar wavelets: with applications (pp. 7-20). Cham: Springer International Publishing.