15.10 Haar wavelet basis

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This module gives an overview of wavelets and their usefulness as a basis in image processing. In particular we look at the properties of the Haar wavelet basis.

Introduction

Fourier series is a useful orthonormal representation on ${L}^{2}(\left[0 , T\right])$ especiallly for inputs into LTI systems. However, it is ill suited for some applications, i.e. image processing (recall Gibb's phenomena ).

Wavelets , discovered in the last 15 years, are another kind of basis for ${L}^{2}(\left[0 , T\right])$ and have many nice properties.

Basis comparisons

Fourier series - ${c}_{n}$ give frequency information. Basis functions last the entire interval.

Wavelets - basis functions give frequency info but are local in time.

In Fourier basis, the basis functions are harmonic multiples of $e^{i{\omega }_{0}t}$

In Haar wavelet basis , the basis functions are scaled and translated versions of a "mother wavelet" $\psi (t)$ .

Basis functions $\{{\psi }_{j,k}(t)\}$ are indexed by a scale j and a shift k.

Let $\forall , 0\le t< T\colon \phi (t)=1$ Then $\{\phi (t)2^{\left(\frac{j}{2}\right)}\psi (2^{j}t-k)\colon j\in ℤ\land (k=0,1,2,\dots ,{2}^{j}-1)\}$

$\psi (t)=\begin{cases}1 & \text{if 0\le t< \frac{T}{2}}\\ -1 & \text{if 0\le \frac{T}{2}< T}\end{cases}$

Let ${\psi }_{j,k}(t)=2^{\left(\frac{j}{2}\right)}\psi (2^{j}t-k)$

Larger $j$ → "skinnier" basis function, $j=\{0, 1, 2, \dots \}$ , $2^{j}$ shifts at each scale: $k=0,1,\dots ,{2}^{j}-1$

Check: each ${\psi }_{j,k}(t)$ has unit energy

$(\int {\psi }_{j,k}(t)^{2}\,d t=1)\implies ({\parallel {\psi }_{j,k}\left(t\right)\parallel }_{2}=1)$

Any two basis functions are orthogonal.

Also, $\{{\psi }_{j,k}, \phi \}$ span ${L}^{2}(\left[0 , T\right])$

Haar wavelet transform

Using what we know about Hilbert spaces : For any $f(t)\in {L}^{2}(\left[0 , T\right])$ , we can write

Synthesis

$f(t)=\sum_{j} \sum_{k} {w}_{j,k}{\psi }_{j,k}(t)+{c}_{0}\phi (t)$

Analysis

${w}_{j,k}=\int_{0}^{T} f(t){\psi }_{j,k}(t)\,d t$
${c}_{0}=\int_{0}^{T} f(t)\phi (t)\,d t$
the ${w}_{j,k}$ are real
The Haar transform is super useful especially in image compression

Haar wavelet demonstration

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