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Haar Wavelet Basis

Module by: Roy Ha, Justin Romberg. E-mail the authors

Summary: 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 0 T L 2 0 T 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 0 T L 2 0 T and have many nice properties.

Basis Comparisons

Fourier series - c n c n give frequency information. Basis functions last the entire interval.

Figure 1: Fourier basis functions
Figure 1 (fig1.png)

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

Figure 2: Wavelet basis functions
Figure 2 (fig2.png)

In Fourier basis, the basis functions are harmonic multiples of ei ω 0 t ω 0 t

Figure 3: basis=1Tei ω 0 nt basis 1 T ω 0 n t
Figure 3 (fig3.png)

In Haar wavelet basis, the basis functions are scaled and translated versions of a "mother wavelet" ψt ψ t .

Figure 4
Figure 4 (fig4.png)

Basis functions ψ j , k t ψ j , k t are indexed by a scale j and a shift k.

Let 0t<T:φt=1 0 t T φ t 1 Then φt 2j2ψ2jtk φt2j2ψ2jtk j(k= 0 , 1 , 2 , , 2 j - 1 ) φ t 2 j 2 ψ 2 j t k j k 0 , 1 , 2 , , 2 j - 1 φ t 2 j 2 ψ 2 j t k

Figure 5
Figure 5 (fig5.png)
ψt={1  if  0t<T2-1  if  0T2<T ψ t 1 0 t T 2 -1 0 T 2 T
(1)
Figure 6
Figure 6 (fig6.png)

Let ψ j , k t=2j2ψ2jtk ψ j , k t 2 j 2 ψ 2 j t k

Figure 7
Figure 7 (fig7.png)

Larger jj → "skinnier" basis function, j=012 j 0 1 2 , 2j 2 j shifts at each scale: k= 0 , 1 , , 2 j - 1 k 0 , 1 , , 2 j - 1

Check: each ψ j , k t ψ j , k t has unit energy

Figure 8
Figure 8 (fig8.png)
( ψ j , k 2tdt=1)( ψ j , k ( t ) 2 =1) t ψ j , k t 2 1 ψ j , k ( t ) 2 1
(2)

Any two basis functions are orthogonal.

Figure 9: Integral of product = 0
(a) Same scale(b) Different scale
Figure 9(a) (fig9a.png)Figure 9(b) (fig9b.png)

Also, ψ j , k φ ψ j , k φ span L 2 0 T L 2 0 T

Haar Wavelet Transform

Using what we know about Hilbert spaces: For any ft L 2 0 T f t L 2 0 T , we can write

Synthesis

ft=jk w j , k ψ j , k t+ c 0 φt f t j j k k w j , k ψ j , k t c 0 φ t
(3)

Analysis

w j , k =0Tft ψ j , k tdt w j , k t 0 T f t ψ j , k t
(4)
c 0 =0Tftφtdt c 0 t 0 T f t φ t
(5)

Note:

the w j , k w j , k are real
The Haar transform is super useful especially in image compression

Haar Wavelet Demonstration

Figure 10: Interact (when online) with a Mathematica CDF demonstrating the Haar Wavelet as an Orthonormal Basis.
HaarDemo

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