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Haar Wavelet Basis

Module by: Roy Ha. E-mail the author

Summary: (Blank Abstract)

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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.

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=1T⁢ei⁢ ω 0 ⁢n⁢t 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" ψ⁢t 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 ∀0≤t<T:φ⁢t=1 0 t T φ t 1 Then φ⁢t 2j2⁢ψ⁢2j⁢t−k φ⁢t2j2⁢ψ⁢2j⁢t−k j∈ℤ∧(k= 0 , 1 , 2 , … , 2 j - 1 ) φ t 2 j 2 ψ 2 j t k j ℤ k 0 , 1 , 2 , … , 2 j - 1 φ t 2 j 2 ψ 2 j t k

Figure 5
Figure 5 (fig5.png)
ψ⁢t={1  if  0≤t<T2-1  if  0≤T2<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=2j2⁢ψ⁢2j⁢t−k 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 ] )

Using what we know about Hilbert spaces: For any f⁢t∈ L 2 ( [ 0 , T ] ) f t L 2 ( [ 0 , T ] ) , we can write

Synthesis

f⁢t=∑j∑k w j , k ⁢ ψ j , k ⁢t+ c 0 ⁢φ⁢t f t j j k k w j , k ψ j , k t c 0 φ t
(3)

Analysis

w j , k =∫0Tf⁢t⁢ ψ j , k ⁢tdt w j , k t 0 T f t ψ j , k t
(4)
c 0 =∫0Tf⁢t⁢φ⁢tdt 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

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