Haar Wavelet Basis

Connexions module: m10764 1

Haar Wavelet Basis

Roy Ha Justin Romberg

This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License^†

Abstract

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.

1 Introduction

Fourier series¹is a useful orthonormal representation²on L²([0, T ])especiallly for inputs into LTI systems.

However, it is ill suited for some applications, i.e. image processing (recall Gibb's phenomena³).

Wavelets, discovered in the last 15 years, are another kind of basis for L²([0, T ]) and have many nice properties.

2 Basis Comparisons

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

∗Version 2.7: Jul 19, 2006 4:38 pm GMT-5

†http://creativecommons.org/licenses/by/1.0

1"Fourier Series: Eigenfunction Approach" <http://cnx.org/content/m10496/latest/>

2"Orthonormal Basis Expansions" <http://cnx.org/content/m10760/latest/>

3"Gibbs's Phenomena" <http://cnx.org/content/m10092/latest/>

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Figure 1: Fourier basis functions

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

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Figure 2: Wavelet basis functions

In Fourier basis, the basis functions are harmonic multiples of e^iω0t

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Figure 4

Basis functions {ψj,k(t)} are indexed by a scale j and a shift k.

Let ∀, 0 ≤ t < T : (φ (t) = 1) Then n

φ (t) , 2^j2ψ 2^jt − k
|j ∈ Z ∧ k = 0, 1, 2, . . . , 2^j− 1o

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Figure 5

ψ (t) =







1if 0 ≤ t < ^T₂

−1if 0 ≤ ^T₂ < T (1)

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Figure 6

Let ψj,k(t) = 2^2jψ 2^jt − k

Figure 7

Larger j → "skinnier" basis function, j = {0, 1, 2, . . . }, 2^j shifts at each scale: k = 0, 1, . . . , 2^j− 1 Check: each ψj,k(t)has unit energy

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Figure 8

Z

ψ_j,k²(t) dt = 1 ⇒ k ψ_j,k(t) k₂= 1 (2)

Any two basis functions are orthogonal.

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Also, {ψj,k, φ}span L²([0, T ])

3 Haar Wavelet Transform

Using what we know about Hilbert spaces⁵: For any f (t) ∈ L²([0, T ]), we can write Synthesis

f (t) =X

j

X

k

(w_j,kψ_j,k(t))

!

+ c₀φ (t) (3)

Analysis

w_j,k= Z T

0

f (t) ψ_j,k(t) dt (4)

c0= Z T

0

f (t) φ (t) dt (5)

note: the wj,k are real

The Haar transform is super useful especially in image compression Example 1

This demonstration lets you create a signal by combining Haar basis functions, illustrating the synthesis equation of the Haar Wavelet Transform. See here⁶ for instructions on how to use the demo.

This is an unsupported media type. To view, please see http://cnx.org/content/m10764/latest/HaarSyn.llb

5"Inner Products" <http://cnx.org/content/m10755/latest/>

6"How to use the LabVIEW demos" <http://cnx.org/content/m11550/latest/>