Connexions

You are here: Home » Content » Haar Wavelet Basis

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.

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.

Note: You are viewing an old version of this document. The latest version is available here.

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.

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

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

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

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

ψt={1  if  0t<T2-1  if  0T2<T ψ t 1 0 t T 2 -1 0 T 2 T
(1)

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

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

( 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.

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

Content actions

Give feedback:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks