Skip to content Skip to navigation

OpenStax_CNX

You are here: Home » Content » Example Wavelets

Navigation

Recently Viewed

This feature requires Javascript to be enabled.
Download
x

Download module as:

  • PDF
  • EPUB (what's this?)

    What is an EPUB file?

    EPUB is an electronic book format that can be read on a variety of mobile devices.

    Downloading to a reading device

    For detailed instructions on how to download this content's EPUB to your specific device, click the "(what's this?)" link.

  • More downloads ...
Reuse / Edit
x

Module:

Add to a lens
x

Add module to:

Add to Favorites
x

Add module to:

 

Example Wavelets

Module by: Kileen Cheng. E-mail the author

Summary: Many considerations go into the design of a wavelet system including properties such as orthogonality, compact support, symmetry, and smoothness. Here we will examine several popularly-used wavelet systems and their respective properties.

Design Considerations:

There are several design properties for the construction of a wavelet basis that one would want to be fulfilled.

  1. symmetry: If the wavelets are not symmetric, then the wavelet transform of the mirror of an image is not the mirror of the wavelet transform.
  2. smoothness: This property is determined by the number of vanishing moments. Recall that the primal vanishing moments determine the smoothness of the reconstruction. The dual vanishing moments determine the convergence rate of the multiresolution projections and are necessary for detecting singularities.
  3. orthogonality: This property can be too restrictive at times. Thus the need for biorthgonal wavelet systems.
  4. compact support: This property is a function of the filter length.

Haar Wavelet

The Haar wavelet is the most fundamental of the wavelet systems and is also known as the length-2 Daubechies filter (See Figure 1).

Figure 1: Haar Scaling and Wavelet Functions
Figure 1 (haar.png)

The Haar wavelet system properties are:

  1. Symmetric scaling function
  2. Anti-symmetric wavelet function
  3. One vanishing moment (a minimum)
  4. Orthogonal
  5. Compact support
It is important to note that the Haar wavelet system is the only one that is orthogonal, symmetric, and has compact support.
hn=1212 h n 1 2 1 2
(1)
gn=1212 g n 1 2 1 2
(2)

Sinc Wavelet

The Sinc wavelet is the second fundamental of the wavelet systems (see Figure 2). Recall that the Fourier transform of the sinc is the brick-wall filter (or ideal low-pass filter).

Figure 2: Sinc Scaling and Wavelet Functions
Figure 2 (sinc.png)

The Sinc wavelet system properties are:

  1. Orthogonal
  2. Infinite number of vanishing moments
  3. Infinite support (IIR and non-causal)
Given our scaling function to be the sinc function
φt=sinct=sinπtπt φ t sinc t t t
(3)
we have the following wavelet expression
ψt=2φ2tφt ψ t 2 φ 2 t φ t
(4)
The filter coefficients can then be found as
hn=sincπn2 h n sinc n 2
(5)

Daubechies Wavelet

The Daubechies coefficients for the scaling and wavelet filters are unique in that they have a high degree of smoothness. All N21 N 2 1 degrees of freedom are used to maximize the number of vanishing moments:

i,i0N21: H i π=0 i i 0 N 2 1 H i 0
(6)
The Daubechies wavelet system properties are (see Figure 3:
  1. If length of filter N=2 N 2 , Daubechies = Haar
  2. Anti-symmetric if length of filter N>2 N 2
  3. Compact support
  4. Orthogonal
  5. N21 N 2 1 vanishing moments
  6. Increasing smoothness as NN increases

Figure 3: Length-4 Daubechies Scaling and Wavelet Functions
Figure 3 (daub4.png)

Content actions

Download module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add module to:

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? tag icon

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

Reuse / Edit:

Reuse or edit module (?)

Check out and edit

If you have permission to edit this content, using the "Reuse / Edit" action will allow you to check the content out into your Personal Workspace or a shared Workgroup and then make your edits.

Derive a copy

If you don't have permission to edit the content, you can still use "Reuse / Edit" to adapt the content by creating a derived copy of it and then editing and publishing the copy.