To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
Languages
Recent
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

Orthogonal wavelet

From Wikipedia, the free encyclopedia

An orthogonal wavelet is a wavelet whose associated wavelet transform is orthogonal. That is, the inverse wavelet transform is the adjoint of the wavelet transform. If this condition is weakened one may end up with biorthogonal wavelets.

YouTube Encyclopedic

  • 1/3
    Views:
    1 166
    919
    605
  • Lec 12 Biorthogonal Basic Concept
  • Mod-01 Lec-48 Towards selecting Wavelets through vanishing moments
  • Mod-01 Lec-35 The Lifting Structure & Polyphase Matrices

Transcription

Basics

The scaling function is a refinable function. That is, it is a fractal functional equation, called the refinement equation (twin-scale relation or dilation equation):

,

where the sequence of real numbers is called a scaling sequence or scaling mask. The wavelet proper is obtained by a similar linear combination,

,

where the sequence of real numbers is called a wavelet sequence or wavelet mask.

A necessary condition for the orthogonality of the wavelets is that the scaling sequence is orthogonal to any shifts of it by an even number of coefficients:

,

where is the Kronecker delta.

In this case there is the same number M=N of coefficients in the scaling as in the wavelet sequence, the wavelet sequence can be determined as . In some cases the opposite sign is chosen.

Vanishing moments, polynomial approximation and smoothness

A necessary condition for the existence of a solution to the refinement equation is that there exists a positive integer A such that (see Z-transform):

The maximally possible power A is called polynomial approximation order (or pol. app. power) or number of vanishing moments. It describes the ability to represent polynomials up to degree A-1 with linear combinations of integer translates of the scaling function.

In the biorthogonal case, an approximation order A of corresponds to A vanishing moments of the dual wavelet , that is, the scalar products of with any polynomial up to degree A-1 are zero. In the opposite direction, the approximation order à of is equivalent to à vanishing moments of . In the orthogonal case, A and à coincide.

A sufficient condition for the existence of a scaling function is the following: if one decomposes , and the estimate

holds for some , then the refinement equation has a n times continuously differentiable solution with compact support.

Examples

  • Suppose then , and the estimate holds for n=A-2. The solutions are Schoenbergs B-splines of order A-1, where the (A-1)-th derivative is piecewise constant, thus the (A-2)-th derivative is Lipschitz-continuous. A=1 corresponds to the index function of the unit interval.
  • A=2 and p linear may be written as
Expansion of this degree 3 polynomial and insertion of the 4 coefficients into the orthogonality condition results in The positive root gives the scaling sequence of the D4-wavelet, see below.

References

This page was last edited on 20 October 2022, at 19:03
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.