Wavelet and multiresolution process
This presentation is the property of its rightful owner.
Sponsored Links
1 / 37

Wavelet and multiresolution process PowerPoint PPT Presentation


  • 49 Views
  • Uploaded on
  • Presentation posted in: General

Wavelet and multiresolution process. Pei Wu 5.Nov 2012. Mathematical preliminaries: Some topology. Open set: any point A in the set must have a open ball O( r,A ) contained in the set. Closed set: complement of open set.

Download Presentation

Wavelet and multiresolution process

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Wavelet and multiresolution process

Wavelet and multiresolution process

Pei Wu

5.Nov 2012


Mathematical preliminaries some topology

Mathematical preliminaries: Some topology

Open set: any point A in the set must have a open ball O(r,A) contained in the set.

Closed set: complement of open set.

Intersection of closed set is always closed. Union of open set is always open

Compact: if we put infinite point in the set it must have infinity point “gather” around some point in the set.

Complete: a “converge” sequence must converge at a point in the set.


Mathematical preliminaries hilbert space

Mathematical preliminaries: Hilbert space

  • Hilbert space is a space…

    • linear

    • complete

    • with norm

    • with inner product

  • Example: Euclidean space, L2 space, …


Mathematical preliminaries orthonormal basis

Mathematical preliminaries: orthonormal basis

f,g is orthogonal iff <f,g>=0

f is normalized iff <f,f>=1

Orthonormal basis: e1, e2, e3,… s.t.

a set of basis is called complete if


Equivalent condition for orthonormal

equivalent condition for orthonormal

A set of element {ei} is orthonormal if and only if:

A orthonormal set induces isometric mapping between Hilbert space and l2.


Motivation

Motivation

in context of Fourier transform we suppose the frequency spectrum is invariant across time:

However in many cases we want:


Example music

Example: Music


Windowed fourier transform

Windowed Fourier Transform


Analyze of windowed fourier transform

Analyze of Windowed Fourier transform

A function cannot be localized in both time and frequency (uncertainty principle).

High frequency resolution means low time resolving power.


Trade off between frequency resolution and time resolution

Trade-off between frequency resolution and time resolution


Adaptive resolution

Adaptive resolution

Use big ruler to measure big thing, small ruler to measure small thing.


Wavelet

Wavelet

Use scale transform to construct ruler with different resolution.


Cwt continuous wavelet transform

CWT(continuous wavelet transform)


Proof 1

Proof (1)


Proof 2

Proof (2)


Discretizing cwt

Discretizing CWT

a,b take only discrete number:

And we want them to be orthogonal:


Example for wavelet

Example for wavelet

(a)Meyer

(b,c)Battle-Lemarie


Example for wavelet 2

Example for wavelet (2)

(d) Haar

(e,f)Daubechies


Constructing orthogonal wavelet

Constructing orthogonal wavelet

Multiresolution analysis

A series of linear subspace {Vi} that:


Example

Example


From scaling function to wavelet

From scaling function to wavelet

Firstly we find a set of orthonormal basis in V0:

hn would play important role in discrete analysis


Example haar wavelet

Example: Haar wavelet


Relaxing orthogonal condition

Relaxing orthogonal condition

is linearly independent but not orthogonal.

is orthonormal basis of V0


Example battle lemarie wavelet

Example: Battle-Lemarie Wavelet

Use spline to get continuous function


Meyer wavelet compact support

Meyer Wavelet: compact support


Fast wavelet transform

Fast Wavelet transform

Mallat algorithm : top-down

Given c1 how can we get c0 and d0?

Given c0 and d0 how to reconstruct c1 ?


Mallat algorithm 2

Mallat algorithm (2)


Mallat algorithm 3 frequency domain perspect

Mallat algorithm (3):frequency domain perspect

Subband coding


Adaptive resolution1

Adaptive resolution


2d wavelet

2D Wavelet

Wavelet expansion of 2D function

Basis for 2D function:


Mallet algorithm

Mallet algorithm


Frequency domain decomposition

Frequency Domain Decomposition


Denoise using wavelet

Denoise using wavelet


Wavelet packet

Wavelet packet

We can carry on decomposition on high-frequency part

Adaptive approach to decide decompose or not.


Demo finger print image

Demo: finger-print image


Demo finger print image1

Demo: finger-print image


Thank you

Thank You!!


  • Login