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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.

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Wavelet and multiresolution process

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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:




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.



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.





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:



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



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



2d wavelet
2D Wavelet

Wavelet expansion of 2D function

Basis for 2D function:




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.





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