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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.



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.



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



Example for wavelet 2

Example for wavelet (2)

(d) Haar


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

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!!

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