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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

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.

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

• Hilbert space is a space…

• linear

• complete

• with norm

• with inner product

• Example: Euclidean space, L2 space, …

### 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

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

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

### 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

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

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

### Wavelet

Use scale transform to construct ruler with different resolution.

### Discretizing CWT

a,b take only discrete number:

And we want them to be orthogonal:

### Example for wavelet

(a)Meyer

(b,c)Battle-Lemarie

(d) Haar

(e,f)Daubechies

### Constructing orthogonal wavelet

Multiresolution analysis

A series of linear subspace {Vi} that:

### From scaling function to wavelet

Firstly we find a set of orthonormal basis in V0:

hn would play important role in discrete analysis

### Relaxing orthogonal condition

is linearly independent but not orthogonal.

is orthonormal basis of V0

### Example: Battle-Lemarie Wavelet

Use spline to get continuous function

### Fast Wavelet transform

Mallat algorithm : top-down

Given c1 how can we get c0 and d0?

Given c0 and d0 how to reconstruct c1 ?

Subband coding

### 2D Wavelet

Wavelet expansion of 2D function

Basis for 2D function:

### Wavelet packet

We can carry on decomposition on high-frequency part

Adaptive approach to decide decompose or not.