Wavelet and multiresolution process

Pei Wu5.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:

Example: Music

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

Adaptive resolution Use big ruler to measure big thing,

small ruler to measure small thing.

Wavelet Use scale transform to construct

ruler with different resolution.

CWT(continuous wavelet transform)

Proof (1)

Proof (2)

Discretizing CWT a,b take only discrete number:

And we want them to be orthogonal:

Example for wavelet (a)Meyer (b,c)Battle-Lemarie

Example for wavelet (2) (d) Haar (e,f)Daubechies

Constructing orthogonal wavelet Multiresolution analysis A series of linear subspace {Vi} that:

Example

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

Relaxing orthogonal condition is linearly independent

but not orthogonal.

is orthonormal basis of V0

Example: Battle-Lemarie Wavelet Use spline to get continuous function

Meyer Wavelet: compact support

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 (3):frequency domain perspect Subband coding

Adaptive resolution

2D Wavelet Wavelet expansion of 2D function Basis for 2D function:

Mallet algorithm

Frequency Domain Decomposition

Denoise using wavelet

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

Thank You!!