Wavelet Transform A Presentation

By Subash Chandra Nayak01EC3010

IIT Kharagpur INDIA

Introduction to the world of transform

• What are transforms :-

• A mathematical operation that takes a function or sequence and maps into another one

• General Form :-

• Examples :-

Laplace, Fourier, DTFT, DFT, FFT, z-transform

dwxwKwFxf ),()()(

j

iijFKjf

Fourier Transform

dfefXtx

dtetxfX

ftj

ftj

2

2

)()(

)()(

Mathematical Form :-

Notes :- Fourier transform identifies all spectral components present in the signal, however it does not provide any information regarding the temporal (time) localization of the components

Fourier Transform :: Limitations

• Signals are of two types

# Stationary

# Non – Stationary

• Non stationary signals are those who have got time varying spectral components ... FT gives only provides the existence of the spectral components of the signal ... But does not provide any information on the time occurrence of spectral components

• Explanation

The basis function e-jwt stretches to infinity , Hence only analyzes the signal globally

In order to obtain time-localization of spectral components , the signal need to be analyzed locally

Time – Frequency Representation

• Instantaneous frequency :-Instantaneous frequency :-

• Group delay :-Group delay :-

• Disadvantages of above expressionsDisadvantages of above expressions

These equations though have a huge theoretical significance but are not easy to implement easily

)(2

1)( tx

df

dtf x

)(2

1)( fX

df

dftx

Short-time Fourier Transform

• Also known as a STFT

• given as

• w(t) :- windowing function generally a Gaussian pulse is used, other choices are rectangular , elliptic etc..

• Maps 1D function to 2D time-frequency domain

• Advantages :-

# Gives us time-frequency description of the signal

# Overcomes the difficulties of Fourier transform by use of windowing functions

dtetwtxwSTFT jwt

t

wx

)]()([),(

STFT :: Disadvantages• Heisenberg Principle :-Heisenberg Principle :- One can not get infinite time and

frequency resolution beyond Heisenberg’s Limit

• Trade offs :-Trade offs :-• Wider window

Good frequency resolution , Poor time resolution

• Narrower window

Good time resolution , Poor frequency resolution

4

1. ft

Wavelet Transform

• Overcomes the shortcoming of STFT by using variable length windows :: i.e. Narrower window for high frequency thereby giving better time resolution and Wider window at low frequency thereby giving better frequency resolution

• Heisenberg’s Principle still holds

• Mathematical form:-

where x(t) = given signal tau = translation parameter s = scaling parameter = 1/f phi(t) = Mother wavelet , All kernels are obtained by scaling

and/or translating mother wavelet

dts

ttx

sssCWT

t

x )(*)(||

1),(),(

Continuous Wavelet transform

• The kernel functions used in wavelet transform are all obtained from one prototype function known as mother wavelet , by scaling and/or translating it

• Here

a = scale parameter

b = translation parameter

• Continuous Wavelet transform

)(1

)(, a

bt

atba

)()(0,1 tt

dtttxa

baW ba )()(1

),( ,

CWT (Contd..)

• In order to become a wavelet a function must satisfy the above two conditions

dtt

dtt

2|)(|

0)(

Inverse wavelet transform

0)(

||

|)(|

)(),(11

)( ,2

dtt

provided

dww

wC

where

dadbtbaWaC

tx ba

Examples of wavelets

Constant Q-filtering

• CWT can be rewritten as

• A special property of the above filter defined by the mother wavelet is that they are Constant-Q filters

• Q factor = Center frequency/Bandwidth• Hence the filter defined by wavelet increases their Bandwidth

as scale increases ( i.e. center frequency increases )• This boils down to filter bank implementation of discrete

wavelet transform

)(*)(),( *0, bbxbaW a

Filter Banks :: General Structure

• Condition for Perfect Reconstruction

delayl

zHzFzHzF

zzHzFzHzF l

0)()()()(

2)()()()(

1100

1100

Filter bank (Contd..)

• Product filter

Now let’s define product filter as :-

P0(z) = F0(z)H0(z)

And Normalized Product filter as

P(z) = zL P0(z) where L = delay in total process

So the PR condition boils down to this realationship

P(z) – P(-z) = 2

Harr Filter Bank

Note that f0(n) and f1(n) are non-causal ... Hence here Unit delay is required to implement it hence here

L = 1

Product filter

P(w) is said to be halfband filter because of its symmetry Also

P(w) + P(w + pi) = 2

Product filter II

• P(w) should be as flat as possible around 0 and pi . The more is the flatness of P(w) around 0 and pi the better the Product filter is . Hence P(w) is always tried to be designed as a MAXFLAT Filter

• Order of filter :: p

p = (L+1)/2 ; L = number of delay elements• Methods of determination of P(z)

# Duabechies method

# Meyer methods• Both of the above methods give us P(z) for a given order “p” .

The higher is the order the better is the filter but at the same time it will require more hardware complexity

Spectral factorization

• The spectral factorization is the problem of finding h0(z) once P(z) is known

• Linear Phase Factorization

H0(z) and F0(z) are of different degree. Gives filter with linear phase

• Orthogonal Factorization

H0(z) and F0(z) are of same degree. Gives filter with non-linear phase. Daubechies family of filters belongs to this category. For orthogonal filter

)()(

)()(

)()1()(

11

00

01

nNhnf

nNhnf

nNhnh n

For orthogonal

filter

Discrete wavelet Transform

• Discrete domain counterpart of CWT

• Implemented using Filter banks satisfying PR condition

• Represents the given signal by discrete coefficients {dk,n}

• DWT is given by

)2(2)( 2, ntt knk

k

1||)(|| , tnk

dtttxxd

tdtx

nknknk

nkk n

nk

)()(,

)()(

*,,,

,,

Scaling Function

• These are functions used to approximate the signal up to a particular level of detail

• For Harr System

Harr Scaling Function10

1)(

t

t

)2(2)( 2, ntt knk

k

Refinement equation and wavelet Equation

• Refinement equation is an equation relating to scaling function and filter coefficients

• Wavelet equation is an equation relating to wavelet function and filter coefficients

• By solving the above two we can obtain the scaling and wavelet function for a given filter bank structure

Refinement Equation

Wavelet Equation

N

k

ktkht0

0 )2(][2)(

N

k

ktkht0

0 )2(][2)(

DWT Implementation

g`[n]

h`[n]

2

2 g`[n]

h`[n]

2

2

2

2

g[n]

h[n]

+

2

2

g[n]

h[n]

+

a(k,n)

d(k+1,n)

a(k+1,n)

a(k+2,n)

d(k+2,n)

a(k+1,n)

a(k,n)

Decomposition Reconstruction

We have only shown the above implementation for the Haar Wavelet, however, as we willsee later, this implementation – subband coding – is applicable in general.

DWT Sub-band Decomposition

x[n] Length: 512B: 0 ~

g[n] h[n]

g[n] h[n]

g[n] h[n]

2

d1: Level 1 DWTCoeff.

Length: 256B: 0 ~ /2 Hz

Length: 256B: /2 ~ Hz

Length: 128B: 0 ~ /4 HzLength: 128

B: /4 ~ /2 Hz

d2: Level 2 DWTCoeff.

d3: Level 3 DWTCoeff.

…….

Length: 64B: 0 ~ /8 HzLength: 64

B: /8 ~ /4 Hz

2

2 2

22

|H(jw)|

w/2-/2

|G(jw)|

w- /2-/2

Sub-band coding

Some Important properties of wavelets

• Compact Support :-• Finite duration wavelets are called compactly supported in time

domain but are not band-limited in frequency. Can be implemented using FIR filters

• Examples

Harr, Daubechies, Symlets , Coiflets • Narrow band wavelets are called compactly supported in

frequency domain. Can be implemented using IIR filters• Examples

Meyer’s wavelet

Some Important properties of wavelets

• Symmetry

• Symmetric / Antisymetric wavelets have got liner-phase

• Orthogonal wavelets are asymmetric and have a non-linear phase

• Biorthogonal wavelets are asymmetric but have got linear phase can be implemented using FIR filters

• Vanishing Moment

• pth vanishing moment is defined as

• The more the number of moments of a wavelets are zero the more is its compressive power

dtttM pp )(

Some Important properties of wavelets

• Smoothness• is roughly the number of times a function can be differentiated

at any given point• Closely related to vanishing Moments• Smoothness provides better numerical stability• It also provides better reconstruction propertiy

2D DWT

• Generalization of concept to 2D• 2D functions images f(x,y) I[m,n] intensity function

• Why would we want to take 2D-DWT of an image anyway?– Compression– Denoising– Feature extraction

• Mathematical form

),(),,(),(

),(),(),(

jyixsyxfjia

jyixsjiayxf

o

i joo

)()(),( yxyxs )()(),( yxyxs

Implementation of 2D-DWT

INPUTIMAGE…

…

……

RO

WS

COLUMNSH~ 2 1

G~ 2 1

H~ 1 2

G~ 1 2

H~ 1 2

G~ 1 2

ROWS

ROWS

COLUMNS

COLUMNS

COLUMNS

LL

LH

HL

HH

)(1hkD

)(1vkD

)(1dkD

1kA

INPUTIMAGE

LL LH

HL HH

LLLH

HL HH

LHH

LLH

LHL

LLLLH

HL HH

LHH

LLH

LHL

Up and Down … Up and Down

2 1Downsample columns along the rows: For each row, keep the even indexed columns, discard the odd indexed columns

1 2Downsample columns along the rows: For each column, keep the even indexed rows, discard the odd indexed rows

2 1

1 2

Upsample columns along the rows: For each row, insert zeros at between every other sample (column)

Upsample rows along the columns: For each column, insert zeros at between every other sample (row)

Reconstruction

)(1hkD

)(1vkD

)(1dkD

1kA 1 2

1 2

1 2

1 2

H

G

H

G

2 1

2 1

H

G

ORIGINALIMAGE

LL

LH

HL

HH

DWT in Work

DWT at work

DWT at work

Applications of wavelets

• There are a lots of uses of wavelets .... The most prominent application of wavelets are

• Computer and Human Vision

• FBI Finger Print compression

• Image compression

• Denoising Noisy data

• Detecting self-similar behavior in noisy data

• Musical Notes synthesis

• Animations

Things that I didn’t Cover

• Different algorithms for getting Product filter, max-flat filter realization, spectral factorization , solutions for refinement and wavelet equations and many more

• Noble Identity, modulation matrix, Polyphase matrix forms• MRA , Mallat’s pyramidal algorithm• Lifting• Basic Vector algebra needed for wavelet analysis ( It’s too

mathematical to present )• Orthogonality, biorthogonality, frames• Vector algebra approach for wavelets• Wavelet for denoising and many more .................