Wavelet Transform

     

What Are Wavelets?

In general, a family of representations using:

• hierarchical (nested) basis functions

• finite (“compact”) support

• basis functions often orthogonal

• fast transforms, often linear-time

MULTIRESOLUTION ANALYSIS (MRA)

• Wavelet Transform– An alternative approach to the short time Fourier transform to

overcome the resolution problem – Similar to STFT: signal is multiplied with a function

• Multiresolution Analysis – Analyze the signal at different frequencies with different

resolutions– Good time resolution and poor frequency resolution at high

frequencies– Good frequency resolution and poor time resolution at low

frequencies– More suitable for short duration of higher frequency; and longer

duration of lower frequency components

PRINCIPLES OF WAVELET TRANSFORM

• Split Up the Signal into a Bunch of Signals

• Representing the Same Signal, but all Corresponding to Different Frequency Bands

• Only Providing What Frequency Bands Exists at What Time Intervals

Wavelet Transform (WT)

• Wavelet transform decomposes a signal into a set of basis functions.• These basis functions are called wavelets• Wavelets are obtained from a single prototype wavelet (t) called

mother wavelet by dilations and shifting:

where a is the scaling parameter and b is the shifting parameter

)(1

)(, a

bt

atba

• The continuous wavelet transform (CWT) of a function f is defined as

• If is such that

f can be reconstructed by an inverse wavelet transform:

dta

bttf

afbaTf ba )()(

1,),( *

,

dC2

)(

20

,1 )(),()(

a

dadbtbaTfCtf ba

SCALE

• Scale– a>1: dilate the signal

– a<1: compress the signal

• Low Frequency -> High Scale -> Non-detailed Global View of Signal -> Span Entire Signal

• High Frequency -> Low Scale -> Detailed View Last in Short Time

• Only Limited Interval of Scales is Necessary

Wavelet transform vs. Fourier Transform

• The standard Fourier Transform (FT) decomposes the signal into individual frequency components.

• The Fourier basis functions are infinite in extent.

• FT can never tell when or where a frequency occurs.

• Any abrupt changes in time in the input signal f(t) are spread out over the whole frequency axis in the transform output F() and vice versa.

• WT uses short window at high frequencies and long window at low frequencies (recall a and b in (1)). It can localize abrupt changes in both time and frequency domains.

RESOLUTION OF TIME & FREQUENCY

Time

Frequency

Better time resolution;Poor frequency resolution

Better frequency resolution;Poor time resolution

• Each box represents a equal portion • Resolution in STFT is selected once for entire analysis

Discrete Wavelet Transform

• Discrete wavelets

• In reality, we often choose

• In the discrete case, the wavelets can be generated from dilation equations, for example,

(t)h(0)(2t) + h(1)(2t-1) + h(2)(2t-2) + h(3)(2t-3)]• Solving equation (2), one may get the so called scaling function (t).

• Use different sets of parameters h(i)one may get different scaling functions.

),( 02

0, ktaa jj

kj ., Zkj

.20 a

2

Discrete WT Continued

• The corresponding wavelet can be generated by the following equation

(t)[h(3)(2t) - h(2)(2t-1) + h(1)(2t-2) - h(0)(2t-3)]. (3)

• When and

equation (3) generates the D4 (Daubechies) wavelets.

2

,24/)31()0( h ,24/)33()1( h ,24/)33()2( h

24/)31()3( h

Discrete WT continued

• In general, consider h(n) as a low pass filter and g(n) as a high-pass filter where

• g is called the mirror filter of h. g and h are called quadrature mirror filters (QMF).

• Redefine– Scaling function

).1()1()( nNhng n

).2()(2)( nxnhxn

Discrete Formula

– Wavelet function

• Decomposition and reconstruction of a signal by the QMF.

where and is down-sampling and is up-sampling

).2()(2)( nxngxn

2

2

2

2

f(n) +

f(n)

)(ng

)(nh )(nh

)(ng

)()( nhnh ).()( ngng

Generalized Definition

• Let be matrices, where are positive integers

is the low-pass filter and is the high-pass filter.

• If there are matrices and which satisfy:

where is an identity matrix. Gi and Hi are called a discrete wavelet pair.

• If and

The wavelet pair is said to be orthonormal.

,...)2,1(, iHG ii...2,1,0, iN i1 ii NN

ii NN 1iG

iH

iiiii IGGHH

iI11 ii NN

Tii

Tii GGHH , i

Tii

Tii IHHGG 0 T

iiTii GHHG

ii GH ,

iH iG

• For signal let and• One may have

• The above is called the generalized Discrete Wavelet Transform (DWT) up to the scale

is called the smooth part of the DWT andis called the DWT at scale

• In terms of equation

),,...,( 21 Nffff NN 0 .,...2,1 Ji

.J

fHHH JJ 11....

fHHG JJ 11... .J

),,,......,....( 1121231111 fGfHGfHHGfHHGfHHH JJJJ

).2()()(1

0

12

0, ktrttf j

p

j kkj

j

Multilevel Decomposition

• A block diagram

2

2

2

f(n) 2

)(nh

)(nh

)(ng

)(ng

2

2

)(nh

)(ng

Haar Wavelets 

1 0 0 1

Scaling Function Wavelet

]2

1,

2

1[)( nh ]

2

1,

2

1[)( ng

Example: Haar Wavelet

[18]

Summary on Haar Transform• Two major sub-operations

– Scaling captures info. at different frequencies– Translation captures info. at different locations

• Can be represented by filtering and downsampling• Relatively poor energy compaction

1

x

2D Wavelet Transform

• We perform the 2-D wavelet transform by applying 1-D wavelet transform first on rows and then on columns.

Rows Columns LL

f(m, n) LH

HL

HH

H 2

G

2 G

2 H

2

2

G

H

2

 

Applications

• Signal processing– Target identification.

– Seismic and geophysical signal processing.

– Medical and biomedical signal and image processing.

• Image compression (very good result for high compression ratio).

• Video compression (very good result for high compression ratio).

• Audio compression (a challenge for high-quality audio).

• Signal de-noising.

Original Video Sequence Reconstructed Video Sequence

3-D Wavelet Transform for Video Compression