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Wavelet Transform Theory

Prof. Mark Fowler

Department of Electrical Engineering

State University of New York at Binghamton

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What is a Wavelet Transform?

Decomposition of a signal into constituent parts Note that there are many ways to do this. Some are:

Fourier series: harmonic sinusoids; single integer index

Fourier transform (FT): nonharmonic sinusoids; single real index

Walsh decomposition: harmonic square waves; single integer index Karhunen-Loeve decomp: eigenfunctions of covariance; single real index

Short-Time FT (STFT): windowed, nonharmonic sinusoids; double index

provides time-frequency viewpoint

Wavelet Transform: time-compacted waves; double index Wavelet transform also provides time-frequency view

Decomposes signal in terms of duration-limited, band-pass components

high-frequency components are short-duration, wide-band

low-frequency components are longer-duration, narrow-band Can provide combo of good time-frequency localization and orthogonality

the STFT cant do this

More precisely, wavelets give time-scale viewpoint

this is connected to the multi-resolution viewpoint of wavelets

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General Characteristics of Wavelet Systems

Signal decomposition: build signals from building blocks, where thebuilding blocks (i.e. basis functions) are doubly indexed.

The components of the decomposition (i.e. the basis functions) are

localized in time-frequency

ON can be achieved w/o sacrificing t-f localization

The coefficients of the decomposition can be computed efficiently (e.g.,using O(N) operations).

Specific Characteristics of Wavelet Systems

Basis functions are generated from a single wavelet or scaling function

by scaling and translation Exhibit multiresolution characteristics: dilating the scaling functions

provides a higher resolution space that includes the original

Lower resolution coefficients can be computed from higher resolution

coefficients through a filter bank structure

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Fourier Development vs. Wavelet Development

Fourier and others:

expansion functions are chosen, then properties of transform are found

Wavelets

desired properties are mathematical imposed

the needed expansion functions are then derived

Why are there so many different wavelets the basic desired property constraints dont use all the degrees of freedom

remaining degrees of freedom are used to achieve secondary properties

these secondary properties are usually application-specific the primary properties are generally application-nonspecific

What kinds of signals are wavelets and Fourier good for?

Wavelets are good for transients

localization property allows wavelets to give efficient rep. of transients Fourier is good for periodic or stationary signals

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Why are Wavelets Effective?

Provide unconditional basis for large signal class

wavelet coefficients drop-off rapidly

thus, good for compression, denoising, detection/recognition

goal of any expansion is have the coefficients provide more info about signal than time-domain

have most of the coefficients be very small (sparserepresentation)

FT is not sparse for transients

Accurate local description and separation of signal characteristics Fourier puts localization info in the phase in a complicated way

STFT cant give localization andorthogonality

Wavelets can be adjusted or adapted to application

remaining degrees of freedom are used to achieve goals Computation of wavelet coefficient is well-suited to computer

no derivatives of integrals needed

turns out to be a digital filter bank

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Multiresolution Viewpoint

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Multiresolution Approach

Stems from image processing field

consider finer and finer approximations to an image

Define a nested set of signal spaces

We build these spaces as follows:

Let be the space spanned by the integer translations of a

fundamental signal (t), called the scaling function:

that is, if f(t) is in then it can be represented by:

So far we can use just about any function (t), but well see that to getthe nesting only certain scaling functions can be used.

221012 LVVVVV !!

0V

0V

= kk ktatf )()(

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Multiresolution Analysis (MRA) Equation

Now that we have how do we make the others and ensure that they

are nested?

If we let be the space spanned by integer translates of (2t) we getthe desired property that is indeed a space of functions havinghigher resolution.

Now how do we get the nesting?

We need that any function in also be in ; in particular we need

that the scaling function (which is in ) be in , which the requiresthat

where the expansion coefficient is This is the requirement on the scaling function to ensure nesting: it

must satisfy this equation

called the multiresolution analysis (MRA) equation

this is like a differential equation that the scaling function is the solution to

0V

1V

1V

0V 1V

0V

1V

=n

ntnht )2(2)()(

2)(nh

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The h(n) Specify the Scaling Function

Thus, the coefficients h(n) determine the scaling function

for a given set of h(n), (t)

may or may not exist

may or may not be unique

Want to find conditions on h(n) for (t) to exist and be unique, and also: to be orthogonal (because that leads to an ON wavelet expansion)

to give wavelets that have desirable properties

h(n)

n

(t)

tMRA

Equation

h(n) mustsatisfyconditions

h(n) must

satisfyconditions

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Whence the Wavelets?

The spaces Vj represent increasingly higher resolution spaces

To go from Vj to higher resolution Vj+1 requires the addition of details

These details are the part of Vj+1 not able to be represented in Vj

This can be captured through the orthogonal complement of Vj w.r.t Vj+1 Call this orthogonal complement space Wj

all functions in Wjare orthogonal to all functions in Vj

That is:

Consider that V0 is the lowest resolution of interest

How do we characterize the space W0?

we need to find an ON basis for W0, say where the basis functionsarise from translating a single function (well worry about the scaling part

later):

Z=>=< lkjdttttt ljkjljkj ,,0)()()(),( ,,,,

)(,0 tk

)()(,0 kttk =

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Finding the Wavelets

The wavelets are the basis functions for the Wjspaces

thus, they lie in Vj+1

In particular, the function lies in the space V1 so it can be

expanded as

This is a fundamental result linking the scaling function and the wavelet

the h 1(n) specify the wavelet, via the specified scaling function

)(t

=n

nntnht Z),2(2)()( 1

h1(n)

n

(t)

t

h1(n) must

satisfyconditions

h1(n) must

satisfyconditions

Wavelet

Equation(WE)

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Wavelet-Scaling Function Connection

There is a fundamental connection between the scaling function and itscoefficients h(n) , the wavelet function and its coefficients h1(n):

h1(n)

n

(t)

t

h(n)

n

(t)

t

How are h1(n) andh(n) related?

WaveletEquation

(WE)

MREquation

(MRE)

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Relationship Between h1(n) and h(n)

We state here the conditions for the important special case of

finite number N of nonzero h(n)

ON within V0:

ON between V0

and W0

:

Given the h(n) that define the desired scaling function, then the h1(n)that define the wavelet function are given by

Much of wavelet theory addresses the origin, characteristics, and

ramifications of this relationship between h1(n) and h(n)

requirements on h(n) and h1(n) to achieve ON expansions

how the MRE and WE lead to a filter bank structure requirements on h(n) and h1(n) to achieve other desired properties

extensions beyond the ON case

= )()()( kdtktt

= )()()( kdtktt

)1()1()(1 nNhnhn =

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The Resulting Expansions

Let f(t) be in L2

(R) There are three ways of interest that we can expand f(t)

1 We can give an limited resolution approximation to f(t) via

increasing j gives a better (i.e., higher resolution) approximation

this is in general not the most useful expansion

=k

jjkj ktatf )2(2)(

2/

221012 LVVVVV !!

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The Resulting Expansions (cont.)

2 A low-resolution approximation plus its wavelet details

Choosing j0 sets the level of the coarse approximation

This is most useful in practice: j0 is usually chosen according to application Also in practice, the upper value of j is chosen to be finite

=+=

0

00

0 )2(2)()2(2)()(2/2/

jj

jj

jkk

jj

j ktkdktkctf

!= ++ 212

0000 jjjjWWWVL

Low-Resolution

Approximation

Wavelet Details

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The Resulting Expansions (cont.)

3 Only the wavelet details

Choosing j0=- eliminates the coarse approximation leaving only details

This is most similar to the true wavelet decomposition as it was originallydeveloped

This is not that useful in practice: j0 is usually chosen to be finite according

to application

==

j

jjj

k

ktkdtf )2(2)()( 2/

!! = 210122

WWWWWL

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The Expansion Coefficients cj0(k) and dj(k)

We consider here only the simple, but important, case of ON expansion

i.e., the s are ON, the s are ON, and the s are ON to the s

Then we can use standard ON expansion theory:

We will see how to compute these without resorting to computing innerproducts

we will use the coefficients h1(n) and h(n) instead of the wavelet and scaling

function, respectively we look at a relationship between the expansion coefficients at one level

and those at the next level of resolution

dtttfttfkc kjkjj == )()()(),()( ,, 000

dtttfttfkd kjkjj == )()()(),()( ,,

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Summary of Multiresolution View

Nested Resolution spaces:

Wavelet Spaces provide orthogonal complement between resolutions

Wavelet Series Expansion of a continuous-time signal f(t):

MR equation (MRE) provides link between the scaling functions atsuccessive levels of resolution:

Wavelet equation (WE) provides link between a resolution level and itscomplement

=

+=

0

00

0)2(2)()2(2)()( 2/

2/

jj

jjj

kk

jjj ktkdktkctf

221012 LVVVVV !!

!= ++ 212

0000 jjjjWWWVL

Z= nntnhtn

,)2(2)()(

=n

nntnht Z),2(2)()( 1

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Summary of Multiresolution View (cont.)

There is a fundamental connection between the scaling function and its

coefficients h(n) , the wavelet function and its coefficients h1(n):

h1(n)

n

(t)

t

h(n)

n

(t)

tMREquation

(MRE)

How are h1(n) and

h(n) related?

WaveletEquation

(WE)

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Filter Banks and DWT

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Generalizing the MRE and WE

Here again are the MRE and the WE:

We get:

=n

ntnht )2(2)()( =n

ntnht )2(2)()( 1

scale & translate: replace ktt j 2

= +m

jj mtkmhkt )2(2)2()2(1

Connects Vj to Vj+1

MRE

= +m

jj mtkmhkt )2(2)2()2(1

1

Connects Wj to Vj+1

WE

Linking Expansion Coefficients Between Scales

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Linking Expansion Coefficients Between Scales

Start with the Generalized MRA and WE:

= +m

jjmtkmhkt )2(2)2()2(

1 = +m


11

)(),()( , ttfkc kjj = )(),()( , ttfkd kjj =

)()2()( 1 mckmhkc jm

j +

=

= ++m

jjj mttfkmhkc )2(2),()2()(

12/)1(

)()2()( 11 mckmhkd jm

j +

=

= ++m

jjj mttfkmhkd )2(2),()2()(

12/)1(1

)(1 mcj+

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Convolution-Decimation Structure

)()2()( 10 mckmhkc jm

j + = )()2()( 11 mckmhkd jm

j + =New Notation For Convenience: h(n) h0(n)

)()(

)()()(

10

010

mcnmh

nhncny

j

m

j

+

+

==

)()(

)()()(

11

111

mcnmh

nhncny

j

m

j

+

+

==

Convolution

Decimation

k=0 1 2 3 4

n=0 1 2 3 4 5 6 7 8 9

n = 2k =0 2 4 6 8

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Summary of Progression to Convolution-Decimation Structure

= +m


1

MRE

= +m


11

WE

)()2()( 1 mckmhkc j

m

j + = )()2()( 11 mckmhkd jm

j + =

LPF

h0(-n)

HPFh1(-n)

cj+1(k)

cj(k)

dj(k)

2

2

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Computing The Expansion Coefficients

The above structure can be cascaded:

given the scaling function coefficients at a specified level all the lower

resolution cs and ds can be computed using the filter structure

LPFh0(-n)

HPFh1(-n)

cj+1(k)

cj(k)

dj(k)2

2

LPF

h0(-n)

HPFh1(-n)

cj-1(k)

2

2

LPF

h0(-n)

HPF

h1(-n)

cj-2(k)

2

2

dj-1(k)

dj-2(k)

Vj+1

Vj

Vj-1

Vj-2

Wj

Wj-1

Wj-2

Filter Bank Generation of the Spaces

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Filter Bank Generation of the Spaces

LPFh0(-n)

HPF

h1(-n)2

2

LPF

h0(-n)

HPFh1(-n)

2

2

LPFh0(-n)

HPF

h1(-n)2

2

Vj+1

VjVj-1

Vj-2

Wj

Wj-1

Wj-2

WjWj-1Wj-2Vj-2

Vj+1

/2/4/8

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Time

Freq

DISCRETE FOURIERTRANSFORM

WAVELETTRANSFORM

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Time

F

req

WAVELETTRANSFORM

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Q256

Q1

Q28-Level

Wavelet

Transform

h1(1) to h1(8)

h1(9) to h1(16)

l8(1) to l8(8)

h1q(1) to h1q(8)

h1q(9) to h1q(16)

l8q(1) to l8q(8)

Side Info

s(t)

Bits allocated to quantizers to minimize MSE Then allocations less than Bmin are set to zero

Eliminates negligible cells

Side info sent to describe allocations

WT-BASED COMPRESSION EXAMPLE

Time

F

requency

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