Wavelets and Multiresolution Processing Thinh Nguyen - - PowerPoint PPT Presentation

Wavelets and Multiresolution Processing

Thinh Nguyen

Multiresolution Multiresolution Analysis (MRA) Analysis (MRA)

A scaling function

scaling function is used to create a series of approximations of a function or image, each differing by a factor of 2 from its neighboring approximations.

Additional functions called wavelets

wavelets are then used to encode the difference in information between adjacent approximations.

Express a signal as If the expansion is unique, the are

called basis functions basis functions, and the expansion set is called a basis basis

Series Expansions Series Expansions

( ) f x ( ) ( )

k k k

f x x α ϕ =∑

expansion coefficients expansion functions

( )

k x

ϕ

{ }

( )

k x

ϕ

All the functions expressible with this basis

form a function space function space which is referred to as the closed span closed span of the expansion set

If , then is in the closed span

• f

and can be expressed as

Series Expansions Series Expansions

{ }

( )

k k

V Span x ϕ = ( ) f x V ∈ ( ) f x

{ }

( )

k x

ϕ ( ) ( )

k k k

f x x α ϕ =∑

The expansion functions form an

• rthonormal basis for V

The basis and its dual are equivalent, i.e.,

and

Orthonormal Orthonormal Basis Basis

( ), ( ) 1

j k jk

j k x x j k ϕ ϕ δ ≠ ⎧ = = ⎨ = ⎩ ( ) ( )

k k

x x ϕ ϕ = % ( ), ( ) ( ) ( )

k k k

x f x x f x dx α ϕ ϕ∗ = = ∫

Consider the set of expansion functions composed

• f integer translations and binary scalings of the

real square-integrable function defined by for all and

By choosing the scaling function wisely,

can be made to span

Scaling Functions Scaling Functions

( ) x ϕ

{ } {

}

/ 2 , ( )

2 (2 )

j j j k x

x k ϕ ϕ = − , j k ∈฀

2

( ) ( ) x L ϕ ∈ ฀

{ }

, ( ) j k x

ϕ

2( )

L ฀ ( ) x ϕ

Index k determines the position of

along the x-axis, index j determines its width; controls its height or amplitude.

By restricting j to a specific value the

resulting expansion set is a subset of

One can write

, ( ) j k x

ϕ

/ 2

2 j

{ } {

}

/ 2 , ( )

2 (2 )

j j j k x

x k ϕ ϕ = −

• j

j =

{ }

, ( ) j k x

ϕ

{ }

, ( )

• j k x

ϕ

{ }

, ( )

• j

j k k

V Span x ϕ =

Example: The Example: The Haar Haar Scaling Function Scaling Function

1,0 1,1 1,4

( ) 0.5 ( ) ( ) 0.25 ( ) f x x x x ϕ ϕ ϕ = + −

0, 1,2 1,2 1

1 1 ( ) ( ) ( ) 2 2

k k k

x x x ϕ ϕ ϕ

+

= + 1

V V ⊂

1 1 ( ) x x

• therwise

ϕ ≤ < ⎧ = ⎨ ⎩

1.

The scaling function is orthogonal to its integer translates

2.

The subspaces spanned by the scaling function at low scales are nested within those spanned at higher scales:

MRA Requirements MRA Requirements

1 1 2

V V V V V V

−∞ − ∞

⊂ ⊂ ⊂ ⊂ ⊂ ⊂ ⊂ L L

Given a scaling function which satisfies the

MRA requirements, one can define a wavelet wavelet function function which, together with its integer translates and binary scalings, spans the difference between any two adjacent scaling subspaces and

Wavelet Functions Wavelet Functions

( ) x ψ

j

V

1 j

V +

Define the wavelet set

for all that spans the spaces

We write

and, if

{ } {

}

/ 2 , ( )

2 (2 )

j j j k x

x k ψ ψ = −

Wavelet Functions Wavelet Functions

k ∈฀

j

W

{ }

, ( ) j j k k

W Span x ψ = ( )

j

f x W ∈

,

( ) ( )

k j k k

f x x α ψ =∑

This implies that

for all appropriate

We can write

and also

Orthogonality Orthogonality: :

1 j j j

V V W

+ =

⊕

1 j j j

V V W

+ =

⊕

, ,

( ), ( )

j k j l

x x ϕ ψ = , , j k l ∈฀

2 1

( ) L V W W = ⊕ ⊕ ⊕ ฀ L

2 2 1 1 2

( ) L W W W W W

− −

= ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ฀ L L

(no need for scaling functions, only wavelets!)

Example: Example: Haar Haar Wavelet Functions in Wavelet Functions in W W0

0 and

and W W1

1

( ) ( ) ( )

a d

f x f x f x = + low frequencies high frequencies

A function can be expressed as

Wavelet Series Expansions Wavelet Series Expansions

, ,

( ) ( ( ) ) ( ) ( )

j j j j k k k k j j

f x c k x x k d ψ ϕ

∞ =

= +

∑ ∑ ∑

2

( ) ( ) f x L ∈ ฀

approximation or scaling coefficients approximation or scaling coefficients detail or wavelet coefficients detail or wavelet coefficients

,

( ) ( ), ( )

j j k x

c k f x ϕ =

,

( ) ( ), ( )

j k j

d x x k f ψ =

2 1

( )

j j j

W V W L

+

= ⊕ ⊕ ⊕ ฀ L

Consider If , the expansion coefficients are

Example: The Example: The Haar Haar Wavelet Series Wavelet Series Expansion of Expansion of y y= = x x 2

2

2

1 x x y

• therwise

⎧ ≤ < = ⎨ ⎩

1 1 2 2 1 1 2 2 1 1 0,0 1,0 , 0 0 1,1

1 1 (0) (0) 3 4 2 3 2 (0) (1) 32 ( ) ( ( ) ) ) ( 32 c x x dx d x dx x d x dx d x x x d x ψ ψ ψ ϕ = = = = − = = − = = −

∫ ∫ ∫ ∫

1 1 2 1 1 1

0,0 1, 1 ,1 , ( )

1 1 2 3 2 3 4 ( ) ( ) ( 2 32 ) 3

V W W V V W V V W V W W

x x x x y ψ ψ ψ ϕ

= ⊕ = ⊕ = ⊕ ⊕

⎡ ⎤ ⎡ ⎤ = + − + − − + ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ L 1 4 24 3 14 4 244 3 14444 4 244444 3 1444 4 24444 3 1444444444 4 24444444444 3

j =

Example: The Example: The Haar Haar Wavelet Series Wavelet Series Expansion of Expansion of y y= = x x 2

2

Let denote a discrete

function

Its DWT is defined as Let and so that

The Discrete Wavelet Transform (DWT) The Discrete Wavelet Transform (DWT)

( ), 0,1, , 1 f x x M = − K

, ,

1 ( , ) ( ) 1 ( , ) ( ( ) ( ) )

x j k x k j

W j k f x M W j k f x x M x

ϕ ψ

ψ ϕ = =

∑ ∑

j j ≥

, ,

( ) 1 1 ( ) ( ) ( , ) ( , )

j k k j j k j k

f x W j k W j k M M x x

ϕ ψ

ψ ϕ

∞ =

= +

∑ ∑ ∑

approximation approximation coefficients coefficients detail detail coefficients coefficients

2J M =

• j =

0,1, , 1, 0,1, , 1, 0,1, ,2 1

j

x M j J k = − ⎧ ⎪ = − ⎨ ⎪ = − ⎩ K K K

Consider the discrete function It is The summations are performed over

and for and for

Use the Haar scaling and wavelet

functions

Example: Computing the DWT Example: Computing the DWT

(0) 1, (1) 4, (2) 3, (3) f f f f = = = − =

2

4 2 2 M J = = ⎯⎯ → = 0,1,2,3 x = k = j = 0,1 k = 1 j =

Example: Computing the DWT Example: Computing the DWT

[ ] [ ]

3 3 3 0,0 1 0, 3 , 1, 1

1 (0,0) ( ) 1 4 3 1 2 1 (0,0) ( ) 1 4 3 ( ) 0 ( ) 4 2 1 (1,0) ( ) 1 1 ( ) 1 1 1 1 2 1 ( ) 2 2 1 ( ) 1 1 1 4 ( ) 3 1.5 2 2 1 (1,1) ( ) 1 2 1 ( ) 2 1 2 3 2 4 0 ( 2

x x x x

x x W f x W f x W f x W f x x x

ϕ ψ ψ ψ

ψ ψ ψ ϕ

= = = =

= = ⋅ + ⋅ − ⋅ + ⋅ = = = ⋅ + ⋅ − ⋅ + ⋅ = ⎡ ⎤ = = ⋅ + ⋅ − ⋅ + ⋅ = − ⎣ ⎦ = = ⋅ + ⋅ − ⋅ + ⋅ − − − −

∑ ∑ ∑ ∑

) 1 2 .5 2 ⎡ ⎤ = − ⎣ ⎦

The DWT of the 4-sample function

relative to the Haar wavelet and scaling functions thus is

The original function can be

reconstructed as for

Example: Computing the DWT Example: Computing the DWT

{ }

1,4, 1.5 2, 1.5 2 − −

0,0 1,0 1 0, ,1

( ) 1 ( ) (0,0) (0,0) 2 (1 ( ,0) (1,1 ( ) ) ) ( ) f x W W W W x x x x

ϕ ψ ψ ψ

ψ ϕ ψ ψ ⎡ = + + ⎣ ⎤ + ⎦

0,1,2,3 x =

In 2-D, one needs one scaling function

and three wavelets

• is a 1-D scaling function and

is its corresponding wavelet

Wavelet Transform in 2 Wavelet Transform in 2-

• D

D

( , ) ( ) ( ) x y x y ϕ ϕ ϕ = ( , ) ( ) ( ) ( , ) ( ) ( ) ( , ) ( ) ( )

H V D

x y x y x y x y x y x y ψ ψ ϕ ψ ϕ ψ ψ ψ ψ ⎧ = ⎪ = ⎨ ⎪ = ⎩

• detects horizontal details
• detects vertical details
• detects diagonal details

(.) ϕ (.) ψ

Define the scaled and translated basis

functions

Then

2 2-

• D DWT: Definition

D DWT: Definition

{ }

/ 2 / , 2 , , ,

2 (2 ,2 ) 2 (2 ,2 ), ( , ( ) , , ) ,

i j m n j j j j i j j m n j

x m y n x m y x n i H V D x y y ϕ ϕ ψ ψ = − − = − − =

{ }

1 1 1 1 , , , ,

( , ( , ) 1 ( , , ) ( , ) 1 ( , , ) ( , , , ) , )

M N x y j m M N i x i j n y m n

W j m n f x y MN W j x y m n f x y x y i H V D MN

ϕ ψ

ψ ϕ

− − = = − − = =

= = =

∑ ∑ ∑ ∑

, , , , , ,

1 ( , ) ( , ( ( , ) , ) , ) 1 ( , , )

m n i i j m H V D j j m n m n i j n

f x y W j m n MN W j m n M x y x N y

ϕ ψ

ψ ϕ

∞ = =

= +

∑∑ ∑ ∑ ∑∑

Filter bank implementation of 2 Filter bank implementation of 2-

• D

D wavelet wavelet

analysis FB analysis FB synthesis FB synthesis FB resulting resulting decomposition decomposition

( , ) f x y

LL LH HL HH

HH HL LH LL

Example: A Three Example: A Three-

• Scale FWT

Scale FWT

Analysis and Synthesis Filters Analysis and Synthesis Filters

scaling scaling function function wavelet wavelet function function analysis analysis filters filters synthesis synthesis filters filters

“An Introduction to Wavelets,” by Amara Graps Amara’s Wavelet Page (with many links to other

resources) http://www.amara.com/current/wavelet.html

“Wavelets for Kids,” (A Tutorial Introduction), by B.

Vidakovic and P. Mueller

Gilbert Strang’s tutorial papers from his MIT webpage

http://www-math.mit.edu/~gs/

W avelets and Subband Coding, by Jelena Kovacevic

and Martin Vetterli, Prentice Hall, 2000.

Want to Learn More About Wavelets? Want to Learn More About Wavelets?