wavelets and multiresolution processing (wavelet transforms)

University of Ioannina - Department of Computer Science

Wavelets and Multiresolution Processing(Wavelet Transforms)

Christophoros [email protected]

Digital Image Processing

mailto:[email protected]

2

C. Nikou – Digital Image Processing (E12)

Contents

– Image pyramids– Subband coding– The Haar transform– Multiresolution analysis

• Series expansion• Scaling functions• Wavelet functions

– Wavelet series– Discrete wavelet transform (DWT)– Fast wavelet transform (FWT)– Wavelet packets

3


1-D Wavelet TransformsThe Wavelet Series

A continuous signal may be represented by a scaling function in a subspace and some number of wavelet functions in subspaces

0 0

0

, ,( ) ( ) ( ) ( ) ( )j j k j j kk j j k

f x c k x d k xφ ψ∞

=

= +∑ ∑∑

Scaling coefficients

0jV

0 0 01 2,, , ...j j jW W W+ +

0 0 0, ,( ) ( ), ( ) ( ) ( )j j k j kc k f x x f x x dxφ φ= = ∫

Detail (wavelet) coefficients

, ,( ) ( ), ( ) ( ) ( )j j k j kd k f x x f x x dxψ ψ= = ∫

4


1-D Wavelet TransformsThe Wavelet Series (cont…)

Example: using Haar wavelets and starting from j0=0, compute the wavelet series of

Scaling coefficients1 1

2 20 0,0

0 0

1(0) ( )3

c x x dx x dxφ= = =∫ ∫

There is only one scaling coefficient for k=0. Integer translations of the scaling function do not overlap with the signal.

2 , 0 1( )

0, otherwisex x

f x⎧ ≤ ≤

= ⎨⎩

5



Example (continued): Detail (wavelet) coefficients

1 0.5 12 2 2

0 0,00 0 0.5

1(0) ( )4

d x x dx x dx x dxψ= = − = −∫ ∫ ∫

1 0.25 0.52 2 2

1 1,00 0 0.25

2(0) ( ) 2 232

d x x dx x dx x dxψ= = − = −∫ ∫ ∫

1 0.75 12 2 2

1 1,10 0.5 0.75

3 2(1) ( ) 2 232

d x x dx x dx x dxψ= = − = −∫ ∫ ∫

6



Example (continued): Substituting these values:

0 0 1

1 0 0

2 1 1 0 0 1

0,0 0,0 1,0 1,11 1 2 2( ) ( ) ( ) ( ) ( ) ...3 4 32 32

V W W

V V W

V V W V W W

f x x x x xφ ψ ψ ψ

= ⊕

= ⊕ = ⊕ ⊕

⎡ ⎤⎡ ⎤= + − + − − +⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

7



Example (continued): The scaling function approximates the signal by its average value.

Each wavelet subspace adds a level of detail in the wavelet series representation of the signal.

8


1-D Wavelet TransformsThe Discrete Wavelet Transform

If the signal is discrete, of length M, we compute its DWT.

0

0

0 , ,1 1( ) ( , ) ( ) ( , ) ( )j k j k

k j j kf n W j k n W j k n

M Mφ ψφ ψ∞

=

= +∑ ∑∑

Scaling coefficients

Detail coefficients

0

1

0 ,0

1( , ) ( ) ( )M

j kn

W j k f n nMφ φ

−

=

= ∑

0

1

, 00

1( , ) ( ) ( ), forM

j kn

W j k f n n j jMψ ψ

−

=

= ≥∑

9


1-D Wavelet TransformsThe Discrete Wavelet Transform (cont…)

We take M equally spaced samples over the support of the scaling and wavelet functions.Normally, we let j0=0 and M=2J.

Therefore, the summations are performed over

0

0

0 , ,1 1( ) ( , ) ( ) ( , ) ( )j k j k

k j j k

f n W j k n W j k nM Mφ ψφ ψ

∞

=

= +∑ ∑∑

0, 1, 2,..., 1, 0, 1, 2,..., 1, 0, 1, 2,..., 2 1jn M j J k= − = − = −

For Haar wavelets, the discretized scaling and wavelet functions correspond to the rows of the MxM Haarmatrix.

10



Example: f(n)={1, 4, -3, 0}.

We will compute the DWT of the signal using the Haar scaling function and the corresponding wavelet functions.

Here, M=4=2J, J=2 and with j0=0 the summations are performed over n=0, 1, 2, 3, j=0, 1 and k=0 for j=0 or k=0, 1 for j=1. The values of the sampled scaling and wavelet functions are the elements of the rows of H4.

11



Example (continued): f(n)={1, 4, -3, 0}.

00

1 3

0 , 0,00, 00 0

1 1( , ) ( ) ( ) ( ) ( )2

M

j k j kn n

W j k f n n f n nMφ φ φ

−

= == =

= = ⇔∑ ∑

1(0,0) [1 1 4 1 3 ( 1) 0 ( 1)] 42

Wψ = ⋅ + ⋅ − ⋅ − + ⋅ − =

1(0,0) [1 1 4 1 3 1 0 1] 12

Wφ = ⋅ + ⋅ − ⋅ + ⋅ =

1(1,0) [1 2 4 ( 2) 3 0 0 0] 1.5 22

Wψ = ⋅ + ⋅ − − ⋅ + ⋅ = −

1(1,1) [1 0 4 0 3 2 0 ( 2)] 1.5 22

Wψ = ⋅ + ⋅ − ⋅ + ⋅ − = −

12



Example (continued): f(n)={1, 4, -3, 0}. The DWT relative to the Haar scaling and wavelet functions is

{ }1, 4, 1.5 2, 1.5 2− −

To reconstruct the signal, we compute

0

0

0 , ,

0,0 0,0 1,0 1,1

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

1 (0,0) ( ) (0,0) ( ) (1,0) ( ) (1,1) ( )2

j k j kk j j k

f n W j k n W j k nM M

W n W n W n W n

φ ψ

φ ψ ψ ψ

φ ψ

φ ψ ψ ψ

∞

=

= +

⎡ ⎤= + + +⎣ ⎦

∑ ∑∑

for n=0, 1, 2, 3. Notice that we could have started from a different approximation level j0∫0.

13


1-D Wavelet TransformsThe Fast Wavelet Transform

A computationally efficient implementation of the DWT [Mallat 1989]. It resembles subband coding.Consider the multiresolution refinement equation

Scaling x by 2j, translating it by k and letting m=2k+n:

( ) ( ) 2 (2 )n

x h n x nφφ φ= −∑

( )(2 ) ( ) 2 2(2 )j j

nx k h n x k nφφ φ− = − − =∑

1( ) 2 (2 2 )j

nh n x k nφ φ += − −∑

1( 2 ) 2 (2 )j

mh m k x mφ φ += − −∑

14


1-D Wavelet TransformsThe Fast Wavelet Transform (cont…)

The scaling vector hφ may be considered as weights used to expand φ(2jx-k) as a sum of scale j+1 scaling functions.

A similar sequence of operations leads to

1(2 ) ( 2 ) 2 (2 )j j

mx k h m k x mφφ φ +− = − −∑

1(2 ) ( 2 ) 2 (2 )j j

mx k h m k x mψψ φ +− = − −∑

15



Injecting these expressions into the DWT formulas1

,0

1( , ) ( ) ( )M

j kn

W j k f n nMφ φ

−

=

= ∑

0

1

, 00

1( , ) ( ) ( ), forM

j kn

W j k f n n j jMψ ψ

−

=

= ≥∑yields the following important equations relating the DWT coefficients of adjacent scales.

( , ) ( 2 ) ( 1, )m

W j k h m k W j mφ φ φ= − +∑( , ) ( 2 ) ( 1, )

m

W j k h m k W j mψ ψ φ= − +∑

16



Both the scaling and the wavelet coefficients of a certain scale jmay be obtained by

• convolution of the scaling coefficients of the next scale j+1(the finer detail scale), with the order-reversed scaling and wavelet vectors hφ(-n) and hψ(-n).• subsampling the result.

( , ) ( 2 ) ( 1, )m

W j k h m k W j mφ φ φ= − +∑( , ) ( 2 ) ( 1, )

m

W j k h m k W j mψ ψ φ= − +∑

The complexity is O(M).

17



The convolutions are evaluated at non-negative even indices. This is equivalent to filtering and downsampling by 2.

2 , 0( , ) ( ) ( 1, )

n k kW j k h n W j nφ φ φ = ≥

= − ∗ +

2 , 0( , ) ( ) ( 1, )

n k kW j k h n W j nψ ψ φ = ≥

= − ∗ +

18



The procedure may be iterated to create multistage structures of more scales.The highest scale coefficients are assumed to be the values of the signal itself ( , ) ( )W J n f nφ =

19



The corresponding frequency splitting characteristics for the two-stage procedure.

20



21



Reconstruction may be obtained by the IFWT• upsampling by 2 (inserting zeros).• convolution of the scaling coefficients by hφ(n) and the wavelet coefficients by hψ(n) and summation.

2 2

0( 1, ) ( ) ( , ) ( ) ( , )

kW j k h k W j k h k W j kφ φ φ ψ ψ

↑ ↑

≥+ = ∗ + ∗

22



It can also be extended to multiple stages.

23



24


1-D Wavelet TransformsRelation to the Fourier Transform

• The Fourier basis functions guarantee the existence of the transform for energy signals.• The wavelet transform depends upon the availability of scaling functions for a given wavelet function.• The wavelet transform depends on the orthonormality (biorthogonality) of the scaling and wavelet functions.• The F.T. informs us about the frequency content of a signal. It does not inform us on the specific time instant that a certain frequency occurs.•The W.T. provides information on “when the frequency occurs”.

25


1-D Wavelet TransformsRelation to the Fourier Transform (cont…)

Heisenberg cells.• The width of each rectangle in (a) represents one time instant.• The height of each rectangle in (b) represents a single frequency.

Signal DFT DWT

26



Time domain pinpoints the instance an event occurs but has no frequency information.DFT domain pinpoints the frequencies that are present in the events but provides no time resolution (when a certain frequency appears).

Signal DFT DWT

27



In DWT the time-frequency resolution varies but the area of each tile is the same. There is a compromise between time and frequency resolutions.At low frequencies the tiles are shorter (better frequency resolution, less ambiguity regarding frequency) but wider (more ambiguity regarding time).At high frequencies the opposite happens (time resolution is improved).

Signal DFT DWT

28


2-D Discrete Wavelet Transform

Extension from 1-D wavelet transforms.A 2-D scaling function and three 2-D wavelet functions are required.They are separable. They are the product of two 1-D functions.

( , ) ( ) ( )( , ) ( ) ( )( , ) ( ) ( )( , ) ( ) ( )

H

V

D

x y x yx y x yx y x yx y x y

φ φ φ

ψ ψ φ

ψ φ ψ

ψ ψ ψ

=

=

=

=

Variations along columns

Variations along rows

Variations along diagonals

29


2-D Discrete Wavelet Transform (cont…)

Given separable scaling and wavelet functions the scaled and translated basis

, ,

/ 2, ,

/ 2

( , ) 2 (2 ,2 )

( , ) 2 (2 ,2 ), { , , }j m n

j j jj m n

i j i j j

x y x m y n

x y x m y n i H V D

φ φ

ψ ψ

= − −

= − − =

where i is an index and there are two translations m and n.

30



The DWT is given by

And the inverse transformation is

0

1 1

0 , ,0 0

1( , , ) ( , ) ( , )M N

j m nm n

W j m n f m n m nMNφ φ

− −

= =

= ∑∑

, .0

1 1

0 0

1( , , ) ( , ) ( , ), { , , }j m n

M Ni i

m n

W j m n f m n m n i H V DMNψ ψ

− −

= =

= =∑∑

0

0

0 , ,

, ,, ,

1( , ) ( , , ) ( , )

1 ( , , , ) ( , )

j m nm n

i ij m n

i H V D j j m n

f m n W j m n m nMN

W j m n k m nMN

φ

ψ

φ

ψ∞

= =

=

+

∑∑

∑ ∑∑∑

31



We take M equally spaced samples over the support of the scaling and wavelet functions.We normally, we let j0=0 and select N=M=2J

so that0, 1, 2,..., 1, 0, 1, 2,..., 1, 0, 1, 2,..., 2 1jm n M j J k= = − = − = −

Like the 1-D transform it can be implemented using filtering and downsampling.We take the 1-D FWT of the rows followed by the 1-D FWT of the resulting columns.

32



33



Reconstruction (inverse 2-D FWT)

34



35



36



a) Noisy CT image. We wish to remove the noise by manipulating the DWT.

b) Thresholding all of the detail coefficients of the two scale DWT. Significant edge information is eliminated.

c) Zeroing the highest resolution detail coefficients (not thresholding the lowest resolution details).

d) Information loss in c) with respect to the original image. The edges are slightly disturbed.

e) Zeroing all of the detail coefficients of the two scale DWT.

d) Information loss in e) with respect to the original image. Significant edge information is removed.

37


Wavelet Packets

The DWT, as it was defined, decomposes a signal into a sum of scaling and wavelet functions whose bandwidths are logarithmically related.• low frequencies use narrow bandwidths.• high frequencies use wider bandwidths.For greater flexibility of the partitioning of the time-frequency plane the DWT must be generalized.

Signal DFT DWT

38


Wavelet Packets (cont…)

Coefficient tree and subspace analysis tree for the two-scale (three levels) FWT filter bank.The subspace analysis tree provides a more compact representation of the decomposition in terms of subspaces.

11 JJ JV V W−−= ⊕

39



Subspace analysis tree and spectrum splitting for the three-scale (four levels) FWT filter bank.Here, we have three options for the expansion:

1

2

3

1

2 1

3 2 1

J

J

J

J J

J J J

J J J J

V V W

V V W W

V V W W W

−

−

−

−

− −

− − −

= ⊕

= ⊕ ⊕

= ⊕ ⊕ ⊕

40



Wavelet packets are conventional wavelet transforms in which the details are filtered iteratively. For example, the three scale wavelet packet tree is

Indices A (approximation) and D (detail) denote the path from the parent to the node.

41


Wavelet Packets (cont…)Filter bank and spectrum (three scale wavelet packet).

42



The cost of this generalization increases the computational complexity of the transform from O(M)to O(MlogM).Also, the three scale packet almost triples the number of decompositions available from the three low pass bands.

43



The subspace of the signal may be expanded as

33 2, 2, 1, 1, 1, 1,JJ J J A J D J AA J AD J DA J DDV V W W W W W W W−− − − − − − −= ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕

with spectrum:

44



It can also be expanded as

1 1, 1, 1,J J J A J DA J DDV V W W W− − − −= ⊕ ⊕ ⊕

with spectrum:

45



In general, a P-scale 1-D wavelet packet (associated with P+1 level analysis trees) supports

[ ]2( 1) ( ) 1D P D P+ = +

unique decompositions, where D(1)=1 is the initial signal.

For instance, D(4)=26 and D(5)=677.

46



The problem increases dramatically in 2-D.a P-scale 2-D wavelet packet (associated with P+1 level analysis trees) supports

[ ]4( 1) ( ) 1D P D P+ = +

unique decompositions, where D(1)=1 is the initial signal.

For instance, D(4)=83.522 possible decompositions.How do we select among them? Impractical to examine each one of them for a given application.

47


Wavelet Packets (cont…)A common application of wavelets is image compression. Here we will examine a simplified example.We seek to compress the fingerprint image by selecting the “best” three-scale wavelet packet decomposition.

48



A criterion for the decomposition is the image energy2( ) ( , )

m n

E f f m n=∑∑

AlgorithmFor each node of the analysis tree, from the root to the leaves:Step 1: Compute the energy of the node (EP) and the energy of its offsprings (EA, EH, EV, ED).Step 2: If EA+EH+EV+ED <EP include the offsprings in the analysis tree as they reduce the initial energy.Otherwise, keep only the parent which is a leaf of the tree.

49



Many of the 64 initial subbands are eliminated.

wavelets and multiresolution processing (wavelet transforms)

Documents